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      <text bytes="735" sha1="a5wehuldd0go2uniagwvx66n6c80irq" xml:space="preserve">&lt;strong&gt;MediaWiki has been installed.&lt;/strong&gt;

Consult the [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Contents User's Guide] for information on using the wiki software.

== Getting started ==
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Configuration_settings Configuration settings list]
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:FAQ MediaWiki FAQ]
* [https://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Localisation#Translation_resources Localise MediaWiki for your language]
* [https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:Combating_spam Learn how to combat spam on your wiki]</text>
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Check out [[Migration_FAQ|this list of anticipated Frequently Ask Questions]] regarding the recent migration of this wiki.

==Welcome to the Xenharmonic Wiki!==

The Xenharmonic wiki is an open resource for information about [[Musical tuning|Musical Tuning]].&lt;span style="line-height: 1.5;"&gt; This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.&lt;/span&gt;

&lt;span style="line-height: 1.5;"&gt;This wiki is created by volunteers. Relevant new content is most welcome! You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.&lt;/span&gt;

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
** [[EDF|Equal divisions of the perfect fifth]]
** [[edVI through edX|Equal divisions of sixths through tenths]]
*** [[edVI|Equal divisions of sixths]]
*** [[edVII|Equal divisions of sevenths]]
*** [[EDO#Individual pages for EDOs|Equal divisions of the octave]]
*** [[edIX|Equal divisions of ninths]]
*** [[edX|Equal divisions of tenths]]
** [[edXI and beyond|Equal divisions of elevenths and beyond]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function in the side bar to the left! &lt;/span&gt;</text>
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Check out [[Migration_FAQ|this list of anticipated Frequently Ask Questions]] regarding the recent migration of this wiki.

==Welcome to the Xenharmonic Wiki!==

The Xenharmonic wiki is an open resource for information about [[Musical tuning|Musical Tuning]].&lt;span style="line-height: 1.5;"&gt; This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.&lt;/span&gt;

&lt;span style="line-height: 1.5;"&gt;This wiki is created by volunteers. Relevant new content is most welcome! You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.&lt;/span&gt;

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
** [[EDF|Equal divisions of the perfect fifth]]
** [[edVI through edX|Equal divisions of sixths through tenths]]
*** [[edVI|Equal divisions of sixths]]
*** [[edVII|Equal divisions of sevenths]]
*** [[EDO#Individual pages for EDOs|Equal divisions of the octave]]
*** [[edIX|Equal divisions of ninths]]
*** [[edX|Equal divisions of tenths]]
** [[edXI and beyond|Equal divisions of elevenths and beyond]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function in the side bar to the left! &lt;/span&gt;</text>
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Check out [[Migration_FAQ|this list of anticipated Frequently Ask Questions]] regarding the recent migration of this wiki.

==Welcome to the Xenharmonic Wiki!==

The Xenharmonic wiki is an open resource for information about [[Musical tuning|Musical Tuning]].&lt;span style="line-height: 1.5;"&gt; This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.&lt;/span&gt;

&lt;span style="line-height: 1.5;"&gt;This wiki is created by volunteers. Relevant new content is most welcome! You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.&lt;/span&gt;

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
** [[EDF|Equal divisions of the perfect fifth]]
** [[edVI through edX|Equal divisions of sixths through tenths]]
*** [[edVI|Equal divisions of sixths]]
*** [[edVII|Equal divisions of sevenths]]
*** [[EDO#Individual pages for EDOs|Equal divisions of the octave]]
*** [[edIX|Equal divisions of ninths]]
*** [[edX|Equal divisions of tenths]]
** [[edXI and beyond|Equal divisions of elevenths and beyond]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>1i8lgano4iavyui3ninflai26xtz5zo</sha1>
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Check out [[Migration_FAQ|this list of anticipated Frequently Ask Questions]] regarding the recent migration of this wiki.

==Welcome to the Xenharmonic Wiki!==

The Xenharmonic wiki is an open resource for information about [[Musical tuning|Musical Tuning]].&lt;span style="line-height: 1.5;"&gt; This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.&lt;/span&gt;

&lt;span style="line-height: 1.5;"&gt;This wiki is created by volunteers. Relevant new content is most welcome! You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.&lt;/span&gt; You can also [[crash]] this tab.

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
** [[EDF|Equal divisions of the perfect fifth]]
** [[edVI through edX|Equal divisions of sixths through tenths]]
*** [[edVI|Equal divisions of sixths]]
*** [[edVII|Equal divisions of sevenths]]
*** [[EDO#Individual pages for EDOs|Equal divisions of the octave]]
*** [[edIX|Equal divisions of ninths]]
*** [[edX|Equal divisions of tenths]]
** [[edXI and beyond|Equal divisions of elevenths and beyond]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>jvefbljeenec1i89skysokq9goe2vzo</sha1>
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Check out [[Migration_FAQ|this list of anticipated Frequently Ask Questions]] regarding the recent migration of this wiki.

==Welcome to the {{SITENAME}}!==

The {{SITENAME}} is an open resource for information about [[Musical tuning|Musical Tuning]].&lt;span style="line-height: 1.5;"&gt; This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.&lt;/span&gt;

&lt;span style="line-height: 1.5;"&gt;This wiki is created by volunteers. Relevant new content is most welcome! You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.&lt;/span&gt; You can also [[crash]] this tab.

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
** [[EDF|Equal divisions of the perfect fifth]]
** [[edVI through edX|Equal divisions of sixths through tenths]]
*** [[edVI|Equal divisions of sixths]]
*** [[edVII|Equal divisions of sevenths]]
*** [[EDO#Individual pages for EDOs|Equal divisions of the octave]]
*** [[edIX|Equal divisions of ninths]]
*** [[edX|Equal divisions of tenths]]
** [[edXI and beyond|Equal divisions of elevenths and beyond]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>28o2xjk2j8kxtltvqdndklhxug1sw1b</sha1>
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      <text bytes="3067" sha1="m8h9noz5fhn1quarek9qamiqp0qgijl" xml:space="preserve">'''&lt;big&gt;&lt;big&gt;&lt;big&gt;THANKS FOR MIGRATING FROM [http://xenharmonic.wikispaces.com XENHARMONIC.WIKISPACES.COM]. BUT WE HAVE A DIFFERENT BIG TEXT! &lt;br&gt;&lt;big&gt;&lt;big&gt;&lt;big&gt;&lt;big&gt;&lt;big&gt;THIS WIKI IS [[Wikifuture#Complete_incomplete_articles|INCOMPLETE]]'''&lt;small&gt;&lt;small&gt;&lt;small&gt;

__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Ask Questions]] regarding the recent migration of this wiki.

==Welcome to the {{SITENAME}}!==

The {{SITENAME}} is an open resource for information about [[Musical tuning|Musical Tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. Relevant new content is most welcome! You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts. You can also [[crash]] this tab.

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
** [[EDF|Equal divisions of the perfect fifth]]
** [[edVI through edX|Equal divisions of sixths through tenths]]
*** [[edVI|Equal divisions of sixths]]
*** [[edVII|Equal divisions of sevenths]]
*** [[EDO#Individual pages for EDOs|Equal divisions of the octave]]
*** [[edIX|Equal divisions of ninths]]
*** [[edX|Equal divisions of tenths]]
** [[edXI and beyond|Equal divisions of elevenths and beyond]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>m8h9noz5fhn1quarek9qamiqp0qgijl</sha1>
    </revision>
    <revision>
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      <format>text/x-wiki</format>
      <text bytes="2856" sha1="1i8lgano4iavyui3ninflai26xtz5zo" xml:space="preserve">__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Ask Questions]] regarding the recent migration of this wiki.

==Welcome to the Xenharmonic Wiki!==

The Xenharmonic wiki is an open resource for information about [[Musical tuning|Musical Tuning]].&lt;span style="line-height: 1.5;"&gt; This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.&lt;/span&gt;

&lt;span style="line-height: 1.5;"&gt;This wiki is created by volunteers. Relevant new content is most welcome! You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.&lt;/span&gt;

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
** [[EDF|Equal divisions of the perfect fifth]]
** [[edVI through edX|Equal divisions of sixths through tenths]]
*** [[edVI|Equal divisions of sixths]]
*** [[edVII|Equal divisions of sevenths]]
*** [[EDO#Individual pages for EDOs|Equal divisions of the octave]]
*** [[edIX|Equal divisions of ninths]]
*** [[edX|Equal divisions of tenths]]
** [[edXI and beyond|Equal divisions of elevenths and beyond]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>1i8lgano4iavyui3ninflai26xtz5zo</sha1>
    </revision>
    <revision>
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Check out [[Migration_FAQ|this list of anticipated Frequently Ask Questions]] regarding the recent migration of this wiki.

==Welcome to the Xenharmonic Wiki!==

The Xenharmonic wiki is an open resource for information about [[Musical tuning|Musical Tuning]].&lt;span style="line-height: 1.5;"&gt; This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.&lt;/span&gt;

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us keep developing it. Relevant new content is always welcome,. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
** [[EDF|Equal divisions of the perfect fifth]]
** [[edVI through edX|Equal divisions of sixths through tenths]]
*** [[edVI|Equal divisions of sixths]]
*** [[edVII|Equal divisions of sevenths]]
*** [[EDO#Individual pages for EDOs|Equal divisions of the octave]]
*** [[edIX|Equal divisions of ninths]]
*** [[edX|Equal divisions of tenths]]
** [[edXI and beyond|Equal divisions of elevenths and beyond]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>gg0u9worsv4l0rbvzhpuvuohi8wcp9o</sha1>
    </revision>
    <revision>
      <id>32589</id>
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Check out [[Migration_FAQ|this list of anticipated Frequently Ask Questions]] regarding the recent migration of this wiki.

==Welcome to the Xenharmonic Wiki!==

The Xenharmonic wiki is an open resource for information about [[Musical tuning|Musical Tuning]].&lt;span style="line-height: 1.5;"&gt; This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.&lt;/span&gt;

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us keep developing it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
** [[EDF|Equal divisions of the perfect fifth]]
** [[edVI through edX|Equal divisions of sixths through tenths]]
*** [[edVI|Equal divisions of sixths]]
*** [[edVII|Equal divisions of sevenths]]
*** [[EDO#Individual pages for EDOs|Equal divisions of the octave]]
*** [[edIX|Equal divisions of ninths]]
*** [[edX|Equal divisions of tenths]]
** [[edXI and beyond|Equal divisions of elevenths and beyond]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>ousk3339axdph06q2bgllm2ce6y1kkj</sha1>
    </revision>
    <revision>
      <id>32590</id>
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Check out [[Migration_FAQ|this list of anticipated Frequently Ask Questions]] regarding the recent migration of this wiki.

==Welcome to the Xenharmonic Wiki!==

The Xenharmonic wiki is an open resource for information about [[Musical tuning|Musical Tuning]].&lt;span style="line-height: 1.5;"&gt; This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.&lt;/span&gt;

This wiki is created by volunteers. '''It is a work in progress''', and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
** [[EDF|Equal divisions of the perfect fifth]]
** [[edVI through edX|Equal divisions of sixths through tenths]]
*** [[edVI|Equal divisions of sixths]]
*** [[edVII|Equal divisions of sevenths]]
*** [[EDO#Individual pages for EDOs|Equal divisions of the octave]]
*** [[edIX|Equal divisions of ninths]]
*** [[edX|Equal divisions of tenths]]
** [[edXI and beyond|Equal divisions of elevenths and beyond]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>t3f4iidec41ut4c5gn60n85zp498yvp</sha1>
    </revision>
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Check out [[Migration_FAQ|this list of anticipated Frequently Ask Questions]] regarding the recent migration of this wiki.

==Welcome to the Xenharmonic Wiki!==

The Xenharmonic wiki is an open resource for information about [[Musical tuning|Musical Tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
** [[EDF|Equal divisions of the perfect fifth]]
** [[edVI through edX|Equal divisions of sixths through tenths]]
*** [[edVI|Equal divisions of sixths]]
*** [[edVII|Equal divisions of sevenths]]
*** [[EDO#Individual pages for EDOs|Equal divisions of the octave]]
*** [[edIX|Equal divisions of ninths]]
*** [[edX|Equal divisions of tenths]]
** [[edXI and beyond|Equal divisions of elevenths and beyond]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>rkwcjchfrb8fbils57l26uv95aix6s7</sha1>
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Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki.

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>eulfhuo8e6n9tnexl09n29c30a9h5rl</sha1>
    </revision>
    <revision>
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Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. If 6ou see something wrong, pray to [[The Xen]]!

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>epyzh96v3nq22fj020mlydgecyhvmyg</sha1>
    </revision>
    <revision>
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Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. If you see something wrong, pray to [[The Xen]]!

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>ddsmeiumuxw5sfemytrr7hpotbu8r82</sha1>
    </revision>
    <revision>
      <id>33167</id>
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Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. '''If you see something wrong, pray to [[The Xen]]!'''

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>h7uo49zk685djnki2zwjp6z0d662ubw</sha1>
    </revision>
    <revision>
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      <format>text/x-wiki</format>
      <text bytes="2789" sha1="q2zt5acp1g774tm5vbp9o3jq7e1k0lm" xml:space="preserve">__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki.

&lt;big&gt;'''If you see something wrong, pray to [[The Xen]]!'''&lt;small&gt;

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>q2zt5acp1g774tm5vbp9o3jq7e1k0lm</sha1>
    </revision>
    <revision>
      <id>33195</id>
      <parentid>33190</parentid>
      <timestamp>2018-09-21T13:07:32Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <origin>33195</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2871" sha1="k6dh2uhkteuqhgzo7ag0gz0bketx7iq" xml:space="preserve">__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki.

&lt;big&gt;'''If you see something wrong, pray to [[The Xen]]!'''&lt;small&gt; &lt;!-- this wiki is religious, so it's supposed to encourage that. don't delete --&gt;

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>k6dh2uhkteuqhgzo7ag0gz0bketx7iq</sha1>
    </revision>
    <revision>
      <id>33196</id>
      <parentid>33195</parentid>
      <timestamp>2018-09-21T13:07:56Z</timestamp>
      <contributor>
        <username>Tyler Henthorn</username>
        <id>1</id>
      </contributor>
      <minor/>
      <comment>Reverted edits by [[Special:Contributions/PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) to last revision by [[User:Spt3125|Spt3125]]</comment>
      <origin>33196</origin>
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      <format>text/x-wiki</format>
      <text bytes="2721" sha1="eulfhuo8e6n9tnexl09n29c30a9h5rl" xml:space="preserve">__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki.

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>eulfhuo8e6n9tnexl09n29c30a9h5rl</sha1>
    </revision>
    <revision>
      <id>33197</id>
      <parentid>33196</parentid>
      <timestamp>2018-09-21T13:09:58Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <comment>this wiki is religious, so it's supposed to encourage that. don't delete</comment>
      <origin>33197</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2871" sha1="k6dh2uhkteuqhgzo7ag0gz0bketx7iq" xml:space="preserve">__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki.

&lt;big&gt;'''If you see something wrong, pray to [[The Xen]]!'''&lt;small&gt; &lt;!-- this wiki is religious, so it's supposed to encourage that. don't delete --&gt;

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>k6dh2uhkteuqhgzo7ag0gz0bketx7iq</sha1>
    </revision>
    <revision>
      <id>33198</id>
      <parentid>33197</parentid>
      <timestamp>2018-09-21T13:12:20Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
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      <format>text/x-wiki</format>
      <text bytes="2942" sha1="7ol2r6hf0lfb31comlfnv86qqe6yt00" xml:space="preserve">__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki.

&lt;big&gt;'''If you see something wrong, pray to [[The Xen]]!'''&lt;small&gt; &lt;!-- this wiki is religious, so it's supposed to encourage that. don't delete, see https://dev.xen.wiki/w/Main_Page for another reference to The Xen --&gt;

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>7ol2r6hf0lfb31comlfnv86qqe6yt00</sha1>
    </revision>
    <revision>
      <id>33199</id>
      <parentid>33198</parentid>
      <timestamp>2018-09-21T13:20:20Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <origin>33199</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2986" sha1="f4oqgmthlscfrqqfgz20n5lwkpk3vbe" xml:space="preserve">__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki.

&lt;big&gt;'''If you see [[Wikifuture#Complete incomplete articles|something wrong]], pray to [[The Xen]]!'''&lt;small&gt; &lt;!-- this wiki is religious, so it's supposed to encourage that. don't delete, see https://dev.xen.wiki/w/Main_Page for another reference to The Xen --&gt;

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>f4oqgmthlscfrqqfgz20n5lwkpk3vbe</sha1>
    </revision>
    <revision>
      <id>33203</id>
      <parentid>33199</parentid>
      <timestamp>2018-09-21T14:53:55Z</timestamp>
      <contributor>
        <username>Mike Battaglia</username>
        <id>3</id>
      </contributor>
      <comment>No religious stuff please</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2847" sha1="81eegz99qg173z6mfv3pcd65y7asdy9" xml:space="preserve">__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki.

&lt;big&gt;'''If you see [[Wikifuture#Complete incomplete articles|something wrong]], please join and help us work on it!'''&lt;/big&gt;

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>81eegz99qg173z6mfv3pcd65y7asdy9</sha1>
    </revision>
    <revision>
      <id>33213</id>
      <parentid>33203</parentid>
      <timestamp>2018-09-21T15:33:59Z</timestamp>
      <contributor>
        <username>Tyler Henthorn</username>
        <id>1</id>
      </contributor>
      <comment>Information on fixing migration errors is covered in the Migration FAQ.</comment>
      <origin>33213</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2722" sha1="sox3dsbfzsbusnrks7zoppcs9k8oxd9" xml:space="preserve">__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>sox3dsbfzsbusnrks7zoppcs9k8oxd9</sha1>
    </revision>
    <revision>
      <id>33335</id>
      <parentid>33213</parentid>
      <timestamp>2018-09-22T08:10:34Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <comment>Redirected page to [[dev:Main Page]]</comment>
      <origin>33335</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2752" sha1="hbww7e32xorozmmn2i7smnli38gnljc" xml:space="preserve">#REDIRECT [[:dev:Main Page]]

__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
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        <username>PiotrGrochowski</username>
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      <comment>Changed redirect target from [[dev:Main Page]] to [[Special:Block/PiotrGrochowski]]</comment>
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      <text bytes="2767" sha1="1f5q7siznfyev01coin8uemwja6m7zh" xml:space="preserve">#REDIRECT [[Special:Block/PiotrGrochowski]]

__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>1f5q7siznfyev01coin8uemwja6m7zh</sha1>
    </revision>
    <revision>
      <id>33359</id>
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      <timestamp>2018-09-22T10:20:55Z</timestamp>
      <contributor>
        <username>Tyler Henthorn</username>
        <id>1</id>
      </contributor>
      <comment>What was the purpose of this Piotr? Please do not vandalize the wiki.</comment>
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      <text bytes="2722" sha1="sox3dsbfzsbusnrks7zoppcs9k8oxd9" xml:space="preserve">__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>sox3dsbfzsbusnrks7zoppcs9k8oxd9</sha1>
    </revision>
    <revision>
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      <contributor>
        <username>Tyler Henthorn</username>
        <id>1</id>
      </contributor>
      <minor/>
      <comment>Protected "[[Main Page]]": Excessive vandalism: Let and admin know if you need to make a change to this page ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))</comment>
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Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>sox3dsbfzsbusnrks7zoppcs9k8oxd9</sha1>
    </revision>
    <revision>
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        <username>Keenan Pepper</username>
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      <comment>/* Welcome to the Xenharmonic Wiki! */ link to [[Xenharmonic Wiki:Introduction]]</comment>
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Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>hv7t250vgsboe9mhsnm5t3kpmqmxadp</sha1>
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| ja = ようこそ！Xenharmonicの世界へ
}}
__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>ieyioql0lhufao39bc67z8sjar5lhji</sha1>
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| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
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    </revision>
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| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__
Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>8an3ovujfoeyh0rqfrt3puzla47xxkg</sha1>
    </revision>
    <revision>
      <id>38196</id>
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      <contributor>
        <username>TallKite</username>
        <id>23</id>
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      <comment>added a link to the EDO page in "Links to popular areas"</comment>
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| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__

Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[EDO|EDOs]] and other [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page! &lt;/span&gt;</text>
      <sha1>s9ufteflt0kzkntiqrnxydhzb679qka</sha1>
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        <username>Tyler Henthorn</username>
        <id>1</id>
      </contributor>
      <minor/>
      <comment>Innocuous change.</comment>
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| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__

Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[Musical tuning|musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox|Sandbox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[wikifuture|Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning|Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* Here are [[MicrotonalListeningList|links to lots]] of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation|Just Intonation]]
** [[Gallery of Just Intervals|Gallery of Just Intervals]]
* [[EDO|EDOs]] and other [[Equal-step_Tuning|Equal Temperaments]]
* [[Tour_of_Regular_Temperaments|Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[historical temperaments|Historical Temperaments]]
* [[People|People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library|The Library]] - a list of electronically-available published works on microtonal/xenharmonic music&lt;br&gt;

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>g2suwc7kpqtaq4p9k4h795bim4hwg8t</sha1>
    </revision>
    <revision>
      <id>53582</id>
      <parentid>52826</parentid>
      <timestamp>2020-11-27T23:32:25Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
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      <minor/>
      <comment>simplified links</comment>
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| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__

Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* The [[Listen]] page links to lots of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation]]
** [[Gallery of just intervals]]
* [[EDO]]s and other [[Equal-step tuning]]s
* [[Tour of Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[Historical temperaments]]
* [[People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library]] - a list of electronically-available published works on microtonal/xenharmonic music

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>cn7opa1cdiu97u0k6i1xhxexreykg8o</sha1>
    </revision>
    <revision>
      <id>56807</id>
      <parentid>53582</parentid>
      <timestamp>2020-12-16T09:50:40Z</timestamp>
      <contributor>
        <username>Tyler Henthorn</username>
        <id>1</id>
      </contributor>
      <minor/>
      <comment>innocuous change</comment>
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| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__

Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.  

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* The [[Listen]] page links to lots of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation]]
** [[Gallery of just intervals]]
* [[EDO]]s and other [[Equal-step tuning]]s
* [[Tour of Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
* [[Historical temperaments]]
* [[People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library]] - a list of electronically-available published works on microtonal/xenharmonic music

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>dfsit9hmy7l4tj9goh1o0djm41likqo</sha1>
    </revision>
    <revision>
      <id>68314</id>
      <parentid>56807</parentid>
      <timestamp>2021-04-22T20:40:07Z</timestamp>
      <contributor>
        <username>Mike Battaglia</username>
        <id>3</id>
      </contributor>
      <comment>/* Links to some popular areas of the Xenharmonic Universe */ added high-level regular temperament page</comment>
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| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__

Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.  

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* The [[Listen]] page links to lots of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation]]
** [[Gallery of just intervals]]
* [[EDO]]s and other [[Equal-step tuning]]s
* [[Regular temperament]]
** [[Tour of Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
** [[Historical temperaments]]
* [[People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library]] - a list of electronically-available published works on microtonal/xenharmonic music

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>ry83udutns9cve41i7icocbnxbvng05</sha1>
    </revision>
    <revision>
      <id>68315</id>
      <parentid>68314</parentid>
      <timestamp>2021-04-22T21:03:54Z</timestamp>
      <contributor>
        <username>Mike Battaglia</username>
        <id>3</id>
      </contributor>
      <comment>/* Links to some popular areas of the Xenharmonic Universe */</comment>
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| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__

Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.  

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* The [[Listen]] page links to lots of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation]]
** [[Gallery of just intervals]]
* [[EDO]]s and other [[Equal-step tuning]]s
* [[Regular temperament|A High-Level Intro for Regular Temperaments]]
** [[Tour of Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
** [[Historical temperaments]]
* [[People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library]] - a list of electronically-available published works on microtonal/xenharmonic music

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>98qq5dh2nzonm8ojp9gub1ehgqkgo4n</sha1>
    </revision>
    <revision>
      <id>70305</id>
      <parentid>68315</parentid>
      <timestamp>2021-05-21T21:57:48Z</timestamp>
      <contributor>
        <username>Mike Battaglia</username>
        <id>3</id>
      </contributor>
      <minor/>
      <comment>Removed protection from "[[Main Page]]"</comment>
      <origin>68315</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2804" sha1="98qq5dh2nzonm8ojp9gub1ehgqkgo4n" xml:space="preserve">{{interwiki
| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__

Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.  

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* The [[Listen]] page links to lots of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation]]
** [[Gallery of just intervals]]
* [[EDO]]s and other [[Equal-step tuning]]s
* [[Regular temperament|A High-Level Intro for Regular Temperaments]]
** [[Tour of Regular Temperaments]]
** Mike Battaglia's [[Mike's Lectures On Regular Temperament Theory|Lectures on Regular Temperaments]]
** [[Historical temperaments]]
* [[People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library]] - a list of electronically-available published works on microtonal/xenharmonic music

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>98qq5dh2nzonm8ojp9gub1ehgqkgo4n</sha1>
    </revision>
    <revision>
      <id>70306</id>
      <parentid>70305</parentid>
      <timestamp>2021-05-21T22:01:57Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>flatten links to popular areas</comment>
      <origin>70306</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2590" sha1="ny0hj8ee61hs6kkn773s22rex6zbj5p" xml:space="preserve">{{interwiki
| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__

Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.  

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* The [[Listen]] page links to lots of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[Regular temperaments]] and [[historical temperaments]]
* [[People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library]] - a list of electronically-available published works on microtonal/xenharmonic music

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>ny0hj8ee61hs6kkn773s22rex6zbj5p</sha1>
    </revision>
    <revision>
      <id>70307</id>
      <parentid>70306</parentid>
      <timestamp>2021-05-21T22:04:19Z</timestamp>
      <contributor>
        <username>Mike Battaglia</username>
        <id>3</id>
      </contributor>
      <minor/>
      <comment>Protected "[[Main Page]]": Cmloegcmluin wanted to change something so I let him do it ([Edit=Allow only administrators] (indefinite) [Move=Allow only administrators] (indefinite))</comment>
      <origin>70306</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2590" sha1="ny0hj8ee61hs6kkn773s22rex6zbj5p" xml:space="preserve">{{interwiki
| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__

Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.  

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* The [[Listen]] page links to lots of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Universe==

* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[Regular temperaments]] and [[historical temperaments]]
* [[People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library]] - a list of electronically-available published works on microtonal/xenharmonic music

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>ny0hj8ee61hs6kkn773s22rex6zbj5p</sha1>
    </revision>
    <revision>
      <id>70799</id>
      <parentid>70307</parentid>
      <timestamp>2021-05-27T01:18:38Z</timestamp>
      <contributor>
        <username>Xenjacob</username>
        <id>19</id>
      </contributor>
      <comment>/* Links to some popular areas of the Xenharmonic MULTIverse */ this is an important correction but it should stand to the reasonings here.</comment>
      <origin>70799</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2592" sha1="7wum3zizjwp82f3ygzy1r33tius7br4" xml:space="preserve">{{interwiki
| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__

Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.  

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

==If you are new to musical tuning==

* [[Why use alternative tunings?|Why use alternative tunings?]] is an overview of common reasons for learning about tunings other than standard western tuning.
* The [[Approaches to Musical Tuning]] page offers links to various theoretical approaches to exploring alternative intonation systems
* The [[Listen]] page links to lots of alternatively tuned music that has been made, in case you're wondering what it all sounds like.
* Links to xenharmonic websites: '''[[Links#Maps|Maps]]''' | '''[[Links#Forums|Forums]]''' | '''[[Links#Educational websites|Educational websites]]''' | '''[[Links#Selected microtonal labels and concert producers|Labels and concert producers]]'''

==Links to some popular areas of the Xenharmonic Multiverse==

* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[Regular temperaments]] and [[historical temperaments]]
* [[People]] who have contributed to the overall project of xenharmonic exploration
* [[The Library]] - a list of electronically-available published works on microtonal/xenharmonic music

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>7wum3zizjwp82f3ygzy1r33tius7br4</sha1>
    </revision>
    <revision>
      <id>87274</id>
      <parentid>70799</parentid>
      <timestamp>2022-02-26T11:28:22Z</timestamp>
      <contributor>
        <username>Mike Battaglia</username>
        <id>3</id>
      </contributor>
      <comment>Sevish put together this nice template w/ a picture and cleaned up formatting a bit</comment>
      <origin>87274</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2369" sha1="azwab92zqbvsikdqj2a2kxe2dcwygob" xml:space="preserve">{{interwiki
| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__

Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.  

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to sign up and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]
==If you are new to musical tuning==
* [[Why use alternative tunings?|Why use alternative tunings?]]
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[People|Meet the people]] who have contributed to xenharmonic exploration
* [[Approaches to Musical Tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of electronically-available published works about microtonal/xenharmonic music

==Popular topics==
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]

==Practical xenharmonics==
* [[Useful Tools]]
* [[List of microtonal software plugins]]
* [[Microtonal instruments]]

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>azwab92zqbvsikdqj2a2kxe2dcwygob</sha1>
    </revision>
    <revision>
      <id>95370</id>
      <parentid>87274</parentid>
      <timestamp>2022-09-12T20:19:08Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Actually help ppl to "sign up"</comment>
      <origin>95370</origin>
      <model>wikitext</model>
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      <text bytes="2410" sha1="9xetqdg0ayl35vkzc4qge5ty9qskha2" xml:space="preserve">{{interwiki
| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__

Check out [[Migration_FAQ|this list of anticipated Frequently Asked Questions]] regarding the recent migration of this wiki. 

==Welcome to the Xenharmonic Wiki!==

The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.  

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to [[Help:How to Get Your Xenwiki Account|sign up]] and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]
==If you are new to musical tuning==
* [[Why use alternative tunings?|Why use alternative tunings?]]
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[People|Meet the people]] who have contributed to xenharmonic exploration
* [[Approaches to Musical Tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of electronically-available published works about microtonal/xenharmonic music

==Popular topics==
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]

==Practical xenharmonics==
* [[Useful Tools]]
* [[List of microtonal software plugins]]
* [[Microtonal instruments]]

==But what about rhythm?==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>9xetqdg0ayl35vkzc4qge5ty9qskha2</sha1>
    </revision>
    <revision>
      <id>109553</id>
      <parentid>95370</parentid>
      <timestamp>2023-04-26T06:14:52Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>De-emphasise the message about migration. See talk</comment>
      <origin>109553</origin>
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      <text bytes="2377" sha1="okkcjyattqr9x1nbfjco65ykg1id69q" xml:space="preserve">{{interwiki
| de = Hauptseite
| en = Main Page
| es = Bienvenidos
| ja = ようこそ！Xenharmonicの世界へ
}}__NOTOC__
== Welcome to the Xenharmonic Wiki! ==

The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.  

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to [[Help:How to Get Your Xenwiki Account|sign up]] and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]

Check out [[Migration FAQ]] regarding the migration of this wiki from Wikispaces. 

== If you are new to musical tuning ==
* [[Why use alternative tunings?|Why use alternative tunings?]]
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[People|Meet the people]] who have contributed to xenharmonic exploration
* [[Approaches to Musical Tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of electronically-available published works about microtonal/xenharmonic music

== Popular topics ==
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]

== Practical xenharmonics ==
* [[Useful Tools]]
* [[List of microtonal software plugins]]
* [[Microtonal instruments]]

== But what about rhythm? ==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>okkcjyattqr9x1nbfjco65ykg1id69q</sha1>
    </revision>
    <revision>
      <id>159585</id>
      <parentid>109553</parentid>
      <timestamp>2024-10-12T08:05:46Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Update interwiki</comment>
      <origin>159585</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2371" sha1="f8up9xnm1dfl53z96tcl2y33twrqgzx" xml:space="preserve">{{Interwiki
| en = Main Page
| de = Hauptseite
| es = Bienvenidos
| ja = メインページ
| zh = 主页
}}__NOTOC__
== Welcome to the Xenharmonic Wiki! ==

The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.  

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to [[Help:How to Get Your Xenwiki Account|sign up]] and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]

Check out [[Migration FAQ]] regarding the migration of this wiki from Wikispaces. 

== If you are new to musical tuning ==
* [[Why use alternative tunings?|Why use alternative tunings?]]
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[People|Meet the people]] who have contributed to xenharmonic exploration
* [[Approaches to Musical Tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of electronically-available published works about microtonal/xenharmonic music

== Popular topics ==
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]

== Practical xenharmonics ==
* [[Useful Tools]]
* [[List of microtonal software plugins]]
* [[Microtonal instruments]]

== But what about rhythm? ==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>f8up9xnm1dfl53z96tcl2y33twrqgzx</sha1>
    </revision>
    <revision>
      <id>161168</id>
      <parentid>159585</parentid>
      <timestamp>2024-10-19T13:55:05Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+interwiki link</comment>
      <origin>161168</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2396" sha1="2sbb5td57jv2gcu9e0ietkgeg19lc7a" xml:space="preserve">{{Interwiki
| en = Main Page
| de = Hauptseite
| es = Bienvenidos
| ja = メインページ
| ro = Pagina Principală
| zh = 主页
}}__NOTOC__
== Welcome to the Xenharmonic Wiki! ==
The xenharmonic wiki is an open resource for information about [[musical tuning]]. This includes information about systems of tuning, the relationships between those systems, and the theory and math behind them.  

This wiki is created by volunteers. It is a work in progress, and depends on members of the community to help us develop it. Relevant new content is always welcome. You can get the hang of editing in the [[SandBox]], then dive right in. [[wiki help|Wiki Help]] provides some tips as well. [[Wikifuture]] is a wish list of sorts.

'''Remember that this wiki is a work in progress!''' There are '''many''' new pages that need to be created, and old ones that need to be finished or updated. If you see a way to make something better, please feel free to [[Help:How to Get Your Xenwiki Account|sign up]] and help contribute!

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction|Introduction]].

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]

Check out [[Migration FAQ]] regarding the migration of this wiki from Wikispaces. 

== If you are new to musical tuning ==
* [[Why use alternative tunings?|Why use alternative tunings?]]
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[People|Meet the people]] who have contributed to xenharmonic exploration
* [[Approaches to Musical Tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of electronically-available published works about microtonal/xenharmonic music

== Popular topics ==
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]

== Practical xenharmonics ==
* [[Useful Tools]]
* [[List of microtonal software plugins]]
* [[Microtonal instruments]]

== But what about rhythm? ==
If you're interested in the study and practice of world rhythms, irrational time signatures, and harmonic polyrhythms, visit the [[http://xenrhythmic.wikia.com/wiki/The_Xenrhythm_Wikia Xenrhythm wiki]]

&lt;span style="font-size: 120%;"&gt;Don't forget about the search function at the top of the page!&lt;/span&gt;</text>
      <sha1>2sbb5td57jv2gcu9e0ietkgeg19lc7a</sha1>
    </revision>
    <revision>
      <id>190056</id>
      <parentid>161168</parentid>
      <timestamp>2025-04-06T15:43:28Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Implement part of the changes proposed on [[Talk:Main Page/Revision proposals]]</comment>
      <origin>190056</origin>
      <model>wikitext</model>
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      <text bytes="1731" sha1="hxjems9weriiwsyihxbpidphxh2twuh" xml:space="preserve">{{Interwiki
| en = Main Page
| de = Hauptseite
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| ja = メインページ
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| zh = 主页
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== Welcome to the Xenharmonic Wiki! ==
&lt;big&gt;The Xenharmonic Wiki is an open resource for information about [[musical tuning]]. This information includes tuning systems and their relationships, [[General theory|theory]], [[Mathematical theory|math]] and [[practice]].&lt;/big&gt;

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction]].

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]
== If you are new to musical tuning ==
* [[Why use alternative tunings?]]
* What is [[microtonal music]]? How is it related to [[xenharmonic music]]?
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[People|Meet the people]] who have contributed to xenharmonic exploration
* [[List of approaches to musical tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of published works about microtonal/xenharmonic music

== Popular topics ==
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]

== Practical xenharmonics ==
* [[List of music software]]
* [[List of microtonal software plugins]]
* [[Instruments]]

== Contributing to the Xenharmonic Wiki ==
This wiki is created by volunteers. It is a perpetual work in progress, depending on members of the community to help us develop it. We welcome relevant new content and constructive updates to existing pages, so please feel free to [[Help:How to Get Your Xenwiki Account|sign up and contribute]]!</text>
      <sha1>hxjems9weriiwsyihxbpidphxh2twuh</sha1>
    </revision>
    <revision>
      <id>190180</id>
      <parentid>190056</parentid>
      <timestamp>2025-04-08T04:21:13Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Remove "big" text tag in lead section (issues on mobile); improve wording for microtonal and xenharmonic music; remove link to People and mention XA instead; link to Guides; misc. edits</comment>
      <origin>190180</origin>
      <model>wikitext</model>
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      <text bytes="1730" sha1="e0y6f39vnd0gz3bbbard2s0t64pt7v4" xml:space="preserve">{{Interwiki
| en = Main Page
| de = Hauptseite
| es = Bienvenidos
| ja = メインページ
| ro = Pagina Principală
| zh = 主页
}}__NOTOC__
== Welcome to the Xenharmonic Wiki! ==
The [[Xenharmonic Wiki]] is an open resource for information about [[musical tuning]]. This information includes tuning systems and their relationships, [[General theory|theory]], [[Mathematical theory|math]] and [[practice]].

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction]].

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]
== If you are new to musical tuning ==
* [[Why use alternative tunings?]]
* [[Microtonal music|What are microtonality and xenharmony]]?
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[List of approaches to musical tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of published works about microtonal/xenharmonic music
* Learn about the [[Xenharmonic Alliance]], a social group of xenharmonic musicians

== Popular topics ==
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]

== Practical xenharmonics ==
* [[List of music software]]
* [[List of microtonal software plugins]]
* [[Instruments]]
* [[Guides]]

== Contributing to the Xenharmonic Wiki ==
This wiki is created by volunteers. It is a perpetual work in progress, depending on members of the community to help us develop it. We welcome relevant new content and constructive updates to existing pages, so please feel free to [[Help:How to Get Your Xenwiki Account|sign up and contribute]]!</text>
      <sha1>e0y6f39vnd0gz3bbbard2s0t64pt7v4</sha1>
    </revision>
    <revision>
      <id>190191</id>
      <parentid>190180</parentid>
      <timestamp>2025-04-08T11:06:19Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>adding the word "music" benefits the bullet point imo</comment>
      <origin>190191</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1734" sha1="4x18lyodnyptd7ddtyqt2cdu4ehyxdh" xml:space="preserve">{{Interwiki
| en = Main Page
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| es = Bienvenidos
| ja = メインページ
| ro = Pagina Principală
| zh = 主页
}}__NOTOC__
== Welcome to the Xenharmonic Wiki! ==
The [[Xenharmonic Wiki]] is an open resource for information about [[musical tuning]]. This information includes tuning systems and their relationships, [[General theory|theory]], [[Mathematical theory|math]] and [[practice]].

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction]].

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]
== If you are new to musical tuning ==
* [[Why use alternative tunings?]]
* [[Microtonal music|What are microtonal and xenharmonic music]]?
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[List of approaches to musical tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of published works about microtonal/xenharmonic music
* Learn about the [[Xenharmonic Alliance]], a social group of xenharmonic musicians

== Popular topics ==
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]

== Practical xenharmonics ==
* [[List of music software]]
* [[List of microtonal software plugins]]
* [[Instruments]]
* [[Guides]]

== Contributing to the Xenharmonic Wiki ==
This wiki is created by volunteers. It is a perpetual work in progress, depending on members of the community to help us develop it. We welcome relevant new content and constructive updates to existing pages, so please feel free to [[Help:How to Get Your Xenwiki Account|sign up and contribute]]!</text>
      <sha1>4x18lyodnyptd7ddtyqt2cdu4ehyxdh</sha1>
    </revision>
    <revision>
      <id>212306</id>
      <parentid>190191</parentid>
      <timestamp>2025-10-08T03:22:19Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Rework opening sentence, based on the proposition on the talk page by [[User:BudjarnLambeth]] and others</comment>
      <origin>212306</origin>
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| ja = メインページ
| ro = Pagina Principală
| zh = 主页
}}__NOTOC__
== Welcome to the Xenharmonic Wiki! ==
The Xenharmonic Wiki is an open resource dedicated to [[musical tuning]] systems, focusing on [[xenharmonic music]] while also documenting [[historical tunings]] and tuning practices from [[world musical traditions|world traditions]]. It covers the [[theory]] and [[practice|practical applications]] of these systems.

For a lengthier introduction, see [[Xenharmonic Wiki:Introduction]].

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]
== If you are new to musical tuning ==
* [[Why use alternative tunings?]]
* [[Microtonal music|What are microtonal and xenharmonic music]]?
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[List of approaches to musical tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of published works about microtonal/xenharmonic music
* Learn about the [[Xenharmonic Alliance]], a social group of xenharmonic musicians

== Popular topics ==
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]

== Practical xenharmonics ==
* [[List of music software]]
* [[List of microtonal software plugins]]
* [[Instruments]]
* [[Guides]]

== Contributing to the Xenharmonic Wiki ==
This wiki is created by volunteers. It is a perpetual work in progress, depending on members of the community to help us develop it. We welcome relevant new content and constructive updates to existing pages, so please feel free to [[Help:How to Get Your Xenwiki Account|sign up and contribute]]!</text>
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== Welcome to the Xenharmonic Wiki! ==
The Xenharmonic Wiki is an open resource dedicated to musical [[tuning system]]s, focusing on [[xenharmonic music]] while also documenting [[historical tunings]] and tuning practices from [[world musical traditions|world traditions]]. It covers the [[theory]] and [[practice|practical applications]] of these systems.

For a lengthier introduction, see [[Xenharmonic Wiki: Introduction]].

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]
== If you are new to musical tuning ==
* [[Why use alternative tunings?]]
* [[Microtonal music|What are microtonal and xenharmonic music]]?
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[List of approaches to musical tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of published works about microtonal/xenharmonic music
* Learn about the [[Xenharmonic Alliance]], a social group of xenharmonic musicians

== Popular topics ==
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]

== Practical xenharmonics ==
* [[List of music software]]
* [[List of microtonal software plugins]]
* [[Instruments]]
* [[Guides]]

== Contributing to the Xenharmonic Wiki ==
This wiki is created by volunteers. It is a perpetual work in progress, depending on members of the community to help us develop it. We welcome relevant new content and constructive updates to existing pages, so please feel free to [[Help: How to get your Xenharmonic Wiki account|sign up and contribute]]!</text>
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== Welcome to the Xenharmonic Wiki! ==
The [[Xenharmonic Wiki]] is an open resource dedicated to musical [[tuning system]]s, focusing on [[xenharmonic music]] while also documenting [[historical tunings]] and tuning practices from [[world musical traditions|world traditions]]. It covers the [[theory]] and [[practice|practical applications]] of these systems.

For a lengthier introduction, see [[Xenharmonic Wiki: Introduction]].

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb|right]]
== If you are new to musical tuning ==
* [[Why use alternative tunings?]]
* [[Microtonal music|What are microtonal and xenharmonic music]]?
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[List of approaches to musical tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of published works about microtonal/xenharmonic music
* Learn about the [[Xenharmonic Alliance]], a social group of xenharmonic musicians

== Popular topics ==
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]

== Practical xenharmonics ==
* [[List of music software]]
* [[List of microtonal software plugins]]
* [[Instruments]]
* [[Guides]]

== Contributing to the Xenharmonic Wiki ==
This wiki is created by volunteers. It is a perpetual work in progress, depending on members of the community to help us develop it. We welcome relevant new content and constructive updates to existing pages, so please feel free to [[Help: How to get your Xenharmonic Wiki account|sign up and contribute]]!</text>
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== Welcome to the Xenharmonic Wiki! ==
The [[Xenharmonic Wiki]] is an open resource dedicated to musical [[tuning system]]s, focusing on [[xenharmonic music]] while also documenting [[historical tunings]] and tuning practices from [[world musical traditions|world traditions]]. It covers the [[theory]] and [[practice|practical applications]] of these systems.

For a lengthier introduction, see [[Xenharmonic Wiki: Introduction]].

[[File:Pts-2-3-5-e2-twtop-tlin.jpg|thumb]]
== If you are new to musical tuning ==
* [[Why use alternative tunings?]]
* [[Microtonal music|What are microtonal and xenharmonic music]]?
* [[Listen]] to alternatively tuned music, in case you're wondering what it all sounds like.
* [[List of approaches to musical tuning|Discover approaches to musical tuning]]
* [[Links|Explore links]] to xenharmonic websites
* [[The Library|Browse the library]] of published works about microtonal/xenharmonic music
* Learn about the [[Xenharmonic Alliance]], a social group of xenharmonic musicians

== Popular topics ==
* [[Just intonation]]
* [[EDO]]s and other [[equal-step tuning]]s
* [[MOS scale|Moment of Symmetry (MOS) scales]]
* [[Regular temperaments]]
* [[Historical temperaments]]

== Practical xenharmonics ==
* [[List of music software]]
* [[List of microtonal software plugins]]
* [[Instruments]]
* [[Guides]]

== Contributing to the Xenharmonic Wiki ==
This wiki is created by volunteers. It is a perpetual work in progress, depending on members of the community to help us develop it. We welcome relevant new content and constructive updates to existing pages, so please feel free to [[Help: How to get your Xenharmonic Wiki account|sign up and contribute]]!</text>
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|-
! | g (deg/edt)
! | G (deg/edt)
! | cardinality
! | edt
|-
| | 33/133
| | 1/4
| | 137
| | 532
|-
| | 32/129
| | 1/4
| | 133
| | 516
|-
| | 31/125
| | 1/4
| | 129
| | 500
|-
| | 30/121
| | 1/4
| | 125
| | 484
|-
| | 29/117
| | 1/4
| | 121
| | 468
|-
| | 28/113
| | 1/4
| | 117
| | 452
|-
| | 27/109
| | 1/4
| | 113
| | 436
|-
| | 26/105
| | 1/4
| | 109
| | 420
|-
| | 25/101
| | 1/4
| | 105
| | 404
|-
| | 24/97
| | 1/4
| | 101
| | 388
|-
| | 23/93
| | 1/4
| | 97
| | 372
|-
| | 22/89
| | 1/4
| | 93
| | 356
|-
| | 21/85
| | 1/4
| | 89
| | 340
|-
| | 20/81
| | 1/4
| | 85
| | 324
|-
| | 19/77
| | 1/4
| | 81
| | 308
|-
| | 18/73
| | 1/4
| | 77
| | 292
|-
| | 17/69
| | 1/4
| | 73
| | 276
|-
| | 16/65
| | 1/4
| | 69
| | 260
|-
| | 15/61
| | 1/4
| | 65
| | 244
|-
| | 14/57
| | 1/4
| | 61
| | 228
|-
| | 13/53
| | 1/4
| | 57
| | 212
|-
| | 12/49
| | 1/4
| | 53
| | 196
|-
| | 11/45
| | 1/4
| | 49
| | 180
|-
| | 10/41
| | 1/4
| | 45
| | 164
|-
| | 9/37
| | 1/4
| | 41
| | 148
|-
| | 8/33
| | 1/4
| | 37
| | 132
|-
| | 7/29
| | 1/4
| | 33
| | 116
|-
| | 6/25
| | 1/4
| | 29
| | 100
|-
| | 5/21
| | 1/4
| | 25
| | 84
|-
| | 4/17
| | 1/4
| | 21
| | 68
|-
| | 3/13
| | 1/4
| | 17
| | 52
|-
| | 2/9
| | 1/4
| | 13
| | 36
|-
| | 1/5
| | 1/4
| | 9
| | 20
|-
| | 1/5
| | 2/9
| | 14
| | 45
|-
| | 1/5
| | 3/14
| | 19
| | 70
|-
| | 1/5
| | 4/19
| | 24
| | 95
|-
| | 1/5
| | 5/24
| | 29
| | 120
|-
| | 1/5
| | 6/29
| | 34
| | 145
|-
| | 1/5
| | 7/34
| | 39
| | 170
|-
| | 1/5
| | 8/39
| | 44
| | 195
|-
| | 1/5
| | 9/44
| | 49
| | 220
|-
| | 1/5
| | 10/49
| | 54
| | 245
|-
| | 1/5
| | 11/54
| | 59
| | 270
|-
| | 1/5
| | 12/59
| | 64
| | 295
|-
| | 1/5
| | 13/64
| | 69
| | 320
|-
| | 1/5
| | 14/69
| | 74
| | 345
|-
| | 1/5
| | 15/74
| | 79
| | 370
|-
| | 1/5
| | 16/79
| | 84
| | 395
|-
| | 1/5
| | 17/84
| | 89
| | 420
|-
| | 1/5
| | 18/89
| | 94
| | 445
|-
| | 1/5
| | 19/94
| | 99
| | 470
|-
| | 1/5
| | 20/99
| | 104
| | 495
|-
| | 1/5
| | 21/104
| | 109
| | 520
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
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|| 32/129 || 1/4 || 133 || 516 ||
|| 31/125 || 1/4 || 129 || 500 ||
|| 30/121 || 1/4 || 125 || 484 ||
|| 29/117 || 1/4 || 121 || 468 ||
|| 28/113 || 1/4 || 117 || 452 ||
|| 27/109 || 1/4 || 113 || 436 ||
|| 26/105 || 1/4 || 109 || 420 ||
|| 25/101 || 1/4 || 105 || 404 ||
|| 24/97 || 1/4 || 101 || 388 ||
|| 23/93 || 1/4 || 97 || 372 ||
|| 22/89 || 1/4 || 93 || 356 ||
|| 21/85 || 1/4 || 89 || 340 ||
|| 20/81 || 1/4 || 85 || 324 ||
|| 19/77 || 1/4 || 81 || 308 ||
|| 18/73 || 1/4 || 77 || 292 ||
|| 17/69 || 1/4 || 73 || 276 ||
|| 16/65 || 1/4 || 69 || 260 ||
|| 15/61 || 1/4 || 65 || 244 ||
|| 14/57 || 1/4 || 61 || 228 ||
|| 13/53 || 1/4 || 57 || 212 ||
|| 12/49 || 1/4 || 53 || 196 ||
|| 11/45 || 1/4 || 49 || 180 ||
|| 10/41 || 1/4 || 45 || 164 ||
|| 9/37 || 1/4 || 41 || 148 ||
|| 8/33 || 1/4 || 37 || 132 ||
|| 7/29 || 1/4 || 33 || 116 ||
|| 6/25 || 1/4 || 29 || 100 ||
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|| 3/13 || 1/4 || 17 || 52 ||
|| 2/9 || 1/4 || 13 || 36 ||
|| 1/5 || 1/4 || 9 || 20 ||
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|| 1/5 || 11/54 || 59 || 270 ||
|| 1/5 || 12/59 || 64 || 295 ||
|| 1/5 || 13/64 || 69 || 320 ||
|| 1/5 || 14/69 || 74 || 345 ||
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|| 1/5 || 17/84 || 89 || 420 ||
|| 1/5 || 18/89 || 94 || 445 ||
|| 1/5 || 19/94 || 99 || 470 ||
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&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21/104&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;520&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>tfnuseskrb0oo4pzetylkxu5w23lcut</sha1>
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    <revision>
      <id>108864</id>
      <parentid>2</parentid>
      <timestamp>2023-04-23T19:07:35Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Categorised this uncategorised page</comment>
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      <format>text/x-wiki</format>
      <text bytes="2023" sha1="dp9n6u8darm3o4c934x32d63ra0802l" xml:space="preserve">{| class="wikitable"
|-
! | g (deg/edt)
! | G (deg/edt)
! | cardinality
! | edt
|-
| | 33/133
| | 1/4
| | 137
| | 532
|-
| | 32/129
| | 1/4
| | 133
| | 516
|-
| | 31/125
| | 1/4
| | 129
| | 500
|-
| | 30/121
| | 1/4
| | 125
| | 484
|-
| | 29/117
| | 1/4
| | 121
| | 468
|-
| | 28/113
| | 1/4
| | 117
| | 452
|-
| | 27/109
| | 1/4
| | 113
| | 436
|-
| | 26/105
| | 1/4
| | 109
| | 420
|-
| | 25/101
| | 1/4
| | 105
| | 404
|-
| | 24/97
| | 1/4
| | 101
| | 388
|-
| | 23/93
| | 1/4
| | 97
| | 372
|-
| | 22/89
| | 1/4
| | 93
| | 356
|-
| | 21/85
| | 1/4
| | 89
| | 340
|-
| | 20/81
| | 1/4
| | 85
| | 324
|-
| | 19/77
| | 1/4
| | 81
| | 308
|-
| | 18/73
| | 1/4
| | 77
| | 292
|-
| | 17/69
| | 1/4
| | 73
| | 276
|-
| | 16/65
| | 1/4
| | 69
| | 260
|-
| | 15/61
| | 1/4
| | 65
| | 244
|-
| | 14/57
| | 1/4
| | 61
| | 228
|-
| | 13/53
| | 1/4
| | 57
| | 212
|-
| | 12/49
| | 1/4
| | 53
| | 196
|-
| | 11/45
| | 1/4
| | 49
| | 180
|-
| | 10/41
| | 1/4
| | 45
| | 164
|-
| | 9/37
| | 1/4
| | 41
| | 148
|-
| | 8/33
| | 1/4
| | 37
| | 132
|-
| | 7/29
| | 1/4
| | 33
| | 116
|-
| | 6/25
| | 1/4
| | 29
| | 100
|-
| | 5/21
| | 1/4
| | 25
| | 84
|-
| | 4/17
| | 1/4
| | 21
| | 68
|-
| | 3/13
| | 1/4
| | 17
| | 52
|-
| | 2/9
| | 1/4
| | 13
| | 36
|-
| | 1/5
| | 1/4
| | 9
| | 20
|-
| | 1/5
| | 2/9
| | 14
| | 45
|-
| | 1/5
| | 3/14
| | 19
| | 70
|-
| | 1/5
| | 4/19
| | 24
| | 95
|-
| | 1/5
| | 5/24
| | 29
| | 120
|-
| | 1/5
| | 6/29
| | 34
| | 145
|-
| | 1/5
| | 7/34
| | 39
| | 170
|-
| | 1/5
| | 8/39
| | 44
| | 195
|-
| | 1/5
| | 9/44
| | 49
| | 220
|-
| | 1/5
| | 10/49
| | 54
| | 245
|-
| | 1/5
| | 11/54
| | 59
| | 270
|-
| | 1/5
| | 12/59
| | 64
| | 295
|-
| | 1/5
| | 13/64
| | 69
| | 320
|-
| | 1/5
| | 14/69
| | 74
| | 345
|-
| | 1/5
| | 15/74
| | 79
| | 370
|-
| | 1/5
| | 16/79
| | 84
| | 395
|-
| | 1/5
| | 17/84
| | 89
| | 420
|-
| | 1/5
| | 18/89
| | 94
| | 445
|-
| | 1/5
| | 19/94
| | 99
| | 470
|-
| | 1/5
| | 20/99
| | 104
| | 495
|-
| | 1/5
| | 21/104
| | 109
| | 520
|}
[[Category:Table]]
[[Category:Tritave]]</text>
      <sha1>dp9n6u8darm3o4c934x32d63ra0802l</sha1>
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    <revision>
      <id>108906</id>
      <parentid>108864</parentid>
      <timestamp>2023-04-23T19:31:46Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Table]] to [[Category:Tables]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
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|-
! | g (deg/edt)
! | G (deg/edt)
! | cardinality
! | edt
|-
| | 33/133
| | 1/4
| | 137
| | 532
|-
| | 32/129
| | 1/4
| | 133
| | 516
|-
| | 31/125
| | 1/4
| | 129
| | 500
|-
| | 30/121
| | 1/4
| | 125
| | 484
|-
| | 29/117
| | 1/4
| | 121
| | 468
|-
| | 28/113
| | 1/4
| | 117
| | 452
|-
| | 27/109
| | 1/4
| | 113
| | 436
|-
| | 26/105
| | 1/4
| | 109
| | 420
|-
| | 25/101
| | 1/4
| | 105
| | 404
|-
| | 24/97
| | 1/4
| | 101
| | 388
|-
| | 23/93
| | 1/4
| | 97
| | 372
|-
| | 22/89
| | 1/4
| | 93
| | 356
|-
| | 21/85
| | 1/4
| | 89
| | 340
|-
| | 20/81
| | 1/4
| | 85
| | 324
|-
| | 19/77
| | 1/4
| | 81
| | 308
|-
| | 18/73
| | 1/4
| | 77
| | 292
|-
| | 17/69
| | 1/4
| | 73
| | 276
|-
| | 16/65
| | 1/4
| | 69
| | 260
|-
| | 15/61
| | 1/4
| | 65
| | 244
|-
| | 14/57
| | 1/4
| | 61
| | 228
|-
| | 13/53
| | 1/4
| | 57
| | 212
|-
| | 12/49
| | 1/4
| | 53
| | 196
|-
| | 11/45
| | 1/4
| | 49
| | 180
|-
| | 10/41
| | 1/4
| | 45
| | 164
|-
| | 9/37
| | 1/4
| | 41
| | 148
|-
| | 8/33
| | 1/4
| | 37
| | 132
|-
| | 7/29
| | 1/4
| | 33
| | 116
|-
| | 6/25
| | 1/4
| | 29
| | 100
|-
| | 5/21
| | 1/4
| | 25
| | 84
|-
| | 4/17
| | 1/4
| | 21
| | 68
|-
| | 3/13
| | 1/4
| | 17
| | 52
|-
| | 2/9
| | 1/4
| | 13
| | 36
|-
| | 1/5
| | 1/4
| | 9
| | 20
|-
| | 1/5
| | 2/9
| | 14
| | 45
|-
| | 1/5
| | 3/14
| | 19
| | 70
|-
| | 1/5
| | 4/19
| | 24
| | 95
|-
| | 1/5
| | 5/24
| | 29
| | 120
|-
| | 1/5
| | 6/29
| | 34
| | 145
|-
| | 1/5
| | 7/34
| | 39
| | 170
|-
| | 1/5
| | 8/39
| | 44
| | 195
|-
| | 1/5
| | 9/44
| | 49
| | 220
|-
| | 1/5
| | 10/49
| | 54
| | 245
|-
| | 1/5
| | 11/54
| | 59
| | 270
|-
| | 1/5
| | 12/59
| | 64
| | 295
|-
| | 1/5
| | 13/64
| | 69
| | 320
|-
| | 1/5
| | 14/69
| | 74
| | 345
|-
| | 1/5
| | 15/74
| | 79
| | 370
|-
| | 1/5
| | 16/79
| | 84
| | 395
|-
| | 1/5
| | 17/84
| | 89
| | 420
|-
| | 1/5
| | 18/89
| | 94
| | 445
|-
| | 1/5
| | 19/94
| | 99
| | 470
|-
| | 1/5
| | 20/99
| | 104
| | 495
|-
| | 1/5
| | 21/104
| | 109
| | 520
|}
[[Category:Tables]]
[[Category:Tritave]]</text>
      <sha1>jjgo1brgru2ubs12vqbdxv88ts6rzjd</sha1>
    </revision>
    <revision>
      <id>154928</id>
      <parentid>108906</parentid>
      <timestamp>2024-09-13T15:08:07Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>154928</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1779" sha1="6mef3lwen75501anjxe422g45ppazh5" xml:space="preserve">&lt;onlyinclude&gt;{| class="wikitable"
! g (deg/edt) !! G (deg/edt) !! Cardinality !! Edt
|-
| 33/133 || 1/4 || 137 || 532
|-
| 32/129 || 1/4 || 133 || 516
|-
| 31/125 || 1/4 || 129 || 500
|-
| 30/121 || 1/4 || 125 || 484
|-
| 29/117 || 1/4 || 121 || 468
|-
| 28/113 || 1/4 || 117 || 452
|-
| 27/109 || 1/4 || 113 || 436
|-
| 26/105 || 1/4 || 109 || 420
|-
| 25/101 || 1/4 || 105 || 404
|-
| 24/97 || 1/4 || 101 || 388
|-
| 23/93 || 1/4 || 97 || 372
|-
| 22/89 || 1/4 || 93 || 356
|-
| 21/85 || 1/4 || 89 || 340
|-
| 20/81 || 1/4 || 85 || 324
|-
| 19/77 || 1/4 || 81 || 308
|-
| 18/73 || 1/4 || 77 || 292
|-
| 17/69 || 1/4 || 73 || 276
|-
| 16/65 || 1/4 || 69 || 260
|-
| 15/61 || 1/4 || 65 || 244
|-
| 14/57 || 1/4 || 61 || 228
|-
| 13/53 || 1/4 || 57 || 212
|-
| 12/49 || 1/4 || 53 || 196
|-
| 11/45 || 1/4 || 49 || 180
|-
| 10/41 || 1/4 || 45 || 164
|-
| 9/37 || 1/4 || 41 || 148
|-
| 8/33 || 1/4 || 37 || 132
|-
| 7/29 || 1/4 || 33 || 116
|-
| 6/25 || 1/4 || 29 || 100
|-
| 5/21 || 1/4 || 25 || 84
|-
| 4/17 || 1/4 || 21 || 68
|-
| 3/13 || 1/4 || 17 || 52
|-
| 2/9 || 1/4 || 13 || 36
|-
| 1/5 || 1/4 || 9 || 20
|-
| 1/5 || 2/9 || 14 || 45
|-
| 1/5 || 3/14 || 19 || 70
|-
| 1/5 || 4/19 || 24 || 95
|-
| 1/5 || 5/24 || 29 || 120
|-
| 1/5 || 6/29 || 34 || 145
|-
| 1/5 || 7/34 || 39 || 170
|-
| 1/5 || 8/39 || 44 || 195
|-
| 1/5 || 9/44 || 49 || 220
|-
| 1/5 || 10/49 || 54 || 245
|-
| 1/5 || 11/54 || 59 || 270
|-
| 1/5 || 12/59 || 64 || 295
|-
| 1/5 || 13/64 || 69 || 320
|-
| 1/5 || 14/69 || 74 || 345
|-
| 1/5 || 15/74 || 79 || 370
|-
| 1/5 || 16/79 || 84 || 395
|-
| 1/5 || 17/84 || 89 || 420
|-
| 1/5 || 18/89 || 94 || 445
|-
| 1/5 || 19/94 || 99 || 470
|-
| 1/5 || 20/99 || 104 || 495
|-
| 1/5 || 21/104 || 109 || 520
|}&lt;/onlyinclude&gt;

[[Category:Tables]]
[[Category:Tritave]]</text>
      <sha1>6mef3lwen75501anjxe422g45ppazh5</sha1>
    </revision>
    <revision>
      <id>154929</id>
      <parentid>154928</parentid>
      <timestamp>2024-09-13T15:12:59Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
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      <minor/>
      <origin>154929</origin>
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      <text bytes="1904" sha1="7dbu2qd5ary9tgytrdmqqdfzw3nngbh" xml:space="preserve">&lt;onlyinclude&gt;{| class="wikitable"
! g (deg/edt) !! G (deg/edt) !! Cardinality !! Edt
|-
| 33/133 || 1/4 || 137 || 532
|-
| 32/129 || 1/4 || 133 || 516
|-
| 31/125 || 1/4 || 129 || 500
|-
| 30/121 || 1/4 || 125 || 484
|-
| 29/117 || 1/4 || 121 || 468
|-
| 28/113 || 1/4 || 117 || 452
|-
| 27/109 || 1/4 || 113 || 436
|-
| 26/105 || 1/4 || 109 || 420
|-
| 25/101 || 1/4 || 105 || 404
|-
| 24/97 || 1/4 || 101 || 388
|-
| 23/93 || 1/4 || 97 || 372
|-
| 22/89 || 1/4 || 93 || 356
|-
| 21/85 || 1/4 || 89 || 340
|-
| 20/81 || 1/4 || 85 || 324
|-
| 19/77 || 1/4 || 81 || 308
|-
| 18/73 || 1/4 || 77 || 292
|-
| 17/69 || 1/4 || 73 || 276
|-
| 16/65 || 1/4 || 69 || 260
|-
| 15/61 || 1/4 || 65 || 244
|-
| 14/57 || 1/4 || 61 || 228
|-
| 13/53 || 1/4 || 57 || 212
|-
| 12/49 || 1/4 || 53 || 196
|-
| 11/45 || 1/4 || 49 || 180
|-
| 10/41 || 1/4 || 45 || 164
|-
| 9/37 || 1/4 || 41 || 148
|-
| 8/33 || 1/4 || 37 || 132
|-
| 7/29 || 1/4 || 33 || 116
|-
| 6/25 || 1/4 || 29 || 100
|-
| 5/21 || 1/4 || 25 || 84
|-
| 4/17 || 1/4 || 21 || 68
|-
| 3/13 || 1/4 || 17 || 52
|-
| 2/9 || 1/4 || 13 || 36
|-
| 1/5 || 1/4 || 9 || 20
|-
| 1/5 || 2/9 || 14 || 45
|-
| 1/5 || 3/14 || 19 || 70
|-
| 1/5 || 4/19 || 24 || 95
|-
| 1/5 || 5/24 || 29 || 120
|-
| 1/5 || 6/29 || 34 || 145
|-
| 1/5 || 7/34 || 39 || 170
|-
| 1/5 || 8/39 || 44 || 195
|-
| 1/5 || 9/44 || 49 || 220
|-
| 1/5 || 10/49 || 54 || 245
|-
| 1/5 || 11/54 || 59 || 270
|-
| 1/5 || 12/59 || 64 || 295
|-
| 1/5 || 13/64 || 69 || 320
|-
| 1/5 || 14/69 || 74 || 345
|-
| 1/5 || 15/74 || 79 || 370
|-
| 1/5 || 16/79 || 84 || 395
|-
| 1/5 || 17/84 || 89 || 420
|-
| 1/5 || 18/89 || 94 || 445
|-
| 1/5 || 19/94 || 99 || 470
|-
| 1/5 || 20/99 || 104 || 495
|-
| 1/5 || 21/104 || 109 || 520
&lt;includeonly&gt;
|- style="text-align: center;"
| colspan="4" | {{Navbar|(Anti Bohlen-Pierce)|text=This table:}}
&lt;/includeonly&gt;
|}&lt;/onlyinclude&gt;

[[Category:Tables]]
[[Category:Tritave]]</text>
      <sha1>7dbu2qd5ary9tgytrdmqqdfzw3nngbh</sha1>
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    <revision>
      <id>154940</id>
      <parentid>154929</parentid>
      <timestamp>2024-09-13T15:29:08Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
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      <text bytes="1900" sha1="dy6bwo79k59mtd7ijt1zaqhqm3tx673" xml:space="preserve">{| class="wikitable"
! g (deg/edt) !! G (deg/edt) !! Cardinality !! Edt
|-
| 33/133 || 1/4 || 137 || 532
|-
| 32/129 || 1/4 || 133 || 516
|-
| 31/125 || 1/4 || 129 || 500
|-
| 30/121 || 1/4 || 125 || 484
|-
| 29/117 || 1/4 || 121 || 468
|-
| 28/113 || 1/4 || 117 || 452
|-
| 27/109 || 1/4 || 113 || 436
|-
| 26/105 || 1/4 || 109 || 420
|-
| 25/101 || 1/4 || 105 || 404
|-
| 24/97 || 1/4 || 101 || 388
|-
| 23/93 || 1/4 || 97 || 372
|-
| 22/89 || 1/4 || 93 || 356
|-
| 21/85 || 1/4 || 89 || 340
|-
| 20/81 || 1/4 || 85 || 324
|-
| 19/77 || 1/4 || 81 || 308
|-
| 18/73 || 1/4 || 77 || 292
|-
| 17/69 || 1/4 || 73 || 276
|-
| 16/65 || 1/4 || 69 || 260
|-
| 15/61 || 1/4 || 65 || 244
|-
| 14/57 || 1/4 || 61 || 228
|-
| 13/53 || 1/4 || 57 || 212
|-
| 12/49 || 1/4 || 53 || 196
|-
| 11/45 || 1/4 || 49 || 180
|-
| 10/41 || 1/4 || 45 || 164
|-
| 9/37 || 1/4 || 41 || 148
|-
| 8/33 || 1/4 || 37 || 132
|-
| 7/29 || 1/4 || 33 || 116
|-
| 6/25 || 1/4 || 29 || 100
|-
| 5/21 || 1/4 || 25 || 84
|-
| 4/17 || 1/4 || 21 || 68
|-
| 3/13 || 1/4 || 17 || 52
|-
| 2/9 || 1/4 || 13 || 36
|-
| 1/5 || 1/4 || 9 || 20
|-
| 1/5 || 2/9 || 14 || 45
|-
| 1/5 || 3/14 || 19 || 70
|-
| 1/5 || 4/19 || 24 || 95
|-
| 1/5 || 5/24 || 29 || 120
|-
| 1/5 || 6/29 || 34 || 145
|-
| 1/5 || 7/34 || 39 || 170
|-
| 1/5 || 8/39 || 44 || 195
|-
| 1/5 || 9/44 || 49 || 220
|-
| 1/5 || 10/49 || 54 || 245
|-
| 1/5 || 11/54 || 59 || 270
|-
| 1/5 || 12/59 || 64 || 295
|-
| 1/5 || 13/64 || 69 || 320
|-
| 1/5 || 14/69 || 74 || 345
|-
| 1/5 || 15/74 || 79 || 370
|-
| 1/5 || 16/79 || 84 || 395
|-
| 1/5 || 17/84 || 89 || 420
|-
| 1/5 || 18/89 || 94 || 445
|-
| 1/5 || 19/94 || 99 || 470
|-
| 1/5 || 20/99 || 104 || 495
|-
| 1/5 || 21/104 || 109 || 520
&lt;includeonly&gt;
|- style="text-align: center;"
| colspan="4" | {{Navbar|(Anti Bohlen-Pierce)|text=This table:}}
&lt;/includeonly&gt;
|}&lt;noinclude&gt;
[[Category:Tables]]
[[Category:Tritave]]
&lt;/noinclude&gt;</text>
      <sha1>dy6bwo79k59mtd7ijt1zaqhqm3tx673</sha1>
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    <revision>
      <id>177301</id>
      <parentid>154940</parentid>
      <timestamp>2025-01-20T14:35:39Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
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      <minor/>
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      <text bytes="1903" sha1="2phi0gncn44tfngymif5h0vrzciaaos" xml:space="preserve">{| class="wikitable"
|-
! g (deg/edt) !! G (deg/edt) !! Cardinality !! EDT
|-
| 33/133 || 1/4 || 137 || 532
|-
| 32/129 || 1/4 || 133 || 516
|-
| 31/125 || 1/4 || 129 || 500
|-
| 30/121 || 1/4 || 125 || 484
|-
| 29/117 || 1/4 || 121 || 468
|-
| 28/113 || 1/4 || 117 || 452
|-
| 27/109 || 1/4 || 113 || 436
|-
| 26/105 || 1/4 || 109 || 420
|-
| 25/101 || 1/4 || 105 || 404
|-
| 24/97 || 1/4 || 101 || 388
|-
| 23/93 || 1/4 || 97 || 372
|-
| 22/89 || 1/4 || 93 || 356
|-
| 21/85 || 1/4 || 89 || 340
|-
| 20/81 || 1/4 || 85 || 324
|-
| 19/77 || 1/4 || 81 || 308
|-
| 18/73 || 1/4 || 77 || 292
|-
| 17/69 || 1/4 || 73 || 276
|-
| 16/65 || 1/4 || 69 || 260
|-
| 15/61 || 1/4 || 65 || 244
|-
| 14/57 || 1/4 || 61 || 228
|-
| 13/53 || 1/4 || 57 || 212
|-
| 12/49 || 1/4 || 53 || 196
|-
| 11/45 || 1/4 || 49 || 180
|-
| 10/41 || 1/4 || 45 || 164
|-
| 9/37 || 1/4 || 41 || 148
|-
| 8/33 || 1/4 || 37 || 132
|-
| 7/29 || 1/4 || 33 || 116
|-
| 6/25 || 1/4 || 29 || 100
|-
| 5/21 || 1/4 || 25 || 84
|-
| 4/17 || 1/4 || 21 || 68
|-
| 3/13 || 1/4 || 17 || 52
|-
| 2/9 || 1/4 || 13 || 36
|-
| 1/5 || 1/4 || 9 || 20
|-
| 1/5 || 2/9 || 14 || 45
|-
| 1/5 || 3/14 || 19 || 70
|-
| 1/5 || 4/19 || 24 || 95
|-
| 1/5 || 5/24 || 29 || 120
|-
| 1/5 || 6/29 || 34 || 145
|-
| 1/5 || 7/34 || 39 || 170
|-
| 1/5 || 8/39 || 44 || 195
|-
| 1/5 || 9/44 || 49 || 220
|-
| 1/5 || 10/49 || 54 || 245
|-
| 1/5 || 11/54 || 59 || 270
|-
| 1/5 || 12/59 || 64 || 295
|-
| 1/5 || 13/64 || 69 || 320
|-
| 1/5 || 14/69 || 74 || 345
|-
| 1/5 || 15/74 || 79 || 370
|-
| 1/5 || 16/79 || 84 || 395
|-
| 1/5 || 17/84 || 89 || 420
|-
| 1/5 || 18/89 || 94 || 445
|-
| 1/5 || 19/94 || 99 || 470
|-
| 1/5 || 20/99 || 104 || 495
|-
| 1/5 || 21/104 || 109 || 520
&lt;includeonly&gt;
|- style="text-align: center;"
| colspan="4" | {{Navbar|(Anti Bohlen-Pierce)|text=This table:}}
&lt;/includeonly&gt;
|}&lt;noinclude&gt;
[[Category:Tables]]
[[Category:Tritave]]
&lt;/noinclude&gt;</text>
      <sha1>2phi0gncn44tfngymif5h0vrzciaaos</sha1>
    </revision>
    <revision>
      <id>211520</id>
      <parentid>177301</parentid>
      <timestamp>2025-09-29T17:22:02Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <origin>211520</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1713" sha1="2whacxlewortuvhyqsyvny9lhphfxen" xml:space="preserve">{| class="wikitable"
|-
! g (deg/edt) !! G (deg/edt) !! Cardinality !! EDT
|-
| 33/133 || 1/4 || 137 || 532
|-
| 32/129 || 1/4 || 133 || 516
|-
| 31/125 || 1/4 || 129 || 500
|-
| 30/121 || 1/4 || 125 || 484
|-
| 29/117 || 1/4 || 121 || 468
|-
| 28/113 || 1/4 || 117 || 452
|-
| 27/109 || 1/4 || 113 || 436
|-
| 26/105 || 1/4 || 109 || 420
|-
| 25/101 || 1/4 || 105 || 404
|-
| 24/97 || 1/4 || 101 || 388
|-
| 23/93 || 1/4 || 97 || 372
|-
| 22/89 || 1/4 || 93 || 356
|-
| 21/85 || 1/4 || 89 || 340
|-
| 20/81 || 1/4 || 85 || 324
|-
| 19/77 || 1/4 || 81 || 308
|-
| 18/73 || 1/4 || 77 || 292
|-
| 17/69 || 1/4 || 73 || 276
|-
| 16/65 || 1/4 || 69 || 260
|-
| 15/61 || 1/4 || 65 || 244
|-
| 14/57 || 1/4 || 61 || 228
|-
| 13/53 || 1/4 || 57 || 212
|-
| 12/49 || 1/4 || 53 || 196
|-
| 11/45 || 1/4 || 49 || 180
|-
| 10/41 || 1/4 || 45 || 164
|-
| 9/37 || 1/4 || 41 || 148
|-
| 8/33 || 1/4 || 37 || 132
|-
| 7/29 || 1/4 || 33 || 116
|-
| 6/25 || 1/4 || 29 || 100
|-
| 5/21 || 1/4 || 25 || 84
|-
| 4/17 || 1/4 || 21 || 68
|-
| 3/13 || 1/4 || 17 || 52
|-
| 2/9 || 1/4 || 13 || 36
|-
| 1/5 || 1/4 || 9 || 20
|-
| 1/5 || 2/9 || 14 || 45
|-
| 1/5 || 3/14 || 19 || 70
|-
| 1/5 || 4/19 || 24 || 95
|-
| 1/5 || 5/24 || 29 || 120
|-
| 1/5 || 6/29 || 34 || 145
|-
| 1/5 || 7/34 || 39 || 170
|-
| 1/5 || 8/39 || 44 || 195
|-
| 1/5 || 9/44 || 49 || 220
|-
| 1/5 || 10/49 || 54 || 245
|-
| 1/5 || 11/54 || 59 || 270
|-
| 1/5 || 12/59 || 64 || 295
|-
| 1/5 || 13/64 || 69 || 320
|-
| 1/5 || 14/69 || 74 || 345
|-
| 1/5 || 15/74 || 79 || 370
|-
| 1/5 || 16/79 || 84 || 395
|-
| 1/5 || 17/84 || 89 || 420
|-
| 1/5 || 18/89 || 94 || 445
|-
| 1/5 || 19/94 || 99 || 470
|-
| 1/5 || 20/99 || 104 || 495
|-
| 1/5 || 21/104 || 109 || 520
|}</text>
      <sha1>2whacxlewortuvhyqsyvny9lhphfxen</sha1>
    </revision>
    <revision>
      <id>211521</id>
      <parentid>211520</parentid>
      <timestamp>2025-09-29T17:22:18Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>FloraC moved page [[(Anti) Bohlen-Pierce]] to [[User:Moremajorthanmajor/(Anti) Bohlen-Pierce]] without leaving a redirect: Page is understood by no one</comment>
      <origin>211520</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1713" sha1="2whacxlewortuvhyqsyvny9lhphfxen" xml:space="preserve">{| class="wikitable"
|-
! g (deg/edt) !! G (deg/edt) !! Cardinality !! EDT
|-
| 33/133 || 1/4 || 137 || 532
|-
| 32/129 || 1/4 || 133 || 516
|-
| 31/125 || 1/4 || 129 || 500
|-
| 30/121 || 1/4 || 125 || 484
|-
| 29/117 || 1/4 || 121 || 468
|-
| 28/113 || 1/4 || 117 || 452
|-
| 27/109 || 1/4 || 113 || 436
|-
| 26/105 || 1/4 || 109 || 420
|-
| 25/101 || 1/4 || 105 || 404
|-
| 24/97 || 1/4 || 101 || 388
|-
| 23/93 || 1/4 || 97 || 372
|-
| 22/89 || 1/4 || 93 || 356
|-
| 21/85 || 1/4 || 89 || 340
|-
| 20/81 || 1/4 || 85 || 324
|-
| 19/77 || 1/4 || 81 || 308
|-
| 18/73 || 1/4 || 77 || 292
|-
| 17/69 || 1/4 || 73 || 276
|-
| 16/65 || 1/4 || 69 || 260
|-
| 15/61 || 1/4 || 65 || 244
|-
| 14/57 || 1/4 || 61 || 228
|-
| 13/53 || 1/4 || 57 || 212
|-
| 12/49 || 1/4 || 53 || 196
|-
| 11/45 || 1/4 || 49 || 180
|-
| 10/41 || 1/4 || 45 || 164
|-
| 9/37 || 1/4 || 41 || 148
|-
| 8/33 || 1/4 || 37 || 132
|-
| 7/29 || 1/4 || 33 || 116
|-
| 6/25 || 1/4 || 29 || 100
|-
| 5/21 || 1/4 || 25 || 84
|-
| 4/17 || 1/4 || 21 || 68
|-
| 3/13 || 1/4 || 17 || 52
|-
| 2/9 || 1/4 || 13 || 36
|-
| 1/5 || 1/4 || 9 || 20
|-
| 1/5 || 2/9 || 14 || 45
|-
| 1/5 || 3/14 || 19 || 70
|-
| 1/5 || 4/19 || 24 || 95
|-
| 1/5 || 5/24 || 29 || 120
|-
| 1/5 || 6/29 || 34 || 145
|-
| 1/5 || 7/34 || 39 || 170
|-
| 1/5 || 8/39 || 44 || 195
|-
| 1/5 || 9/44 || 49 || 220
|-
| 1/5 || 10/49 || 54 || 245
|-
| 1/5 || 11/54 || 59 || 270
|-
| 1/5 || 12/59 || 64 || 295
|-
| 1/5 || 13/64 || 69 || 320
|-
| 1/5 || 14/69 || 74 || 345
|-
| 1/5 || 15/74 || 79 || 370
|-
| 1/5 || 16/79 || 84 || 395
|-
| 1/5 || 17/84 || 89 || 420
|-
| 1/5 || 18/89 || 94 || 445
|-
| 1/5 || 19/94 || 99 || 470
|-
| 1/5 || 20/99 || 104 || 495
|-
| 1/5 || 21/104 || 109 || 520
|}</text>
      <sha1>2whacxlewortuvhyqsyvny9lhphfxen</sha1>
    </revision>
  </page>
  <page>
    <title>0edo</title>
    <ns>0</ns>
    <id>4</id>
    <redirect title="Single-pitch tuning" />
    <revision>
      <id>4</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>4</origin>
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      <text bytes="1013" sha1="oiaj777zzhxkk5jccvggo650u1293bk" xml:space="preserve">=Zero equal divisions of the octave=

There are two ways to approach this idea.

Given that ''n''''-''edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[Trivial_temperaments|trivial temperament]], 0edo tempers out all commas and is consistent in all limits.
[[Category:edo]]
[[Category:limiting_case]]
[[Category:philosophy]]</text>
      <sha1>oiaj777zzhxkk5jccvggo650u1293bk</sha1>
    </revision>
    <revision>
      <id>3958</id>
      <timestamp>2017-07-22T19:05:56Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 615831721 - Original comment: added link (deorphaning)**</comment>
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      <text bytes="3538" sha1="bxem8i7yjd68rmwfjwx65bt2nvhka0i" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2017-07-22 19:05:56 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;615831721&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;added link (deorphaning)&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=Zero equal divisions of the octave=

There are two ways to approach this idea.

Given that //n////-//edo means that you are dividing the octave into 1///n// equal divisions and that 1/0 is undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as //n// gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As //n// gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperaments|trivial temperament]], 0edo tempers out all commas and is consistent in all limits.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;0edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Zero equal divisions of the octave"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Zero equal divisions of the octave&amp;lt;/h1&amp;gt;
&amp;lt;br /&amp;gt;
There are two ways to approach this idea.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Given that &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;-&amp;lt;/em&amp;gt;edo means that you are dividing the octave into 1&amp;lt;em&amp;gt;/n&amp;lt;/em&amp;gt; equal divisions and that 1/0 is undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The other way of looking at it is to see what happens as &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
An example of a &amp;lt;a class="wiki_link" href="/trivial%20temperaments"&amp;gt;trivial temperament&amp;lt;/a&amp;gt;, 0edo tempers out all commas and is consistent in all limits.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>3959</id>
      <timestamp>2016-09-16T03:59:30Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 592319876 - Original comment: **</comment>
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      <text bytes="3396" sha1="nxqnnhgjhcakjrhqv6wyrpxxohrocqy" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-09-16 03:59:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;592319876&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;

There are two ways to approach this idea.

Given that //n////-//edo means that you are dividing the octave into 1///n// equal divisions and that 1/0 is undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as //n// gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As //n// gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

It tempers out all commas and is consistent in all limits.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;0edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
There are two ways to approach this idea.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Given that &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;-&amp;lt;/em&amp;gt;edo means that you are dividing the octave into 1&amp;lt;em&amp;gt;/n&amp;lt;/em&amp;gt; equal divisions and that 1/0 is undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The other way of looking at it is to see what happens as &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It tempers out all commas and is consistent in all limits.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>3960</id>
      <timestamp>2016-09-16T03:57:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 592319816 - Original comment: **</comment>
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      <text bytes="3252" sha1="kkclv68axtlxd80djkzq90y5xlyhcpn" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-09-16 03:57:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;592319816&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;

There are two ways to approach this idea.

Given that //n////-//edo means that you are dividing the octave into 1///n// equal divisions and that 1/0 is undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as //n// gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As //n// gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;0edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
There are two ways to approach this idea.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Given that &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;-&amp;lt;/em&amp;gt;edo means that you are dividing the octave into 1&amp;lt;em&amp;gt;/n&amp;lt;/em&amp;gt; equal divisions and that 1/0 is undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The other way of looking at it is to see what happens as &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>3961</id>
      <timestamp>2016-09-03T06:46:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 590996208 - Original comment: Definitely not in standard mathematics**</comment>
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      <text bytes="3168" sha1="49qtmt31sen2e1oe8wqyab4qio3klgi" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-09-03 06:46:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;590996208&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;Definitely not in standard mathematics&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;

There are two ways to approach this idea.

Given that //n////-//edo means that you are dividing the octave into 1///n// equal divisions and that 1/0 is undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all.

The other way of looking at it is to see what happens as //n// gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As //n// gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;0edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
There are two ways to approach this idea.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Given that &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;-&amp;lt;/em&amp;gt;edo means that you are dividing the octave into 1&amp;lt;em&amp;gt;/n&amp;lt;/em&amp;gt; equal divisions and that 1/0 is undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The other way of looking at it is to see what happens as &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>3962</id>
      <timestamp>2016-09-02T21:28:29Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;davethecomposer</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 590988684 - Original comment: **</comment>
      <origin>3962</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3178" sha1="gysyagj03l6qsyiz3g57wv84bajxeua" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:davethecomposer|davethecomposer]] and made on &lt;tt&gt;2016-09-02 21:28:29 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;590988684&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;

There are two ways to approach this idea.

Given that //n////-//edo means that you are dividing the octave into 1///n// equal divisions and that 1/0 is undefined in standard mathematics, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all.

The other way of looking at it is to see what happens as //n// gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As //n// gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;0edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
There are two ways to approach this idea.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Given that &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;-&amp;lt;/em&amp;gt;edo means that you are dividing the octave into 1&amp;lt;em&amp;gt;/n&amp;lt;/em&amp;gt; equal divisions and that 1/0 is undefined in standard mathematics, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The other way of looking at it is to see what happens as &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>3963</id>
      <timestamp>2016-08-30T11:38:11Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 590525358 - Original comment: **</comment>
      <origin>3963</origin>
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      <text bytes="3558" sha1="329hrk3n5cuo9utoowgwum0c20a5hts" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-08-30 11:38:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;590525358&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;

There are two ways to approach this idea.

Given that //n////-//edo means that you are dividing the octave into 1///n// equal divisions and that 1/0 is undefined in kid's mathematics, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all. However, in standard mathematics, 1/0 is Infinity, therefore one step is infinite. However, it cannot be heard because it's outside human hearing range. So only one sound of 0edo can be heard.

The other way of looking at it is to see what happens as //n// gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As //n// gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;0edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
There are two ways to approach this idea.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Given that &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;-&amp;lt;/em&amp;gt;edo means that you are dividing the octave into 1&amp;lt;em&amp;gt;/n&amp;lt;/em&amp;gt; equal divisions and that 1/0 is undefined in kid's mathematics, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all. However, in standard mathematics, 1/0 is Infinity, therefore one step is infinite. However, it cannot be heard because it's outside human hearing range. So only one sound of 0edo can be heard.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The other way of looking at it is to see what happens as &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    </revision>
    <revision>
      <id>3964</id>
      <timestamp>2016-08-22T00:05:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 589904488 - Original comment: **</comment>
      <origin>3964</origin>
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      <format>text/x-wiki</format>
      <text bytes="3174" sha1="7q3lq7c4f9t8cbduup3tax4ibgndp1c" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2016-08-22 00:05:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;589904488&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;

There are two ways to approach this idea.

Given that //n////-//edo means that you are dividing the octave into 1///n// equal divisions and that 1/0 is undefined in standard mathematics, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all.

The other way of looking at it is to see what happens as //n// gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As //n// gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;0edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
There are two ways to approach this idea.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Given that &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;-&amp;lt;/em&amp;gt;edo means that you are dividing the octave into 1&amp;lt;em&amp;gt;/n&amp;lt;/em&amp;gt; equal divisions and that 1/0 is undefined in standard mathematics, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The other way of looking at it is to see what happens as &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>3965</id>
      <timestamp>2016-08-20T17:12:21Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;davethecomposer</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 589844690 - Original comment: **</comment>
      <origin>3965</origin>
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      <text bytes="3174" sha1="jc7y966as4p6wkdhsk43gzjv3nupbdm" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:davethecomposer|davethecomposer]] and made on &lt;tt&gt;2016-08-20 17:12:21 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;589844690&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;

There are two ways to approach this idea.

Given that //n////-//edo means that you are dividing the octave into 1///n// equal divisions and that 1/0 is undefined in standard mathematics, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all.

The other way of looking at it is to see what happens as //n// gets smaller. At 1-edo you have one note per octave. At .5-edo you have 1/.5 which is one note every two octaves. As //n// gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;0edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
There are two ways to approach this idea.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Given that &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;-&amp;lt;/em&amp;gt;edo means that you are dividing the octave into 1&amp;lt;em&amp;gt;/n&amp;lt;/em&amp;gt; equal divisions and that 1/0 is undefined in standard mathematics, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The other way of looking at it is to see what happens as &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller. At 1-edo you have one note per octave. At .5-edo you have 1/.5 which is one note every two octaves. As &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>jc7y966as4p6wkdhsk43gzjv3nupbdm</sha1>
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    <revision>
      <id>3966</id>
      <timestamp>2016-08-19T20:33:29Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;davethecomposer</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 589607592 - Original comment: **</comment>
      <origin>3966</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3182" sha1="95rma8700ub7ma8l1in8wrw0ywfwyec" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:davethecomposer|davethecomposer]] and made on &lt;tt&gt;2016-08-19 20:33:29 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;589607592&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;

There are two ways to approach this idea.

Given that //n////-//edo means that you are dividing the octave into 1///n// equal divisions and that 1/0 is undefined in standard mathematics, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all.

The other way of looking at it is to see what happens as you //n// gets smaller. At 1-edo you have one note per octave. At .5-edo you have 1/.5 which is one note every two octaves. As //n// gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;0edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
There are two ways to approach this idea.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Given that &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;-&amp;lt;/em&amp;gt;edo means that you are dividing the octave into 1&amp;lt;em&amp;gt;/n&amp;lt;/em&amp;gt; equal divisions and that 1/0 is undefined in standard mathematics, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The other way of looking at it is to see what happens as you &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller. At 1-edo you have one note per octave. At .5-edo you have 1/.5 which is one note every two octaves. As &amp;lt;em&amp;gt;n&amp;lt;/em&amp;gt; gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2015-03-11T11:20:53Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;davethecomposer</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 543740824 - Original comment: **</comment>
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      <text bytes="1803" sha1="l01k8f679aeffjwfp5h51eg9lmlnyx5" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:davethecomposer|davethecomposer]] and made on &lt;tt&gt;2015-03-11 11:20:53 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;543740824&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;

Presumably a single note and nothing else, but oddly and/or paradoxically perhaps absolute silence or the fretless continuum.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;0edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Presumably a single note and nothing else, but oddly and/or paradoxically perhaps absolute silence or the fretless continuum.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>3968</id>
      <timestamp>2014-05-30T12:27:01Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 512138434 - Original comment: **</comment>
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      <text bytes="1797" sha1="fvdu4q0ujnxgcjb81m1macmzbywtpnp" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2014-05-30 12:27:01 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;512138434&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;

Presumably a single note and nothing else, bu oddly and/or paradoxically perhaps absolute silence or the fretless continuum.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;0edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Presumably a single note and nothing else, bu oddly and/or paradoxically perhaps absolute silence or the fretless continuum.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>3969</id>
      <timestamp>2014-05-28T14:16:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 511746962 - Original comment: **</comment>
      <origin>3969</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1741" sha1="sy70tbanelxxoyf3sc3qhztaf5hw1to" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2014-05-28 14:16:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;511746962&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;

Oddly and/or paradoxically, 0-EDD (EDO) might be absolute silence and/or the fretless continuum.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;0edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="color: #00abda; font-family: 'Times New Roman',Times,serif; font-size: 149%;"&amp;gt;0 Equal Divisions per Ditave (Octave)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Oddly and/or paradoxically, 0-EDD (EDO) might be absolute silence and/or the fretless continuum.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>sy70tbanelxxoyf3sc3qhztaf5hw1to</sha1>
    </revision>
    <revision>
      <id>32407</id>
      <parentid>4</parentid>
      <timestamp>2018-09-17T12:47:43Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <origin>32407</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1034" sha1="aovbk2umghlx1k58s4eommrruhn9iis" xml:space="preserve">=Zero equal divisions of the octave=

There are two ways to approach this idea.

Given that ''n''''-''edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[Trivial_temperaments|trivial temperament]], 0edo tempers out all commas and is consistent in all limits.
[[Category:edo]]
[[Category:limiting_case]]
[[Category:philosophy]]</text>
      <sha1>aovbk2umghlx1k58s4eommrruhn9iis</sha1>
    </revision>
    <revision>
      <id>32408</id>
      <parentid>32407</parentid>
      <timestamp>2018-09-17T12:48:40Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <comment>parsing disrepancy</comment>
      <origin>32408</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1030" sha1="2g568fh1oc8fqe80rpf3z324n8m4xbt" xml:space="preserve">=Zero equal divisions of the octave=

There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[Trivial_temperaments|trivial temperament]], 0edo tempers out all commas and is consistent in all limits.
[[Category:edo]]
[[Category:limiting_case]]
[[Category:philosophy]]</text>
      <sha1>2g568fh1oc8fqe80rpf3z324n8m4xbt</sha1>
    </revision>
    <revision>
      <id>54942</id>
      <parentid>32408</parentid>
      <timestamp>2020-12-04T23:13:04Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54942</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1056" sha1="ohd765e9gbpzduiaxzdkph88gt0g1r4" xml:space="preserve">=Zero equal divisions of the octave=

There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[Trivial_temperaments|trivial temperament]], 0edo tempers out all commas and is consistent in all limits.
[[Category:Equal divisions of the octave]]
[[Category:limiting_case]]
[[Category:philosophy]]</text>
      <sha1>ohd765e9gbpzduiaxzdkph88gt0g1r4</sha1>
    </revision>
    <revision>
      <id>70685</id>
      <parentid>54942</parentid>
      <timestamp>2021-05-25T07:21:57Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>seems to be not "philosophy"</comment>
      <origin>70685</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1012" sha1="nc82bexomi2fxbqo20fdgrpyb52c8g0" xml:space="preserve">=Zero equal divisions of the octave=

There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

[[Category:Equal divisions of the octave]]
[[Category:Limiting case]]</text>
      <sha1>nc82bexomi2fxbqo20fdgrpyb52c8g0</sha1>
    </revision>
    <revision>
      <id>74960</id>
      <parentid>70685</parentid>
      <timestamp>2021-07-23T19:15:00Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Reorganized page, added Music section with 1 track</comment>
      <origin>74960</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1460" sha1="29f76qojxs2t81z6zdqk2z5p4q2n0c6" xml:space="preserve">'''0 equal divisions of the octave''' (or '''0edo''') is a [[tuning system|tuning]] that contains a single note.

== Theory ==

There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

== Music ==
{| class="wikitable sortable"
!Title
!Composer
!Year
!Genre
!Additional links
|-
|[https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood'']
|Cryptovolans, Reuben Gingrich
|2021
|Metal
|[https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Album (Bandcamp)]

|}


[[Category:Equal divisions of the octave]]
[[Category:Limiting case]]</text>
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    </revision>
    <revision>
      <id>74962</id>
      <parentid>74960</parentid>
      <timestamp>2021-07-23T19:46:25Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Grammar typo</comment>
      <origin>74962</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1459" sha1="dz8pk26jrtqgp4satdi24b8kwwzpw75" xml:space="preserve">'''0 equal division of the octave''' (or '''0edo''') is a [[tuning system|tuning]] that contains a single note.

== Theory ==

There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

== Music ==
{| class="wikitable sortable"
!Title
!Composer
!Year
!Genre
!Additional links
|-
|[https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood'']
|Cryptovolans, Reuben Gingrich
|2021
|Metal
|[https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Album (Bandcamp)]

|}


[[Category:Equal divisions of the octave]]
[[Category:Limiting case]]</text>
      <sha1>dz8pk26jrtqgp4satdi24b8kwwzpw75</sha1>
    </revision>
    <revision>
      <id>74964</id>
      <parentid>74962</parentid>
      <timestamp>2021-07-23T20:03:25Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Slightly rephrased introduction</comment>
      <origin>74964</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1454" sha1="bg8n03fc4s5iz7jdjtbucbhth4xhd62" xml:space="preserve">'''0 equal division of the octave''' (or '''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==

There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

== Music ==
{| class="wikitable sortable"
!Title
!Composer
!Year
!Genre
!Additional links
|-
|[https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood'']
|Cryptovolans, Reuben Gingrich
|2021
|Metal
|[https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Album (Bandcamp)]

|}


[[Category:Equal divisions of the octave]]
[[Category:Limiting case]]</text>
      <sha1>bg8n03fc4s5iz7jdjtbucbhth4xhd62</sha1>
    </revision>
    <revision>
      <id>75009</id>
      <parentid>74964</parentid>
      <timestamp>2021-07-24T16:32:32Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Typography edit</comment>
      <origin>75009</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1451" sha1="cupisj9jvp7er1pkgrk7vudhj9vefod" xml:space="preserve">'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==

There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

== Music ==
{| class="wikitable sortable"
!Title
!Composer
!Year
!Genre
!Additional links
|-
|[https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood'']
|Cryptovolans, Reuben Gingrich
|2021
|Metal
|[https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Album (Bandcamp)]

|}


[[Category:Equal divisions of the octave]]
[[Category:Limiting case]]</text>
      <sha1>cupisj9jvp7er1pkgrk7vudhj9vefod</sha1>
    </revision>
    <revision>
      <id>92562</id>
      <parentid>75009</parentid>
      <timestamp>2022-06-14T13:17:55Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92562</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1475" sha1="b5c0mw8njwlew42lkhrmpxoukmxjeic" xml:space="preserve">'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

== Music ==
{| class="wikitable sortable"
!Title
!Composer
!Year
!Genre
!Additional links
|-
|[https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood'']
|Cryptovolans, Reuben Gingrich
|2021
|Metal
|[https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Album (Bandcamp)]

|}

[[Category:Equal divisions of the octave|#]] &lt;!-- 1-digit number --&gt;
[[Category:Limiting case]]</text>
      <sha1>b5c0mw8njwlew42lkhrmpxoukmxjeic</sha1>
    </revision>
    <revision>
      <id>95648</id>
      <parentid>92562</parentid>
      <timestamp>2022-09-22T02:24:14Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>/* Music */ Simpler format</comment>
      <origin>95648</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1383" sha1="0va45q2x64zxlisryjc1bfl5q1qp6tv" xml:space="preserve">'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

== Music ==
'''Cryptovolans, Reuben Gingrich'''
* ''[https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben Many Birds Pecking On Wood]'' (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

[[Category:Equal divisions of the octave|#]] &lt;!-- 1-digit number --&gt;
[[Category:Limiting case]]</text>
      <sha1>0va45q2x64zxlisryjc1bfl5q1qp6tv</sha1>
    </revision>
    <revision>
      <id>95649</id>
      <parentid>95648</parentid>
      <timestamp>2022-09-22T03:09:05Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>95649</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1609" sha1="anr2aj0oazoobew42284elklcp2pwqz" xml:space="preserve">'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note or is not defined as an [[EDO|edo]].

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there). Or they are, but they are unequal (e. g. [[Well temperament|well temperaments]]) or not primarily compared to the octave (e. g. the former perfect ambitus of a mode was a ninth or a tenth).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

== Music ==
'''Cryptovolans, Reuben Gingrich'''
* ''[https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben Many Birds Pecking On Wood]'' (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

[[Category:Equal divisions of the octave|#]] &lt;!-- 1-digit number --&gt;
[[Category:Limiting case]]</text>
      <sha1>anr2aj0oazoobew42284elklcp2pwqz</sha1>
    </revision>
    <revision>
      <id>95650</id>
      <parentid>95649</parentid>
      <timestamp>2022-09-22T03:27:07Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Undo revision 95649 by [[Special:Contributions/Moremajorthanmajor|Moremajorthanmajor]] ([[User talk:Moremajorthanmajor|talk]]) -- being trivial doesn't mean it may contradict the very definition of edo</comment>
      <origin>95650</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1383" sha1="0va45q2x64zxlisryjc1bfl5q1qp6tv" xml:space="preserve">'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

== Music ==
'''Cryptovolans, Reuben Gingrich'''
* ''[https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben Many Birds Pecking On Wood]'' (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

[[Category:Equal divisions of the octave|#]] &lt;!-- 1-digit number --&gt;
[[Category:Limiting case]]</text>
      <sha1>0va45q2x64zxlisryjc1bfl5q1qp6tv</sha1>
    </revision>
    <revision>
      <id>95727</id>
      <parentid>95650</parentid>
      <timestamp>2022-09-26T03:04:52Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Markup</comment>
      <origin>95727</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1383" sha1="bd4k1y76mevyzue668bnb49regtnh0o" xml:space="preserve">'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

== Music ==
'''Cryptovolans, Reuben Gingrich'''
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

[[Category:Equal divisions of the octave|#]] &lt;!-- 1-digit number --&gt;
[[Category:Limiting case]]</text>
      <sha1>bd4k1y76mevyzue668bnb49regtnh0o</sha1>
    </revision>
    <revision>
      <id>96342</id>
      <parentid>95727</parentid>
      <timestamp>2022-10-04T21:04:55Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
      <origin>96342</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1398" sha1="hlemdhi71kpvn55oayiiweplx68eg4v" xml:space="preserve">{{Infobox ET}}
'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

== Music ==
'''Cryptovolans, Reuben Gingrich'''
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

[[Category:Equal divisions of the octave|#]] &lt;!-- 1-digit number --&gt;
[[Category:Limiting case]]</text>
      <sha1>hlemdhi71kpvn55oayiiweplx68eg4v</sha1>
    </revision>
    <revision>
      <id>113751</id>
      <parentid>96342</parentid>
      <timestamp>2023-05-19T19:00:51Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Music */ Added music by request</comment>
      <origin>113751</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1520" sha1="5v0coqe6cgsuywpch1sfvlkii551l12" xml:space="preserve">{{Infobox ET}}
'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

== Music ==
'''Cryptovolans, Reuben Gingrich'''
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

'''Elliott Carter'''
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

[[Category:Equal divisions of the octave|#]] &lt;!-- 1-digit number --&gt;
[[Category:Limiting case]]</text>
      <sha1>5v0coqe6cgsuywpch1sfvlkii551l12</sha1>
    </revision>
    <revision>
      <id>115592</id>
      <parentid>113751</parentid>
      <timestamp>2023-06-03T05:56:27Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>/* Music */ Markup</comment>
      <origin>115592</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1512" sha1="bejtpw75vaswf45cvttuw1pffzz6elp" xml:space="preserve">{{Infobox ET}}
'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.

== Music ==
; Cryptovolans, Reuben Gingrich
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

; Elliott Carter
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

[[Category:Equal divisions of the octave|#]] &lt;!-- 1-digit number --&gt;
[[Category:Limiting case]]</text>
      <sha1>bejtpw75vaswf45cvttuw1pffzz6elp</sha1>
    </revision>
    <revision>
      <id>128989</id>
      <parentid>115592</parentid>
      <timestamp>2023-12-13T03:30:20Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>128989</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1612" sha1="qw9nfyk5lx823lssn0cui2rwlcta37p" xml:space="preserve">{{Infobox ET}}
'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

An example of a [[trivial temperament]], 0edo tempers out all commas and is consistent in all limits.


As a result of the step size of 0edo being infinite, the relative error of all intervals is zero.

== Music ==
; Cryptovolans, Reuben Gingrich
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

; Elliott Carter
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

[[Category:Equal divisions of the octave|#]] &lt;!-- 1-digit number --&gt;
[[Category:Limiting case]]</text>
      <sha1>qw9nfyk5lx823lssn0cui2rwlcta37p</sha1>
    </revision>
    <revision>
      <id>129017</id>
      <parentid>128989</parentid>
      <timestamp>2023-12-13T16:36:29Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>-redundant whitespaces; -redundant category; +links</comment>
      <origin>129017</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1605" sha1="5po5ro9fmjvms2f21rm74wghqk1oocj" xml:space="preserve">{{Infobox ET}}
'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

Being an example of a [[trivial temperament]], 0edo [[tempering out|tempers out]] all [[comma]]s and is [[consistent]] in all [[limit]]s. As a result of the step size of 0edo being infinite, the [[relative interval error|relative error]] of all intervals is zero.

== Music ==
; Cryptovolans, Reuben Gingrich
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

; Elliott Carter
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

[[Category:Limiting case]]</text>
      <sha1>5po5ro9fmjvms2f21rm74wghqk1oocj</sha1>
    </revision>
    <revision>
      <id>130596</id>
      <parentid>129017</parentid>
      <timestamp>2024-01-04T00:23:38Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added links</comment>
      <origin>130596</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1617" sha1="e3kw8m5368izpj9h72emofiom740abe" xml:space="preserve">{{Infobox ET}}
'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1''/n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus would comprise no sounds at all (or intervals from unison are undefined, so 1 note is there).

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one note without any octaves.

Being an example of a [[trivial temperament]], 0edo [[tempering out|tempers out]] all [[comma]]s and is [[consistent]] in all [[limit]]s. As a result of the step size of 0edo being infinite, the [[relative interval error|relative error]] of all intervals is zero.

== Music ==
; [[Cryptovolans]], [[Reuben Gingrich]]
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

; [[Elliott Carter]]
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

[[Category:Limiting case]]</text>
      <sha1>e3kw8m5368izpj9h72emofiom740abe</sha1>
    </revision>
    <revision>
      <id>131005</id>
      <parentid>130596</parentid>
      <timestamp>2024-01-06T15:03:45Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Undefined != unpitched or silence</comment>
      <origin>131005</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1565" sha1="oaiqubr9xojrc2ng4o8ljinhxytyxji" xml:space="preserve">{{Infobox ET}}
'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1/''n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus one could not use it as a tuning system.

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one pitch without any octaves.

Being an example of a [[trivial temperament]], 0edo [[tempering out|tempers out]] all [[comma]]s and is [[consistent]] in all [[limit]]s. As a result of the step size of 0edo being infinite, the [[relative interval error|relative error]] of all intervals is zero.

== Music ==
; [[Cryptovolans]], [[Reuben Gingrich]]
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

; [[Elliott Carter]]
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

[[Category:Limiting case]]</text>
      <sha1>oaiqubr9xojrc2ng4o8ljinhxytyxji</sha1>
    </revision>
    <revision>
      <id>135285</id>
      <parentid>131005</parentid>
      <timestamp>2024-02-12T08:40:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Equivalency to 0ed-p</comment>
      <origin>135285</origin>
      <model>wikitext</model>
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      <text bytes="1637" sha1="t557rznifl9nyd8zxm243wq4ywc4dv4" xml:space="preserve">{{Infobox ET}}
'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1/''n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus one could not use it as a tuning system.

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one pitch without any octaves.

Being an example of a [[trivial temperament]], 0edo [[tempering out|tempers out]] all [[comma]]s and is [[consistent]] in all [[limit]]s. As a result of the step size of 0edo being infinite, the [[relative interval error|relative error]] of all intervals is zero.

0edo is equivalent to 0ed-''p'' of any positive, finite number ''p''. 

== Music ==
; [[Cryptovolans]], [[Reuben Gingrich]]
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

; [[Elliott Carter]]
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

[[Category:Limiting case]]</text>
      <sha1>t557rznifl9nyd8zxm243wq4ywc4dv4</sha1>
    </revision>
    <revision>
      <id>146188</id>
      <parentid>135285</parentid>
      <timestamp>2024-06-12T12:34:19Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Music */ +music</comment>
      <origin>146188</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1734" sha1="tu2yky19c88l2zs12ro5jd2b1hiwq8m" xml:space="preserve">{{Infobox ET}}
'''0 equal division of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1/''n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus one could not use it as a tuning system.

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one pitch without any octaves.

Being an example of a [[trivial temperament]], 0edo [[tempering out|tempers out]] all [[comma]]s and is [[consistent]] in all [[limit]]s. As a result of the step size of 0edo being infinite, the [[relative interval error|relative error]] of all intervals is zero.

0edo is equivalent to 0ed-''p'' of any positive, finite number ''p''. 

== Music ==
; [[Cryptovolans]], [[Reuben Gingrich]]
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

; [[Elliott Carter]]
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

; [[No Clue Music]]
* [https://www.youtube.com/watch?v=g-WXe2mvAPk ''0 EDO Experiment''] (2024)

[[Category:Limiting case]]</text>
      <sha1>tu2yky19c88l2zs12ro5jd2b1hiwq8m</sha1>
    </revision>
    <revision>
      <id>148252</id>
      <parentid>146188</parentid>
      <timestamp>2024-07-21T06:16:06Z</timestamp>
      <contributor>
        <username>Tristanbay</username>
        <id>418</id>
      </contributor>
      <comment>Pluralized "division"</comment>
      <origin>148252</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1735" sha1="77sp5a3j00mq8i14efrk4du9u843zwt" xml:space="preserve">{{Infobox ET}}
'''0 equal divisions of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1/''n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus one could not use it as a tuning system.

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one pitch without any octaves.

Being an example of a [[trivial temperament]], 0edo [[tempering out|tempers out]] all [[comma]]s and is [[consistent]] in all [[limit]]s. As a result of the step size of 0edo being infinite, the [[relative interval error|relative error]] of all intervals is zero.

0edo is equivalent to 0ed-''p'' of any positive, finite number ''p''. 

== Music ==
; [[Cryptovolans]], [[Reuben Gingrich]]
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

; [[Elliott Carter]]
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

; [[No Clue Music]]
* [https://www.youtube.com/watch?v=g-WXe2mvAPk ''0 EDO Experiment''] (2024)

[[Category:Limiting case]]</text>
      <sha1>77sp5a3j00mq8i14efrk4du9u843zwt</sha1>
    </revision>
    <revision>
      <id>168427</id>
      <parentid>148252</parentid>
      <timestamp>2024-12-03T11:30:45Z</timestamp>
      <contributor>
        <username>AraMax</username>
        <id>466</id>
      </contributor>
      <minor/>
      <comment>Added romanian language page on the interwiki section</comment>
      <origin>168427</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1773" sha1="8s7k9closk2f7gwmtrhqny7ym3ex97l" xml:space="preserve">{{interwiki
| ro = Sistem monocord
}}
{{Infobox ET}}
'''0 equal divisions of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1/''n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus one could not use it as a tuning system.

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one pitch without any octaves.

Being an example of a [[trivial temperament]], 0edo [[tempering out|tempers out]] all [[comma]]s and is [[consistent]] in all [[limit]]s. As a result of the step size of 0edo being infinite, the [[relative interval error|relative error]] of all intervals is zero.

0edo is equivalent to 0ed-''p'' of any positive, finite number ''p''. 

== Music ==
; [[Cryptovolans]], [[Reuben Gingrich]]
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

; [[Elliott Carter]]
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

; [[No Clue Music]]
* [https://www.youtube.com/watch?v=g-WXe2mvAPk ''0 EDO Experiment''] (2024)

[[Category:Limiting case]]</text>
      <sha1>8s7k9closk2f7gwmtrhqny7ym3ex97l</sha1>
    </revision>
    <revision>
      <id>169269</id>
      <parentid>168427</parentid>
      <timestamp>2024-12-06T09:59:20Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Music */ add music</comment>
      <origin>169269</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1867" sha1="sy6a0d0di1es2r43io38p3y3ty1n2ld" xml:space="preserve">{{interwiki
| ro = Sistem monocord
}}
{{Infobox ET}}
'''0 equal divisions of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1/''n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus one could not use it as a tuning system.

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one pitch without any octaves.

Being an example of a [[trivial temperament]], 0edo [[tempering out|tempers out]] all [[comma]]s and is [[consistent]] in all [[limit]]s. As a result of the step size of 0edo being infinite, the [[relative interval error|relative error]] of all intervals is zero.

0edo is equivalent to 0ed-''p'' of any positive, finite number ''p''. 

== Music ==
; [[Cryptovolans]], [[Reuben Gingrich]]
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

; [[Elliott Carter]]
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

; [[Herman Miller]]
* ''[https://soundcloud.com/morphosyntax-1/lost-at-c Lost at C]'' (2021)

; [[No Clue Music]]
* [https://www.youtube.com/watch?v=g-WXe2mvAPk ''0 EDO Experiment''] (2024)

[[Category:Limiting case]]</text>
      <sha1>sy6a0d0di1es2r43io38p3y3ty1n2ld</sha1>
    </revision>
    <revision>
      <id>180541</id>
      <parentid>169269</parentid>
      <timestamp>2025-02-13T06:13:39Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>180541</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1905" sha1="nfmjn2siu2ok3vxjcgli032lhlf0emp" xml:space="preserve">{{interwiki
| ro = Sistem monocord
}}
''See also: [[Single-pitch tuning]]''
{{Infobox ET}}
'''0 equal divisions of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1/''n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus one could not use it as a tuning system.

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one pitch without any octaves.

Being an example of a [[trivial temperament]], 0edo [[tempering out|tempers out]] all [[comma]]s and is [[consistent]] in all [[limit]]s. As a result of the step size of 0edo being infinite, the [[relative interval error|relative error]] of all intervals is zero.

0edo is equivalent to 0ed-''p'' of any positive, finite number ''p''. 

== Music ==
; [[Cryptovolans]], [[Reuben Gingrich]]
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

; [[Elliott Carter]]
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

; [[Herman Miller]]
* ''[https://soundcloud.com/morphosyntax-1/lost-at-c Lost at C]'' (2021)

; [[No Clue Music]]
* [https://www.youtube.com/watch?v=g-WXe2mvAPk ''0 EDO Experiment''] (2024)

[[Category:Limiting case]]</text>
      <sha1>nfmjn2siu2ok3vxjcgli032lhlf0emp</sha1>
    </revision>
    <revision>
      <id>186671</id>
      <parentid>180541</parentid>
      <timestamp>2025-03-17T22:13:29Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <origin>186671</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1867" sha1="mehrfc5yk8oe7mubpp6n3d1zsnmt2q2" xml:space="preserve">''See also: [[Single-pitch tuning]]''
{{Infobox ET}}
'''0 equal divisions of the octave''' ('''0edo''') is the [[tuning system]] that contains a single note.

== Theory ==
There are two ways to approach this idea.

Given that ''n''-edo means that you are dividing the octave into 1/''n'' equal divisions and that 1/0 is sometimes considered undefined, it would follow that 0edo would be similarly undefined and thus one could not use it as a tuning system.

The other way of looking at it is to see what happens as ''n'' gets smaller. At 1-edo you have one note per octave. At 0.5-edo you have 1/0.5 which is one note every two octaves. As ''n'' gets smaller you reach a point where you only have one note within an audible octave range and any other notes outside of this range. Taking this to its conclusion, and assuming you want 0edo to be defined, you would conclude that 0edo is just one pitch without any octaves.

Being an example of a [[trivial temperament]], 0edo [[tempering out|tempers out]] all [[comma]]s and is [[consistent]] in all [[limit]]s. As a result of the step size of 0edo being infinite, the [[relative interval error|relative error]] of all intervals is zero.

0edo is equivalent to 0ed-''p'' of any positive, finite number ''p''. 

== Music ==
; [[Cryptovolans]], [[Reuben Gingrich]]
* [https://soundcloud.com/sexytoadsandfrogsfriendcircle/0-cryptovolans-reuben ''Many Birds Pecking On Wood''] (2021) ([https://sexytoadsandfrogsfriendcircle.bandcamp.com/album/staffcirc-vol-7-terra-octava Bandcamp])

; [[Elliott Carter]]
* [https://www.youtube.com/watch?v=WOjH42CwWFU ''8 Etudes and a Fantasy: No. 7. Intensely''] (1950)

; [[Herman Miller]]
* ''[https://soundcloud.com/morphosyntax-1/lost-at-c Lost at C]'' (2021)

; [[No Clue Music]]
* [https://www.youtube.com/watch?v=g-WXe2mvAPk ''0 EDO Experiment''] (2024)

[[Category:Limiting case]]</text>
      <sha1>mehrfc5yk8oe7mubpp6n3d1zsnmt2q2</sha1>
    </revision>
    <revision>
      <id>186673</id>
      <parentid>186671</parentid>
      <timestamp>2025-03-17T22:14:19Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <comment>redirected page to "single-pitch tuning"</comment>
      <origin>186673</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="61" sha1="pf144yr42swxrzvjbd0idxw97fin2i8" xml:space="preserve">#redirect [[Single-pitch tuning]]

[[Category:Limiting case]]</text>
      <sha1>pf144yr42swxrzvjbd0idxw97fin2i8</sha1>
    </revision>
    <revision>
      <id>186928</id>
      <parentid>186673</parentid>
      <timestamp>2025-03-18T13:57:35Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <origin>186928</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="33" sha1="lp7h8fnvi7huairls6l00g78ls7eog1" xml:space="preserve">#redirect [[Single-pitch tuning]]</text>
      <sha1>lp7h8fnvi7huairls6l00g78ls7eog1</sha1>
    </revision>
    <revision>
      <id>234025</id>
      <parentid>186928</parentid>
      <timestamp>2026-07-16T06:26:18Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Categories</comment>
      <origin>234025</origin>
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      <format>text/x-wiki</format>
      <text bytes="103" sha1="4p1wdgksn3gtdtdsnwiv1fabs1k2v23" xml:space="preserve">#REDIRECT [[Single-pitch tuning]]

[[Category:Equal divisions of the octave|#]] &lt;!-- 1-digit number --&gt;</text>
      <sha1>4p1wdgksn3gtdtdsnwiv1fabs1k2v23</sha1>
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  </page>
  <page>
    <title>1/2-comma meantone</title>
    <ns>0</ns>
    <id>5</id>
    <revision>
      <id>5</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>5</origin>
      <model>wikitext</model>
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      <text bytes="533" sha1="3tqgnugwmztxpllk0ldlk5255tup7h9" xml:space="preserve">In 1/2-comma [[Meantone|meantone]] temperament, each perfect fifth is tempered by a half of a [[Syntonic_Comma|syntonic comma]] from its just value of [[3/2|3/2]]. This results in minor sevenths being exactly 9/5 (and major seconds being exactly 10/9).

In this system, the "major thirds" are exactly 100/81 or approximately 365 cents, thus bordering on neutral thirds. The fifths of this temperament are even narrower than those of [[26edo|26edo]], which is the most likely candidate for a closed system approximating this meantime.</text>
      <sha1>3tqgnugwmztxpllk0ldlk5255tup7h9</sha1>
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    <revision>
      <id>3970</id>
      <timestamp>2015-12-09T15:38:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;RicardoMatosinhos</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 569631497 - Original comment: **</comment>
      <origin>3970</origin>
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      <text bytes="2518" sha1="7muvp2w4qv54z98x3kesnu642vrurz9" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:RicardoMatosinhos|RicardoMatosinhos]] and made on &lt;tt&gt;2015-12-09 15:38:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;569631497&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In 1/2-comma [[meantone]] temperament, each perfect fifth is tempered by a half of a [[Syntonic Comma|syntonic comma]] from its just value of [[3_2|3/2]]. This results in minor sevenths being exactly 9/5 (and major seconds being exactly 10/9).

In this system, the "major thirds" are exactly 100/81 or approximately 365 cents, thus bordering on neutral thirds. The fifths of this temperament are even narrower than those of [[26edo]], which is the most likely candidate for a closed system approximating this meantime. &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
In this system, the &amp;amp;quot;major thirds&amp;amp;quot; are exactly 100/81 or approximately 365 cents, thus bordering on neutral thirds. The fifths of this temperament are even narrower than those of &amp;lt;a class="wiki_link" href="/26edo"&amp;gt;26edo&amp;lt;/a&amp;gt;, which is the most likely candidate for a closed system approximating this meantime.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 260054158 - Original comment: **</comment>
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      <text bytes="2507" sha1="9wchhqwelhu022pqxcpyu20iq8uiy21" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on &lt;tt&gt;2011-09-30 13:15:57 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;260054158&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In 1/2-comma [[meantone]] temperament, each perfect fifth is tempered by a half of a [[Syntonic Comma|syntonic comma]] from its just value of [[3_2|3/2]]. This results in minor sevenths being exactly 9/5 (and major seconds being exactly 10/9).

In this system, the "major thirds" are exactly 100/81 or approximately 365 cents, thus bordering on neutral thirds. The fifths of this temperament are even narrower than those of [[26edo]], which is the most likely candidate for a closed system approximating this meantone.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-2 syntonic comma meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In 1/2-comma &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; temperament, each perfect fifth is tempered by a half of a &amp;lt;a class="wiki_link" href="/Syntonic%20Comma"&amp;gt;syntonic comma&amp;lt;/a&amp;gt; from its just value of &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. This results in minor sevenths being exactly 9/5 (and major seconds being exactly 10/9).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In this system, the &amp;amp;quot;major thirds&amp;amp;quot; are exactly 100/81 or approximately 365 cents, thus bordering on neutral thirds. The fifths of this temperament are even narrower than those of &amp;lt;a class="wiki_link" href="/26edo"&amp;gt;26edo&amp;lt;/a&amp;gt;, which is the most likely candidate for a closed system approximating this meantone.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9wchhqwelhu022pqxcpyu20iq8uiy21</sha1>
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    <revision>
      <id>3972</id>
      <timestamp>2011-09-30T09:19:05Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;glennobrien</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 259961950 - Original comment: **</comment>
      <origin>3972</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1866" sha1="cd1tykrxbyon7x2gfooz2elwat8f3ai" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:glennobrien|glennobrien]] and made on &lt;tt&gt;2011-09-30 09:19:05 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;259961950&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In 1/2-comma [[meantone]] temperament, each perfect fifth is tempered by a half of a [[Syntonic Comma|syntonic comma]] from its just value of [[3_2|3/2]]. This results in minor sevenths being exactly 9/5 (and major seconds being exactly 10/9).&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-2 syntonic comma meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In 1/2-comma &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; temperament, each perfect fifth is tempered by a half of a &amp;lt;a class="wiki_link" href="/Syntonic%20Comma"&amp;gt;syntonic comma&amp;lt;/a&amp;gt; from its just value of &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. This results in minor sevenths being exactly 9/5 (and major seconds being exactly 10/9).&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>cd1tykrxbyon7x2gfooz2elwat8f3ai</sha1>
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    <revision>
      <id>37771</id>
      <parentid>5</parentid>
      <timestamp>2018-10-24T21:24:29Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>Xenwolf moved page [[1-2 syntonic comma meantone]] to [[1/2 syntonic comma meantone]]: changed into intended lemma (that was not possible in Wikispaces)</comment>
      <origin>37771</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="533" sha1="3tqgnugwmztxpllk0ldlk5255tup7h9" xml:space="preserve">In 1/2-comma [[Meantone|meantone]] temperament, each perfect fifth is tempered by a half of a [[Syntonic_Comma|syntonic comma]] from its just value of [[3/2|3/2]]. This results in minor sevenths being exactly 9/5 (and major seconds being exactly 10/9).

In this system, the "major thirds" are exactly 100/81 or approximately 365 cents, thus bordering on neutral thirds. The fifths of this temperament are even narrower than those of [[26edo|26edo]], which is the most likely candidate for a closed system approximating this meantime.</text>
      <sha1>3tqgnugwmztxpllk0ldlk5255tup7h9</sha1>
    </revision>
    <revision>
      <id>37787</id>
      <parentid>37771</parentid>
      <timestamp>2018-10-24T21:46:03Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>lemma bold, added links, added cat</comment>
      <origin>37787</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="544" sha1="jjl9bw6nlis82uwdcqdlywi0ewmaw78" xml:space="preserve">In '''1/2-comma [[meantone]]''' temperament, each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the "major thirds" are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament are even narrower than those of [[26edo]], which is the most likely candidate for a closed system approximating this meantime.

[[Category:Meantone]]</text>
      <sha1>jjl9bw6nlis82uwdcqdlywi0ewmaw78</sha1>
    </revision>
    <revision>
      <id>47449</id>
      <parentid>37787</parentid>
      <timestamp>2020-07-02T20:15:06Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <comment>Correct math and example. 33 is actually a closer approximation than 26.</comment>
      <origin>47449</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="571" sha1="nue3c00neh91p8antc6cls8holdhn4o" xml:space="preserve">In '''1/2-comma [[meantone]]''' temperament, each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the "major thirds" are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo], but closer to 33, which is the best small number candidate for a closed system approximating this meantime.

[[Category:Meantone]]</text>
      <sha1>nue3c00neh91p8antc6cls8holdhn4o</sha1>
    </revision>
    <revision>
      <id>47450</id>
      <parentid>47449</parentid>
      <timestamp>2020-07-02T20:15:32Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <minor/>
      <comment>fix link</comment>
      <origin>47450</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="572" sha1="giuux5vzqwr8r9x2boadiz9o6abbn2p" xml:space="preserve">In '''1/2-comma [[meantone]]''' temperament, each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the "major thirds" are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantime.

[[Category:Meantone]]</text>
      <sha1>giuux5vzqwr8r9x2boadiz9o6abbn2p</sha1>
    </revision>
    <revision>
      <id>47451</id>
      <parentid>47450</parentid>
      <timestamp>2020-07-02T21:48:59Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>typo</comment>
      <origin>47451</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="575" sha1="6m51f38ibbxrsq3laviosinb8qstjan" xml:space="preserve">In '''1/2-comma [[meantone]]''' temperament, each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the "major thirds" are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantoneaaa.

[[Category:Meantone]]</text>
      <sha1>6m51f38ibbxrsq3laviosinb8qstjan</sha1>
    </revision>
    <revision>
      <id>47452</id>
      <parentid>47451</parentid>
      <timestamp>2020-07-02T21:49:08Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>47452</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="572" sha1="no00ylfj8m1yngzetww6ylrj26mu2co" xml:space="preserve">In '''1/2-comma [[meantone]]''' temperament, each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the "major thirds" are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone.

[[Category:Meantone]]</text>
      <sha1>no00ylfj8m1yngzetww6ylrj26mu2co</sha1>
    </revision>
    <revision>
      <id>61920</id>
      <parentid>47452</parentid>
      <timestamp>2021-02-19T10:39:26Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <comment>Expand on convergent math.</comment>
      <origin>61920</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="760" sha1="olz2nwmre11aul1q5l4gfdkmljeoszc" xml:space="preserve">In '''1/2-comma [[meantone]]''' temperament, each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the "major thirds" are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, [[125edo]] using the bb val flat 5th is audibly indistinguishable from a closed system of half comma meantone, with an error measured in thousandths of a cent.

[[Category:Meantone]]</text>
      <sha1>olz2nwmre11aul1q5l4gfdkmljeoszc</sha1>
    </revision>
    <revision>
      <id>109719</id>
      <parentid>61920</parentid>
      <timestamp>2023-04-27T01:11:23Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added to category "historical"</comment>
      <origin>109719</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="784" sha1="t2w8n8k2o2v7ewlvbxhf4mxu9pkikq5" xml:space="preserve">In '''1/2-comma [[meantone]]''' temperament, each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the "major thirds" are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, [[125edo]] using the bb val flat 5th is audibly indistinguishable from a closed system of half comma meantone, with an error measured in thousandths of a cent.

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>t2w8n8k2o2v7ewlvbxhf4mxu9pkikq5</sha1>
    </revision>
    <revision>
      <id>124896</id>
      <parentid>109719</parentid>
      <timestamp>2023-09-30T07:07:32Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>FloraC moved page [[1/2 syntonic comma meantone]] to [[1/2-comma meantone]]: Syntonic comma is assumed</comment>
      <origin>109719</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="784" sha1="t2w8n8k2o2v7ewlvbxhf4mxu9pkikq5" xml:space="preserve">In '''1/2-comma [[meantone]]''' temperament, each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the "major thirds" are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, [[125edo]] using the bb val flat 5th is audibly indistinguishable from a closed system of half comma meantone, with an error measured in thousandths of a cent.

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>t2w8n8k2o2v7ewlvbxhf4mxu9pkikq5</sha1>
    </revision>
    <revision>
      <id>127728</id>
      <parentid>124896</parentid>
      <timestamp>2023-11-21T04:44:45Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+tuning profile</comment>
      <origin>127728</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1308" sha1="q7et346gatbm10gf0hm9dtndsluhnaf" xml:space="preserve">'''1/2-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the major thirds are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, [[125edo]] using the bb [[val]] flat 5th is audibly indistinguishable from a closed system of half-comma meantone, with an error measured in thousandths of a cent.

For a tuning with so flat a fifth, the 7-limit [[extension]] is [[flattone]] instead of septimal meantone, or [[7/6]] and [[8/7]] would swap their sizes. 

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 2 || 4 || -1 || ]
|-
| ⟨ || 0 || -1 || -4 || 9 || ]
|-
| ⟨ || 0 || 1/2 || 2 || -9/2 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1891.2019 2764.8074 3379.1833 }}

[[Mistuning map]]: {{val| 0 -10.7531 -21.5063 +10.3574 }}

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>q7et346gatbm10gf0hm9dtndsluhnaf</sha1>
    </revision>
    <revision>
      <id>137471</id>
      <parentid>127728</parentid>
      <timestamp>2024-02-28T23:14:34Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <comment>I don't think this is a historical system like third-comma or less eccentric, it's very xen</comment>
      <origin>137471</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1284" sha1="oe0l19y4yej1iqh41zclxgakbc200dm" xml:space="preserve">'''1/2-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the major thirds are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, [[125edo]] using the bb [[val]] flat 5th is audibly indistinguishable from a closed system of half-comma meantone, with an error measured in thousandths of a cent.

For a tuning with so flat a fifth, the 7-limit [[extension]] is [[flattone]] instead of septimal meantone, or [[7/6]] and [[8/7]] would swap their sizes. 

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 2 || 4 || -1 || ]
|-
| ⟨ || 0 || -1 || -4 || 9 || ]
|-
| ⟨ || 0 || 1/2 || 2 || -9/2 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1891.2019 2764.8074 3379.1833 }}

[[Mistuning map]]: {{val| 0 -10.7531 -21.5063 +10.3574 }}

[[Category:Meantone]]</text>
      <sha1>oe0l19y4yej1iqh41zclxgakbc200dm</sha1>
    </revision>
    <revision>
      <id>149722</id>
      <parentid>137471</parentid>
      <timestamp>2024-08-07T10:54:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Adopt flattertone mapping</comment>
      <origin>149722</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1423" sha1="gaytmuw1a2u5x2msk71te7pm6uveh3n" xml:space="preserve">'''1/2-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the major thirds are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, [[125edo]] using the bb [[val]] flat 5th is audibly indistinguishable from a closed system of half-comma meantone, with an error measured in thousandths of a cent.

For a tuning with so flat a fifth, the 7-limit [[extension]] that preserves [[7-odd-limit]] [[diamond monotone]] is [[flattertone]]. Otherwise the septimal meantone mapping has the sizes of [[7/6]] and [[8/7]] swapped, and the flattone mapping has the sizes of [[7/5]] and [[10/7]] swapped. 

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 2 || 4 || 10 || ]
|-
| ⟨ || 0 || -1 || -4 || -17 || ]
|-
| ⟨ || 0 || 1/2 || 2 || 17/2 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1891.2019 2764.8074 3350.4316 }}

[[Mistuning map]]: {{val| 0 -10.7531 -21.5063 -18.3944 }}

[[Category:Meantone]]</text>
      <sha1>gaytmuw1a2u5x2msk71te7pm6uveh3n</sha1>
    </revision>
    <revision>
      <id>149748</id>
      <parentid>149722</parentid>
      <timestamp>2024-08-08T01:16:36Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <comment>Readd historical because of Fogalino or whoever it was proposing it in the 1500s</comment>
      <origin>149748</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1447" sha1="son7m73bvlq124lscclsusn4a47oujq" xml:space="preserve">'''1/2-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the major thirds are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, [[125edo]] using the bb [[val]] flat 5th is audibly indistinguishable from a closed system of half-comma meantone, with an error measured in thousandths of a cent.

For a tuning with so flat a fifth, the 7-limit [[extension]] that preserves [[7-odd-limit]] [[diamond monotone]] is [[flattertone]]. Otherwise the septimal meantone mapping has the sizes of [[7/6]] and [[8/7]] swapped, and the flattone mapping has the sizes of [[7/5]] and [[10/7]] swapped. 

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 2 || 4 || 10 || ]
|-
| ⟨ || 0 || -1 || -4 || -17 || ]
|-
| ⟨ || 0 || 1/2 || 2 || 17/2 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1891.2019 2764.8074 3350.4316 }}

[[Mistuning map]]: {{val| 0 -10.7531 -21.5063 -18.3944 }}

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>son7m73bvlq124lscclsusn4a47oujq</sha1>
    </revision>
    <revision>
      <id>164626</id>
      <parentid>149748</parentid>
      <timestamp>2024-11-07T14:27:32Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Mistuning -&gt; error</comment>
      <origin>164626</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1443" sha1="8ut2wzaxn30mm0736cmp0tkxqvbqr81" xml:space="preserve">'''1/2-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a half of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor sevenths being exactly [[9/5]] (and major seconds being exactly [[10/9]]).

In this system, the major thirds are exactly [[100/81]] or approximately 365 [[cent]]s, thus bordering on neutral thirds. The fifths of this temperament fall between those of [[26edo]] and [[33edo]], but closer to 33, which is the best small number candidate for a closed system approximating this meantone. If you want more precision, [[125edo]] using the bb [[val]] flat 5th is audibly indistinguishable from a closed system of half-comma meantone, with an error measured in thousandths of a cent.

For a tuning with so flat a fifth, the 7-limit [[extension]] that preserves [[7-odd-limit]] [[diamond monotone]] is [[flattertone]]. Otherwise the septimal meantone mapping has the sizes of [[7/6]] and [[8/7]] swapped, and the flattone mapping has the sizes of [[7/5]] and [[10/7]] swapped. 

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 2 || 4 || 10 || ]
|-
| ⟨ || 0 || -1 || -4 || -17 || ]
|-
| ⟨ || 0 || 1/2 || 2 || 17/2 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1891.2019 2764.8074 3350.4316 }}

[[Error map]]: {{val| 0 -10.7531 -21.5063 -18.3944 }}

[[Category:Meantone]]
[[Category:Historical]]</text>
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  </page>
  <page>
    <title>1/3-comma meantone</title>
    <ns>0</ns>
    <id>6</id>
    <revision>
      <id>6</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
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      <text bytes="927" sha1="bn32jxk4wndcvosu4k0fwe3cj2dt4z9" xml:space="preserve">In '''1/3-comma [[Meantone|meantone]]''' temperament, each perfect fifth is tempered by a third of a [[Syntonic_Comma|syntonic comma]] from its just value of [[3/2|3/2]]. This results in minor thirds being exactly [[6/5|6/5]] (and major sixths being exactly [[5/3|5/3]]).

In other words, 1/3-comma meantone is the specific tuning of meantone temperament with an [[Fractional_monzos|eigenmonzo]] of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents, which is about 7 cents flatter than in [[Quarter-comma_meantone|1/4-comma meantone]].

1/3-comma meantone is practically identical to the closed system of [[19edo|19edo]] (although technically there is a slight difference).

=External links=
[http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html Die mitteltönige Stimmung mit reinen kleinen Terzen] (explanation of 1/3 comma meantone, in german)
[[Category:definition]]
[[Category:meantone]]</text>
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    <revision>
      <id>3973</id>
      <timestamp>2013-10-23T04:22:04Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 462294670 - Original comment: **</comment>
      <origin>3973</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3650" sha1="9tx017yewn3b6iiov43oph6t3k4k22f" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2013-10-23 04:22:04 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;462294670&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In **1/3-comma [[meantone]]** temperament, each perfect fifth is tempered by a third of a [[Syntonic Comma|syntonic comma]] from its just value of [[3_2|3/2]]. This results in minor thirds being exactly [[6_5|6/5]] (and major sixths being exactly [[5_3|5/3]]).
In other words, 1/3-comma meantone is the specific tuning of meantone temperament with an [[Fractional monzos|eigenmonzo]] of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents, which is about 7 cents flatter than in [[Quarter-comma meantone|1/4-comma meantone]].

1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).
=External links= 
[[http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html|Die mitteltönige Stimmung mit reinen kleinen Terzen]] (explanation of 1/3 comma meantone, in german)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-3 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;strong&amp;gt;1/3-comma &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; temperament, each perfect fifth is tempered by a third of a &amp;lt;a class="wiki_link" href="/Syntonic%20Comma"&amp;gt;syntonic comma&amp;lt;/a&amp;gt; from its just value of &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. This results in minor thirds being exactly &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt; (and major sixths being exactly &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt;).&amp;lt;br /&amp;gt;
In other words, 1/3-comma meantone is the specific tuning of meantone temperament with an &amp;lt;a class="wiki_link" href="/Fractional%20monzos"&amp;gt;eigenmonzo&amp;lt;/a&amp;gt; of 6/5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The fifth is 694.786 cents in size, and the major third 379.145 cents, which is about 7 cents flatter than in &amp;lt;a class="wiki_link" href="/Quarter-comma%20meantone"&amp;gt;1/4-comma meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
1/3-comma meantone is practically identical to the closed system of &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt; (although technically there is a slight difference).&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="External links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;External links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html" rel="nofollow"&amp;gt;Die mitteltönige Stimmung mit reinen kleinen Terzen&amp;lt;/a&amp;gt; (explanation of 1/3 comma meantone, in german)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>3974</id>
      <timestamp>2012-01-26T07:23:31Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 295493550 - Original comment: **</comment>
      <origin>3974</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3617" sha1="b57zg55mrpykj9ko1kiyoohpsoq2soj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2012-01-26 07:23:31 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;295493550&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In 1/3-comma [[meantone]] temperament, each perfect fifth is tempered by a third of a [[Syntonic Comma|syntonic comma]] from its just value of [[3_2|3/2]]. This results in minor thirds being exactly [[6_5|6/5]] (and major sixths being exactly [[5_3|5/3]]).
In other words, 1/3-comma meantone is the specific tuning of meantone temperament with an [[Fractional monzos|eigenmonzo]] of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents, which is about 7 cents flatter than in [[Quarter-comma meantone|1/4-comma meantone]].

1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).
=External links= 
[[http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html|Die mitteltönige Stimmung mit reinen kleinen Terzen]] (explanation of 1/3 comma meantone, in german)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-3 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In 1/3-comma &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; temperament, each perfect fifth is tempered by a third of a &amp;lt;a class="wiki_link" href="/Syntonic%20Comma"&amp;gt;syntonic comma&amp;lt;/a&amp;gt; from its just value of &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. This results in minor thirds being exactly &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt; (and major sixths being exactly &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt;).&amp;lt;br /&amp;gt;
In other words, 1/3-comma meantone is the specific tuning of meantone temperament with an &amp;lt;a class="wiki_link" href="/Fractional%20monzos"&amp;gt;eigenmonzo&amp;lt;/a&amp;gt; of 6/5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The fifth is 694.786 cents in size, and the major third 379.145 cents, which is about 7 cents flatter than in &amp;lt;a class="wiki_link" href="/Quarter-comma%20meantone"&amp;gt;1/4-comma meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
1/3-comma meantone is practically identical to the closed system of &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt; (although technically there is a slight difference).&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="External links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;External links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html" rel="nofollow"&amp;gt;Die mitteltönige Stimmung mit reinen kleinen Terzen&amp;lt;/a&amp;gt; (explanation of 1/3 comma meantone, in german)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>3975</id>
      <timestamp>2012-01-26T07:21:16Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 295493168 - Original comment: **</comment>
      <origin>3975</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3583" sha1="s02vcxrcv47na69v5phy425pg9huslg" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2012-01-26 07:21:16 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;295493168&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In 1/3-comma [[meantone]] temperament, each perfect fifth is tempered by a third of a [[Syntonic Comma|syntonic comma]] from its just value of [[3_2|3/2]]. This results in minor thirds being exactly [[6_5|6/5]] (and major sixths being exactly [[5_3|5/3]]).
In other words, 1/3-comma meantone is the specific tuning of meantone temperament with an [[Fractional monzos|eigenmonzo]] of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents - both flatter than in [[Quarter-comma meantone|1/4-comma meantone]].

1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).
=External links= 
[[http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html|Die mitteltönige Stimmung mit reinen kleinen Terzen]] (explanation of 1/3 comma meantone, in german)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-3 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In 1/3-comma &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; temperament, each perfect fifth is tempered by a third of a &amp;lt;a class="wiki_link" href="/Syntonic%20Comma"&amp;gt;syntonic comma&amp;lt;/a&amp;gt; from its just value of &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. This results in minor thirds being exactly &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt; (and major sixths being exactly &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt;).&amp;lt;br /&amp;gt;
In other words, 1/3-comma meantone is the specific tuning of meantone temperament with an &amp;lt;a class="wiki_link" href="/Fractional%20monzos"&amp;gt;eigenmonzo&amp;lt;/a&amp;gt; of 6/5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The fifth is 694.786 cents in size, and the major third 379.145 cents - both flatter than in &amp;lt;a class="wiki_link" href="/Quarter-comma%20meantone"&amp;gt;1/4-comma meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
1/3-comma meantone is practically identical to the closed system of &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt; (although technically there is a slight difference).&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="External links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;External links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html" rel="nofollow"&amp;gt;Die mitteltönige Stimmung mit reinen kleinen Terzen&amp;lt;/a&amp;gt; (explanation of 1/3 comma meantone, in german)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>s02vcxrcv47na69v5phy425pg9huslg</sha1>
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    <revision>
      <id>3976</id>
      <timestamp>2011-09-19T22:35:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;keenanpepper</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 255918310 - Original comment: **</comment>
      <origin>3976</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3230" sha1="0emudqt1w9xvhq4jt7xk2gn0939kz51" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on &lt;tt&gt;2011-09-19 22:35:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;255918310&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In 1/3-comma [[meantone]] temperament, each perfect fifth is tempered by a third of a [[Syntonic Comma|syntonic comma]] from its just value of [[3_2|3/2]]. This results in minor thirds being exactly [[6_5|6/5]] (and major sixths being exactly [[5_3|5/3]]).
In other words, 1/3-comma meantone is the specific tuning of meantone temperament with an [[Fractional monzos|eigenmonzo]] of 6/5.
1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).
=External links= 
[[http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html|Die mitteltönige Stimmung mit reinen kleinen Terzen]] (explanation of 1/3 comma meantone, in german)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-3 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In 1/3-comma &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; temperament, each perfect fifth is tempered by a third of a &amp;lt;a class="wiki_link" href="/Syntonic%20Comma"&amp;gt;syntonic comma&amp;lt;/a&amp;gt; from its just value of &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. This results in minor thirds being exactly &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt; (and major sixths being exactly &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt;).&amp;lt;br /&amp;gt;
In other words, 1/3-comma meantone is the specific tuning of meantone temperament with an &amp;lt;a class="wiki_link" href="/Fractional%20monzos"&amp;gt;eigenmonzo&amp;lt;/a&amp;gt; of 6/5.&amp;lt;br /&amp;gt;
1/3-comma meantone is practically identical to the closed system of &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt; (although technically there is a slight difference).&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="External links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;External links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html" rel="nofollow"&amp;gt;Die mitteltönige Stimmung mit reinen kleinen Terzen&amp;lt;/a&amp;gt; (explanation of 1/3 comma meantone, in german)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>0emudqt1w9xvhq4jt7xk2gn0939kz51</sha1>
    </revision>
    <revision>
      <id>3977</id>
      <timestamp>2011-06-20T05:43:23Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 237681909 - Original comment: **</comment>
      <origin>3977</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1840" sha1="fpkhvpyb8gzfkqs1euo0rntto1uifaa" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2011-06-20 05:43:23 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;237681909&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=External links= 
[[http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html|Die mitteltönige Stimmung mit reinen kleinen Terzen]] (explanation of 1/3 comma meantone, in german)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-3 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="External links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;External links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html" rel="nofollow"&amp;gt;Die mitteltönige Stimmung mit reinen kleinen Terzen&amp;lt;/a&amp;gt; (explanation of 1/3 comma meantone, in german)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>fpkhvpyb8gzfkqs1euo0rntto1uifaa</sha1>
    </revision>
    <revision>
      <id>36434</id>
      <parentid>6</parentid>
      <timestamp>2018-10-08T16:12:06Z</timestamp>
      <contributor>
        <username>Hstraub</username>
        <id>11</id>
      </contributor>
      <comment>A</comment>
      <origin>36434</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="960" sha1="qq9c4786gkef4i7g4ycs6cjq82if71f" xml:space="preserve">[[de:Drittelkomma-mitteltönig]]
In '''1/3-comma [[Meantone|meantone]]''' temperament, each perfect fifth is tempered by a third of a [[Syntonic_Comma|syntonic comma]] from its just value of [[3/2|3/2]]. This results in minor thirds being exactly [[6/5|6/5]] (and major sixths being exactly [[5/3|5/3]]).

In other words, 1/3-comma meantone is the specific tuning of meantone temperament with an [[Fractional_monzos|eigenmonzo]] of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents, which is about 7 cents flatter than in [[Quarter-comma_meantone|1/4-comma meantone]].

1/3-comma meantone is practically identical to the closed system of [[19edo|19edo]] (although technically there is a slight difference).

=External links=
[http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html Die mitteltönige Stimmung mit reinen kleinen Terzen] (explanation of 1/3 comma meantone, in german)
[[Category:definition]]
[[Category:meantone]]</text>
      <sha1>qq9c4786gkef4i7g4ycs6cjq82if71f</sha1>
    </revision>
    <revision>
      <id>37773</id>
      <parentid>36434</parentid>
      <timestamp>2018-10-24T21:25:04Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>Xenwolf moved page [[1-3 Syntonic Comma Meantone]] to [[1/3 syntonic comma meantone]]: changed into intended lemma (that was not possible in Wikispaces), changed into [[Sentence case]]</comment>
      <origin>37773</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="960" sha1="qq9c4786gkef4i7g4ycs6cjq82if71f" xml:space="preserve">[[de:Drittelkomma-mitteltönig]]
In '''1/3-comma [[Meantone|meantone]]''' temperament, each perfect fifth is tempered by a third of a [[Syntonic_Comma|syntonic comma]] from its just value of [[3/2|3/2]]. This results in minor thirds being exactly [[6/5|6/5]] (and major sixths being exactly [[5/3|5/3]]).

In other words, 1/3-comma meantone is the specific tuning of meantone temperament with an [[Fractional_monzos|eigenmonzo]] of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents, which is about 7 cents flatter than in [[Quarter-comma_meantone|1/4-comma meantone]].

1/3-comma meantone is practically identical to the closed system of [[19edo|19edo]] (although technically there is a slight difference).

=External links=
[http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html Die mitteltönige Stimmung mit reinen kleinen Terzen] (explanation of 1/3 comma meantone, in german)
[[Category:definition]]
[[Category:meantone]]</text>
      <sha1>qq9c4786gkef4i7g4ycs6cjq82if71f</sha1>
    </revision>
    <revision>
      <id>43866</id>
      <parentid>37773</parentid>
      <timestamp>2020-04-10T13:42:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Add the amount by which thirds and fifths are off</comment>
      <origin>43866</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="951" sha1="9jyo9m6we95ko0uusk0k95lgvr5l3hf" xml:space="preserve">[[de:Drittelkomma-mitteltönig]]
In '''1/3-comma [[Meantone|meantone]]''' temperament, each perfect fifth is tempered by a third of a [[Syntonic_Comma|syntonic comma]] from its just value of [[3/2|3/2]]. This results in minor thirds being exactly [[6/5|6/5]] (and major sixths being exactly [[5/3|5/3]]).

In other words, 1/3-comma meantone is the specific tuning of meantone temperament with an [[Fractional_monzos|eigenmonzo]] of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 

1/3-comma meantone is practically identical to the closed system of [[19edo|19edo]] (although technically there is a slight difference).

=External links=
[http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html Die mitteltönige Stimmung mit reinen kleinen Terzen] (explanation of 1/3 comma meantone, in german)
[[Category:definition]]
[[Category:meantone]]</text>
      <sha1>9jyo9m6we95ko0uusk0k95lgvr5l3hf</sha1>
    </revision>
    <revision>
      <id>46014</id>
      <parentid>43866</parentid>
      <timestamp>2020-06-07T18:41:19Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>interwiki via template, wiki markup simplified</comment>
      <origin>46014</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="983" sha1="6fwca6eta8z1kslg6ta42rj3nvm02h0" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3 syntonic comma meantone
| es = 
| ja = 
}}
In '''1/3-comma meantone''' temperament, each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]).

In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[Fractional_monzos|eigenmonzo]] of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 

1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).

== External links ==

* [http://www.lehrklaenge.de/html/mitteltonige_stimmung_iii.html Die mitteltönige Stimmung mit reinen kleinen Terzen] (explanation of 1/3 comma meantone, in german)

[[Category:Definition]]
[[Category:Meantone]]</text>
      <sha1>6fwca6eta8z1kslg6ta42rj3nvm02h0</sha1>
    </revision>
    <revision>
      <id>46015</id>
      <parentid>46014</parentid>
      <timestamp>2020-06-07T18:54:54Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>linkfix</comment>
      <origin>46015</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="985" sha1="pg5jckc0g9ubwkzwfawjxj7zcam1yf5" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3 syntonic comma meantone
| es = 
| ja = 
}}
In '''1/3-comma meantone''' temperament, each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]).

In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[Fractional_monzos|eigenmonzo]] of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 

1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).

== External links ==

* [http://www.lehrklaenge.de/PHP/Tonsystem/MitteltoenigeStimmung3.php Die mitteltönige Stimmung mit reinen kleinen Terzen] (explanation of 1/3 comma meantone, German)

[[Category:Definition]]
[[Category:Meantone]]</text>
      <sha1>pg5jckc0g9ubwkzwfawjxj7zcam1yf5</sha1>
    </revision>
    <revision>
      <id>72579</id>
      <parentid>46015</parentid>
      <timestamp>2021-06-13T23:32:32Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>link directly to eigenmonzo now that it has its own page</comment>
      <origin>72579</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="967" sha1="0s4q779et2jkfh20hv8vpxfjt4qyosl" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3 syntonic comma meantone
| es = 
| ja = 
}}
In '''1/3-comma meantone''' temperament, each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]).

In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[eigenmonzo]] of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 

1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).

== External links ==

* [http://www.lehrklaenge.de/PHP/Tonsystem/MitteltoenigeStimmung3.php Die mitteltönige Stimmung mit reinen kleinen Terzen] (explanation of 1/3 comma meantone, German)

[[Category:Definition]]
[[Category:Meantone]]</text>
      <sha1>0s4q779et2jkfh20hv8vpxfjt4qyosl</sha1>
    </revision>
    <revision>
      <id>104767</id>
      <parentid>72579</parentid>
      <timestamp>2023-03-06T21:52:07Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Definition]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>104767</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="943" sha1="cfpxjl69y9yqzdkfnrg2i75g4x2km33" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3 syntonic comma meantone
| es = 
| ja = 
}}
In '''1/3-comma meantone''' temperament, each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]).

In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[eigenmonzo]] of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 

1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).

== External links ==

* [http://www.lehrklaenge.de/PHP/Tonsystem/MitteltoenigeStimmung3.php Die mitteltönige Stimmung mit reinen kleinen Terzen] (explanation of 1/3 comma meantone, German)

[[Category:Meantone]]</text>
      <sha1>cfpxjl69y9yqzdkfnrg2i75g4x2km33</sha1>
    </revision>
    <revision>
      <id>109721</id>
      <parentid>104767</parentid>
      <timestamp>2023-04-27T01:12:04Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added to category "historical"</comment>
      <origin>109721</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="967" sha1="16vrcm91x3orbxwlb14to3jdifv0f4w" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3 syntonic comma meantone
| es = 
| ja = 
}}
In '''1/3-comma meantone''' temperament, each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]).

In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[eigenmonzo]] of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 

1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).

== External links ==

* [http://www.lehrklaenge.de/PHP/Tonsystem/MitteltoenigeStimmung3.php Die mitteltönige Stimmung mit reinen kleinen Terzen] (explanation of 1/3 comma meantone, German)

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>16vrcm91x3orbxwlb14to3jdifv0f4w</sha1>
    </revision>
    <revision>
      <id>112148</id>
      <parentid>109721</parentid>
      <timestamp>2023-05-05T23:41:41Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>add unchanged-intervals along with eigenmonzo, per prior agreement on Xenwiki Work Group</comment>
      <origin>112148</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="988" sha1="fy4o0o4dhjsb6fhmlvzct3awcb4x5pu" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3 syntonic comma meantone
| es = 
| ja = 
}}
In '''1/3-comma meantone''' temperament, each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]).

In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[eigenmonzo]] (unchanged-interval) of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 

1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).

== External links ==

* [http://www.lehrklaenge.de/PHP/Tonsystem/MitteltoenigeStimmung3.php Die mitteltönige Stimmung mit reinen kleinen Terzen] (explanation of 1/3 comma meantone, German)

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>fy4o0o4dhjsb6fhmlvzct3awcb4x5pu</sha1>
    </revision>
    <revision>
      <id>124894</id>
      <parentid>112148</parentid>
      <timestamp>2023-09-30T07:06:58Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>FloraC moved page [[1/3 syntonic comma meantone]] to [[1/3-comma meantone]] over redirect: Syntonic comma is assumed</comment>
      <origin>112148</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="988" sha1="fy4o0o4dhjsb6fhmlvzct3awcb4x5pu" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3 syntonic comma meantone
| es = 
| ja = 
}}
In '''1/3-comma meantone''' temperament, each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]).

In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[eigenmonzo]] (unchanged-interval) of 6/5.

The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 

1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).

== External links ==

* [http://www.lehrklaenge.de/PHP/Tonsystem/MitteltoenigeStimmung3.php Die mitteltönige Stimmung mit reinen kleinen Terzen] (explanation of 1/3 comma meantone, German)

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>fy4o0o4dhjsb6fhmlvzct3awcb4x5pu</sha1>
    </revision>
    <revision>
      <id>124918</id>
      <parentid>124894</parentid>
      <timestamp>2023-10-01T08:30:09Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+tuning profile</comment>
      <origin>124918</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1398" sha1="su649u89fs2rqivzndwgy3movb374nn" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3 syntonic comma meantone
| es = 
| ja = 
}}
'''1/3-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]). In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[Eigenmonzo|eigenmonzo (unchanged-interval)]] of 6/5. The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 4/3 || 4/3 || 1/3 || ]
|-
| ⟨ || 0 || -1/3 || -4/3 || -10/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1894.7862 2779.1450 3347.8624 }}

[[Mistuning map]]: {{val| 0 -7.1688 -7.1688 -20.9635 }}

== External links ==
* [http://www.lehrklaenge.de/PHP/Tonsystem/MitteltoenigeStimmung3.php Lehrklaenge.de | ''Die mitteltönige Stimmung mit reinen kleinen Terzen''] (explanation of 1/3 comma meantone, German)

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>su649u89fs2rqivzndwgy3movb374nn</sha1>
    </revision>
    <revision>
      <id>146832</id>
      <parentid>124918</parentid>
      <timestamp>2024-07-01T00:57:55Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>External link</comment>
      <origin>146832</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1542" sha1="o5smumuvxbcgqdvojvzuajb1m0r6il6" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3 syntonic comma meantone
| es = 
| ja = 
}}
'''1/3-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]). In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[Eigenmonzo|eigenmonzo (unchanged-interval)]] of 6/5. The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 4/3 || 4/3 || 1/3 || ]
|-
| ⟨ || 0 || -1/3 || -4/3 || -10/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1894.7862 2779.1450 3347.8624 }}

[[Mistuning map]]: {{val| 0 -7.1688 -7.1688 -20.9635 }}

== External links ==
* [http://www.lehrklaenge.de/PHP/Tonsystem/MitteltoenigeStimmung3.php Lehrklaenge.de | ''Die mitteltönige Stimmung mit reinen kleinen Terzen''] (explanation of 1/3 comma meantone, German)
* [http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2] on [[Tonalsoft Encyclopedia]]

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>o5smumuvxbcgqdvojvzuajb1m0r6il6</sha1>
    </revision>
    <revision>
      <id>164627</id>
      <parentid>146832</parentid>
      <timestamp>2024-11-07T14:27:55Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Mistuning -&gt; error</comment>
      <origin>164627</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1538" sha1="dy2bdeadlrknb491l6evehuwv0rredr" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3 syntonic comma meantone
| es = 
| ja = 
}}
'''1/3-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]). In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[Eigenmonzo|eigenmonzo (unchanged-interval)]] of 6/5. The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 4/3 || 4/3 || 1/3 || ]
|-
| ⟨ || 0 || -1/3 || -4/3 || -10/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1894.7862 2779.1450 3347.8624 }}

[[Error map]]: {{val| 0 -7.1688 -7.1688 -20.9635 }}

== External links ==
* [http://www.lehrklaenge.de/PHP/Tonsystem/MitteltoenigeStimmung3.php Lehrklaenge.de | ''Die mitteltönige Stimmung mit reinen kleinen Terzen''] (explanation of 1/3 comma meantone, German)
* [http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2] on [[Tonalsoft Encyclopedia]]

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>dy2bdeadlrknb491l6evehuwv0rredr</sha1>
    </revision>
    <revision>
      <id>200239</id>
      <parentid>164627</parentid>
      <timestamp>2025-06-21T06:59:38Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "(unchanged-interval)" to "(unchanged interval)"</comment>
      <origin>200239</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1538" sha1="cg5vuukrchji4ld6ujrnwlo7pcfx730" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3 syntonic comma meantone
| es = 
| ja = 
}}
'''1/3-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]). In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[Eigenmonzo|eigenmonzo (unchanged interval)]] of 6/5. The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference).

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 4/3 || 4/3 || 1/3 || ]
|-
| ⟨ || 0 || -1/3 || -4/3 || -10/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1894.7862 2779.1450 3347.8624 }}

[[Error map]]: {{val| 0 -7.1688 -7.1688 -20.9635 }}

== External links ==
* [http://www.lehrklaenge.de/PHP/Tonsystem/MitteltoenigeStimmung3.php Lehrklaenge.de | ''Die mitteltönige Stimmung mit reinen kleinen Terzen''] (explanation of 1/3 comma meantone, German)
* [http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2] on [[Tonalsoft Encyclopedia]]

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>cg5vuukrchji4ld6ujrnwlo7pcfx730</sha1>
    </revision>
    <revision>
      <id>222253</id>
      <parentid>200239</parentid>
      <timestamp>2026-01-23T18:43:32Z</timestamp>
      <contributor>
        <username>Hotcrystal0</username>
        <id>614</id>
      </contributor>
      <comment>clarify in “practically identical”</comment>
      <origin>222253</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1577" sha1="ovo3wmic7h21v9ll3ip58pmzf2txiyq" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3 syntonic comma meantone
| es = 
| ja = 
}}
'''1/3-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]). In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[Eigenmonzo|eigenmonzo (unchanged interval)]] of 6/5. The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference as 19edo's fifth is 0.05 cents flatter).

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 4/3 || 4/3 || 1/3 || ]
|-
| ⟨ || 0 || -1/3 || -4/3 || -10/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1894.7862 2779.1450 3347.8624 }}

[[Error map]]: {{val| 0 -7.1688 -7.1688 -20.9635 }}

== External links ==
* [http://www.lehrklaenge.de/PHP/Tonsystem/MitteltoenigeStimmung3.php Lehrklaenge.de | ''Die mitteltönige Stimmung mit reinen kleinen Terzen''] (explanation of 1/3 comma meantone, German)
* [http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2] on [[Tonalsoft Encyclopedia]]

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>ovo3wmic7h21v9ll3ip58pmzf2txiyq</sha1>
    </revision>
    <revision>
      <id>234263</id>
      <parentid>222253</parentid>
      <timestamp>2026-07-17T03:20:22Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Interwiki</comment>
      <origin>234263</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1599" sha1="3joa3uxwnm1sy9qevllry62mxsmezfh" xml:space="preserve">{{interwiki
| de = Drittelkomma-mitteltönig
| en = 1/3-comma meantone
| es = 
| ja = 1/3-コンマミーントーン
}}
'''1/3-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a third of a [[syntonic comma]] from its just value of [[3/2]]. This results in minor thirds being exactly [[6/5]] (and major sixths being exactly [[5/3]]). In other words, 1/3-comma meantone is the specific tuning of [[meantone]] temperament with an [[Eigenmonzo|eigenmonzo (unchanged interval)]] of 6/5. The fifth is 694.786 cents in size, and the major third 379.145 cents. Both are flat by 1/3 syntonic comma (''c''. 7.2 cents) than that in the JI. 1/3-comma meantone is practically identical to the closed system of [[19edo]] (although technically there is a slight difference as 19edo's fifth is 0.05 cents flatter).

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 4/3 || 4/3 || 1/3 || ]
|-
| ⟨ || 0 || -1/3 || -4/3 || -10/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1894.7862 2779.1450 3347.8624 }}

[[Error map]]: {{val| 0 -7.1688 -7.1688 -20.9635 }}

== External links ==
* [http://www.lehrklaenge.de/PHP/Tonsystem/MitteltoenigeStimmung3.php Lehrklaenge.de | ''Die mitteltönige Stimmung mit reinen kleinen Terzen''] (explanation of 1/3 comma meantone, German)
* [http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2] on [[Tonalsoft Encyclopedia]]

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>3joa3uxwnm1sy9qevllry62mxsmezfh</sha1>
    </revision>
  </page>
  <page>
    <title>1-3-7-11 tetrads</title>
    <ns>0</ns>
    <id>7</id>
    <revision>
      <id>7</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
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      <text bytes="269" sha1="e7n69o2u89rjp73m7oow8wbruxyzjyq" xml:space="preserve">These are the only 11-limit just intonation tetrads which have scale steps all larger than 11/10; they are the otonal, 1-9/8-11/8-7/4, with steps 9/8-11/9-14/11-8/7, and the utonal, 11/9-11/8-11/7-2, with steps 11/9-9/8-8/7-14/11.
[[Category:chord]]
[[Category:tetrad]]</text>
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    <revision>
      <id>3978</id>
      <timestamp>2011-08-04T02:09:50Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 244255573 - Original comment: **</comment>
      <origin>3978</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1699" sha1="jxshuwnorwh3r9yxdgby8xikund877u" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-04 02:09:50 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;244255573&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;These are the only 11-limit just intonation tetrads which have scale steps all larger than 11/10; they are the otonal, 1-9/8-11/8-7/4, with steps 9/8-11/9-14/11-8/7, and the utonal, 11/9-11/8-11/7-2, with steps 11/9-9/8-8/7-14/11.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-3-7-11 tetrads&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;These are the only 11-limit just intonation tetrads which have scale steps all larger than 11/10; they are the otonal, 1-9/8-11/8-7/4, with steps 9/8-11/9-14/11-8/7, and the utonal, 11/9-11/8-11/7-2, with steps 11/9-9/8-8/7-14/11.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>42010</id>
      <parentid>7</parentid>
      <timestamp>2019-06-30T00:32:14Z</timestamp>
      <contributor>
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        <id>117</id>
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      <comment>/* top */re-categorisation (AWB)</comment>
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      <text bytes="270" sha1="do2zcenecq0sbkmj3s1w2308569fl0q" xml:space="preserve">These are the only 11-limit just intonation tetrads which have scale steps all larger than 11/10; they are the otonal, 1-9/8-11/8-7/4, with steps 9/8-11/9-14/11-8/7, and the utonal, 11/9-11/8-11/7-2, with steps 11/9-9/8-8/7-14/11.
[[Category:Chords]]
[[Category:Tetrad]]</text>
      <sha1>do2zcenecq0sbkmj3s1w2308569fl0q</sha1>
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    <revision>
      <id>47494</id>
      <parentid>42010</parentid>
      <timestamp>2020-07-06T18:26:35Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
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      <text bytes="274" sha1="au8e59j2xq8jmlnfl925chqcwxaobgp" xml:space="preserve">These are the only 11-odd limit just intonation tetrads which have scale steps all larger than 11/10; they are the otonal, 1-9/8-11/8-7/4, with steps 9/8-11/9-14/11-8/7, and the utonal, 11/9-11/8-11/7-2, with steps 11/9-9/8-8/7-14/11.
[[Category:Chords]]
[[Category:Tetrad]]</text>
      <sha1>au8e59j2xq8jmlnfl925chqcwxaobgp</sha1>
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    <revision>
      <id>114409</id>
      <parentid>47494</parentid>
      <timestamp>2023-05-23T12:02:55Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Recategorize</comment>
      <origin>114409</origin>
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      <format>text/x-wiki</format>
      <text bytes="276" sha1="dd1od82ih9mhwbncp6cn98gijmntkrc" xml:space="preserve">These are the only 11-odd limit just intonation tetrads which have scale steps all larger than 11/10; they are the otonal, 1-9/8-11/8-7/4, with steps 9/8-11/9-14/11-8/7, and the utonal, 11/9-11/8-11/7-2, with steps 11/9-9/8-8/7-14/11.

[[Category:Chords]]
[[Category:Tetrads]]</text>
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    <revision>
      <id>131300</id>
      <parentid>114409</parentid>
      <timestamp>2024-01-09T03:44:05Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Added links to dead end page</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="300" sha1="ga5wt6jwsg0eju0btzflx2ao6rxx3fs" xml:space="preserve">These are the only 11-[[odd limit]] [[just intonation]] [[tetrad]]s which have scale steps all larger than [[11/10]]; they are the [[otonal]], 1-9/8-11/8-7/4, with steps 9/8-11/9-14/11-8/7, and the [[utonal]], 11/9-11/8-11/7-2, with steps 11/9-9/8-8/7-14/11.

[[Category:Chords]]
[[Category:Tetrads]]</text>
      <sha1>ga5wt6jwsg0eju0btzflx2ao6rxx3fs</sha1>
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  </page>
  <page>
    <title>1/5-comma meantone</title>
    <ns>0</ns>
    <id>9</id>
    <revision>
      <id>9</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <text bytes="552" sha1="imgr8vsvat9k48t6hyuxrgx6cyienr0" xml:space="preserve">In 1/5-comma [[Meantone|meantone]] temperament, each perfect fifth is tempered by a fifth of a [[Syntonic_Comma|syntonic comma]] from its just value of [[3/2|3/2]]. This results in major sevenths being exactly 15/8 (and minor seconds being exactly 16/15), and in fact this defines 1/5-comma; it is the regular tuning which tempers out 81/80 and tunes 2 and 16/15 purely.

1/5-comma meantone is practically identical to the closed system of [[43edo|43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)</text>
      <sha1>imgr8vsvat9k48t6hyuxrgx6cyienr0</sha1>
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    <revision>
      <id>3981</id>
      <timestamp>2011-09-30T12:15:16Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 260032358 - Original comment: **</comment>
      <origin>3981</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2529" sha1="3zroqz7rpa8y88e4jv7nrk6bos3ecj4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-09-30 12:15:16 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;260032358&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In 1/5-comma [[meantone]] temperament, each perfect fifth is tempered by a fifth of a [[Syntonic Comma|syntonic comma]] from its just value of [[3_2|3/2]]. This results in major sevenths being exactly 15/8 (and minor seconds being exactly 16/15), and in fact this defines 1/5-comma; it is the regular tuning which tempers out 81/80 and tunes 2 and 16/15 purely.

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-5 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In 1/5-comma &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; temperament, each perfect fifth is tempered by a fifth of a &amp;lt;a class="wiki_link" href="/Syntonic%20Comma"&amp;gt;syntonic comma&amp;lt;/a&amp;gt; from its just value of &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. This results in major sevenths being exactly 15/8 (and minor seconds being exactly 16/15), and in fact this defines 1/5-comma; it is the regular tuning which tempers out 81/80 and tunes 2 and 16/15 purely.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
1/5-comma meantone is practically identical to the closed system of &amp;lt;a class="wiki_link" href="/43edo"&amp;gt;43edo&amp;lt;/a&amp;gt; (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>3zroqz7rpa8y88e4jv7nrk6bos3ecj4</sha1>
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    <revision>
      <id>3982</id>
      <timestamp>2011-09-30T12:14:24Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 260032072 - Original comment: **</comment>
      <origin>3982</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2531" sha1="neo1czv7apoa3ywr1xcfbn8n22zgofa" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-09-30 12:14:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;260032072&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In 1/5-comma [[meantone]] temperament, each perfect fifth is tempered by a fifth of a [[Syntonic Comma|syntonic comma]] from its just value of [[3_2|3/2]]. This results in major sevenths being exactly 15/8 (and minor seconds being exactly 16/15), and in fact this defines 1/5-comma; it is the meantone tuning which tempers out 81/80 and tunes 2 and 16/15 purely.

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-5 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In 1/5-comma &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; temperament, each perfect fifth is tempered by a fifth of a &amp;lt;a class="wiki_link" href="/Syntonic%20Comma"&amp;gt;syntonic comma&amp;lt;/a&amp;gt; from its just value of &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. This results in major sevenths being exactly 15/8 (and minor seconds being exactly 16/15), and in fact this defines 1/5-comma; it is the meantone tuning which tempers out 81/80 and tunes 2 and 16/15 purely.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
1/5-comma meantone is practically identical to the closed system of &amp;lt;a class="wiki_link" href="/43edo"&amp;gt;43edo&amp;lt;/a&amp;gt; (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>neo1czv7apoa3ywr1xcfbn8n22zgofa</sha1>
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    <revision>
      <id>3983</id>
      <timestamp>2011-09-30T09:29:02Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;glennobrien</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 259965388 - Original comment: **</comment>
      <origin>3983</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2152" sha1="2ple8bvznrcavxsrhowzn13995aw28t" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:glennobrien|glennobrien]] and made on &lt;tt&gt;2011-09-30 09:29:02 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;259965388&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In 1/5-comma [[meantone]] temperament, each perfect fifth is tempered by a fifth of a [[Syntonic Comma|syntonic comma]] from its just value of [[3_2|3/2]]. This results in major sevenths being exactly 15/8 (and minor seconds being exactly 16/15).

1/5-comma meantone is practically identical to the closed system of 43edo (although technically there is a slight difference).&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-5 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In 1/5-comma &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; temperament, each perfect fifth is tempered by a fifth of a &amp;lt;a class="wiki_link" href="/Syntonic%20Comma"&amp;gt;syntonic comma&amp;lt;/a&amp;gt; from its just value of &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. This results in major sevenths being exactly 15/8 (and minor seconds being exactly 16/15).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
1/5-comma meantone is practically identical to the closed system of 43edo (although technically there is a slight difference).&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>2ple8bvznrcavxsrhowzn13995aw28t</sha1>
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    <revision>
      <id>3984</id>
      <timestamp>2011-09-30T09:24:25Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;glennobrien</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 259963796 - Original comment: **</comment>
      <origin>3984</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1872" sha1="knpn3u2t42pv14er7g8dexckji92ptt" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:glennobrien|glennobrien]] and made on &lt;tt&gt;2011-09-30 09:24:25 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;259963796&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In 1/5-comma [[meantone]] temperament, each perfect fifth is tempered by a fifth of a [[Syntonic Comma|syntonic comma]] from its just value of [[3_2|3/2]]. This results in major sevenths being exactly 15/8 (and minor seconds being exactly 16/15).&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-5 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In 1/5-comma &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; temperament, each perfect fifth is tempered by a fifth of a &amp;lt;a class="wiki_link" href="/Syntonic%20Comma"&amp;gt;syntonic comma&amp;lt;/a&amp;gt; from its just value of &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. This results in major sevenths being exactly 15/8 (and minor seconds being exactly 16/15).&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>knpn3u2t42pv14er7g8dexckji92ptt</sha1>
    </revision>
    <revision>
      <id>37776</id>
      <parentid>9</parentid>
      <timestamp>2018-10-24T21:31:28Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>Xenwolf moved page [[1-5 Syntonic Comma Meantone]] to [[1/5 syntonic comma meantone]]: changed into intended lemma (that was not possible in Wikispaces), changed into [[Sentence case]]</comment>
      <origin>37776</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="552" sha1="imgr8vsvat9k48t6hyuxrgx6cyienr0" xml:space="preserve">In 1/5-comma [[Meantone|meantone]] temperament, each perfect fifth is tempered by a fifth of a [[Syntonic_Comma|syntonic comma]] from its just value of [[3/2|3/2]]. This results in major sevenths being exactly 15/8 (and minor seconds being exactly 16/15), and in fact this defines 1/5-comma; it is the regular tuning which tempers out 81/80 and tunes 2 and 16/15 purely.

1/5-comma meantone is practically identical to the closed system of [[43edo|43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)</text>
      <sha1>imgr8vsvat9k48t6hyuxrgx6cyienr0</sha1>
    </revision>
    <revision>
      <id>37784</id>
      <parentid>37776</parentid>
      <timestamp>2018-10-24T21:40:08Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>lemma bold, added links, added cat</comment>
      <origin>37784</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="563" sha1="gxuenye5s00zebfkxequ5iq7dh2sa25" xml:space="preserve">In '''1/5-comma [[meantone]]''' temperament, each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes 2 and 16/15 purely.

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 [[cent]]s sharper.)

[[Category:Meantone]]</text>
      <sha1>gxuenye5s00zebfkxequ5iq7dh2sa25</sha1>
    </revision>
    <revision>
      <id>43869</id>
      <parentid>37784</parentid>
      <timestamp>2020-04-10T14:03:09Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Add the amount by which thirds and fifths are off</comment>
      <origin>43869</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="724" sha1="ot60obabm6vpcvbcwjgyclkn295vqb9" xml:space="preserve">In '''1/5-comma [[meantone]]''' temperament, each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes 2 and 16/15 purely. 

The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[Cent|cents]]), with the major third sharp and the fifth flat. 

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)

[[Category:Meantone]]</text>
      <sha1>ot60obabm6vpcvbcwjgyclkn295vqb9</sha1>
    </revision>
    <revision>
      <id>52588</id>
      <parentid>43869</parentid>
      <timestamp>2020-11-10T15:24:06Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>Xenwolf moved page [[1/5 syntonic comma meantone]] to [[1/5-comma meantone]]: meantone implies "syntonic", so no need for ist in the lemma</comment>
      <origin>52588</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="724" sha1="ot60obabm6vpcvbcwjgyclkn295vqb9" xml:space="preserve">In '''1/5-comma [[meantone]]''' temperament, each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes 2 and 16/15 purely. 

The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[Cent|cents]]), with the major third sharp and the fifth flat. 

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)

[[Category:Meantone]]</text>
      <sha1>ot60obabm6vpcvbcwjgyclkn295vqb9</sha1>
    </revision>
    <revision>
      <id>109728</id>
      <parentid>52588</parentid>
      <timestamp>2023-04-27T01:14:41Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added to category "historical"</comment>
      <origin>109728</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="748" sha1="ik3ma8tviid696d3vcji8jeevr2tsa5" xml:space="preserve">In '''1/5-comma [[meantone]]''' temperament, each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes 2 and 16/15 purely. 

The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[Cent|cents]]), with the major third sharp and the fifth flat. 

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>ik3ma8tviid696d3vcji8jeevr2tsa5</sha1>
    </revision>
    <revision>
      <id>124916</id>
      <parentid>109728</parentid>
      <timestamp>2023-10-01T08:18:17Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+tuning profile</comment>
      <origin>124916</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1124" sha1="15ukfmb2gtt2htc2sq31da6ndjdy360" xml:space="preserve">'''1/5-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes [[2/1|2]] and [[16/15]] purely. 

The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[cent]]s), with the major third sharp and the fifth flat. 

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 4/5 || -4/5 || -5 || ]
|-
| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
|-
| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1897.6537 2790.6150 3376.5374 }}

[[Mistuning map]]: {{val| 0 -4.3013 +4.3013 +7.7115 }}

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>15ukfmb2gtt2htc2sq31da6ndjdy360</sha1>
    </revision>
    <revision>
      <id>155159</id>
      <parentid>124916</parentid>
      <timestamp>2024-09-16T15:48:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Something to note with septimal meantone</comment>
      <origin>155159</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1272" sha1="j92lv3esnr3uj3ict1144ptj6059cn8" xml:space="preserve">'''1/5-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes [[2/1|2]] and [[16/15]] purely. 

The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[cent]]s), with the major third sharp and the fifth flat. The augmented sixth is exactly [[225/128]], and for septimal meantone, this means the [[7/4]] is sharp by a [[225/224|septimal kleisma (225/224)]]. 

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 4/5 || -4/5 || -5 || ]
|-
| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
|-
| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1897.6537 2790.6150 3376.5374 }}

[[Mistuning map]]: {{val| 0 -4.3013 +4.3013 +7.7115 }}

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>j92lv3esnr3uj3ict1144ptj6059cn8</sha1>
    </revision>
    <revision>
      <id>164629</id>
      <parentid>155159</parentid>
      <timestamp>2024-11-07T14:30:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Mistuning -&gt; error</comment>
      <origin>164629</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1268" sha1="rmmpszivazhvydgzojhz9pfb695w4ky" xml:space="preserve">'''1/5-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes [[2/1|2]] and [[16/15]] purely. 

The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[cent]]s), with the major third sharp and the fifth flat. The augmented sixth is exactly [[225/128]], and for septimal meantone, this means the [[7/4]] is sharp by a [[225/224|septimal kleisma (225/224)]]. 

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 4/5 || -4/5 || -5 || ]
|-
| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
|-
| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1897.6537 2790.6150 3376.5374 }}

[[Error map]]: {{val| 0 -4.3013 +4.3013 +7.7115 }}

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>rmmpszivazhvydgzojhz9pfb695w4ky</sha1>
    </revision>
    <revision>
      <id>210348</id>
      <parentid>164629</parentid>
      <timestamp>2025-09-21T13:47:19Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Music</comment>
      <origin>210348</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1354" sha1="4q2qqf6wh5c0ht2b1w1jo9ni4h2wyz0" xml:space="preserve">'''1/5-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes [[2/1|2]] and [[16/15]] purely. 

The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[cent]]s), with the major third sharp and the fifth flat. The augmented sixth is exactly [[225/128]], and for septimal meantone, this means the [[7/4]] is sharp by a [[225/224|septimal kleisma (225/224)]]. 

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 4/5 || -4/5 || -5 || ]
|-
| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
|-
| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1897.6537 2790.6150 3376.5374 }}

[[Error map]]: {{val| 0 -4.3013 +4.3013 +7.7115 }}

== Music ==
; [[septimal lemon]]
* [https://youtu.be/jlbpQqahZh0 ''Bubbles''] (2025)

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>4q2qqf6wh5c0ht2b1w1jo9ni4h2wyz0</sha1>
    </revision>
    <revision>
      <id>229532</id>
      <parentid>210348</parentid>
      <timestamp>2026-05-03T08:29:21Z</timestamp>
      <contributor>
        <username>Hstraub</username>
        <id>11</id>
      </contributor>
      <comment>Link to German</comment>
      <origin>229532</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1429" sha1="8jgfp0a21av5zxvbs144guvqwfbs2e0" xml:space="preserve">{{Interwiki
| en = 1/5-comma mesntone
| de = Fünftelkomma-mitteltönig
}}
'''1/5-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes [[2/1|2]] and [[16/15]] purely. 

The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[cent]]s), with the major third sharp and the fifth flat. The augmented sixth is exactly [[225/128]], and for septimal meantone, this means the [[7/4]] is sharp by a [[225/224|septimal kleisma (225/224)]]. 

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 4/5 || -4/5 || -5 || ]
|-
| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
|-
| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1897.6537 2790.6150 3376.5374 }}

[[Error map]]: {{val| 0 -4.3013 +4.3013 +7.7115 }}

== Music ==
; [[septimal lemon]]
* [https://youtu.be/jlbpQqahZh0 ''Bubbles''] (2025)

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>8jgfp0a21av5zxvbs144guvqwfbs2e0</sha1>
    </revision>
    <revision>
      <id>229533</id>
      <parentid>229532</parentid>
      <timestamp>2026-05-03T08:30:22Z</timestamp>
      <contributor>
        <username>Hstraub</username>
        <id>11</id>
      </contributor>
      <origin>229533</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1429" sha1="p1untb4frpxyjn87aipc3or9avt2r6p" xml:space="preserve">{{Interwiki
| en = 1/5-comma meantone
| de = Fünftelkomma-mitteltönig
}}
'''1/5-comma meantone''' is the tuning of [[meantone]] temperament in which each perfect fifth is tempered by a fifth of a [[syntonic comma]] from its just value of [[3/2]]. This results in major sevenths being exactly [[15/8]] (and minor seconds being exactly [[16/15]]), and in fact this defines 1/5-comma; it is the regular tuning which tempers out [[81/80]] and tunes [[2/1|2]] and [[16/15]] purely. 

The major third ([[5/4]]) and the fifth are equally off from JI by 1/5 syntonic comma (''c''. 4.3 [[cent]]s), with the major third sharp and the fifth flat. The augmented sixth is exactly [[225/128]], and for septimal meantone, this means the [[7/4]] is sharp by a [[225/224|septimal kleisma (225/224)]]. 

1/5-comma meantone is practically identical to the closed system of [[43edo]] (although technically there is a slight difference, as the fifth of 43 is 0.0207 cents sharper.)

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 4/5 || -4/5 || -5 || ]
|-
| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
|-
| ⟨ || 0 || 1/5 || 4/5 || 2 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1897.6537 2790.6150 3376.5374 }}

[[Error map]]: {{val| 0 -4.3013 +4.3013 +7.7115 }}

== Music ==
; [[septimal lemon]]
* [https://youtu.be/jlbpQqahZh0 ''Bubbles''] (2025)

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>p1untb4frpxyjn87aipc3or9avt2r6p</sha1>
    </revision>
  </page>
  <page>
    <title>1/6-comma meantone</title>
    <ns>0</ns>
    <id>10</id>
    <revision>
      <id>10</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>10</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1292" sha1="r2bwjy6r7ap6cflbw8kn3pqow0yy6nv" xml:space="preserve">1/6 comma meantone is the tuning of [[Meantone_family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone 45/32 is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. It is closely related to [[55edo|55edo]] in terms of its tuning.

=Fractional projection matrix=
The [[Fractional_monzos|fractional projection map]] defining 7-limit 1/6 comma meantone is

{| class="wikitable"
|-
| | [1
| | 0
| | 0
| | 0&amp;gt;
|-
| | [2/3
| | 1/3
| | 1/6
| | 0&amp;gt;
|-
| | [-4/3
| | 4/3
| | 2/3
| | 0&amp;gt;
|-
| | [-19/3
| | 10/3
| | 5/3
| | 0&amp;gt;
|}

=Links=
[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone] by Ross W. Duffin [http://www.webcitation.org/5zW8FuybZ permalink]

[http://sonic-arts.org/monzo/55edo/55edo.htm Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone] by [[Joe_Monzo|Joe Monzo]] [http://www.webcitation.org/5zW910Jax permalink]      [[Category:1/6-comma]]
[[Category:meantone]]</text>
      <sha1>r2bwjy6r7ap6cflbw8kn3pqow0yy6nv</sha1>
    </revision>
    <revision>
      <id>3985</id>
      <timestamp>2011-09-30T13:11:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 260052498 - Original comment: **</comment>
      <origin>3985</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5278" sha1="d0nvl6npbz993pv1tqsvxcdx400f62m" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-09-30 13:11:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;260052498&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;1/6 comma meantone is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone 45/32 is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. It is closely related to [[55edo]] in terms of its tuning.

=Fractional projection matrix=
The [[Fractional monzos|fractional projection map]] defining 7-limit 1/6 comma meantone is
|| [1 || 0 || 0 || 0&amp;gt; ||
|| [2/3 || 1/3 || 1/6 || 0&amp;gt; ||
|| [-4/3 || 4/3 || 2/3 || 0&amp;gt; ||
|| [-19/3 || 10/3 || 5/3 || 0&amp;gt; ||

=Links=
[[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html|Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone]] by Ross W. Duffin [[http://www.webcitation.org/5zW8FuybZ|permalink]]
[[http://sonic-arts.org/monzo/55edo/55edo.htm|Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone]] by [[Joe Monzo]] [[http://www.webcitation.org/5zW910Jax|permalink]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-6 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;1/6 comma meantone is the tuning of &amp;lt;a class="wiki_link" href="/Meantone%20family"&amp;gt;meantone temperament&amp;lt;/a&amp;gt; which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone 45/32 is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. It is closely related to &amp;lt;a class="wiki_link" href="/55edo"&amp;gt;55edo&amp;lt;/a&amp;gt; in terms of its tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Fractional projection matrix"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Fractional projection matrix&amp;lt;/h1&amp;gt;
The &amp;lt;a class="wiki_link" href="/Fractional%20monzos"&amp;gt;fractional projection map&amp;lt;/a&amp;gt; defining 7-limit 1/6 comma meantone is&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[-4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[-19/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.case.edu/~rwd/baroquetemp/XMT.intro.html" rel="nofollow"&amp;gt;Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone&amp;lt;/a&amp;gt; by Ross W. Duffin &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW8FuybZ" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/monzo/55edo/55edo.htm" rel="nofollow"&amp;gt;Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Joe%20Monzo"&amp;gt;Joe Monzo&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW910Jax" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>d0nvl6npbz993pv1tqsvxcdx400f62m</sha1>
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    <revision>
      <id>3986</id>
      <timestamp>2011-09-19T21:44:54Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 255901392 - Original comment: **</comment>
      <origin>3986</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4930" sha1="6x7xpgckgkskj5095qvhgldgyw6tp2g" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-09-19 21:44:54 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;255901392&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;1/6 comma meantone is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to [[55edo]] in terms of its tuning.

=Fractional projection matrix=
The [[Fractional monzos|fractional projection map]] defining 7-limit 1/6 comma meantone is
|| [1 || 0 || 0 || 0&amp;gt; ||
|| [2/3 || 1/3 || 1/6 || 0&amp;gt; ||
|| [-4/3 || 4/3 || 2/3 || 0&amp;gt; ||
|| [-19/3 || 10/3 || 5/3 || 0&amp;gt; ||

=Links=
[[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html|Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone]] by Ross W. Duffin [[http://www.webcitation.org/5zW8FuybZ|permalink]]
[[http://sonic-arts.org/monzo/55edo/55edo.htm|Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone]] by [[Joe Monzo]] [[http://www.webcitation.org/5zW910Jax|permalink]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-6 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;1/6 comma meantone is the tuning of &amp;lt;a class="wiki_link" href="/Meantone%20family"&amp;gt;meantone temperament&amp;lt;/a&amp;gt; which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to &amp;lt;a class="wiki_link" href="/55edo"&amp;gt;55edo&amp;lt;/a&amp;gt; in terms of its tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Fractional projection matrix"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Fractional projection matrix&amp;lt;/h1&amp;gt;
The &amp;lt;a class="wiki_link" href="/Fractional%20monzos"&amp;gt;fractional projection map&amp;lt;/a&amp;gt; defining 7-limit 1/6 comma meantone is&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[-4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[-19/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.case.edu/~rwd/baroquetemp/XMT.intro.html" rel="nofollow"&amp;gt;Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone&amp;lt;/a&amp;gt; by Ross W. Duffin &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW8FuybZ" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/monzo/55edo/55edo.htm" rel="nofollow"&amp;gt;Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Joe%20Monzo"&amp;gt;Joe Monzo&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW910Jax" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>6x7xpgckgkskj5095qvhgldgyw6tp2g</sha1>
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    <revision>
      <id>3987</id>
      <timestamp>2011-06-17T17:48:58Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 237373087 - Original comment: **</comment>
      <origin>3987</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4945" sha1="pmc6hm8dtb5vpb2waw4x4kn8gioemyi" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-17 17:48:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;237373087&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;1/6 comma meantone is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to [[55edo]] in terms of its tuning.

=Fractional projection matrix=
The [[Fractional monzos|fractional projection map]] defining 7-limit 1/6 comma meantone is
|| [1 || 0 || 0 || 0&amp;gt; ||
|| [2/3 || 1/3 || 1/6 || 0&amp;gt; ||
|| [-4/3 || 4/3 || 2/3 || 0&amp;gt; || 


|| [-19/3 || 10/3 || 5/3 || 0&amp;gt; ||

=Links=
[[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html|Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone]] by Ross W. Duffin [[http://www.webcitation.org/5zW8FuybZ|permalink]]
[[http://sonic-arts.org/monzo/55edo/55edo.htm|Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone]] by [[Joe Monzo]] [[http://www.webcitation.org/5zW910Jax|permalink]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-6 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;1/6 comma meantone is the tuning of &amp;lt;a class="wiki_link" href="/Meantone%20family"&amp;gt;meantone temperament&amp;lt;/a&amp;gt; which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to &amp;lt;a class="wiki_link" href="/55edo"&amp;gt;55edo&amp;lt;/a&amp;gt; in terms of its tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Fractional projection matrix"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Fractional projection matrix&amp;lt;/h1&amp;gt;
The &amp;lt;a class="wiki_link" href="/Fractional%20monzos"&amp;gt;fractional projection map&amp;lt;/a&amp;gt; defining 7-limit 1/6 comma meantone is&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[-4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;[-19/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.case.edu/~rwd/baroquetemp/XMT.intro.html" rel="nofollow"&amp;gt;Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone&amp;lt;/a&amp;gt; by Ross W. Duffin &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW8FuybZ" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/monzo/55edo/55edo.htm" rel="nofollow"&amp;gt;Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Joe%20Monzo"&amp;gt;Joe Monzo&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW910Jax" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>pmc6hm8dtb5vpb2waw4x4kn8gioemyi</sha1>
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    <revision>
      <id>3988</id>
      <timestamp>2011-06-17T17:33:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 237371211 - Original comment: **</comment>
      <origin>3988</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4943" sha1="idhwp66ribsu7k267rgoftk9mj4hwh2" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-17 17:33:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;237371211&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;1/6 comma meantone is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to [[55edo]] in terms of its tuning.

=Fractional projection matrix=
The [[Fractional monzos|fractional projection map]] defining 7-limit 1/6 comma meantone is
|| [1 || 0 || 0 || 0&amp;gt; ||
|| [2/3 || 1/3 || 1/6 || 0&amp;gt; ||
|| [-4/3 || 4/3 || 2/3 || 0&amp;gt; || 
|| [-19/3 || 10/3 || 5/3 || 0&amp;gt; ||

=Links=
[[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html|Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone]] by Ross W. Duffin [[http://www.webcitation.org/5zW8FuybZ|permalink]]
[[http://sonic-arts.org/monzo/55edo/55edo.htm|Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone]] by [[Joe Monzo]] [[http://www.webcitation.org/5zW910Jax|permalink]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-6 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;1/6 comma meantone is the tuning of &amp;lt;a class="wiki_link" href="/Meantone%20family"&amp;gt;meantone temperament&amp;lt;/a&amp;gt; which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to &amp;lt;a class="wiki_link" href="/55edo"&amp;gt;55edo&amp;lt;/a&amp;gt; in terms of its tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Fractional projection matrix"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Fractional projection matrix&amp;lt;/h1&amp;gt;
The &amp;lt;a class="wiki_link" href="/Fractional%20monzos"&amp;gt;fractional projection map&amp;lt;/a&amp;gt; defining 7-limit 1/6 comma meantone is&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[-4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;[-19/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.case.edu/~rwd/baroquetemp/XMT.intro.html" rel="nofollow"&amp;gt;Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone&amp;lt;/a&amp;gt; by Ross W. Duffin &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW8FuybZ" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/monzo/55edo/55edo.htm" rel="nofollow"&amp;gt;Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Joe%20Monzo"&amp;gt;Joe Monzo&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW910Jax" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>idhwp66ribsu7k267rgoftk9mj4hwh2</sha1>
    </revision>
    <revision>
      <id>3989</id>
      <timestamp>2011-06-17T17:29:54Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 237370859 - Original comment: **</comment>
      <origin>3989</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4403" sha1="526r57zk38lqqbgrc1jlfp3nmh8lpag" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-17 17:29:54 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;237370859&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;1/6 comma meantone is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to [[55edo]] in terms of its tuning.

=Fractional projection matrix=
The [[Fractional monzos|fractional projection map]] defining 7-limit 1/6 comma meantone is
|| [1 0 0 0&amp;gt; ||
|| [2/3 1/3 1/6 0&amp;gt; ||
|| [-4/3 4/3 2/3 0&amp;gt; || 
|| [-19/3 10/3 5/3 0&amp;gt; ||

=Links=
[[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html|Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone]] by Ross W. Duffin [[http://www.webcitation.org/5zW8FuybZ|permalink]]
[[http://sonic-arts.org/monzo/55edo/55edo.htm|Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone]] by [[Joe Monzo]] [[http://www.webcitation.org/5zW910Jax|permalink]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-6 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;1/6 comma meantone is the tuning of &amp;lt;a class="wiki_link" href="/Meantone%20family"&amp;gt;meantone temperament&amp;lt;/a&amp;gt; which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to &amp;lt;a class="wiki_link" href="/55edo"&amp;gt;55edo&amp;lt;/a&amp;gt; in terms of its tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Fractional projection matrix"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Fractional projection matrix&amp;lt;/h1&amp;gt;
The &amp;lt;a class="wiki_link" href="/Fractional%20monzos"&amp;gt;fractional projection map&amp;lt;/a&amp;gt; defining 7-limit 1/6 comma meantone is&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[1 0 0 0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;[-4/3 4/3 2/3 0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;[-19/3 10/3 5/3 0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.case.edu/~rwd/baroquetemp/XMT.intro.html" rel="nofollow"&amp;gt;Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone&amp;lt;/a&amp;gt; by Ross W. Duffin &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW8FuybZ" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/monzo/55edo/55edo.htm" rel="nofollow"&amp;gt;Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Joe%20Monzo"&amp;gt;Joe Monzo&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW910Jax" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>526r57zk38lqqbgrc1jlfp3nmh8lpag</sha1>
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    <revision>
      <id>3990</id>
      <timestamp>2011-06-17T17:28:30Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 237370707 - Original comment: **</comment>
      <origin>3990</origin>
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      <text bytes="4288" sha1="e5ectwhkxkm8n7otq4mygd3unerqoxj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-17 17:28:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;237370707&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;1/6 comma meantone is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to [[55edo]] in terms of its tuning.

=Fractional projection matrix=
The [[Fractional monzos|fractional projection map]] defining 7-limit 1/6 comma meantone is
|| [1 0 0 0&amp;gt; || [2/3 1/3 1/6 0&amp;gt; || [-4/3 4/3 2/3 0&amp;gt; || [-19/3 10/3 5/3 0&amp;gt; ||

=Links=
[[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html|Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone]] by Ross W. Duffin [[http://www.webcitation.org/5zW8FuybZ|permalink]]
[[http://sonic-arts.org/monzo/55edo/55edo.htm|Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone]] by [[Joe Monzo]] [[http://www.webcitation.org/5zW910Jax|permalink]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-6 Syntonic Comma Meantone&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;1/6 comma meantone is the tuning of &amp;lt;a class="wiki_link" href="/Meantone%20family"&amp;gt;meantone temperament&amp;lt;/a&amp;gt; which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to &amp;lt;a class="wiki_link" href="/55edo"&amp;gt;55edo&amp;lt;/a&amp;gt; in terms of its tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Fractional projection matrix"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Fractional projection matrix&amp;lt;/h1&amp;gt;
The &amp;lt;a class="wiki_link" href="/Fractional%20monzos"&amp;gt;fractional projection map&amp;lt;/a&amp;gt; defining 7-limit 1/6 comma meantone is&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;[-19/3 10/3 5/3 0&amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
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&amp;lt;a class="wiki_link_ext" href="http://music.case.edu/~rwd/baroquetemp/XMT.intro.html" rel="nofollow"&amp;gt;Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone&amp;lt;/a&amp;gt; by Ross W. Duffin &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW8FuybZ" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/monzo/55edo/55edo.htm" rel="nofollow"&amp;gt;Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Joe%20Monzo"&amp;gt;Joe Monzo&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW910Jax" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2011-06-17T17:26:41Z</timestamp>
      <contributor>
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        <id>0</id>
      </contributor>
      <comment>**Imported revision 237370461 - Original comment: **</comment>
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      <text bytes="4081" sha1="qoopommfl6bvcw9hnuvg8dcjdetd4dh" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-17 17:26:41 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;237370461&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;1/6 comma meantone is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to [[55edo]] in terms of its tuning.

=Fractional projection matrix=
The [[Fractional monzos|fractional projection map]] defining 7-limit 1/6 comma meantone is
[|1 0 0 0&amp;gt; 
|2/3 1/3 1/6 0&amp;gt; 
|-4/3 4/3 2/3 0&amp;gt; 
|-19/3 10/3 5/3 0&amp;gt;]

=Links=
[[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html|Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone]] by Ross W. Duffin [[http://www.webcitation.org/5zW8FuybZ|permalink]]
[[http://sonic-arts.org/monzo/55edo/55edo.htm|Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone]] by [[Joe Monzo]] [[http://www.webcitation.org/5zW910Jax|permalink]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
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The &amp;lt;a class="wiki_link" href="/Fractional%20monzos"&amp;gt;fractional projection map&amp;lt;/a&amp;gt; defining 7-limit 1/6 comma meantone is&amp;lt;br /&amp;gt;
[|1 0 0 0&amp;amp;gt; &amp;lt;br /&amp;gt;
|2/3 1/3 1/6 0&amp;amp;gt; &amp;lt;br /&amp;gt;
|-4/3 4/3 2/3 0&amp;amp;gt; &amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.case.edu/~rwd/baroquetemp/XMT.intro.html" rel="nofollow"&amp;gt;Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone&amp;lt;/a&amp;gt; by Ross W. Duffin &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW8FuybZ" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/monzo/55edo/55edo.htm" rel="nofollow"&amp;gt;Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Joe%20Monzo"&amp;gt;Joe Monzo&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW910Jax" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2011-06-17T17:10:46Z</timestamp>
      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 237368571 - Original comment: **</comment>
      <origin>3992</origin>
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      <text bytes="3355" sha1="2qfid2wr7zk5mskegltwi5ob0sr375p" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-17 17:10:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;237368571&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;1/6 comma meantone is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to [[55edo]] in terms of its tuning.

=Links=
[[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html|Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone]] by Ross W. Duffin [[http://www.webcitation.org/5zW8FuybZ|permalink]]
[[http://sonic-arts.org/monzo/55edo/55edo.htm|Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone]] by [[Joe Monzo]] [[http://www.webcitation.org/5zW910Jax|permalink]]&lt;/pre&gt;&lt;/div&gt;
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&amp;lt;br /&amp;gt;
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&amp;lt;a class="wiki_link_ext" href="http://music.case.edu/~rwd/baroquetemp/XMT.intro.html" rel="nofollow"&amp;gt;Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone&amp;lt;/a&amp;gt; by Ross W. Duffin &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW8FuybZ" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/monzo/55edo/55edo.htm" rel="nofollow"&amp;gt;Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Joe%20Monzo"&amp;gt;Joe Monzo&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW910Jax" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2011-06-17T17:08:45Z</timestamp>
      <contributor>
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        <id>0</id>
      </contributor>
      <comment>**Imported revision 237368297 - Original comment: **</comment>
      <origin>3993</origin>
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      <text bytes="3369" sha1="qqw5fohq5waqvzcd1fxc6em9xpcac28" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-17 17:08:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;237368297&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;1/6 comma meantone is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to [[55edo]] in terms of its tuning.

=Links=
[[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html|Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone]] by Ross W. Duffin [[http://www.webcitation.org/5zW8FuybZ|permalink]]

[[http://sonic-arts.org/monzo/55edo/55edo.htm|Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone]] by [[Joe Monzo]] [[http://www.webcitation.org/5zW910Jax|permalink]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Links&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.case.edu/~rwd/baroquetemp/XMT.intro.html" rel="nofollow"&amp;gt;Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone&amp;lt;/a&amp;gt; by Ross W. Duffin &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW8FuybZ" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/monzo/55edo/55edo.htm" rel="nofollow"&amp;gt;Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Joe%20Monzo"&amp;gt;Joe Monzo&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW910Jax" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2011-06-17T16:57:22Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 237366911 - Original comment: **</comment>
      <origin>3994</origin>
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      <text bytes="2806" sha1="lg3br9x8qlbhiojkzu4kchotxlv8tvc" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-17 16:57:22 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;237366911&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;1/6 comma meantone is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to [[55edo]] in terms of its tuning.

=Links=
[[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html|Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone]] by Ross W. Duffin [[http://www.webcitation.org/5zW8FuybZ|permalink]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
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&amp;lt;a class="wiki_link_ext" href="http://music.case.edu/~rwd/baroquetemp/XMT.intro.html" rel="nofollow"&amp;gt;Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone&amp;lt;/a&amp;gt; by Ross W. Duffin &amp;lt;a class="wiki_link_ext" href="http://www.webcitation.org/5zW8FuybZ" rel="nofollow"&amp;gt;permalink&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 237366617 - Original comment: **</comment>
      <origin>3995</origin>
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      <text bytes="2645" sha1="2l6l1w4z3d0ngde10uu8usu4pru8vgj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-17 16:54:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;237366617&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;1/6 comma meantone is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. It is closely related to [[55edo]] in terms of its tuning.

=Links=
[[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html|Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone]] by Ross W. Duffin&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
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    <revision>
      <id>37778</id>
      <parentid>10</parentid>
      <timestamp>2018-10-24T21:31:49Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>Xenwolf moved page [[1-6 Syntonic Comma Meantone]] to [[1/6 syntonic comma meantone]]: changed into intended lemma (that was not possible in Wikispaces), changed into [[Sentence case]]</comment>
      <origin>37778</origin>
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      <text bytes="1292" sha1="r2bwjy6r7ap6cflbw8kn3pqow0yy6nv" xml:space="preserve">1/6 comma meantone is the tuning of [[Meantone_family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 cents. This means the fifth is flattened by 1/6 of the syntonic comma (81/80 ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone 45/32 is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. It is closely related to [[55edo|55edo]] in terms of its tuning.

=Fractional projection matrix=
The [[Fractional_monzos|fractional projection map]] defining 7-limit 1/6 comma meantone is

{| class="wikitable"
|-
| | [1
| | 0
| | 0
| | 0&amp;gt;
|-
| | [2/3
| | 1/3
| | 1/6
| | 0&amp;gt;
|-
| | [-4/3
| | 4/3
| | 2/3
| | 0&amp;gt;
|-
| | [-19/3
| | 10/3
| | 5/3
| | 0&amp;gt;
|}

=Links=
[http://music.case.edu/~rwd/baroquetemp/XMT.intro.html Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone] by Ross W. Duffin [http://www.webcitation.org/5zW8FuybZ permalink]

[http://sonic-arts.org/monzo/55edo/55edo.htm Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone] by [[Joe_Monzo|Joe Monzo]] [http://www.webcitation.org/5zW910Jax permalink]      [[Category:1/6-comma]]
[[Category:meantone]]</text>
      <sha1>r2bwjy6r7ap6cflbw8kn3pqow0yy6nv</sha1>
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    <revision>
      <id>37785</id>
      <parentid>37778</parentid>
      <timestamp>2018-10-24T21:43:34Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>lemma bold, simplified and added some links</comment>
      <origin>37785</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1273" sha1="f460zwszy9543drssb54dh3oijzndqq" xml:space="preserve">'''1/6 comma meantone''' is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the syntonic comma ([[81/80]] ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. It is closely related to [[55edo]] in terms of its tuning.

== Fractional projection matrix ==
The [[Fractional monzos|fractional projection map]] defining [[7-limit]] 1/6 comma meantone is

{| class="wikitable"
|-
| [1
| 0
| 0
| 0&amp;gt;
|-
| [2/3
| 1/3
| 1/6
| 0&amp;gt;
|-
| [-4/3
| 4/3
| 2/3
| 0&amp;gt;
|-
| [-19/3
| 10/3
| 5/3
| 0&amp;gt;
|}

== Links ==
* [http://music.case.edu/~rwd/baroquetemp/XMT.intro.html Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone] by Ross W. Duffin [http://www.webcitation.org/5zW8FuybZ permalink]
* [http://sonic-arts.org/monzo/55edo/55edo.htm Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone] by [[Joe Monzo]] [http://www.webcitation.org/5zW910Jax permalink]

[[Category:1/6-comma]]
[[Category:Meantone]]</text>
      <sha1>f460zwszy9543drssb54dh3oijzndqq</sha1>
    </revision>
    <revision>
      <id>80347</id>
      <parentid>37785</parentid>
      <timestamp>2021-11-06T20:55:18Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>projection map → projection matrix</comment>
      <origin>80347</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1276" sha1="b2fx1une586ops86nssg68e24a55yfd" xml:space="preserve">'''1/6 comma meantone''' is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the syntonic comma ([[81/80]] ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. It is closely related to [[55edo]] in terms of its tuning.

== Fractional projection matrix ==
The [[Fractional monzos|fractional projection matrix]] defining [[7-limit]] 1/6 comma meantone is

{| class="wikitable"
|-
| [1
| 0
| 0
| 0&amp;gt;
|-
| [2/3
| 1/3
| 1/6
| 0&amp;gt;
|-
| [-4/3
| 4/3
| 2/3
| 0&amp;gt;
|-
| [-19/3
| 10/3
| 5/3
| 0&amp;gt;
|}

== Links ==
* [http://music.case.edu/~rwd/baroquetemp/XMT.intro.html Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone] by Ross W. Duffin [http://www.webcitation.org/5zW8FuybZ permalink]
* [http://sonic-arts.org/monzo/55edo/55edo.htm Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone] by [[Joe Monzo]] [http://www.webcitation.org/5zW910Jax permalink]

[[Category:1/6-comma]]
[[Category:Meantone]]</text>
      <sha1>b2fx1une586ops86nssg68e24a55yfd</sha1>
    </revision>
    <revision>
      <id>109646</id>
      <parentid>80347</parentid>
      <timestamp>2023-04-26T10:56:14Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <comment>Improve linking, prune nonexistent category.</comment>
      <origin>109646</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1324" sha1="rsdxy6p798fdlglno9o7z2o4sc4sl90" xml:space="preserve">'''1/6 comma meantone''' is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the syntonic comma ([[81/80]] ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it sharply and flatly, respectively, while [[122edo]] using the c val does so near perfectly.

== Fractional projection matrix ==
The [[Fractional monzos|fractional projection matrix]] defining [[7-limit]] 1/6 comma meantone is

{| class="wikitable"
|-
| [1
| 0
| 0
| 0&amp;gt;
|-
| [2/3
| 1/3
| 1/6
| 0&amp;gt;
|-
| [-4/3
| 4/3
| 2/3
| 0&amp;gt;
|-
| [-19/3
| 10/3
| 5/3
| 0&amp;gt;
|}

== Links ==
* [http://music.case.edu/~rwd/baroquetemp/XMT.intro.html Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone] by Ross W. Duffin [http://www.webcitation.org/5zW8FuybZ permalink]
* [http://sonic-arts.org/monzo/55edo/55edo.htm Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone] by [[Joe Monzo]] [http://www.webcitation.org/5zW910Jax permalink]

[[Category:Meantone]]</text>
      <sha1>rsdxy6p798fdlglno9o7z2o4sc4sl90</sha1>
    </revision>
    <revision>
      <id>109647</id>
      <parentid>109646</parentid>
      <timestamp>2023-04-26T10:57:34Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <minor/>
      <comment>Wrong way around.</comment>
      <origin>109647</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1324" sha1="fshnw8uxzrufh3m35mfuclqrhlska2m" xml:space="preserve">'''1/6 comma meantone''' is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the syntonic comma ([[81/80]] ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it flatly and sharply, respectively, while [[122edo]] using the c val does so near perfectly.

== Fractional projection matrix ==
The [[Fractional monzos|fractional projection matrix]] defining [[7-limit]] 1/6 comma meantone is

{| class="wikitable"
|-
| [1
| 0
| 0
| 0&amp;gt;
|-
| [2/3
| 1/3
| 1/6
| 0&amp;gt;
|-
| [-4/3
| 4/3
| 2/3
| 0&amp;gt;
|-
| [-19/3
| 10/3
| 5/3
| 0&amp;gt;
|}

== Links ==
* [http://music.case.edu/~rwd/baroquetemp/XMT.intro.html Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone] by Ross W. Duffin [http://www.webcitation.org/5zW8FuybZ permalink]
* [http://sonic-arts.org/monzo/55edo/55edo.htm Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone] by [[Joe Monzo]] [http://www.webcitation.org/5zW910Jax permalink]

[[Category:Meantone]]</text>
      <sha1>fshnw8uxzrufh3m35mfuclqrhlska2m</sha1>
    </revision>
    <revision>
      <id>109731</id>
      <parentid>109647</parentid>
      <timestamp>2023-04-27T01:15:26Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added to category "historical"</comment>
      <origin>109731</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1348" sha1="cz95m0bhh9gg5tcnupfkephf2kds2pv" xml:space="preserve">'''1/6 comma meantone''' is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the syntonic comma ([[81/80]] ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it flatly and sharply, respectively, while [[122edo]] using the c val does so near perfectly.

== Fractional projection matrix ==
The [[Fractional monzos|fractional projection matrix]] defining [[7-limit]] 1/6 comma meantone is

{| class="wikitable"
|-
| [1
| 0
| 0
| 0&amp;gt;
|-
| [2/3
| 1/3
| 1/6
| 0&amp;gt;
|-
| [-4/3
| 4/3
| 2/3
| 0&amp;gt;
|-
| [-19/3
| 10/3
| 5/3
| 0&amp;gt;
|}

== Links ==
* [http://music.case.edu/~rwd/baroquetemp/XMT.intro.html Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone] by Ross W. Duffin [http://www.webcitation.org/5zW8FuybZ permalink]
* [http://sonic-arts.org/monzo/55edo/55edo.htm Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone] by [[Joe Monzo]] [http://www.webcitation.org/5zW910Jax permalink]

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>cz95m0bhh9gg5tcnupfkephf2kds2pv</sha1>
    </revision>
    <revision>
      <id>121829</id>
      <parentid>109731</parentid>
      <timestamp>2023-08-09T08:38:49Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Corrected Joe Monzo to Joseph Monzo</comment>
      <origin>121829</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1351" sha1="jghdxuxjbg03cbmvt44soquy2saa2r4" xml:space="preserve">'''1/6 comma meantone''' is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the syntonic comma ([[81/80]] ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it flatly and sharply, respectively, while [[122edo]] using the c val does so near perfectly.

== Fractional projection matrix ==
The [[Fractional monzos|fractional projection matrix]] defining [[7-limit]] 1/6 comma meantone is

{| class="wikitable"
|-
| [1
| 0
| 0
| 0&amp;gt;
|-
| [2/3
| 1/3
| 1/6
| 0&amp;gt;
|-
| [-4/3
| 4/3
| 2/3
| 0&amp;gt;
|-
| [-19/3
| 10/3
| 5/3
| 0&amp;gt;
|}

== Links ==
* [http://music.case.edu/~rwd/baroquetemp/XMT.intro.html Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone] by Ross W. Duffin [http://www.webcitation.org/5zW8FuybZ permalink]
* [http://sonic-arts.org/monzo/55edo/55edo.htm Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone] by [[Joseph Monzo]] [http://www.webcitation.org/5zW910Jax permalink]

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>jghdxuxjbg03cbmvt44soquy2saa2r4</sha1>
    </revision>
    <revision>
      <id>124900</id>
      <parentid>121829</parentid>
      <timestamp>2023-09-30T07:08:27Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>FloraC moved page [[1/6 syntonic comma meantone]] to [[1/6-comma meantone]]: Syntonic comma is assumed</comment>
      <origin>121829</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1351" sha1="jghdxuxjbg03cbmvt44soquy2saa2r4" xml:space="preserve">'''1/6 comma meantone''' is the tuning of [[Meantone family|meantone temperament]] which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the syntonic comma ([[81/80]] ratio) of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning tempering out 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it flatly and sharply, respectively, while [[122edo]] using the c val does so near perfectly.

== Fractional projection matrix ==
The [[Fractional monzos|fractional projection matrix]] defining [[7-limit]] 1/6 comma meantone is

{| class="wikitable"
|-
| [1
| 0
| 0
| 0&amp;gt;
|-
| [2/3
| 1/3
| 1/6
| 0&amp;gt;
|-
| [-4/3
| 4/3
| 2/3
| 0&amp;gt;
|-
| [-19/3
| 10/3
| 5/3
| 0&amp;gt;
|}

== Links ==
* [http://music.case.edu/~rwd/baroquetemp/XMT.intro.html Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone] by Ross W. Duffin [http://www.webcitation.org/5zW8FuybZ permalink]
* [http://sonic-arts.org/monzo/55edo/55edo.htm Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone] by [[Joseph Monzo]] [http://www.webcitation.org/5zW910Jax permalink]

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>jghdxuxjbg03cbmvt44soquy2saa2r4</sha1>
    </revision>
    <revision>
      <id>124915</id>
      <parentid>124900</parentid>
      <timestamp>2023-10-01T08:03:33Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Rework</comment>
      <origin>124915</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1420" sha1="k8ciy8k6fvpkehdfxl9j6gzxcdr3zs7" xml:space="preserve">'''1/6-comma meantone''' is the tuning of [[meantone]] temperament which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the [[81/80|syntonic comma (81/80)]] of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning [[tempering out]] 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it flatly and sharply, respectively, while [[122edo]] using the c val does so near perfectly.

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 2/3 || -4/3 || -19/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 1/6 || 2/3 || 5/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1898.3706 2793.4825 3383.7062 }}

[[Mistuning map]]: {{val| 0 -3.5844 +7.1688 +14.8803 }}

== Links ==
* ''Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone'' by [[Ross W. Duffin]] ([https://www.webcitation.org/5zW8FuybZ WebCite])
* ''Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone'' by [[Joseph Monzo]] ([https://web.archive.org/web/20120214163510/sonic-arts.org/monzo/55edo/55edo.htm Internet Archive] [https://www.webcitation.org/5zW910Jax WebCite])

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>k8ciy8k6fvpkehdfxl9j6gzxcdr3zs7</sha1>
    </revision>
    <revision>
      <id>164630</id>
      <parentid>124915</parentid>
      <timestamp>2024-11-07T14:30:16Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Mistuning -&gt; error</comment>
      <origin>164630</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1416" sha1="34uom6facmdp8p9rxg5tj1wkxn3d26n" xml:space="preserve">'''1/6-comma meantone''' is the tuning of [[meantone]] temperament which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the [[81/80|syntonic comma (81/80)]] of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning [[tempering out]] 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it flatly and sharply, respectively, while [[122edo]] using the c val does so near perfectly.

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 2/3 || -4/3 || -19/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 1/6 || 2/3 || 5/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1898.3706 2793.4825 3383.7062 }}

[[Error map]]: {{val| 0 -3.5844 +7.1688 +14.8803 }}

== Links ==
* ''Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone'' by [[Ross W. Duffin]] ([https://www.webcitation.org/5zW8FuybZ WebCite])
* ''Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone'' by [[Joseph Monzo]] ([https://web.archive.org/web/20120214163510/sonic-arts.org/monzo/55edo/55edo.htm Internet Archive] [https://www.webcitation.org/5zW910Jax WebCite])

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>34uom6facmdp8p9rxg5tj1wkxn3d26n</sha1>
    </revision>
    <revision>
      <id>167693</id>
      <parentid>164630</parentid>
      <timestamp>2024-11-26T21:05:10Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>Add reference Mozart's Teaching of Intonation, References section</comment>
      <origin>167693</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2064" sha1="fbgjr1kcg2zuvrcslfdf7nf6axwjto1" xml:space="preserve">'''1/6-comma meantone''' is the tuning of [[meantone]] temperament which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the [[81/80|syntonic comma (81/80)]] of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning [[tempering out]] 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it flatly and sharply, respectively, while [[122edo]] using the c val does so near perfectly. This tuning (or something very close to it) was favored by Leopold Mozart and Wolfgang Amadeus Mozart, with a subset and approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.&lt;ref&gt;''Mozart's Teaching of Intonation'' (John Chesnut, 1977), '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]&lt;/ref&gt;

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 2/3 || -4/3 || -19/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 1/6 || 2/3 || 5/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1898.3706 2793.4825 3383.7062 }}

[[Error map]]: {{val| 0 -3.5844 +7.1688 +14.8803 }}

== Links ==
* ''Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone'' by [[Ross W. Duffin]] ([https://www.webcitation.org/5zW8FuybZ WebCite])
* ''Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone'' by [[Joseph Monzo]] ([https://web.archive.org/web/20120214163510/sonic-arts.org/monzo/55edo/55edo.htm Internet Archive] [https://www.webcitation.org/5zW910Jax WebCite])

== References ==
&lt;references /&gt;

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>fbgjr1kcg2zuvrcslfdf7nf6axwjto1</sha1>
    </revision>
    <revision>
      <id>167694</id>
      <parentid>167693</parentid>
      <timestamp>2024-11-26T21:07:48Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>The Mozarts favored 55EDO or something close to it</comment>
      <origin>167694</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2088" sha1="ad5ms5ir9dg3j3pbg94twftha80qv9g" xml:space="preserve">'''1/6-comma meantone''' is the tuning of [[meantone]] temperament which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the [[81/80|syntonic comma (81/80)]] of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning [[tempering out]] 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it flatly and sharply, respectively, while [[122edo]] using the c val does so near perfectly. This tuning (implemented as 55edo or something close to it) was favored by Leopold Mozart and Wolfgang Amadeus Mozart, with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.&lt;ref&gt;''Mozart's Teaching of Intonation'' (John Chesnut, 1977), '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]&lt;/ref&gt;

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 2/3 || -4/3 || -19/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 1/6 || 2/3 || 5/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1898.3706 2793.4825 3383.7062 }}

[[Error map]]: {{val| 0 -3.5844 +7.1688 +14.8803 }}

== Links ==
* ''Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone'' by [[Ross W. Duffin]] ([https://www.webcitation.org/5zW8FuybZ WebCite])
* ''Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone'' by [[Joseph Monzo]] ([https://web.archive.org/web/20120214163510/sonic-arts.org/monzo/55edo/55edo.htm Internet Archive] [https://www.webcitation.org/5zW910Jax WebCite])

== References ==
&lt;references /&gt;

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>ad5ms5ir9dg3j3pbg94twftha80qv9g</sha1>
    </revision>
    <revision>
      <id>167695</id>
      <parentid>167694</parentid>
      <timestamp>2024-11-26T21:12:40Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>Improve wording</comment>
      <origin>167695</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2085" sha1="ldk1jbbk4egms49nslnhgihh6mbwlm7" xml:space="preserve">'''1/6-comma meantone''' is the tuning of [[meantone]] temperament which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the [[81/80|syntonic comma (81/80)]] of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning [[tempering out]] 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it flatly and sharply, respectively, while [[122edo]] using the c val does so near perfectly. Leopold Mozart and Wolfgang Amadeus Mozart recommended this tuning (implemented as 55edo or something close to it), with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.&lt;ref&gt;''Mozart's Teaching of Intonation'' (John Chesnut, 1977), '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]&lt;/ref&gt;

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 2/3 || -4/3 || -19/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 1/6 || 2/3 || 5/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1898.3706 2793.4825 3383.7062 }}

[[Error map]]: {{val| 0 -3.5844 +7.1688 +14.8803 }}

== Links ==
* ''Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone'' by [[Ross W. Duffin]] ([https://www.webcitation.org/5zW8FuybZ WebCite])
* ''Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone'' by [[Joseph Monzo]] ([https://web.archive.org/web/20120214163510/sonic-arts.org/monzo/55edo/55edo.htm Internet Archive] [https://www.webcitation.org/5zW910Jax WebCite])

== References ==
&lt;references /&gt;

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>ldk1jbbk4egms49nslnhgihh6mbwlm7</sha1>
    </revision>
    <revision>
      <id>167698</id>
      <parentid>167695</parentid>
      <timestamp>2024-11-26T23:57:11Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>Fix reference formatting</comment>
      <origin>167698</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2085" sha1="k5gvk85wk5s0lkixjrzl3c2b7uuuqnb" xml:space="preserve">'''1/6-comma meantone''' is the tuning of [[meantone]] temperament which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the [[81/80|syntonic comma (81/80)]] of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning [[tempering out]] 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it flatly and sharply, respectively, while [[122edo]] using the c val does so near perfectly. Leopold Mozart and Wolfgang Amadeus Mozart recommended this tuning (implemented as 55edo or something close to it), with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.&lt;ref&gt;Chesnut, John (1977) ''Mozart's Teaching of Intonation'', '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]&lt;/ref&gt;

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 2/3 || -4/3 || -19/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 1/6 || 2/3 || 5/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1898.3706 2793.4825 3383.7062 }}

[[Error map]]: {{val| 0 -3.5844 +7.1688 +14.8803 }}

== Links ==
* ''Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone'' by [[Ross W. Duffin]] ([https://www.webcitation.org/5zW8FuybZ WebCite])
* ''Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone'' by [[Joseph Monzo]] ([https://web.archive.org/web/20120214163510/sonic-arts.org/monzo/55edo/55edo.htm Internet Archive] [https://www.webcitation.org/5zW910Jax WebCite])

== References ==
&lt;references /&gt;

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>k5gvk85wk5s0lkixjrzl3c2b7uuuqnb</sha1>
    </revision>
    <revision>
      <id>180827</id>
      <parentid>167698</parentid>
      <timestamp>2025-02-15T21:24:02Z</timestamp>
      <contributor>
        <username>Hstraub</username>
        <id>11</id>
      </contributor>
      <origin>180827</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2176" sha1="150d28jy5fa59hywcrqqdh51m9iezpn" xml:space="preserve">{{interwiki
| de = Sechstelkomma-mitteltönig
| en = 1/6-comma meantone
| es = 
| ja = 
}}
'''1/6-comma meantone''' is the tuning of [[meantone]] temperament which tunes the fifth as the sixth root of 45/4, or in other words 698.371 [[cent]]s. This means the fifth is flattened by 1/6 of the [[81/80|syntonic comma (81/80)]] of 21.506 cents, which is to say by 3.584 cents, hence the name 1/6-comma meantone. In 1/6-comma meantone, the diatonic tritone [[45/32]] is tuned justly, and it can be characterized fully as the regular tuning [[tempering out]] 81/80 and tuning 2 and 45/32 justly. [[55edo]] and [[67edo]] approximate it flatly and sharply, respectively, while [[122edo]] using the c val does so near perfectly. Leopold Mozart and Wolfgang Amadeus Mozart recommended this tuning (implemented as 55edo or something close to it), with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.&lt;ref&gt;Chesnut, John (1977) ''Mozart's Teaching of Intonation'', '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]&lt;/ref&gt;

== Tuning profile ==
[[Projection map]]: 

{| class="right-all"
|-
| [⟨ || 1 || 2/3 || -4/3 || -19/3 || ]
|-
| ⟨ || 0 || 1/3 || 4/3 || 10/3 || ]
|-
| ⟨ || 0 || 1/6 || 2/3 || 5/3 || ]
|-
| ⟨ || 0 || 0 || 0 || 0 || ]]
|}

[[Tuning map]]: {{val| 1200 1898.3706 2793.4825 3383.7062 }}

[[Error map]]: {{val| 0 -3.5844 +7.1688 +14.8803 }}

== Links ==
* ''Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone'' by [[Ross W. Duffin]] ([https://www.webcitation.org/5zW8FuybZ WebCite])
* ''Mozart's tuning: 55-EDO and its close relative, 1/6-comma meantone'' by [[Joseph Monzo]] ([https://web.archive.org/web/20120214163510/sonic-arts.org/monzo/55edo/55edo.htm Internet Archive] [https://www.webcitation.org/5zW910Jax WebCite])

== References ==
&lt;references /&gt;

[[Category:Meantone]]
[[Category:Historical]]</text>
      <sha1>150d28jy5fa59hywcrqqdh51m9iezpn</sha1>
    </revision>
  </page>
  <page>
    <title>1-EDO</title>
    <ns>0</ns>
    <id>11</id>
    <redirect title="1edo" />
    <revision>
      <id>11</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>11</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="108" sha1="8app0erh28o5dfe0k8u42pu1b31pml3" xml:space="preserve">1-EDO es un temperamento trivial, que resulta de solo tomar un tono y octavas consecutivas.
[[Category:edo]]</text>
      <sha1>8app0erh28o5dfe0k8u42pu1b31pml3</sha1>
    </revision>
    <revision>
      <id>3996</id>
      <timestamp>2018-02-07T11:05:46Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;EIHdzP</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 626084333 - Original comment: **</comment>
      <origin>3996</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1396" sha1="sookqtvdfkiseeie3gq1hg9efaiglae" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:EIHdzP|EIHdzP]] and made on &lt;tt&gt;2018-02-07 11:05:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;626084333&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;1-EDO es un temperamento trivial, que resulta de solo tomar un tono y octavas consecutivas.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1-EDO&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;1-EDO es un temperamento trivial, que resulta de solo tomar un tono y octavas consecutivas.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>sookqtvdfkiseeie3gq1hg9efaiglae</sha1>
    </revision>
    <revision>
      <id>34159</id>
      <parentid>11</parentid>
      <timestamp>2018-09-30T18:02:33Z</timestamp>
      <contributor>
        <username>Keenan Pepper</username>
        <id>25</id>
      </contributor>
      <comment>#redirect [[Trivial temperaments]]</comment>
      <origin>34159</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="34" sha1="fj2rulbculjpbazujjox2f6g4an82iv" xml:space="preserve">#redirect [[Trivial temperaments]]</text>
      <sha1>fj2rulbculjpbazujjox2f6g4an82iv</sha1>
    </revision>
    <revision>
      <id>34160</id>
      <parentid>34159</parentid>
      <timestamp>2018-09-30T18:03:00Z</timestamp>
      <contributor>
        <username>Keenan Pepper</username>
        <id>25</id>
      </contributor>
      <comment>#redirect [[1edo]]</comment>
      <origin>34160</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18" sha1="9k2wfffkzgdq1ck1l4uuro5lhcn750y" xml:space="preserve">#redirect [[1edo]]</text>
      <sha1>9k2wfffkzgdq1ck1l4uuro5lhcn750y</sha1>
    </revision>
    <revision>
      <id>34163</id>
      <parentid>34160</parentid>
      <timestamp>2018-09-30T18:04:06Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <comment>you're so savage</comment>
      <origin>34163</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21" sha1="3r6p38cods0wkjths07d4d77rx1qdi3" xml:space="preserve">#redirect [[es:1edo]]</text>
      <sha1>3r6p38cods0wkjths07d4d77rx1qdi3</sha1>
    </revision>
    <revision>
      <id>34169</id>
      <parentid>34163</parentid>
      <timestamp>2018-09-30T18:25:22Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <comment>Changed redirect target from [[es:1edo]] to [[es:1-EDO]]</comment>
      <origin>34169</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22" sha1="ekedb87bzbdlwsy2bx93tepytuahm0r" xml:space="preserve">#redirect [[es:1-EDO]]</text>
      <sha1>ekedb87bzbdlwsy2bx93tepytuahm0r</sha1>
    </revision>
    <revision>
      <id>34190</id>
      <parentid>34169</parentid>
      <timestamp>2018-09-30T19:10:04Z</timestamp>
      <contributor>
        <username>Keenan Pepper</username>
        <id>25</id>
      </contributor>
      <comment>why should an English page redirect to Spanish??</comment>
      <origin>34190</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18" sha1="9k2wfffkzgdq1ck1l4uuro5lhcn750y" xml:space="preserve">#redirect [[1edo]]</text>
      <sha1>9k2wfffkzgdq1ck1l4uuro5lhcn750y</sha1>
    </revision>
    <revision>
      <id>34192</id>
      <parentid>34190</parentid>
      <timestamp>2018-09-30T19:12:15Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <comment>it's not an English page, it was always Spanish and treating it like an English page will break the links on other Spanish pages on [[:en:]]</comment>
      <origin>34192</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22" sha1="ekedb87bzbdlwsy2bx93tepytuahm0r" xml:space="preserve">#redirect [[es:1-EDO]]</text>
      <sha1>ekedb87bzbdlwsy2bx93tepytuahm0r</sha1>
    </revision>
    <revision>
      <id>35859</id>
      <parentid>34192</parentid>
      <timestamp>2018-10-02T14:30:44Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>+cat</comment>
      <origin>35859</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="50" sha1="ciq1sya01evd8slo842adcbnpg23cn5" xml:space="preserve">#redirect [[es:1-EDO]]

[[Category:Hard redirect]]</text>
      <sha1>ciq1sya01evd8slo842adcbnpg23cn5</sha1>
    </revision>
    <revision>
      <id>74206</id>
      <parentid>35859</parentid>
      <timestamp>2021-07-12T15:28:45Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Changed redirection from the Spanish Wiki to the English Wiki</comment>
      <origin>74206</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18" sha1="9k2wfffkzgdq1ck1l4uuro5lhcn750y" xml:space="preserve">#redirect [[1edo]]</text>
      <sha1>9k2wfffkzgdq1ck1l4uuro5lhcn750y</sha1>
    </revision>
  </page>
  <page>
    <title>100/97</title>
    <ns>0</ns>
    <id>12</id>
    <revision>
      <id>12</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>12</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="497" sha1="1e7bevwuhbsdf60giy1pexo7ysdq7eh" xml:space="preserve">'''100/97'''

= 2&lt;span style="font-size: 80%; vertical-align: super;"&gt;2&lt;/span&gt; * 5&lt;span style="font-size: 80%; vertical-align: super;"&gt;2&lt;/span&gt; * 97&lt;span style="font-size: 80%; vertical-align: super;"&gt;-1&lt;/span&gt;

52.732 cents

[[File:jid_100_97_pluck_adu_dr220.mp3]] [[:File:jid_100_97_pluck_adu_dr220.mp3|sound sample]]

100/97 can be used as a tuning for the [[Diaschismic_family#Shrutar|shrutar]] generator, and in particular is only 0.002 cents sharp of the POTE generator for 19-limit shrutar.</text>
      <sha1>1e7bevwuhbsdf60giy1pexo7ysdq7eh</sha1>
    </revision>
    <revision>
      <id>3997</id>
      <timestamp>2014-06-30T20:40:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 515316148 - Original comment: **</comment>
      <origin>3997</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3725" sha1="o3e6wrounc6gti6kwa2pidgvbyxg1qk" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-30 20:40:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;515316148&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**100/97**
= 2&amp;lt;span style="font-size: 80%; vertical-align: super;"&amp;gt;2&amp;lt;/span&amp;gt; * 5&amp;lt;span style="font-size: 80%; vertical-align: super;"&amp;gt;2&amp;lt;/span&amp;gt; * 97&amp;lt;span style="font-size: 80%; vertical-align: super;"&amp;gt;-1&amp;lt;/span&amp;gt;
52.732 cents
[[media type="file" key="jid_100_97_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_100_97_pluck_adu_dr220.mp3|sound sample]]

100/97 can be used as a tuning for the [[Diaschismic family#Shrutar|shrutar]] generator, and in particular is only 0.002 cents sharp of the POTE generator for 19-limit shrutar.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100_97&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;100/97&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
= 2&amp;lt;span style="font-size: 80%; vertical-align: super;"&amp;gt;2&amp;lt;/span&amp;gt; * 5&amp;lt;span style="font-size: 80%; vertical-align: super;"&amp;gt;2&amp;lt;/span&amp;gt; * 97&amp;lt;span style="font-size: 80%; vertical-align: super;"&amp;gt;-1&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
52.732 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_100_97_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_100_97_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_100_97_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_100_97_pluck_adu_dr220.mp3/515315516/jid_100_97_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_100_97_pluck_adu_dr220.mp3/515315516/jid_100_97_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
100/97 can be used as a tuning for the &amp;lt;a class="wiki_link" href="/Diaschismic%20family#Shrutar"&amp;gt;shrutar&amp;lt;/a&amp;gt; generator, and in particular is only 0.002 cents sharp of the POTE generator for 19-limit shrutar.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>o3e6wrounc6gti6kwa2pidgvbyxg1qk</sha1>
    </revision>
    <revision>
      <id>3998</id>
      <timestamp>2012-06-02T21:09:09Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 342109156 - Original comment: **</comment>
      <origin>3998</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1624" sha1="0od87u5ytdq79kmnb6o8u2r3teqe1dv" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-06-02 21:09:09 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;342109156&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;100/97 can be used as a tuning for the [[Diaschismic family#Shrutar|shrutar]] generator, and in particular is only 0.002 cents sharp of the POTE generator for 19-limit shrutar.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100_97&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;100/97 can be used as a tuning for the &amp;lt;a class="wiki_link" href="/Diaschismic%20family#Shrutar"&amp;gt;shrutar&amp;lt;/a&amp;gt; generator, and in particular is only 0.002 cents sharp of the POTE generator for 19-limit shrutar.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>0od87u5ytdq79kmnb6o8u2r3teqe1dv</sha1>
    </revision>
    <revision>
      <id>64429</id>
      <parentid>12</parentid>
      <timestamp>2021-03-27T03:56:14Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64429</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="546" sha1="b1w4nznj95kry5unjvdpmjf6kmkvh88" xml:space="preserve">'''100/97'''

= 2&lt;span style="font-size: 80%; vertical-align: super;"&gt;2&lt;/span&gt; * 5&lt;span style="font-size: 80%; vertical-align: super;"&gt;2&lt;/span&gt; * 97&lt;span style="font-size: 80%; vertical-align: super;"&gt;-1&lt;/span&gt;

52.732 cents

[[File:jid_100_97_pluck_adu_dr220.mp3]] [[:File:jid_100_97_pluck_adu_dr220.mp3|sound sample]]

100/97 can be used as a tuning for the [[Diaschismic_family#Shrutar|shrutar]] generator, and in particular is only 0.002 cents sharp of the POTE generator for 19-limit shrutar.

[[Category:Pages with internal sound examples]]</text>
      <sha1>b1w4nznj95kry5unjvdpmjf6kmkvh88</sha1>
    </revision>
    <revision>
      <id>86280</id>
      <parentid>64429</parentid>
      <timestamp>2022-02-22T23:46:38Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86280</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="497" sha1="1e7bevwuhbsdf60giy1pexo7ysdq7eh" xml:space="preserve">'''100/97'''

= 2&lt;span style="font-size: 80%; vertical-align: super;"&gt;2&lt;/span&gt; * 5&lt;span style="font-size: 80%; vertical-align: super;"&gt;2&lt;/span&gt; * 97&lt;span style="font-size: 80%; vertical-align: super;"&gt;-1&lt;/span&gt;

52.732 cents

[[File:jid_100_97_pluck_adu_dr220.mp3]] [[:File:jid_100_97_pluck_adu_dr220.mp3|sound sample]]

100/97 can be used as a tuning for the [[Diaschismic_family#Shrutar|shrutar]] generator, and in particular is only 0.002 cents sharp of the POTE generator for 19-limit shrutar.</text>
      <sha1>1e7bevwuhbsdf60giy1pexo7ysdq7eh</sha1>
    </revision>
    <revision>
      <id>93918</id>
      <parentid>86280</parentid>
      <timestamp>2022-08-04T19:12:55Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox</comment>
      <origin>93918</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="587" sha1="oycb3re7tp4vnr3vsmprizbybduo95y" xml:space="preserve">{{Infobox Interval
| Ratio = 100/97
| Monzo = 2 0 2 0&lt;sup&gt;21&lt;/sup&gt; -1
| Cents = 52.732
| Name = shrutar quarter tone
| Color name = 97uyy1, fifosuyoyo unison,&lt;br&gt;ninety-suyoyo unison
| FJS name = A1&lt;sup&gt;5,5&lt;/sup&gt;&lt;sub&gt;97&lt;/sub&gt;
| Sound = jid_100_97_pluck_adu_dr220.mp3
}}

The '''shrutar quarter tone''' ('''100/97''' = 2&lt;sup&gt;2&lt;/sup&gt; · 5&lt;sup&gt;2&lt;/sup&gt; · 97&lt;sup&gt;−1&lt;/sup&gt;) can be used as a tuning for the [[Diaschismic_family#Shrutar|shrutar]] generator, and in particular is only 0.002{{cent}} sharp of the [[POTE]] generator for 19-limit shrutar.

[[Category:97-limit]]
[[Category:Stub]]</text>
      <sha1>oycb3re7tp4vnr3vsmprizbybduo95y</sha1>
    </revision>
    <revision>
      <id>98010</id>
      <parentid>93918</parentid>
      <timestamp>2022-10-25T12:11:52Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>98010</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="451" sha1="98onwjbthyv2inuc2ufu3o6mp3jre8h" xml:space="preserve">{{Infobox Interval
| Name = shrutar quarter tone
| Color name = 97uyy1, fifosuyoyo unison, ninety-suyoyo unison
| Sound = jid_100_97_pluck_adu_dr220.mp3
}}

The '''shrutar quarter tone''' ('''100/97''' = 2&lt;sup&gt;2&lt;/sup&gt; · 5&lt;sup&gt;2&lt;/sup&gt; · 97&lt;sup&gt;−1&lt;/sup&gt;) can be used as a tuning for the [[Diaschismic_family#Shrutar|shrutar]] generator, and in particular is only 0.002{{cent}} sharp of the [[POTE]] generator for 19-limit shrutar.

[[Category:Stub]]</text>
      <sha1>98onwjbthyv2inuc2ufu3o6mp3jre8h</sha1>
    </revision>
    <revision>
      <id>120004</id>
      <parentid>98010</parentid>
      <timestamp>2023-07-11T21:49:17Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Stub]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>120004</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="432" sha1="a8qcykr0r4d973wes28lgb2frdgrxbp" xml:space="preserve">{{Infobox Interval
| Name = shrutar quarter tone
| Color name = 97uyy1, fifosuyoyo unison, ninety-suyoyo unison
| Sound = jid_100_97_pluck_adu_dr220.mp3
}}

The '''shrutar quarter tone''' ('''100/97''' = 2&lt;sup&gt;2&lt;/sup&gt; · 5&lt;sup&gt;2&lt;/sup&gt; · 97&lt;sup&gt;−1&lt;/sup&gt;) can be used as a tuning for the [[Diaschismic_family#Shrutar|shrutar]] generator, and in particular is only 0.002{{cent}} sharp of the [[POTE]] generator for 19-limit shrutar.</text>
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  <page>
    <title>100 97</title>
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    <redirect title="100/97" />
    <revision>
      <id>13</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
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      <text bytes="20" sha1="ijc98r3424w4n8hqhtcwnkhipn8a6r0" xml:space="preserve">#REDIRECT [[100/97]]</text>
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  </page>
  <page>
    <title>100/99</title>
    <ns>0</ns>
    <id>14</id>
    <revision>
      <id>14</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
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      <text bytes="1101" sha1="ecjwtn0j0v96jm07q8vb6ipuut5kzj0" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[:de:100/99 Deutsch]]&lt;/span&gt;

In [[Just_intonation|Just Intonation]],100/99 is a [[superparticular|superparticular]] interval (the ptolemisma) of around 17.4¢ which makes the difference between [[11/10|11/10]], the greater undecimal neutral second of about 165¢, and [[10/9|10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo|22edo]], [[41edo|41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80|81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8|9/8]]. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8|11/8]] and two [[6/5|6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic_triad|ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.

See: [[Gallery_of_Just_Intervals|Galley of Just Intervals]]      [[Category:11-limit]]
[[Category:superparticular]]</text>
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    <revision>
      <id>3999</id>
      <timestamp>2016-01-25T11:06:34Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 572861989 - Original comment: **</comment>
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      <text bytes="3898" sha1="epxwevnpu7coittdh1ueblryyi3xn7n" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2016-01-25 11:06:34 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;572861989&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[xenharmonie/100_99|Deutsch]]
&amp;lt;/span&amp;gt;
In [[Just Intonation]],100/99 is a [[superparticular]] interval (the ptolemisma) of around 17.4¢ which makes the difference between [[11_10|11/10]], the greater undecimal neutral second of about 165¢, and [[10_9|10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81_80|81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9_8|9/8]]. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11_8|11/8]] and two [[6_5|6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.

See: [[Gallery of Just Intervals|Galley of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100_99&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="https://xenharmonie.wikispaces.com/100_99"&amp;gt;Deutsch&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;,100/99 is a &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; interval (the ptolemisma) of around 17.4¢ which makes the difference between &amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;, the greater undecimal neutral second of about 165¢, and &amp;lt;a class="wiki_link" href="/10_9"&amp;gt;10/9&amp;lt;/a&amp;gt;, the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt; and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of &amp;lt;a class="wiki_link" href="/81_80"&amp;gt;81/80&amp;lt;/a&amp;gt;, which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt;. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt; and two &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt;s fall short of an octave, so that tempering out 100/99 allows for the &amp;lt;a class="wiki_link" href="/ptolemismic%20triad"&amp;gt;ptolemismic triad&amp;lt;/a&amp;gt; which has intervals of 11/8, 6/5, 6/5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Galley of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <timestamp>2011-10-19T19:26:43Z</timestamp>
      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 266601380 - Original comment: **</comment>
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      <text bytes="3623" sha1="m6fkdnf83kma16hlqqxeq6pinsiczn8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-10-19 19:26:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;266601380&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[Just Intonation]],100/99 is a [[superparticular]] interval (the ptolemisma) of around 17.4¢ which makes the difference between [[11_10|11/10]], the greater undecimal neutral second of about 165¢, and [[10_9|10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81_80|81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9_8|9/8]]. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11_8|11/8]] and two [[6_5|6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.

See: [[Gallery of Just Intervals|Galley of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100_99&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;,100/99 is a &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; interval (the ptolemisma) of around 17.4¢ which makes the difference between &amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;, the greater undecimal neutral second of about 165¢, and &amp;lt;a class="wiki_link" href="/10_9"&amp;gt;10/9&amp;lt;/a&amp;gt;, the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt; and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of &amp;lt;a class="wiki_link" href="/81_80"&amp;gt;81/80&amp;lt;/a&amp;gt;, which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt;. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt; and two &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt;s fall short of an octave, so that tempering out 100/99 allows for the &amp;lt;a class="wiki_link" href="/ptolemismic%20triad"&amp;gt;ptolemismic triad&amp;lt;/a&amp;gt; which has intervals of 11/8, 6/5, 6/5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Galley of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4001</id>
      <timestamp>2011-10-19T19:10:39Z</timestamp>
      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 266596898 - Original comment: **</comment>
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      <text bytes="3151" sha1="ok00xepv125qx6kv3at2ymu6hx1syxu" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-10-19 19:10:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;266596898&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[Just Intonation]],100/99 is a [[superparticular]] interval (the ptolemisma) of around 17.4¢ which makes the difference between [[11_10|11/10]], the greater undecimal neutral second of about 165¢, and [[10_9|10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81_80|81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9_8|9/8]]. Tempering out 100/99 allows for the [[ptolemismic triad]].

See: [[Gallery of Just Intervals|Galley of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100_99&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;,100/99 is a &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; interval (the ptolemisma) of around 17.4¢ which makes the difference between &amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;, the greater undecimal neutral second of about 165¢, and &amp;lt;a class="wiki_link" href="/10_9"&amp;gt;10/9&amp;lt;/a&amp;gt;, the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt; and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of &amp;lt;a class="wiki_link" href="/81_80"&amp;gt;81/80&amp;lt;/a&amp;gt;, which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt;. Tempering out 100/99 allows for the &amp;lt;a class="wiki_link" href="/ptolemismic%20triad"&amp;gt;ptolemismic triad&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Galley of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4002</id>
      <timestamp>2011-09-14T19:27:40Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 254141930 - Original comment: **</comment>
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      <text bytes="2940" sha1="duob45weeogelcq9qvj9sxf50oqsbmt" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-09-14 19:27:40 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;254141930&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[Just Intonation]],100/99 is a [[superparticular]] interval around 17.4¢ which makes the difference between [[11_10|11/10]], the greater undecimal neutral second of about 165¢, and [[10_9|10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81_80|81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9_8|9/8]].

See: [[Gallery of Just Intervals|Galley of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100_99&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;,100/99 is a &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; interval around 17.4¢ which makes the difference between &amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;, the greater undecimal neutral second of about 165¢, and &amp;lt;a class="wiki_link" href="/10_9"&amp;gt;10/9&amp;lt;/a&amp;gt;, the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt; and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of &amp;lt;a class="wiki_link" href="/81_80"&amp;gt;81/80&amp;lt;/a&amp;gt;, which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Galley of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>32349</id>
      <parentid>14</parentid>
      <timestamp>2018-09-17T00:09:21Z</timestamp>
      <contributor>
        <username>Mike Battaglia</username>
        <id>3</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "(\[\[\:(dev|es|purdal|de|en):[^[:space:]\|]*) (.*])" to "$1|$3"</comment>
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      <text bytes="1101" sha1="ta56coff1ftonvohtm20acu33nxeke1" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[:de:100/99|Deutsch]]&lt;/span&gt;

In [[Just_intonation|Just Intonation]],100/99 is a [[superparticular|superparticular]] interval (the ptolemisma) of around 17.4¢ which makes the difference between [[11/10|11/10]], the greater undecimal neutral second of about 165¢, and [[10/9|10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo|22edo]], [[41edo|41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80|81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8|9/8]]. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8|11/8]] and two [[6/5|6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic_triad|ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.

See: [[Gallery_of_Just_Intervals|Galley of Just Intervals]]      [[Category:11-limit]]
[[Category:superparticular]]</text>
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    <revision>
      <id>47860</id>
      <parentid>32349</parentid>
      <timestamp>2020-07-23T07:48:58Z</timestamp>
      <contributor>
        <username>Hstraub</username>
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      <text bytes="1035" sha1="gjl3kujnud0flcrc93h2xkru1b83795" xml:space="preserve">[[de:100/99]]
In [[Just_intonation|Just Intonation]],100/99 is a [[superparticular|superparticular]] interval (the ptolemisma) of around 17.4¢ which makes the difference between [[11/10|11/10]], the greater undecimal neutral second of about 165¢, and [[10/9|10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo|22edo]], [[41edo|41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80|81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8|9/8]]. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8|11/8]] and two [[6/5|6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic_triad|ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.

See: [[Gallery_of_Just_Intervals|Galley of Just Intervals]]      [[Category:11-limit]]
[[Category:superparticular]]</text>
      <sha1>gjl3kujnud0flcrc93h2xkru1b83795</sha1>
    </revision>
    <revision>
      <id>47863</id>
      <parentid>47860</parentid>
      <timestamp>2020-07-23T13:14:26Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>47863</origin>
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      <format>text/x-wiki</format>
      <text bytes="1156" sha1="m8gzd8x3g79ogaueclmah4f2hla6tjn" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Icon = 
| Ratio = 100/99
| Monzo = 2 -2 2 0 -1
| Cents = 17.4
| Name = Ptolemy's comma, &lt;br&gt;ptolemisma
| Color name = 1uyy1, luyoyo comma
| Sound = 
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.

== See Also ==
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:comma]]
[[Category:superparticular]]</text>
      <sha1>m8gzd8x3g79ogaueclmah4f2hla6tjn</sha1>
    </revision>
    <revision>
      <id>50170</id>
      <parentid>47863</parentid>
      <timestamp>2020-09-25T17:06:24Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>recat, added FJS name, size to 5 digits</comment>
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      <text bytes="1209" sha1="2ybrh8fr0esq07usdrl4o54lf749f4n" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Icon = 
| Ratio = 100/99
| Monzo = 2 -2 2 0 -1
| Cents = 17.39948
| Name = Ptolemy's comma, &lt;br&gt;ptolemisma
| Color name = 1uyy1, luyoyo comma
| FJS name = A1&lt;sup&gt;25&lt;/sup&gt;&lt;sub&gt;11&lt;/sub&gt;
| Sound = 
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.

== See Also ==

* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Small comma]]
[[Category:Superparticular]]</text>
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    </revision>
    <revision>
      <id>88273</id>
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      <timestamp>2022-03-01T10:28:28Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
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      <minor/>
      <comment>+links and categories</comment>
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      <text bytes="1433" sha1="1w6zb0jt1y3q62bwtro8vxt1evju9ho" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Ratio = 100/99
| Monzo = 2 -2 2 0 -1
| Cents = 17.39948
| Name = Ptolemy's comma, &lt;br&gt;ptolemisma
| Color name = 1uyy1, luyoyo comma
| FJS name = A1&lt;sup&gt;25&lt;/sup&gt;&lt;sub&gt;11&lt;/sub&gt;
| Sound = 
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Small comma]]
[[Category:Superparticular]]
[[Category:Ptolemismic]]</text>
      <sha1>1w6zb0jt1y3q62bwtro8vxt1evju9ho</sha1>
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    <revision>
      <id>89822</id>
      <parentid>88273</parentid>
      <timestamp>2022-03-20T21:59:54Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Small comma]] to [[Category:Small commas]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>89822</origin>
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      <text bytes="1434" sha1="8r6l02i45dmk6pkles9tq90lvvbhjtk" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Ratio = 100/99
| Monzo = 2 -2 2 0 -1
| Cents = 17.39948
| Name = Ptolemy's comma, &lt;br&gt;ptolemisma
| Color name = 1uyy1, luyoyo comma
| FJS name = A1&lt;sup&gt;25&lt;/sup&gt;&lt;sub&gt;11&lt;/sub&gt;
| Sound = 
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Small commas]]
[[Category:Superparticular]]
[[Category:Ptolemismic]]</text>
      <sha1>8r6l02i45dmk6pkles9tq90lvvbhjtk</sha1>
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      <id>98069</id>
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        <username>Plumtree</username>
        <id>365</id>
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      <comment>Normalising usage of Infobox Interval</comment>
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      <text bytes="1256" sha1="901862zgu8y6pvct10ipmjs7c8vfj8y" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = Ptolemy's comma, ptolemisma
| Color name = 1uyy1, luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]</text>
      <sha1>901862zgu8y6pvct10ipmjs7c8vfj8y</sha1>
    </revision>
    <revision>
      <id>101141</id>
      <parentid>98069</parentid>
      <timestamp>2023-01-10T02:19:54Z</timestamp>
      <contributor>
        <username>TallKite</username>
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      <text bytes="1271" sha1="indw602hznwffq41c723syekbaqbab9" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = Ptolemy's comma, ptolemisma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5.

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]</text>
      <sha1>indw602hznwffq41c723syekbaqbab9</sha1>
    </revision>
    <revision>
      <id>134290</id>
      <parentid>101141</parentid>
      <timestamp>2024-02-04T02:52:53Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
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      <comment>Clarify</comment>
      <origin>134290</origin>
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      <text bytes="1497" sha1="bulkxupts5b83087qjquh0wy4pjd8x9" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = Ptolemy's comma, ptolemisma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament is a merged form where [[11/10]], [[10/9]] and [[9/8]] are all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. It is also the differene

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]</text>
      <sha1>bulkxupts5b83087qjquh0wy4pjd8x9</sha1>
    </revision>
    <revision>
      <id>134292</id>
      <parentid>134290</parentid>
      <timestamp>2024-02-04T02:53:32Z</timestamp>
      <contributor>
        <username>CompactStar</username>
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      <text bytes="1523" sha1="bfmr6un2kpdh3fvph7vtp60vwmx0zlq" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = Ptolemy's comma, ptolemisma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament merges both and has [[11/10]], [[10/9]] and [[9/8] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. It is also the difference between [[25/24]] and [[33/32]].

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]</text>
      <sha1>bfmr6un2kpdh3fvph7vtp60vwmx0zlq</sha1>
    </revision>
    <revision>
      <id>134293</id>
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      <contributor>
        <username>CompactStar</username>
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      <text bytes="1524" sha1="bz7sy0h1q3ddqzuwmbn5onf48pe2saa" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = Ptolemy's comma, ptolemisma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. It is also the difference between [[25/24]] and [[33/32]].

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]</text>
      <sha1>bz7sy0h1q3ddqzuwmbn5onf48pe2saa</sha1>
    </revision>
    <revision>
      <id>155473</id>
      <parentid>134293</parentid>
      <timestamp>2024-09-19T10:53:55Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
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      <comment>Added information.</comment>
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      <text bytes="1608" sha1="e1djykgma0ut9solmqx99qty1670wny" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = Ptolemy's comma, ptolemisma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]].

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]</text>
      <sha1>e1djykgma0ut9solmqx99qty1670wny</sha1>
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    <revision>
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      <text bytes="1656" sha1="pwdwqke1m9ip8q0f671fmyldpceknte" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = Ptolemy's comma, ptolemisma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament (known as "hypnotone" in the 2.3.5.11 subgroup) merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]].

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]</text>
      <sha1>pwdwqke1m9ip8q0f671fmyldpceknte</sha1>
    </revision>
    <revision>
      <id>159317</id>
      <parentid>155475</parentid>
      <timestamp>2024-10-09T19:12:43Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>sectoning + approximation and relationship to 69edo</comment>
      <origin>159317</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1965" sha1="k9kwz3paoao88jr95n8ob49vbqi2jq3" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = Ptolemy's comma, ptolemisma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. 

== Tempering ==
Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament (known as "hypnotone" in the 2.3.5.11 subgroup) merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]].
== Approximation ==
If the interval is not tempered and instead treated as an identity, it is extremely closely represented by one step of [[69edo]]. The associated unnoticeable comma, [[ptolemi-nicema]], represents the difference between 69 100/99s and the octave, is only 0.56 cents wide.

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]</text>
      <sha1>k9kwz3paoao88jr95n8ob49vbqi2jq3</sha1>
    </revision>
    <revision>
      <id>163706</id>
      <parentid>159317</parentid>
      <timestamp>2024-11-03T10:00:18Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>163706</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2009" sha1="cm5hokkuy3cq1k1hqnpgd19sk4xjuu2" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = Ptolemy's comma, ptolemisma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. 

== Tempering ==
Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament (known as "hypnotone" in the 2.3.5.11 subgroup) merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]].
== Approximation ==
If the interval is not tempered and instead treated as an identity, it is extremely closely represented by one step of [[69edo]]. The associated unnoticeable comma, [[ptolemi-nicema]], represents the difference between 69 100/99s and the octave, is only 0.56 cents wide.

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]
[[Category:Commas named after individuals]]</text>
      <sha1>cm5hokkuy3cq1k1hqnpgd19sk4xjuu2</sha1>
    </revision>
    <revision>
      <id>163999</id>
      <parentid>163706</parentid>
      <timestamp>2024-11-04T01:55:47Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>163999</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2007" sha1="g77wv7mcqt0iact5hjchv1so82g3rd4" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = Ptolemy's comma, ptolemisma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. 

== Tempering ==
Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament (known as "hypnotone" in the 2.3.5.11 subgroup) merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]].
== Approximation ==
If the interval is not tempered and instead treated as an identity, it is extremely closely represented by one step of [[69edo]]. The associated unnoticeable comma, [[ptolemi-nicema]], represents the difference between 69 100/99s and the octave, is only 0.56 cents wide.

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]
[[Category:Commas named after polymaths]]</text>
      <sha1>g77wv7mcqt0iact5hjchv1so82g3rd4</sha1>
    </revision>
    <revision>
      <id>174949</id>
      <parentid>163999</parentid>
      <timestamp>2025-01-07T05:12:51Z</timestamp>
      <contributor>
        <username>Fitzgerald Lee</username>
        <id>503</id>
      </contributor>
      <comment>/* Approximation */:3</comment>
      <origin>174949</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2013" sha1="82brclpcdj7cnajjn4nibmleaaiwfk3" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = Ptolemy's comma, ptolemisma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''Ptolemy's comma''' or '''ptolemisma''', is a [[superparticular]] interval of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. 

== Tempering ==
Temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others; in such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament (known as "hypnotone" in the 2.3.5.11 subgroup) merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]].
== Approximation ==
If the interval is not tempered and instead treated as an identity, it is extremely closely represented by one step of [[69edo]]. The associated unnoticeable comma, [[ptolemi-nicema]], represents the difference between 69 100/99s and the octave, is only 0.56 cents wide. Nice.

== See also ==
* [[Ptolemismic triad]]
* [[Rank-4 temperament #Ptolemismic (100/99)]]
* [[Ptolemismic clan]], the clan of rank-3 temperaments where it is tempered out
* [[Small comma]]
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]
[[Category:Commas named after polymaths]]</text>
      <sha1>82brclpcdj7cnajjn4nibmleaaiwfk3</sha1>
    </revision>
    <revision>
      <id>174976</id>
      <parentid>174949</parentid>
      <timestamp>2025-01-07T08:38:33Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Rework to absorb some of the links in the see also section to the main text</comment>
      <origin>174976</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2131" sha1="5x9y759ixyk5vev3y6owhl33ylkh3c4" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = ptolemisma, Ptolemy's comma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''ptolemisma''' or '''Ptolemy's comma''', is a [[small comma|small]] [[superparticular ratio|superparticular]] [[comma]] of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. 

== Temperaments ==
Tempering out this interval in the 11-limit results in the rank-4 ptolemismic temperament (→ [[Rank-4 temperament #Ptolemismic (100/99)]]). Equal temperaments in which 100/99 is tempered out include [[22edo]], [[41edo]] and others. In such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament (known as "hypnotone" in the 2.3.5.11 subgroup) merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]'s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]]. 

See [[Ptolemismic clan]] for the clan of rank-3 temperaments where it is tempered out. 

== Approximation ==
If the interval is not tempered and instead treated as an identity, it is extremely closely represented by one step of [[69edo]]. The associated unnoticeable comma, [[ptolemi-nicema]], represents the difference between 69 100/99s and the octave, is only 0.56 cents wide. Nice.

== See also ==
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]
[[Category:Commas named after polymaths]]</text>
      <sha1>5x9y759ixyk5vev3y6owhl33ylkh3c4</sha1>
    </revision>
    <revision>
      <id>174977</id>
      <parentid>174976</parentid>
      <timestamp>2025-01-07T08:42:26Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Move stuff to the appropriate sections. Improve linking</comment>
      <origin>174977</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2144" sha1="9ar11bgtb5szf0j6bcdfnhrieiyvz7h" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = ptolemisma, Ptolemy's comma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''ptolemisma''' or '''Ptolemy's comma''', is a [[small comma|small]] [[superparticular ratio|superparticular]] [[comma]] of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]]. 

== Temperaments ==
[[Tempering out]] this interval in the 11-limit results in the rank-4 ptolemismic temperament (→ [[Rank-4 temperament #Ptolemismic (100/99)]]). [[Equal temperament]]s in which 100/99 is tempered out include [[22edo]], [[41edo]] and others. In such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament (known as "hypnotone" in the 2.3.5.11 subgroup) merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]'s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. 

See [[Ptolemismic clan]] for the clan of rank-3 temperaments where it is tempered out. 

== Approximation ==
If the interval is not tempered out and instead treated as an identity, it is extremely closely represented by one step of [[69edo]]. The associated unnoticeable comma, [[ptolemi-nicema]], represents the difference between 69 100/99's and the octave, is only 0.56 cents wide. Nice.

== See also ==
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]

[[Category:Ptolemismic]]
[[Category:Commas named after polymaths]]</text>
      <sha1>9ar11bgtb5szf0j6bcdfnhrieiyvz7h</sha1>
    </revision>
    <revision>
      <id>216207</id>
      <parentid>174977</parentid>
      <timestamp>2025-11-05T08:39:38Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+ link</comment>
      <origin>216207</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2182" sha1="n6g8komrgwfktsiu10exjzq1fhg788s" xml:space="preserve">[[de:100/99]]
{{Infobox Interval
| Name = ptolemisma, Ptolemy's comma
| Color name = 1uyy1, luyoyo 1sn,&lt;br&gt;Luyoyo comma
| Comma = yes
}}
In [[just intonation]], '''100/99''', the '''ptolemisma''' or '''Ptolemy's comma''', is a [[small comma|small]] [[superparticular ratio|superparticular]] [[comma]] of around 17.4¢ which makes the difference between [[11/10]], the greater undecimal neutral second of about 165¢, and [[10/9]], the 5-limit minor whole tone of about 182.4¢. It is also the difference between [[25/24]] and [[33/32]]. In terms of commatic relations it notably factors into [[225/224]] and [[896/891]]. 

== Temperaments ==
[[Tempering out]] this interval in the 11-limit results in the rank-4 ptolemismic temperament (→ [[Rank-4 temperament #Ptolemismic (100/99)]]). [[Equal temperament]]s in which 100/99 is tempered out include [[22edo]], [[41edo]] and others. In such systems, 11/10 and 10/9 are equated. This is in contrast to the more familiar tempering out of [[81/80]], which results in meantone and other temperaments, in which 10/9 is equated, not with 11/10, but with [[9/8]]. However, meantone and ptolemismic are not necessarily mutually exclusive–the [[flattone]] temperament (known as "hypnotone" in the 2.3.5.11 subgroup) merges both and has [[11/10]], [[10/9]] and [[9/8]] all tempered to a 10/9-like interval. Along with being the difference between 11/10 and 10/9, 100/99 is the amount by which an [[11/8]] and two [[6/5]]'s fall short of an octave, so that tempering out 100/99 allows for the [[ptolemismic triad]] which has intervals of 11/8, 6/5, 6/5. 

See [[Ptolemismic clan]] for the clan of rank-3 temperaments where it is tempered out. 

== Approximation ==
If the interval is not tempered out and instead treated as an identity, it is extremely closely represented by one step of [[69edo]]. The associated unnoticeable comma, [[ptolemi-nicema]], represents the difference between 69 100/99's and the octave, is only 0.56 cents wide. Nice.

== See also ==
* [[List of superparticular intervals]]
* [[Gallery of just intervals]]
* [[:File: Demo 100 99 tempered.pdf]]

[[Category:Ptolemismic]]
[[Category:Commas named after polymaths]]</text>
      <sha1>n6g8komrgwfktsiu10exjzq1fhg788s</sha1>
    </revision>
  </page>
  <page>
    <title>100 99</title>
    <ns>0</ns>
    <id>15</id>
    <redirect title="100/99" />
    <revision>
      <id>15</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>15</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20" sha1="11vs1r1ui9mum3v7r5ukxu4slntmdap" xml:space="preserve">#REDIRECT [[100/99]]</text>
      <sha1>11vs1r1ui9mum3v7r5ukxu4slntmdap</sha1>
    </revision>
  </page>
  <page>
    <title>100edo</title>
    <ns>0</ns>
    <id>16</id>
    <revision>
      <id>16</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>16</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7242" sha1="9vq2dpf93r0li77ljlxhnsxcf99c4i5" xml:space="preserve">&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6edo]], [[35edo|35edo]], [[47edo|47edo]] and [[88edo|88edo]], [[100edo|100edo]] possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo|12edo]].

=Scales=
[[greeley8|greeley8]]

[[greeley15|greeley15]]

== ==

=100bddd and the 22-note scales=

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo|22edo]] for [[pajara|pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.

{| class="wikitable"
|-
| | Steps of 22-note MODMOS
| | Interval name (decatonic)
| | Interval name (superpyth diatonic)
| | Pure interval size [multiplicity]
| | Dog interval size [multiplicity]
|-
| | 1
| | Diminished 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor second
| | 60¢ [12]
| | 48¢ [10]
|-
| | 2
| | Minor 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented seventh
| | 108¢ [20]
| | 120¢ [2]
|-
| | 3
| | Major 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented unison
| | 168¢ [14]
| | 156&lt;span style="line-height: 1.5;"&gt;¢ [8]&lt;/span&gt;
|-
| | 4
| | Minor 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major second
| | 216¢ [18]
| | 228¢ [4]
|-
| | 5
| | Major 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor third
| | 276¢ [16]
| | 264¢ [6]
|-
| | 6
| | Minor 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented second
| | 324¢ [16]
| | 336¢ [6]
|-
| | 7
| | Major 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished fourth
| | 384¢ [18]
| | 372¢ [4]
|-
| | 8
| | Augmented 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

&lt;span style="vertical-align: sub;"&gt;Diminished &lt;/span&gt;5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major third
| | 432¢ [14]
| | 444¢ [8]
|-
| | 9
| | Perfect 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Perfect fourth
| | 492¢ [20]
| | 480¢ [2]
|-
| | 10
| | Augmented 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished fifth
| | 540¢ [12]
| | 552¢ [10]
|-
| | 11
| | Perfect 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented third

Diminished sixth
| | 600¢ [20]
| | 588¢ [1]

612¢ [1]
|-
| | 12
| | Augmented 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented fourth
| | 660¢ [12]
| | 648¢ [10]
|-
| | 13
| | Perfect 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Perfect fifth
| | 708¢ [20]
| | 720¢ [2]
|-
| | 14
| | Augmented 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor sixth
| | 768¢ [14]
| | 756¢ [8]
|-
| | 15
| | Minor 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished seventh
| | 816¢ [18]
| | 828¢ [4]
|-
| | 16
| | Major 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented sixth
| | 876¢ [16]
| | 864¢ [6]
|-
| | 17
| | Minor 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major sixth
| | 924¢ [16]
| | 936¢ [6]
|-
| | 18
| | Major 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor seventh
| | 984¢ [18]
| | 972¢ [4]
|-
| | 19
| | Minor 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished octave
| | 1032¢ [14]
| | 1044¢ [8]
|-
| | 20
| | Major 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished second
| | 1092¢ [20]
| | 1080¢ [2]
|-
| | 21
| | Augmented 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major seventh
| | 1140¢ [12]
| | 1152¢ [10]
|-
| | 22
| | 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Octave
| | 1200¢ [22]
| | N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;span style="vertical-align: sub;"&gt;10 &lt;/span&gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [https://en.wikipedia.org/wiki/Augmented-fourths_tuning are tuned in tritones]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian|Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively.
[[Category:100edo]]
[[Category:edo]]
[[Category:todo:add_definition]]</text>
      <sha1>9vq2dpf93r0li77ljlxhnsxcf99c4i5</sha1>
    </revision>
    <revision>
      <id>4003</id>
      <timestamp>2016-09-30T04:33:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 593681356 - Original comment: **</comment>
      <origin>4003</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23447" sha1="tlzmfv215pcomcn5rpm572t3xkaa1j3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-09-30 04:33:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;593681356&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="youtube" key="shcrw2vtmJU" width="560" height="315"]]

The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo]], [[35edo]], [[47edo]] and [[88edo]], [[100edo]] possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].



=Scales= 
[[greeley8]]
[[greeley15]]

== == 
=100bddd and the 22-note scales= 

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [[http://www.anaphoria.com/Secor17puzzle.pdf|favored by George Secor]] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term //dog// rather than wolf for these intervals. Dog intervals frequently provide //closer// matches to intervals involving the 7th and 11th harmonics. Even //if// the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.


|| Steps of 22-note MODMOS || Interval name (decatonic) || Interval name (superpyth diatonic) || Pure interval size [multiplicity] || Dog interval size [multiplicity] ||
|| 1 || Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor second || 60¢ [12] || 48¢ [10] ||
|| 2 || Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented seventh || 108¢ [20] || 120¢ [2] ||
|| 3 || Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented unison || 168¢ [14] || 156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt; ||
|| 4 || Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major second || 216¢ [18] || 228¢ [4] ||
|| 5 || Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor third || 276¢ [16] || 264¢ [6] ||
|| 6 || Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented second || 324¢ [16] || 336¢ [6] ||
|| 7 || Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fourth || 384¢ [18] || 372¢ [4] ||
|| 8 || Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major third || 432¢ [14] || 444¢ [8] ||
|| 9 || Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fourth || 492¢ [20] || 480¢ [2] ||
|| 10 || Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fifth || 540¢ [12] || 552¢ [10] ||
|| 11 || Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented third
Diminished sixth || 600¢ [20] || 588¢ [1]
612¢ [1] ||
|| 12 || Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented fourth || 660¢ [12] || 648¢ [10] ||
|| 13 || Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fifth || 708¢ [20] || 720¢ [2] ||
|| 14 || Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor sixth || 768¢ [14] || 756¢ [8] ||
|| 15 || Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished seventh || 816¢ [18] || 828¢ [4] ||
|| 16 || Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented sixth || 876¢ [16] || 864¢ [6] ||
|| 17 || Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major sixth || 924¢ [16] || 936¢ [6] ||
|| 18 || Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor seventh || 984¢ [18] || 972¢ [4] ||
|| 19 || Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished octave || 1032¢ [14] || 1044¢ [8] ||
|| 20 || Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished second || 1092¢ [20] || 1080¢ [2] ||
|| 21 || Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major seventh || 1140¢ [12] || 1152¢ [10] ||
|| 22 || 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Octave || 1200¢ [22] || N/A ||

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10 &amp;lt;/span&amp;gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[https://en.wikipedia.org/wiki/Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow //almost// all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/youtube/shcrw2vtmJU?h=315&amp;amp;amp;w=560&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaYoutube&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;youtube&amp;amp;amp;quot; key=&amp;amp;amp;quot;shcrw2vtmJU&amp;amp;amp;quot; width=&amp;amp;amp;quot;560&amp;amp;amp;quot; height=&amp;amp;amp;quot;315&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;YouTube Video&amp;amp;quot;height=&amp;amp;quot;315&amp;amp;quot; width=&amp;amp;quot;560&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;iframe width="560" height="315" src="//www.youtube.com/embed/shcrw2vtmJU" frameborder="0" allowfullscreen&amp;gt;&amp;lt;/iframe&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Like &amp;lt;a class="wiki_link" href="/6edo"&amp;gt;6edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/35edo"&amp;gt;35edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/88edo"&amp;gt;88edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/100edo"&amp;gt;100edo&amp;lt;/a&amp;gt; possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/greeley8"&amp;gt;greeley8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/greeley15"&amp;gt;greeley15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt; &amp;lt;/h2&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="x100bddd and the 22-note scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;100bddd and the 22-note scales&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt; temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range &amp;lt;a class="wiki_link_ext" href="http://www.anaphoria.com/Secor17puzzle.pdf" rel="nofollow"&amp;gt;favored by George Secor&amp;lt;/a&amp;gt; for neomedieval compositions.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two &amp;amp;quot;wolf&amp;amp;quot; fifths. Much like meantone, this tuning has &amp;amp;quot;wolf&amp;amp;quot; intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the &amp;amp;quot;wolves&amp;amp;quot; are still usable (albeit xenharmonic), it might be better to use the term &amp;lt;em&amp;gt;dog&amp;lt;/em&amp;gt; rather than wolf for these intervals. Dog intervals frequently provide &amp;lt;em&amp;gt;closer&amp;lt;/em&amp;gt; matches to intervals involving the 7th and 11th harmonics. Even &amp;lt;em&amp;gt;if&amp;lt;/em&amp;gt; the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Steps of 22-note MODMOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (decatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (superpyth diatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Pure interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Dog interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;60¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;108¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;168¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;216¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;228¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;276¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;264¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;324¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;336¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;384¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;372¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;444¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;492¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;540¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented third&amp;lt;br /&amp;gt;
Diminished sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;588¢ [1]&amp;lt;br /&amp;gt;
612¢ [1]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;660¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;708¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;768¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;756¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;816¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;828¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;876¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;864¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;924¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;936¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;984¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;972¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1032¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1044¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1092¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1140¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1152¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200¢ [22]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;N/A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10 &amp;lt;/span&amp;gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they &amp;lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Augmented-fourths_tuning" rel="nofollow"&amp;gt;are tuned in tritones&amp;lt;/a&amp;gt;. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow &amp;lt;em&amp;gt;almost&amp;lt;/em&amp;gt; all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the &amp;lt;a class="wiki_link" href="/Indian"&amp;gt;Indian&amp;lt;/a&amp;gt; sruti system.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Other, &amp;amp;quot;gentle&amp;amp;quot; alternatives to 22edo for pajara include &amp;lt;a class="wiki_link" href="/78edo"&amp;gt;78ddd&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/56edo"&amp;gt;56d&amp;lt;/a&amp;gt;. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>tlzmfv215pcomcn5rpm572t3xkaa1j3</sha1>
    </revision>
    <revision>
      <id>4004</id>
      <timestamp>2016-06-19T23:36:02Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;MasonGreen1</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 585820725 - Original comment: **</comment>
      <origin>4004</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23173" sha1="jy37w9v2g320t4jkhdxf7fs57a4infc" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on &lt;tt&gt;2016-06-19 23:36:02 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;585820725&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="youtube" key="shcrw2vtmJU" width="560" height="315"]]

The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].



=Scales= 
[[greeley8]]
[[greeley15]]

== == 
=100bddd and the 22-note scales= 

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [[http://www.anaphoria.com/Secor17puzzle.pdf|favored by George Secor]] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term //dog// rather than wolf for these intervals. Dog intervals frequently provide //closer// matches to intervals involving the 7th and 11th harmonics. Even //if// the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.


|| Steps of 22-note MODMOS || Interval name (decatonic) || Interval name (superpyth diatonic) || Pure interval size [multiplicity] || Dog interval size [multiplicity] ||
|| 1 || Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor second || 60¢ [12] || 48¢ [10] ||
|| 2 || Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented seventh || 108¢ [20] || 120¢ [2] ||
|| 3 || Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented unison || 168¢ [14] || 156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt; ||
|| 4 || Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major second || 216¢ [18] || 228¢ [4] ||
|| 5 || Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor third || 276¢ [16] || 264¢ [6] ||
|| 6 || Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented second || 324¢ [16] || 336¢ [6] ||
|| 7 || Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fourth || 384¢ [18] || 372¢ [4] ||
|| 8 || Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major third || 432¢ [14] || 444¢ [8] ||
|| 9 || Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fourth || 492¢ [20] || 480¢ [2] ||
|| 10 || Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fifth || 540¢ [12] || 552¢ [10] ||
|| 11 || Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented third
Diminished sixth || 600¢ [20] || 588¢ [1]
612¢ [1] ||
|| 12 || Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented fourth || 660¢ [12] || 648¢ [10] ||
|| 13 || Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fifth || 708¢ [20] || 720¢ [2] ||
|| 14 || Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor sixth || 768¢ [14] || 756¢ [8] ||
|| 15 || Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished seventh || 816¢ [18] || 828¢ [4] ||
|| 16 || Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented sixth || 876¢ [16] || 864¢ [6] ||
|| 17 || Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major sixth || 924¢ [16] || 936¢ [6] ||
|| 18 || Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor seventh || 984¢ [18] || 972¢ [4] ||
|| 19 || Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished octave || 1032¢ [14] || 1044¢ [8] ||
|| 20 || Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished second || 1092¢ [20] || 1080¢ [2] ||
|| 21 || Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major seventh || 1140¢ [12] || 1152¢ [10] ||
|| 22 || 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Octave || 1200¢ [22] || N/A ||

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10 &amp;lt;/span&amp;gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[https://en.wikipedia.org/wiki/Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow //almost// all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/youtube/shcrw2vtmJU?h=315&amp;amp;amp;w=560&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaYoutube&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;youtube&amp;amp;amp;quot; key=&amp;amp;amp;quot;shcrw2vtmJU&amp;amp;amp;quot; width=&amp;amp;amp;quot;560&amp;amp;amp;quot; height=&amp;amp;amp;quot;315&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;YouTube Video&amp;amp;quot;height=&amp;amp;quot;315&amp;amp;quot; width=&amp;amp;quot;560&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;iframe width="560" height="315" src="//www.youtube.com/embed/shcrw2vtmJU" frameborder="0" allowfullscreen&amp;gt;&amp;lt;/iframe&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Like &amp;lt;a class="wiki_link" href="/88edo"&amp;gt;88edo&amp;lt;/a&amp;gt;, 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/greeley8"&amp;gt;greeley8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/greeley15"&amp;gt;greeley15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt; &amp;lt;/h2&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="x100bddd and the 22-note scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;100bddd and the 22-note scales&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt; temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range &amp;lt;a class="wiki_link_ext" href="http://www.anaphoria.com/Secor17puzzle.pdf" rel="nofollow"&amp;gt;favored by George Secor&amp;lt;/a&amp;gt; for neomedieval compositions.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two &amp;amp;quot;wolf&amp;amp;quot; fifths. Much like meantone, this tuning has &amp;amp;quot;wolf&amp;amp;quot; intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the &amp;amp;quot;wolves&amp;amp;quot; are still usable (albeit xenharmonic), it might be better to use the term &amp;lt;em&amp;gt;dog&amp;lt;/em&amp;gt; rather than wolf for these intervals. Dog intervals frequently provide &amp;lt;em&amp;gt;closer&amp;lt;/em&amp;gt; matches to intervals involving the 7th and 11th harmonics. Even &amp;lt;em&amp;gt;if&amp;lt;/em&amp;gt; the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Steps of 22-note MODMOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (decatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (superpyth diatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Pure interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Dog interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;60¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;108¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;168¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;216¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;228¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;276¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;264¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;324¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;336¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;384¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;372¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;444¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;492¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;540¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented third&amp;lt;br /&amp;gt;
Diminished sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;588¢ [1]&amp;lt;br /&amp;gt;
612¢ [1]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;660¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;708¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;768¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;756¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;816¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;828¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;876¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;864¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;924¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;936¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;984¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;972¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1032¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1044¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1092¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1140¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1152¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200¢ [22]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;N/A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10 &amp;lt;/span&amp;gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they &amp;lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Augmented-fourths_tuning" rel="nofollow"&amp;gt;are tuned in tritones&amp;lt;/a&amp;gt;. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow &amp;lt;em&amp;gt;almost&amp;lt;/em&amp;gt; all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the &amp;lt;a class="wiki_link" href="/Indian"&amp;gt;Indian&amp;lt;/a&amp;gt; sruti system.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Other, &amp;amp;quot;gentle&amp;amp;quot; alternatives to 22edo for pajara include &amp;lt;a class="wiki_link" href="/78edo"&amp;gt;78ddd&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/56edo"&amp;gt;56d&amp;lt;/a&amp;gt;. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>jy37w9v2g320t4jkhdxf7fs57a4infc</sha1>
    </revision>
    <revision>
      <id>4005</id>
      <timestamp>2016-06-19T23:25:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;MasonGreen1</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 585820495 - Original comment: **</comment>
      <origin>4005</origin>
      <model>wikitext</model>
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      <text bytes="23060" sha1="nmx6d8iopsfqpfzd2wzziyzht2wtbap" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on &lt;tt&gt;2016-06-19 23:25:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;585820495&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="youtube" key="shcrw2vtmJU" width="560" height="315"]]

The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].



=Scales= 
[[greeley8]]
[[greeley15]]

== == 
=100bddd and the 22-note scales= 

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [[http://www.anaphoria.com/Secor17puzzle.pdf|favored by George Secor]] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term //dog// rather than wolf for these intervals. Dog intervals frequently provide //closer// matches to intervals involving the 7th and 11th harmonics. Even //if// the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.


|| Steps of 22-note MODMOS || Interval name (decatonic) || Interval name (superpyth diatonic) || Pure interval size [multiplicity] || Dog interval size [multiplicity] ||
|| 1 || Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor second || 60¢ [12] || 48¢ [10] ||
|| 2 || Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented seventh || 108¢ [20] || 120¢ [2] ||
|| 3 || Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented unison || 168¢ [14] || 156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt; ||
|| 4 || Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major second || 216¢ [18] || 228¢ [4] ||
|| 5 || Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor third || 276¢ [16] || 264¢ [6] ||
|| 6 || Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented second || 324¢ [16] || 336¢ [6] ||
|| 7 || Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fourth || 384¢ [18] || 372¢ [4] ||
|| 8 || Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major third || 432¢ [14] || 444¢ [8] ||
|| 9 || Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fourth || 492¢ [20] || 480¢ [2] ||
|| 10 || Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fifth || 540¢ [12] || 552¢ [10] ||
|| 11 || Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented third
Diminished sixth || 600¢ [20] || 588¢ [1]
612¢ [1] ||
|| 12 || Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented fourth || 660¢ [12] || 648¢ [10] ||
|| 13 || Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fifth || 708¢ [20] || 720¢ [2] ||
|| 14 || Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor sixth || 768¢ [14] || 756¢ [8] ||
|| 15 || Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished seventh || 816¢ [18] || 828¢ [4] ||
|| 16 || Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented sixth || 876¢ [16] || 864¢ [6] ||
|| 17 || Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major sixth || 924¢ [16] || 936¢ [6] ||
|| 18 || Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor seventh || 984¢ [18] || 972¢ [4] ||
|| 19 || Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished octave || 1032¢ [14] || 1044¢ [8] ||
|| 20 || Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished second || 1092¢ [20] || 1080¢ [2] ||
|| 21 || Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major seventh || 1140¢ [12] || 1152¢ [10] ||
|| 22 || 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Octave || 1200¢ [22] || N/A ||

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10 &amp;lt;/span&amp;gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[https://en.wikipedia.org/wiki/Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow all-but-one frets to be straight if the augmented-fourths tuning is used. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Like &amp;lt;a class="wiki_link" href="/88edo"&amp;gt;88edo&amp;lt;/a&amp;gt;, 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/greeley8"&amp;gt;greeley8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/greeley15"&amp;gt;greeley15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt; &amp;lt;/h2&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="x100bddd and the 22-note scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;100bddd and the 22-note scales&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt; temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range &amp;lt;a class="wiki_link_ext" href="http://www.anaphoria.com/Secor17puzzle.pdf" rel="nofollow"&amp;gt;favored by George Secor&amp;lt;/a&amp;gt; for neomedieval compositions.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two &amp;amp;quot;wolf&amp;amp;quot; fifths. Much like meantone, this tuning has &amp;amp;quot;wolf&amp;amp;quot; intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the &amp;amp;quot;wolves&amp;amp;quot; are still usable (albeit xenharmonic), it might be better to use the term &amp;lt;em&amp;gt;dog&amp;lt;/em&amp;gt; rather than wolf for these intervals. Dog intervals frequently provide &amp;lt;em&amp;gt;closer&amp;lt;/em&amp;gt; matches to intervals involving the 7th and 11th harmonics. Even &amp;lt;em&amp;gt;if&amp;lt;/em&amp;gt; the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Steps of 22-note MODMOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (decatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (superpyth diatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Pure interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Dog interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;60¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;108¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;168¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;216¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;228¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;276¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;264¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;324¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;336¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;384¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;372¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;444¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;492¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;540¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented third&amp;lt;br /&amp;gt;
Diminished sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;588¢ [1]&amp;lt;br /&amp;gt;
612¢ [1]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;660¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;708¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;768¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;756¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;816¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;828¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;876¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;864¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;924¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;936¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;984¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;972¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1032¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1044¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1092¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1140¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1152¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200¢ [22]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;N/A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10 &amp;lt;/span&amp;gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they &amp;lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Augmented-fourths_tuning" rel="nofollow"&amp;gt;are tuned in tritones&amp;lt;/a&amp;gt;. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow all-but-one frets to be straight if the augmented-fourths tuning is used. While the 2MOS is simpler, the MODMOS very closely approximates the &amp;lt;a class="wiki_link" href="/Indian"&amp;gt;Indian&amp;lt;/a&amp;gt; sruti system.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Other, &amp;amp;quot;gentle&amp;amp;quot; alternatives to 22edo for pajara include &amp;lt;a class="wiki_link" href="/78edo"&amp;gt;78ddd&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/56edo"&amp;gt;56d&amp;lt;/a&amp;gt;. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>nmx6d8iopsfqpfzd2wzziyzht2wtbap</sha1>
    </revision>
    <revision>
      <id>4006</id>
      <timestamp>2016-06-17T21:45:48Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;MasonGreen1</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 585770669 - Original comment: **</comment>
      <origin>4006</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22132" sha1="hbu9v7fnlta5pj8tw459ba8kcyt30h4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on &lt;tt&gt;2016-06-17 21:45:48 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;585770669&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="youtube" key="shcrw2vtmJU" width="560" height="315"]]

The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].



=Scales= 
[[greeley8]]
[[greeley15]]

== == 
=100bddd and the 22-note scales= 

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [[http://www.anaphoria.com/Secor17puzzle.pdf|favored by George Secor]] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term //dog// rather than wolf for these intervals. Dog intervals frequently provide //closer// matches to intervals involving the 7th and 11th harmonics. Even //if// the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.


|| Steps of 22-note MODMOS || Interval name (decatonic) || Interval name (superpyth diatonic) || Pure interval size [multiplicity] || Dog interval size [multiplicity] ||
|| 1 || Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor second || 60¢ [12] || 48¢ [10] ||
|| 2 || Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented seventh || 108¢ [20] || 120¢ [2] ||
|| 3 || Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented unison || 168¢ [14] || 156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt; ||
|| 4 || Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major second || 216¢ [18] || 228¢ [4] ||
|| 5 || Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor third || 276¢ [16] || 264¢ [6] ||
|| 6 || Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented second || 324¢ [16] || 336¢ [6] ||
|| 7 || Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fourth || 384¢ [18] || 372¢ [4] ||
|| 8 || Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major third || 432¢ [14] || 444¢ [8] ||
|| 9 || Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fourth || 492¢ [20] || 480¢ [2] ||
|| 10 || Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fifth || 540¢ [12] || 552¢ [10] ||
|| 11 || Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented third
Diminished sixth || 600¢ [20] || 588¢ [1]
612¢ [1] ||
|| 12 || Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented fourth || 660¢ [12] || 648¢ [10] ||
|| 13 || Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fifth || 708¢ [20] || 720¢ [2] ||
|| 14 || Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor sixth || 768¢ [14] || 756¢ [8] ||
|| 15 || Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished seventh || 816¢ [18] || 828¢ [4] ||
|| 16 || Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented sixth || 876¢ [16] || 864¢ [6] ||
|| 17 || Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major sixth || 924¢ [16] || 936¢ [6] ||
|| 18 || Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor seventh || 984¢ [18] || 972¢ [4] ||
|| 19 || Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished octave || 1032¢ [14] || 1044¢ [8] ||
|| 20 || Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished second || 1092¢ [20] || 1080¢ [2] ||
|| 21 || Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major seventh || 1140¢ [12] || 1152¢ [10] ||
|| 22 || 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Octave || 1200¢ [22] || N/A ||

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10 &amp;lt;/span&amp;gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[https://en.wikipedia.org/wiki/Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/youtube/shcrw2vtmJU?h=315&amp;amp;amp;w=560&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaYoutube&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;youtube&amp;amp;amp;quot; key=&amp;amp;amp;quot;shcrw2vtmJU&amp;amp;amp;quot; width=&amp;amp;amp;quot;560&amp;amp;amp;quot; height=&amp;amp;amp;quot;315&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;YouTube Video&amp;amp;quot;height=&amp;amp;quot;315&amp;amp;quot; width=&amp;amp;quot;560&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;iframe width="560" height="315" src="//www.youtube.com/embed/shcrw2vtmJU" frameborder="0" allowfullscreen&amp;gt;&amp;lt;/iframe&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Like &amp;lt;a class="wiki_link" href="/88edo"&amp;gt;88edo&amp;lt;/a&amp;gt;, 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/greeley8"&amp;gt;greeley8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/greeley15"&amp;gt;greeley15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt; &amp;lt;/h2&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="x100bddd and the 22-note scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;100bddd and the 22-note scales&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt; temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range &amp;lt;a class="wiki_link_ext" href="http://www.anaphoria.com/Secor17puzzle.pdf" rel="nofollow"&amp;gt;favored by George Secor&amp;lt;/a&amp;gt; for neomedieval compositions.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two &amp;amp;quot;wolf&amp;amp;quot; fifths. Much like meantone, this tuning has &amp;amp;quot;wolf&amp;amp;quot; intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the &amp;amp;quot;wolves&amp;amp;quot; are still usable (albeit xenharmonic), it might be better to use the term &amp;lt;em&amp;gt;dog&amp;lt;/em&amp;gt; rather than wolf for these intervals. Dog intervals frequently provide &amp;lt;em&amp;gt;closer&amp;lt;/em&amp;gt; matches to intervals involving the 7th and 11th harmonics. Even &amp;lt;em&amp;gt;if&amp;lt;/em&amp;gt; the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Steps of 22-note MODMOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (decatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (superpyth diatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Pure interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Dog interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;60¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;108¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;168¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;216¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;228¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;276¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;264¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;324¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;336¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;384¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;372¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;444¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;492¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;540¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented third&amp;lt;br /&amp;gt;
Diminished sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;588¢ [1]&amp;lt;br /&amp;gt;
612¢ [1]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;660¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;708¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;768¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;756¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;816¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;828¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;876¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;864¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;924¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;936¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;984¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;972¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1032¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1044¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1092¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1140¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1152¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200¢ [22]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;N/A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10 &amp;lt;/span&amp;gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they &amp;lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Augmented-fourths_tuning" rel="nofollow"&amp;gt;are tuned in tritones&amp;lt;/a&amp;gt;. This makes guitar construction much easier compared to other non-equally-tempered scales.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>hbu9v7fnlta5pj8tw459ba8kcyt30h4</sha1>
    </revision>
    <revision>
      <id>4007</id>
      <timestamp>2016-06-17T02:40:06Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;MasonGreen1</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 585712975 - Original comment: **</comment>
      <origin>4007</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20892" sha1="j862mjs0sitgsa8ok45gxzvcapsppu4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on &lt;tt&gt;2016-06-17 02:40:06 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;585712975&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="youtube" key="shcrw2vtmJU" width="560" height="315"]]

The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].



=Scales= 
[[greeley8]]
[[greeley15]]

== == 
=100bddd and the 22-note MODMOS= 

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [[http://www.anaphoria.com/Secor17puzzle.pdf|favored by George Secor]] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term //dog// rather than wolf for these intervals. Dog intervals frequently provide //closer// matches to intervals involving the 7th and 11th harmonics. Even //if// the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.


|| Steps of 22-note MODMOS || Interval name (decatonic) || Interval name (superpyth diatonic) || Pure interval size [multiplicity] || Dog interval size [multiplicity] ||
|| 1 || Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor second || 60¢ [12] || 48¢ [10] ||
|| 2 || Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented seventh || 108¢ [20] || 120¢ [2] ||
|| 3 || Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented unison || 168¢ [14] || 156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt; ||
|| 4 || Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major second || 216¢ [18] || 228¢ [4] ||
|| 5 || Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor third || 276¢ [16] || 264¢ [6] ||
|| 6 || Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented second || 324¢ [16] || 336¢ [6] ||
|| 7 || Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fourth || 384¢ [18] || 372¢ [4] ||
|| 8 || Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major third || 432¢ [14] || 444¢ [8] ||
|| 9 || Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fourth || 492¢ [20] || 480¢ [2] ||
|| 10 || Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fifth || 540¢ [12] || 552¢ [10] ||
|| 11 || Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented third
Diminished sixth || 600¢ [20] || 588¢ [1]
612¢ [1] ||
|| 12 || Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented fourth || 660¢ [12] || 648¢ [10] ||
|| 13 || Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fifth || 708¢ [20] || 720¢ [2] ||
|| 14 || Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor sixth || 768¢ [14] || 756¢ [8] ||
|| 15 || Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished seventh || 816¢ [18] || 828¢ [4] ||
|| 16 || Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented sixth || 876¢ [16] || 864¢ [6] ||
|| 17 || Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major sixth || 924¢ [16] || 936¢ [6] ||
|| 18 || Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor seventh || 984¢ [18] || 972¢ [4] ||
|| 19 || Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished octave || 1032¢ [14] || 1044¢ [8] ||
|| 20 || Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished second || 1092¢ [20] || 1080¢ [2] ||
|| 21 || Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major seventh || 1140¢ [12] || 1152¢ [10] ||
|| 22 || 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Octave || 1200¢ [22] || N/A ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Like &amp;lt;a class="wiki_link" href="/88edo"&amp;gt;88edo&amp;lt;/a&amp;gt;, 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/greeley8"&amp;gt;greeley8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/greeley15"&amp;gt;greeley15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt; &amp;lt;/h2&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="x100bddd and the 22-note MODMOS"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;100bddd and the 22-note MODMOS&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt; temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range &amp;lt;a class="wiki_link_ext" href="http://www.anaphoria.com/Secor17puzzle.pdf" rel="nofollow"&amp;gt;favored by George Secor&amp;lt;/a&amp;gt; for neomedieval compositions.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two &amp;amp;quot;wolf&amp;amp;quot; fifths. Much like meantone, this tuning has &amp;amp;quot;wolf&amp;amp;quot; intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the &amp;amp;quot;wolves&amp;amp;quot; are still usable (albeit xenharmonic), it might be better to use the term &amp;lt;em&amp;gt;dog&amp;lt;/em&amp;gt; rather than wolf for these intervals. Dog intervals frequently provide &amp;lt;em&amp;gt;closer&amp;lt;/em&amp;gt; matches to intervals involving the 7th and 11th harmonics. Even &amp;lt;em&amp;gt;if&amp;lt;/em&amp;gt; the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Steps of 22-note MODMOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (decatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (superpyth diatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Pure interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Dog interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;60¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;108¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;168¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;216¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;228¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;276¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;264¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;324¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;336¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;384¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;372¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;444¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;492¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;540¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented third&amp;lt;br /&amp;gt;
Diminished sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;588¢ [1]&amp;lt;br /&amp;gt;
612¢ [1]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;660¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;708¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;768¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;756¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;816¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;828¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;876¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;864¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;924¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;936¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;984¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;972¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1032¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1044¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1092¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1140¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1152¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200¢ [22]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;N/A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>j862mjs0sitgsa8ok45gxzvcapsppu4</sha1>
    </revision>
    <revision>
      <id>4008</id>
      <timestamp>2016-06-17T02:29:07Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;MasonGreen1</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 585712845 - Original comment: **</comment>
      <origin>4008</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20858" sha1="qdskp2jmd8ubo163hvt9u66aw99hyfy" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on &lt;tt&gt;2016-06-17 02:29:07 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;585712845&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="youtube" key="shcrw2vtmJU" width="560" height="315"]]

The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].



=Scales= 
[[greeley8]]
[[greeley15]]

== == 
=100bddd and the 22-note MODMOS= 

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [[http://www.anaphoria.com/Secor17puzzle.pdf|favored by George Secor]] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of them are still usable (albeit xenharmonic), it might be better to use the term //dog// rather than wolf for these intervals. Dog intervals frequently provide //closer// matches to intervals involving the 7th and 11th harmonics. Even //if// the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.


|| Steps of 22-note MODMOS || Interval name (decatonic) || Interval name (superpyth diatonic) || Pure interval size [multiplicity] || Dog interval size [multiplicity] ||
|| 1 || Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor second || 60¢ [12] || 48¢ [10] ||
|| 2 || Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented seventh || 108¢ [20] || 120¢ [2] ||
|| 3 || Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented unison || 168¢ [14] || 156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt; ||
|| 4 || Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major second || 216¢ [18] || 228¢ [4] ||
|| 5 || Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor third || 276¢ [16] || 264¢ [6] ||
|| 6 || Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented second || 324¢ [16] || 336¢ [6] ||
|| 7 || Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fourth || 384¢ [18] || 372¢ [4] ||
|| 8 || Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major third || 432¢ [14] || 444¢ [8] ||
|| 9 || Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fourth || 492¢ [20] || 480¢ [2] ||
|| 10 || Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fifth || 540¢ [12] || 552¢ [10] ||
|| 11 || Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented third
Diminished sixth || 600¢ [20] || 588¢ [1]
612¢ [1] ||
|| 12 || Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented fourth || 660¢ [12] || 648¢ [10] ||
|| 13 || Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fifth || 708¢ [20] || 720¢ [2] ||
|| 14 || Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor sixth || 768¢ [14] || 756¢ [8] ||
|| 15 || Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished seventh || 816¢ [18] || 828¢ [4] ||
|| 16 || Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented sixth || 876¢ [16] || 864¢ [6] ||
|| 17 || Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major sixth || 924¢ [16] || 936¢ [6] ||
|| 18 || Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor seventh || 984¢ [18] || 972¢ [4] ||
|| 19 || Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished octave || 1032¢ [14] || 1044¢ [8] ||
|| 20 || Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished second || 1092¢ [20] || 1080¢ [2] ||
|| 21 || Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major seventh || 1140¢ [12] || 1152¢ [10] ||
|| 22 || 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Octave || 1200¢ [22] || N/A ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/youtube/shcrw2vtmJU?h=315&amp;amp;amp;w=560&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaYoutube&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;youtube&amp;amp;amp;quot; key=&amp;amp;amp;quot;shcrw2vtmJU&amp;amp;amp;quot; width=&amp;amp;amp;quot;560&amp;amp;amp;quot; height=&amp;amp;amp;quot;315&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;YouTube Video&amp;amp;quot;height=&amp;amp;quot;315&amp;amp;quot; width=&amp;amp;quot;560&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;iframe width="560" height="315" src="//www.youtube.com/embed/shcrw2vtmJU" frameborder="0" allowfullscreen&amp;gt;&amp;lt;/iframe&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Like &amp;lt;a class="wiki_link" href="/88edo"&amp;gt;88edo&amp;lt;/a&amp;gt;, 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/greeley8"&amp;gt;greeley8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/greeley15"&amp;gt;greeley15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt; &amp;lt;/h2&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="x100bddd and the 22-note MODMOS"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;100bddd and the 22-note MODMOS&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt; temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range &amp;lt;a class="wiki_link_ext" href="http://www.anaphoria.com/Secor17puzzle.pdf" rel="nofollow"&amp;gt;favored by George Secor&amp;lt;/a&amp;gt; for neomedieval compositions.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two &amp;amp;quot;wolf&amp;amp;quot; fifths. Much like meantone, this tuning has &amp;amp;quot;wolf&amp;amp;quot; intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of them are still usable (albeit xenharmonic), it might be better to use the term &amp;lt;em&amp;gt;dog&amp;lt;/em&amp;gt; rather than wolf for these intervals. Dog intervals frequently provide &amp;lt;em&amp;gt;closer&amp;lt;/em&amp;gt; matches to intervals involving the 7th and 11th harmonics. Even &amp;lt;em&amp;gt;if&amp;lt;/em&amp;gt; the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Steps of 22-note MODMOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (decatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (superpyth diatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Pure interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Dog interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;60¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;108¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;168¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;216¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;228¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;276¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;264¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;324¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;336¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;384¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;372¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;444¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;492¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;540¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented third&amp;lt;br /&amp;gt;
Diminished sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;588¢ [1]&amp;lt;br /&amp;gt;
612¢ [1]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;660¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;708¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;768¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;756¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;816¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;828¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;876¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;864¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;924¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;936¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;984¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;972¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1032¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1044¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1092¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1140¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1152¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200¢ [22]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;N/A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>qdskp2jmd8ubo163hvt9u66aw99hyfy</sha1>
    </revision>
    <revision>
      <id>4009</id>
      <timestamp>2016-06-17T02:23:01Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;MasonGreen1</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 585712775 - Original comment: **</comment>
      <origin>4009</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20868" sha1="9j7p8xb20g0m2ylznuolbzagy6icin6" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:MasonGreen1|MasonGreen1]] and made on &lt;tt&gt;2016-06-17 02:23:01 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;585712775&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="youtube" key="shcrw2vtmJU" width="560" height="315"]]

The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].



=Scales= 
[[greeley8]]
[[greeley15]]

== == 
=100bddd and the 22-note MODMOS= 

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [[http://www.anaphoria.com/Secor17puzzle.pdf|favored by George Secor]] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of them are still usable (albeit xenharmonic), it might be better to use the term //dog// rather than wolf for these intervals. Dog intervals frequently provide //closer// matches to intervals involving the 7th and 11th harmonics. Even //if// the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 10 different keys, and diatonic (superpyth) music also in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.


|| Steps of 22-note MODMOS || Interval name (decatonic) || Interval name (superpyth diatonic) || Pure interval size [multiplicity] || Dog interval size [multiplicity] ||
|| 1 || Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor second || 60¢ [12] || 48¢ [10] ||
|| 2 || Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented seventh || 108¢ [20] || 120¢ [2] ||
|| 3 || Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented unison || 168¢ [14] || 156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt; ||
|| 4 || Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major second || 216¢ [18] || 228¢ [4] ||
|| 5 || Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor third || 276¢ [16] || 264¢ [6] ||
|| 6 || Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented second || 324¢ [16] || 336¢ [6] ||
|| 7 || Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fourth || 384¢ [18] || 372¢ [4] ||
|| 8 || Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major third || 432¢ [14] || 444¢ [8] ||
|| 9 || Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fourth || 492¢ [20] || 480¢ [2] ||
|| 10 || Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished fifth || 540¢ [12] || 552¢ [10] ||
|| 11 || Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented third
Diminished sixth || 600¢ [20] || 588¢ [1]
612¢ [1] ||
|| 12 || Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented fourth || 660¢ [12] || 648¢ [10] ||
|| 13 || Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Perfect fifth || 708¢ [20] || 720¢ [2] ||
|| 14 || Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor sixth || 768¢ [14] || 756¢ [8] ||
|| 15 || Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished seventh || 816¢ [18] || 828¢ [4] ||
|| 16 || Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Augmented sixth || 876¢ [16] || 864¢ [6] ||
|| 17 || Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major sixth || 924¢ [16] || 936¢ [6] ||
|| 18 || Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Minor seventh || 984¢ [18] || 972¢ [4] ||
|| 19 || Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished octave || 1032¢ [14] || 1044¢ [8] ||
|| 20 || Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Diminished second || 1092¢ [20] || 1080¢ [2] ||
|| 21 || Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Major seventh || 1140¢ [12] || 1152¢ [10] ||
|| 22 || 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt; || Octave || 1200¢ [22] || N/A ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/youtube/shcrw2vtmJU?h=315&amp;amp;amp;w=560&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaYoutube&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;youtube&amp;amp;amp;quot; key=&amp;amp;amp;quot;shcrw2vtmJU&amp;amp;amp;quot; width=&amp;amp;amp;quot;560&amp;amp;amp;quot; height=&amp;amp;amp;quot;315&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;YouTube Video&amp;amp;quot;height=&amp;amp;quot;315&amp;amp;quot; width=&amp;amp;quot;560&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;iframe width="560" height="315" src="//www.youtube.com/embed/shcrw2vtmJU" frameborder="0" allowfullscreen&amp;gt;&amp;lt;/iframe&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Like &amp;lt;a class="wiki_link" href="/88edo"&amp;gt;88edo&amp;lt;/a&amp;gt;, 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/greeley8"&amp;gt;greeley8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/greeley15"&amp;gt;greeley15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt; &amp;lt;/h2&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="x100bddd and the 22-note MODMOS"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;100bddd and the 22-note MODMOS&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt; temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range &amp;lt;a class="wiki_link_ext" href="http://www.anaphoria.com/Secor17puzzle.pdf" rel="nofollow"&amp;gt;favored by George Secor&amp;lt;/a&amp;gt; for neomedieval compositions.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two &amp;amp;quot;wolf&amp;amp;quot; fifths. Much like meantone, this tuning has &amp;amp;quot;wolf&amp;amp;quot; intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of them are still usable (albeit xenharmonic), it might be better to use the term &amp;lt;em&amp;gt;dog&amp;lt;/em&amp;gt; rather than wolf for these intervals. Dog intervals frequently provide &amp;lt;em&amp;gt;closer&amp;lt;/em&amp;gt; matches to intervals involving the 7th and 11th harmonics. Even &amp;lt;em&amp;gt;if&amp;lt;/em&amp;gt; the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 10 different keys, and diatonic (superpyth) music also in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Steps of 22-note MODMOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (decatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval name (superpyth diatonic)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Pure interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Dog interval size [multiplicity]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;60¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;108¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 2nd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;168¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;156&amp;lt;span style="line-height: 1.5;"&amp;gt;¢ [8]&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;216¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;228¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 3rd&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;276¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;264¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;324¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;336¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;384¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;372¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 4th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="vertical-align: sub;"&amp;gt;Diminished &amp;lt;/span&amp;gt;5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;444¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;492¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 5th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;540¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented third&amp;lt;br /&amp;gt;
Diminished sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;588¢ [1]&amp;lt;br /&amp;gt;
612¢ [1]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 6th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;660¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Perfect fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;708¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 7th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;768¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;756¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;816¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;828¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 8th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;876¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;864¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;924¢ [16]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;936¢ [6]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 9th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;984¢ [18]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;972¢ [4]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Minor 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1032¢ [14]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1044¢ [8]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1092¢ [20]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080¢ [2]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented 10th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
Diminished 11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1140¢ [12]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1152¢ [10]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11th&amp;lt;span style="vertical-align: sub;"&amp;gt;10&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200¢ [22]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;N/A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9j7p8xb20g0m2ylznuolbzagy6icin6</sha1>
    </revision>
    <revision>
      <id>4010</id>
      <timestamp>2016-06-04T06:20:18Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenjacob</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 584788901 - Original comment: **</comment>
      <origin>4010</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4096" sha1="53f3r0gj4f3x8j4wmc0pf3xee7wz8n1" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenjacob|xenjacob]] and made on &lt;tt&gt;2016-06-04 06:20:18 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;584788901&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="youtube" key="shcrw2vtmJU" width="560" height="315"]]

The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=Scales= 
[[greeley8]]
[[greeley15]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/youtube/shcrw2vtmJU?h=315&amp;amp;amp;w=560&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaYoutube&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;youtube&amp;amp;amp;quot; key=&amp;amp;amp;quot;shcrw2vtmJU&amp;amp;amp;quot; width=&amp;amp;amp;quot;560&amp;amp;amp;quot; height=&amp;amp;amp;quot;315&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;YouTube Video&amp;amp;quot;height=&amp;amp;quot;315&amp;amp;quot; width=&amp;amp;quot;560&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;iframe width="560" height="315" src="//www.youtube.com/embed/shcrw2vtmJU" frameborder="0" allowfullscreen&amp;gt;&amp;lt;/iframe&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Like &amp;lt;a class="wiki_link" href="/88edo"&amp;gt;88edo&amp;lt;/a&amp;gt;, 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/greeley8"&amp;gt;greeley8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/greeley15"&amp;gt;greeley15&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>53f3r0gj4f3x8j4wmc0pf3xee7wz8n1</sha1>
    </revision>
    <revision>
      <id>4011</id>
      <timestamp>2014-06-28T04:25:31Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Dodecaphobe</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 515154264 - Original comment: **</comment>
      <origin>4011</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3309" sha1="ogh8mcrpla94rbqp1xmcs7szg3sts5i" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Dodecaphobe|Dodecaphobe]] and made on &lt;tt&gt;2014-06-28 04:25:31 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;515154264&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=Scales= 
[[greeley8]]
[[greeley15]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Like &amp;lt;a class="wiki_link" href="/88edo"&amp;gt;88edo&amp;lt;/a&amp;gt;, 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/greeley8"&amp;gt;greeley8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/greeley15"&amp;gt;greeley15&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2012-05-29T16:03:39Z</timestamp>
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        <id>0</id>
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      <comment>**Imported revision 340652774 - Original comment: **</comment>
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      <text bytes="2543" sha1="r3tlpxaer2cgt6y35uqlnomgldliav8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-05-29 16:03:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;340652774&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

=Scales= 
[[greeley8]]
[[greeley15]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;100edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
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      <comment>**Imported revision 238149773 - Original comment: **</comment>
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      <text bytes="1571" sha1="gecnjch6sax9vu704hazfcgaxn3vlh9" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2011-06-22 08:18:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;238149773&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=Scales= 
[[greeley8]]
[[greeley15]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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    <revision>
      <id>40205</id>
      <parentid>16</parentid>
      <timestamp>2019-02-13T01:24:23Z</timestamp>
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        <username>Moremajorthanmajor</username>
        <id>28</id>
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      <text bytes="7266" sha1="faht0t4x3yjgqb3x66b28s7w5cxujjx" xml:space="preserve">&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

The 100 equal temperament divides the octave into 100 equal parts of precisely 12 cents each. It is closely related to 50edo, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6edo]], [[35edo|35edo]], [[47edo|47edo]] and [[88edo|88edo]], [[100edo|100edo]] possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo|12edo]].

=Scales=
[[greeley8|greeley8]]

[[greeley15|greeley15]]

=100bddd and the 22-note scales=

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo|22edo]] for [[pajara|pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.

{| class="wikitable"
|-
| | Steps of 22-note MODMOS
| | Interval name (decatonic)
| | Interval name (superpyth diatonic)
| | Pure interval size [multiplicity]
| | Dog interval size [multiplicity]
|-
| | 1
| | Diminished 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor second
| | 60¢ [12]
| | 48¢ [10]
|-
| | 2
| | Minor 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented seventh
| | 108¢ [20]
| | 120¢ [2]
|-
| | 3
| | Major 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented unison
| | ''168¢ [14]''
| | ''156&lt;span style="line-height: 1.5;"&gt;¢ [8]&lt;/span&gt;''
|-
| | 4
| | Minor 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major second
| | 216¢ [18]
| | 228¢ [4]
|-
| | 5
| | Major 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor third
| | 276¢ [16]
| | 264¢ [6]
|-
| | 6
| | Minor 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Diminished fourth
| | ''324¢ [16]''
| | ''336¢ [6]''
|-
| | 7
| | Major 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented second
| | 384¢ [18]
| | 372¢ [4]
|-
| | 8
| | Augmented 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

&lt;span style="vertical-align: sub;"&gt;Diminished &lt;/span&gt;5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major third
| | 432¢ [14]
| | 444¢ [8]
|-
| | 9
| | Perfect 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Perfect fourth
| | 492¢ [20]
| | 480¢ [2]
|-
| | 10
| | Augmented 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished fifth
| | 540¢ [12]
| | 552¢ [10]
|-
| | 11
| | Perfect 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented third

Diminished sixth
| | 600¢ [20]
| | 588¢ [1]

612¢ [1]
|-
| | 12
| | Augmented 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented fourth
| | 660¢ [12]
| | 648¢ [10]
|-
| | 13
| | Perfect 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Perfect fifth
| | 708¢ [20]
| | 720¢ [2]
|-
| | 14
| | Augmented 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor sixth
| | 768¢ [14]
| | 756¢ [8]
|-
| | 15
| | Minor 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished seventh
| | 816¢ [18]
| | 828¢ [4]
|-
| | 16
| | Major 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented fifth
| | ''876¢ [16]''
| | ''864¢ [6]''
|-
| | 17
| | Minor 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major sixth
| | 924¢ [16]
| | 936¢ [6]
|-
| | 18
| | Major 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor seventh
| | 984¢ [18]
| | 972¢ [4]
|-
| | 19
| | Minor 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished octave
| | ''1032¢ [14]''
| | ''1044¢ [8]''
|-
| | 20
| | Major 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished second
| | 1092¢ [20]
| | 1080¢ [2]
|-
| | 21
| | Augmented 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major seventh
| | 1140¢ [12]
| | 1152¢ [10]
|-
| | 22
| | 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Octave
| | 1200¢ [22]
| | N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;span style="vertical-align: sub;"&gt;10 &lt;/span&gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [https://en.wikipedia.org/wiki/Augmented-fourths_tuning are tuned in tritones]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian|Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively.
[[Category:100edo]]
[[Category:edo]]
[[Category:todo:add_definition]]</text>
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      <id>40774</id>
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      <timestamp>2019-03-10T22:45:33Z</timestamp>
      <contributor>
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        <id>73</id>
      </contributor>
      <comment>minor edit</comment>
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      <text bytes="7260" sha1="9kkzoemfpmusexj03r6mg83dfiibg91" xml:space="preserve">&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

'''100EDO''' is the [[EDO|equal division of the octave]] into 100 parts of exact 12 cents each. It is closely related to 50EDO, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50EDO 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6EDO]], [[35edo|35EDO]], [[47edo|47EDO]] and [[88edo|88EDO]], 100EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo|12EDO]].

=Scales=
[[greeley8|greeley8]]

[[greeley15|greeley15]]

=100bddd and the 22-note scales=

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo|22edo]] for [[pajara|pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.

{| class="wikitable"
|-
| | Steps of 22-note MODMOS
| | Interval name (decatonic)
| | Interval name (superpyth diatonic)
| | Pure interval size [multiplicity]
| | Dog interval size [multiplicity]
|-
| | 1
| | Diminished 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor second
| | 60¢ [12]
| | 48¢ [10]
|-
| | 2
| | Minor 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented seventh
| | 108¢ [20]
| | 120¢ [2]
|-
| | 3
| | Major 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented unison
| | ''168¢ [14]''
| | ''156&lt;span style="line-height: 1.5;"&gt;¢ [8]&lt;/span&gt;''
|-
| | 4
| | Minor 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major second
| | 216¢ [18]
| | 228¢ [4]
|-
| | 5
| | Major 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor third
| | 276¢ [16]
| | 264¢ [6]
|-
| | 6
| | Minor 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Diminished fourth
| | ''324¢ [16]''
| | ''336¢ [6]''
|-
| | 7
| | Major 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented second
| | 384¢ [18]
| | 372¢ [4]
|-
| | 8
| | Augmented 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

&lt;span style="vertical-align: sub;"&gt;Diminished &lt;/span&gt;5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major third
| | 432¢ [14]
| | 444¢ [8]
|-
| | 9
| | Perfect 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Perfect fourth
| | 492¢ [20]
| | 480¢ [2]
|-
| | 10
| | Augmented 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished fifth
| | 540¢ [12]
| | 552¢ [10]
|-
| | 11
| | Perfect 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented third

Diminished sixth
| | 600¢ [20]
| | 588¢ [1]

612¢ [1]
|-
| | 12
| | Augmented 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented fourth
| | 660¢ [12]
| | 648¢ [10]
|-
| | 13
| | Perfect 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Perfect fifth
| | 708¢ [20]
| | 720¢ [2]
|-
| | 14
| | Augmented 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor sixth
| | 768¢ [14]
| | 756¢ [8]
|-
| | 15
| | Minor 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished seventh
| | 816¢ [18]
| | 828¢ [4]
|-
| | 16
| | Major 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented fifth
| | ''876¢ [16]''
| | ''864¢ [6]''
|-
| | 17
| | Minor 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major sixth
| | 924¢ [16]
| | 936¢ [6]
|-
| | 18
| | Major 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor seventh
| | 984¢ [18]
| | 972¢ [4]
|-
| | 19
| | Minor 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished octave
| | ''1032¢ [14]''
| | ''1044¢ [8]''
|-
| | 20
| | Major 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished second
| | 1092¢ [20]
| | 1080¢ [2]
|-
| | 21
| | Augmented 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major seventh
| | 1140¢ [12]
| | 1152¢ [10]
|-
| | 22
| | 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Octave
| | 1200¢ [22]
| | N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;span style="vertical-align: sub;"&gt;10 &lt;/span&gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [https://en.wikipedia.org/wiki/Augmented-fourths_tuning are tuned in tritones]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian|Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively.
[[Category:100edo]]
[[Category:edo]]
[[Category:todo:add_definition]]</text>
      <sha1>9kkzoemfpmusexj03r6mg83dfiibg91</sha1>
    </revision>
    <revision>
      <id>40775</id>
      <parentid>40774</parentid>
      <timestamp>2019-03-10T22:48:42Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>40775</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7215" sha1="nr7t9bxchiu5xjffd1wn7022cdx3v7z" xml:space="preserve">&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

'''100EDO''' is the [[EDO|equal division of the octave]] into 100 parts of exact 12 cents each. It is closely related to 50EDO, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50EDO 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6EDO]], [[35edo|35EDO]], [[47edo|47EDO]] and [[88edo|88EDO]], 100EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo|12EDO]].

=Scales=
*[[Greeley8]]
*[[Greeley15]]

=100bddd and the 22-note scales=

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo|22EDO]] for [[Diaschismic family|pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.

{| class="wikitable"
|-
| | Steps of 22-note MODMOS
| | Interval name (decatonic)
| | Interval name (superpyth diatonic)
| | Pure interval size [multiplicity]
| | Dog interval size [multiplicity]
|-
| | 1
| | Diminished 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor second
| | 60¢ [12]
| | 48¢ [10]
|-
| | 2
| | Minor 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented seventh
| | 108¢ [20]
| | 120¢ [2]
|-
| | 3
| | Major 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented unison
| | ''168¢ [14]''
| | ''156&lt;span style="line-height: 1.5;"&gt;¢ [8]&lt;/span&gt;''
|-
| | 4
| | Minor 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major second
| | 216¢ [18]
| | 228¢ [4]
|-
| | 5
| | Major 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor third
| | 276¢ [16]
| | 264¢ [6]
|-
| | 6
| | Minor 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Diminished fourth
| | ''324¢ [16]''
| | ''336¢ [6]''
|-
| | 7
| | Major 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented second
| | 384¢ [18]
| | 372¢ [4]
|-
| | 8
| | Augmented 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

&lt;span style="vertical-align: sub;"&gt;Diminished &lt;/span&gt;5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major third
| | 432¢ [14]
| | 444¢ [8]
|-
| | 9
| | Perfect 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Perfect fourth
| | 492¢ [20]
| | 480¢ [2]
|-
| | 10
| | Augmented 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished fifth
| | 540¢ [12]
| | 552¢ [10]
|-
| | 11
| | Perfect 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented third

Diminished sixth
| | 600¢ [20]
| | 588¢ [1]

612¢ [1]
|-
| | 12
| | Augmented 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented fourth
| | 660¢ [12]
| | 648¢ [10]
|-
| | 13
| | Perfect 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Perfect fifth
| | 708¢ [20]
| | 720¢ [2]
|-
| | 14
| | Augmented 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor sixth
| | 768¢ [14]
| | 756¢ [8]
|-
| | 15
| | Minor 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished seventh
| | 816¢ [18]
| | 828¢ [4]
|-
| | 16
| | Major 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented fifth
| | ''876¢ [16]''
| | ''864¢ [6]''
|-
| | 17
| | Minor 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major sixth
| | 924¢ [16]
| | 936¢ [6]
|-
| | 18
| | Major 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor seventh
| | 984¢ [18]
| | 972¢ [4]
|-
| | 19
| | Minor 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished octave
| | ''1032¢ [14]''
| | ''1044¢ [8]''
|-
| | 20
| | Major 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished second
| | 1092¢ [20]
| | 1080¢ [2]
|-
| | 21
| | Augmented 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major seventh
| | 1140¢ [12]
| | 1152¢ [10]
|-
| | 22
| | 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Octave
| | 1200¢ [22]
| | N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;span style="vertical-align: sub;"&gt;10 &lt;/span&gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [https://en.wikipedia.org/wiki/Augmented-fourths_tuning are tuned in tritones]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively.

[[Category:100edo]]
[[Category:edo]]</text>
      <sha1>nr7t9bxchiu5xjffd1wn7022cdx3v7z</sha1>
    </revision>
    <revision>
      <id>54944</id>
      <parentid>40775</parentid>
      <timestamp>2020-12-04T23:13:04Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54944</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7241" sha1="472s5ievxxkwbpp2nk0kkm1r4qxzolm" xml:space="preserve">&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

'''100EDO''' is the [[EDO|equal division of the octave]] into 100 parts of exact 12 cents each. It is closely related to 50EDO, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50EDO 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6EDO]], [[35edo|35EDO]], [[47edo|47EDO]] and [[88edo|88EDO]], 100EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo|12EDO]].

=Scales=
*[[Greeley8]]
*[[Greeley15]]

=100bddd and the 22-note scales=

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo|22EDO]] for [[Diaschismic family|pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.

{| class="wikitable"
|-
| | Steps of 22-note MODMOS
| | Interval name (decatonic)
| | Interval name (superpyth diatonic)
| | Pure interval size [multiplicity]
| | Dog interval size [multiplicity]
|-
| | 1
| | Diminished 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor second
| | 60¢ [12]
| | 48¢ [10]
|-
| | 2
| | Minor 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented seventh
| | 108¢ [20]
| | 120¢ [2]
|-
| | 3
| | Major 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented unison
| | ''168¢ [14]''
| | ''156&lt;span style="line-height: 1.5;"&gt;¢ [8]&lt;/span&gt;''
|-
| | 4
| | Minor 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major second
| | 216¢ [18]
| | 228¢ [4]
|-
| | 5
| | Major 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor third
| | 276¢ [16]
| | 264¢ [6]
|-
| | 6
| | Minor 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Diminished fourth
| | ''324¢ [16]''
| | ''336¢ [6]''
|-
| | 7
| | Major 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented second
| | 384¢ [18]
| | 372¢ [4]
|-
| | 8
| | Augmented 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

&lt;span style="vertical-align: sub;"&gt;Diminished &lt;/span&gt;5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major third
| | 432¢ [14]
| | 444¢ [8]
|-
| | 9
| | Perfect 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Perfect fourth
| | 492¢ [20]
| | 480¢ [2]
|-
| | 10
| | Augmented 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished fifth
| | 540¢ [12]
| | 552¢ [10]
|-
| | 11
| | Perfect 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented third

Diminished sixth
| | 600¢ [20]
| | 588¢ [1]

612¢ [1]
|-
| | 12
| | Augmented 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented fourth
| | 660¢ [12]
| | 648¢ [10]
|-
| | 13
| | Perfect 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Perfect fifth
| | 708¢ [20]
| | 720¢ [2]
|-
| | 14
| | Augmented 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor sixth
| | 768¢ [14]
| | 756¢ [8]
|-
| | 15
| | Minor 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished seventh
| | 816¢ [18]
| | 828¢ [4]
|-
| | 16
| | Major 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented fifth
| | ''876¢ [16]''
| | ''864¢ [6]''
|-
| | 17
| | Minor 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major sixth
| | 924¢ [16]
| | 936¢ [6]
|-
| | 18
| | Major 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor seventh
| | 984¢ [18]
| | 972¢ [4]
|-
| | 19
| | Minor 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished octave
| | ''1032¢ [14]''
| | ''1044¢ [8]''
|-
| | 20
| | Major 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished second
| | 1092¢ [20]
| | 1080¢ [2]
|-
| | 21
| | Augmented 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major seventh
| | 1140¢ [12]
| | 1152¢ [10]
|-
| | 22
| | 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Octave
| | 1200¢ [22]
| | N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;span style="vertical-align: sub;"&gt;10 &lt;/span&gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [https://en.wikipedia.org/wiki/Augmented-fourths_tuning are tuned in tritones]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively.

[[Category:100edo]]
[[Category:Equal divisions of the octave]]</text>
      <sha1>472s5ievxxkwbpp2nk0kkm1r4qxzolm</sha1>
    </revision>
    <revision>
      <id>67280</id>
      <parentid>54944</parentid>
      <timestamp>2021-04-14T19:07:30Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>67280</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7611" sha1="212c530xiwt1t6wjcjonvlu8l9pui9i" xml:space="preserve">&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

'''100EDO''' is the [[EDO|equal division of the octave]] into 100 parts of exact 12 cents each. It is closely related to 50EDO, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50EDO 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6EDO]], [[35edo|35EDO]], [[47edo|47EDO]] and [[88edo|88EDO]], 100EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo|12EDO]].

=Scales=
*[[Greeley8]]
*[[Greeley15]]

=100bddd and the 22-note scales=

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo|22EDO]] for [[Diaschismic family|pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.

{| class="wikitable"
|-
| | Steps of 22-note MODMOS
| | Interval name (decatonic)
| | Interval name (superpyth diatonic)
| | Pure interval size [multiplicity]
| | Dog interval size [multiplicity]
|-
| | 1
| | Diminished 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor second
| | 60¢ [12]
| | 48¢ [10]
|-
| | 2
| | Minor 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented seventh
| | 108¢ [20]
| | 120¢ [2]
|-
| | 3
| | Major 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented unison
| | ''168¢ [14]''
| | ''156&lt;span style="line-height: 1.5;"&gt;¢ [8]&lt;/span&gt;''
|-
| | 4
| | Minor 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major second
| | 216¢ [18]
| | 228¢ [4]
|-
| | 5
| | Major 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor third
| | 276¢ [16]
| | 264¢ [6]
|-
| | 6
| | Minor 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Diminished fourth
| | ''324¢ [16]''
| | ''336¢ [6]''
|-
| | 7
| | Major 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented second
| | 384¢ [18]
| | 372¢ [4]
|-
| | 8
| | Augmented 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

&lt;span style="vertical-align: sub;"&gt;Diminished &lt;/span&gt;5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major third
| | 432¢ [14]
| | 444¢ [8]
|-
| | 9
| | Perfect 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Perfect fourth
| | 492¢ [20]
| | 480¢ [2]
|-
| | 10
| | Augmented 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished fifth
| | 540¢ [12]
| | 552¢ [10]
|-
| | 11
| | Perfect 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented third

Diminished sixth
| | 600¢ [20]
| | 588¢ [1]

612¢ [1]
|-
| | 12
| | Augmented 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented fourth
| | 660¢ [12]
| | 648¢ [10]
|-
| | 13
| | Perfect 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Perfect fifth
| | 708¢ [20]
| | 720¢ [2]
|-
| | 14
| | Augmented 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor sixth
| | 768¢ [14]
| | 756¢ [8]
|-
| | 15
| | Minor 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished seventh
| | 816¢ [18]
| | 828¢ [4]
|-
| | 16
| | Major 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Augmented fifth
| | ''876¢ [16]''
| | ''864¢ [6]''
|-
| | 17
| | Minor 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major sixth
| | 924¢ [16]
| | 936¢ [6]
|-
| | 18
| | Major 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Minor seventh
| | 984¢ [18]
| | 972¢ [4]
|-
| | 19
| | Minor 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished octave
| | ''1032¢ [14]''
| | ''1044¢ [8]''
|-
| | 20
| | Major 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished second
| | 1092¢ [20]
| | 1080¢ [2]
|-
| | 21
| | Augmented 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Major seventh
| | 1140¢ [12]
| | 1152¢ [10]
|-
| | 22
| | 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Octave
| | 1200¢ [22]
| | N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;span style="vertical-align: sub;"&gt;10 &lt;/span&gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [https://en.wikipedia.org/wiki/Augmented-fourths_tuning are tuned in tritones]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&amp;[[34edo|34d]] temperaments is also usable. On the other hand, the “rough” alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&amp;[[34edo|34d]] temperaments up to 58d is also usable.

[[Category:100edo]]
[[Category:Equal divisions of the octave]]</text>
      <sha1>212c530xiwt1t6wjcjonvlu8l9pui9i</sha1>
    </revision>
    <revision>
      <id>67988</id>
      <parentid>67280</parentid>
      <timestamp>2021-04-20T01:11:56Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Scales */</comment>
      <origin>67988</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9123" sha1="9ezdf9lp81s0v1ybr8vntbjbdx0owua" xml:space="preserve">&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

'''100EDO''' is the [[EDO|equal division of the octave]] into 100 parts of exact 12 cents each. It is closely related to 50EDO, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50EDO 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6EDO]], [[35edo|35EDO]], [[47edo|47EDO]] and [[88edo|88EDO]], 100EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo|12EDO]].

=Scales=
*[[Greeley8]]
*[[Greeley15]]
Since 100edo has a step of 12 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 100edo
! Tones
!Pattern
! L:s
|-
|5
|[[5edo]]
|equal
|-
| 6
|[[4L 2s]]
|17:16
|-
|7
|[[2L 5s]]
|15:14
|-
|8
|[[4L 4s]]
|13:12
|-
|9
|[[1L 8s]]
|12:11
|-
| 10
|[[10edo]]
|equal
|-
|11
|[[1L 10s]]
|10:9
|-
|12
|[[4L 8s]]
|9:8
|-
|13
|[[9L 4s]]
| rowspan="2" |8:7
|-
|14
|[[2L 12s]]
|-
|15
|[[10L 5s]]
| rowspan="2" |7:6
|-
|16
| 4L 12s
|-
| 17
|[[15L 2s]]
| rowspan="3" |6:5
|-
|18
| 10L 8s
|-
|19
|[[5L 14s]]
|-
|20
|[[20edo]]
|equal
|-
|21
|16L 5s
| rowspan="4" |5:4
|-
| 22
|12L 10s
|-
|23
|8L 15s
|-
|24
|4L 20s
|-
|25
|[[25edo]]
|equal
|-
|26
|22L 4s
| rowspan="8" |4:3
|-
|27
|19L 8s
|-
| 28
|16L 12s
|-
|29
|13L 16s
|-
|30
|10L 20s
|-
|31
|7L 24s
|-
| 32
|4L 28s
|-
|33
|1L 32s
|-
|34
|32L 2s
| rowspan="16" | 3:2
|-
|35
|30L 5s
|-
|36
|28L 8s
|-
|37
|26L 11s
|-
|38
| 24L 14s
|-
|39
| 22L 17s
|-
|40
| 20L 20s
|-
|41
|18L 23s
|-
|42
|16L 26s
|-
| 43
| 14L 29s
|-
| 44
|12L 32s
|-
| 45
|10L 35s
|-
|46
|8L 38s
|-
|47
|6L 41s
|-
|48
|4L 44s
|-
|49
|2L 47s
|-
|50
|[[50edo]]
| equal
|-
| 51
|49L 2s
| rowspan="29" |2:1
|-
|52
|48L 4s
|-
|53
|47L 6s
|-
|54
|46L 8s
|-
| 55
| 45L 10s
|-
|56
|44L 12s
|-
| 57
| 43L 14s
|-
|58
|42L 16s
|-
|59
|41L 18s
|-
|60
|40L 20s
|-
|61
|39L 22s
|-
| 62
|38L 24s
|-
|63
|37L 26s
|-
|64
|36L 28s
|-
| 65
|35L 30s
|-
| 66
|34L 32s
|-
|67
|33L 34s
|-
|68
|32L 36s
|-
|69
|31L 38s
|-
|70
|30L 40s
|-
|71
|29L 42s
|-
|72
|28L 44s
|-
|73
|27L 46s
|-
|74
|26L 48s
|-
|75
|25L 50s
|-
|76
| 24L 52s
|-
|77
| 23L 54s
|-
|78
|22L 56s
|-
|79
|21L 58s
|}
= 100bddd and the 22-note scales=

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo|22EDO]] for [[Diaschismic family|pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.

{| class="wikitable"
|-
| |Steps of 22-note MODMOS
| |Interval name (decatonic)
| |Interval name (superpyth diatonic)
| |Pure interval size [multiplicity]
| |Dog interval size [multiplicity]
|-
| |1
| |Diminished 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Minor second
| |60¢ [12]
| |48¢ [10]
|-
| | 2
| |Minor 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Augmented seventh
| |108¢ [20]
| |120¢ [2]
|-
| |3
| |Major 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Augmented unison
| |''168¢ [14]''
| |''156&lt;span style="line-height: 1.5;"&gt;¢ [8]&lt;/span&gt;''
|-
| |4
| |Minor 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Major second
| | 216¢ [18]
| |228¢ [4]
|-
| |5
| |Major 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Minor third
| |276¢ [16]
| |264¢ [6]
|-
| |6
| |Minor 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Diminished fourth
| |''324¢ [16]''
| |''336¢ [6]''
|-
| |7
| |Major 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Augmented second
| |384¢ [18]
| |372¢ [4]
|-
| |8
| |Augmented 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

&lt;span style="vertical-align: sub;"&gt;Diminished &lt;/span&gt;5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Major third
| |432¢ [14]
| |444¢ [8]
|-
| |9
| |Perfect 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Perfect fourth
| |492¢ [20]
| |480¢ [2]
|-
| |10
| |Augmented 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished fifth
| |540¢ [12]
| |552¢ [10]
|-
| |11
| |Perfect 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Augmented third

Diminished sixth
| |600¢ [20]
| |588¢ [1]

612¢ [1]
|-
| |12
| |Augmented 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Augmented fourth
| |660¢ [12]
| |648¢ [10]
|-
| |13
| |Perfect 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Perfect fifth
| |708¢ [20]
| |720¢ [2]
|-
| |14
| |Augmented 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Minor sixth
| |768¢ [14]
| |756¢ [8]
|-
| |15
| |Minor 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished seventh
| |816¢ [18]
| |828¢ [4]
|-
| |16
| |Major 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Augmented fifth
| |''876¢ [16]''
| |''864¢ [6]''
|-
| | 17
| |Minor 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Major sixth
| |924¢ [16]
| |936¢ [6]
|-
| |18
| |Major 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Minor seventh
| |984¢ [18]
| |972¢ [4]
|-
| |19
| | Minor 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Diminished octave
| |''1032¢ [14]''
| |''1044¢ [8]''
|-
| |20
| | Major 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Diminished second
| |1092¢ [20]
| |1080¢ [2]
|-
| |21
| |Augmented 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Major seventh
| |1140¢ [12]
| |1152¢ [10]
|-
| |22
| |11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Octave
| |1200¢ [22]
| |N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;span style="vertical-align: sub;"&gt;10 &lt;/span&gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[wikipedia:Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&amp;[[34edo|34d]] temperaments is also usable. On the other hand, the “rough” alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&amp;[[34edo|34d]] temperaments up to 58d is also usable.

[[Category:100edo]]
[[Category:Equal divisions of the octave]]</text>
      <sha1>9ezdf9lp81s0v1ybr8vntbjbdx0owua</sha1>
    </revision>
    <revision>
      <id>68170</id>
      <parentid>67988</parentid>
      <timestamp>2021-04-21T21:16:01Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* 100bddd and the 22-note scales */</comment>
      <origin>68170</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9753" sha1="q2yjzyxakoj492whymhdehfirswop29" xml:space="preserve">&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

'''100EDO''' is the [[EDO|equal division of the octave]] into 100 parts of exact 12 cents each. It is closely related to 50EDO, but the patent vals differ on the mapping for 7. It tempers out 6144/6125 in the 7-limit, 99/98 and 441/440 in the 11-limit and 144/143 in the 13-limit, and like 50EDO 81/80 in the 5-limit. It provides the optimal patent val for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6EDO]], [[35edo|35EDO]], [[47edo|47EDO]] and [[88edo|88EDO]], 100EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo|12EDO]].

=Scales=
*[[Greeley8]]
*[[Greeley15]]
Since 100edo has a step of 12 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 100edo
! Tones
!Pattern
! L:s
|-
|5
|[[5edo]]
|equal
|-
| 6
|[[4L 2s]]
|17:16
|-
|7
|[[2L 5s]]
|15:14
|-
|8
|[[4L 4s]]
|13:12
|-
|9
|[[1L 8s]]
|12:11
|-
| 10
|[[10edo]]
|equal
|-
|11
|[[1L 10s]]
|10:9
|-
|12
|[[4L 8s]]
|9:8
|-
|13
|[[9L 4s]]
| rowspan="2" |8:7
|-
|14
|[[2L 12s]]
|-
|15
|[[10L 5s]]
| rowspan="2" |7:6
|-
|16
| 4L 12s
|-
| 17
|[[15L 2s]]
| rowspan="3" |6:5
|-
|18
| 10L 8s
|-
|19
|[[5L 14s]]
|-
|20
|[[20edo]]
|equal
|-
|21
|16L 5s
| rowspan="4" |5:4
|-
| 22
|12L 10s
|-
|23
|8L 15s
|-
|24
|4L 20s
|-
|25
|[[25edo]]
|equal
|-
|26
|22L 4s
| rowspan="8" |4:3
|-
|27
|19L 8s
|-
| 28
|16L 12s
|-
|29
|13L 16s
|-
|30
|10L 20s
|-
|31
|7L 24s
|-
| 32
|4L 28s
|-
|33
|1L 32s
|-
|34
|32L 2s
| rowspan="16" | 3:2
|-
|35
|30L 5s
|-
|36
|28L 8s
|-
|37
|26L 11s
|-
|38
| 24L 14s
|-
|39
| 22L 17s
|-
|40
| 20L 20s
|-
|41
|18L 23s
|-
|42
|16L 26s
|-
| 43
| 14L 29s
|-
| 44
|12L 32s
|-
| 45
|10L 35s
|-
|46
|8L 38s
|-
|47
|6L 41s
|-
|48
|4L 44s
|-
|49
|2L 47s
|-
|50
|[[50edo]]
| equal
|-
| 51
|49L 2s
| rowspan="29" |2:1
|-
|52
|48L 4s
|-
|53
|47L 6s
|-
|54
|46L 8s
|-
| 55
| 45L 10s
|-
|56
|44L 12s
|-
| 57
| 43L 14s
|-
|58
|42L 16s
|-
|59
|41L 18s
|-
|60
|40L 20s
|-
|61
|39L 22s
|-
| 62
|38L 24s
|-
|63
|37L 26s
|-
|64
|36L 28s
|-
| 65
|35L 30s
|-
| 66
|34L 32s
|-
|67
|33L 34s
|-
|68
|32L 36s
|-
|69
|31L 38s
|-
|70
|30L 40s
|-
|71
|29L 42s
|-
|72
|28L 44s
|-
|73
|27L 46s
|-
|74
|26L 48s
|-
|75
|25L 50s
|-
|76
| 24L 52s
|-
|77
| 23L 54s
|-
|78
|22L 56s
|-
|79
|21L 58s
|}
= 100bddd and the 22-note scales=

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo|22EDO]] for [[Diaschismic family|pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12edo.

{| class="wikitable"
|-
| |Steps of 22-note MODMOS
| |Interval name (decatonic)
| |Interval name (superpyth diatonic)
| |Pure interval size [multiplicity] &lt;br /&gt; Difference from 22 edo
| |Dog interval size [multiplicity] &lt;br /&gt; Difference from 22 edo
|-
| |1
| |Diminished 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Minor second
| |60¢ [12] &lt;br /&gt;5.4545¢
| |48¢ [10] &lt;br /&gt;-6.5455¢
|-
| | 2
| |Minor 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Augmented seventh
| |108¢ [20] &lt;br /&gt;-1.091¢
| |120¢ [2] &lt;br /&gt;10.909¢
|-
| |3
| |Major 2nd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Augmented unison
| |''168¢ [14]'' &lt;br /&gt;4.364¢
| |''156¢ [8]'' &lt;br /&gt;-7.636¢
|-
| |4
| |Minor 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Major second
| | 216¢ [18] &lt;br /&gt;-2.182¢
| |228¢ [4] &lt;br /&gt;9.818¢
|-
| |5
| |Major 3rd&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Minor third
| |276¢ [16] &lt;br /&gt;3.273¢
| |264¢ [6] &lt;br /&gt;-8.727¢
|-
| |6
| |Minor 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Diminished fourth
| |''324¢ [16]''&lt;br /&gt;-3.273¢
| |''336¢ [6]''&lt;br /&gt;8.727¢
|-
| |7
| |Major 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Augmented second
| |384¢ [18] &lt;br /&gt;2.182¢
| |372¢ [4] &lt;br /&gt;-9.818¢
|-
| |8
| |Augmented 4th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

&lt;span style="vertical-align: sub;"&gt;Diminished &lt;/span&gt;5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Major third
| |432¢ [14] &lt;br /&gt;-4.364¢
| |444¢ [8] &lt;br /&gt;7.636¢
|-
| |9
| |Perfect 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Perfect fourth
| |492¢ [20] &lt;br /&gt;1.091¢
| |480¢ [2] &lt;br /&gt;-10.909¢
|-
| |10
| |Augmented 5th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished fifth
| |540¢ [12] &lt;br /&gt;-5.4545¢
| |552¢ [10] &lt;br /&gt;6.5455¢
|-
| |11
| |Perfect 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Augmented third

Diminished sixth
| |600¢ [20]
| |588¢ [1] &lt;br /&gt;-12¢

612¢ [1] &lt;br /&gt;12¢
|-
| |12
| |Augmented 6th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Augmented fourth
| |660¢ [12] &lt;br /&gt;6.5455¢
| |648¢ [10] &lt;br /&gt;-5.4545¢
|-
| |13
| |Perfect 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Perfect fifth
| |708¢ [20] &lt;br /&gt;-1.091¢
| |720¢ [2] &lt;br /&gt;10.909¢
|-
| |14
| |Augmented 7th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Minor sixth
| |768¢ [14] &lt;br /&gt;4.364¢
| |756¢ [8] &lt;br /&gt;-7.636¢
|-
| |15
| |Minor 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| | Diminished seventh
| |816¢ [18] &lt;br /&gt;-2.182¢
| |828¢ [4] &lt;br /&gt;9.818¢
|-
| |16
| |Major 8th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Augmented fifth
| |''876¢ [16]''&lt;br /&gt;3.273¢
| |''864¢ [6]''&lt;br /&gt;-8.727¢
|-
| | 17
| |Minor 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Major sixth
| |924¢ [16] &lt;br /&gt;-3.273¢
| |936¢ [6] &lt;br /&gt;8.727¢
|-
| |18
| |Major 9th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Minor seventh
| |984¢ [18] &lt;br /&gt;2.182¢
| |972¢ [4] &lt;br /&gt;-9.818¢
|-
| |19
| | Minor 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Diminished octave
| |''1032¢ [14]''&lt;br /&gt;-4.364¢
| |''1044¢ [8]''&lt;br /&gt;7.636¢
|-
| |20
| | Major 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Diminished second
| |1092¢ [20] &lt;br /&gt;1.091¢
| |1080¢ [2] &lt;br /&gt;-10.909¢
|-
| |21
| |Augmented 10th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;

Diminished 11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Major seventh
| |1140¢ [12] &lt;br /&gt;-5.4545¢
| |1152¢ [10] &lt;br /&gt;6.5455¢
|-
| |22
| |11th&lt;span style="vertical-align: sub;"&gt;10&lt;/span&gt;
| |Octave
| |1200¢ [22]
| |N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;span style="vertical-align: sub;"&gt;10 &lt;/span&gt;is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[wikipedia:Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&amp;[[34edo|34d]] temperaments is also usable. On the other hand, the “rough” alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&amp;[[34edo|34d]] temperaments up to 58d is also usable.

[[Category:100edo]]
[[Category:Equal divisions of the octave]]</text>
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    </revision>
    <revision>
      <id>71933</id>
      <parentid>68170</parentid>
      <timestamp>2021-06-05T14:06:48Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cleanup wiki markup uppercase EDO</comment>
      <origin>71933</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8885" sha1="q1s09eizx7drhfh33mk1fuuys4v560x" xml:space="preserve">&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

'''100 EDO''' is the [[equal division of the octave]] into 100 parts of exact 12 cents each. It is closely related to [[50 EDO]], but the [[patent val]]s differ on the mapping for 7. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50 EDO [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6 EDO]], [[35 EDO]], [[47 EDO]] and [[88 EDO]], 100 EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12 EDO]].

== Scales ==

* [[Greeley8]]
* [[Greeley15]]

Since 100 EDO has a step of 12 cents, it also allows one to use its MOS scales as circulating temperaments.

{| class="wikitable"
|+Circulating temperaments in 100 EDO
! Tones
! Pattern
! L:s
|-
| 5
| [[5 EDO]]
| equal
|-
| 6
| [[4L 2s]]
| 17:16
|-
| 7
| [[2L 5s]]
| 15:14
|-
| 8
| [[4L 4s]]
| 13:12
|-
| 9
| [[1L 8s]]
| 12:11
|-
| 10
| [[10 EDO]]
| equal
|-
| 11
| [[1L 10s]]
| 10:9
|-
| 12
| [[4L 8s]]
| 9:8
|-
| 13
| [[9L 4s]]
| rowspan="2" |8:7
|-
| 14
| [[2L 12s]]
|-
| 15
| [[10L 5s]]
| rowspan="2" |7:6
|-
| 16
| 4L 12s
|-
| 17
| [[15L 2s]]
| rowspan="3" |6:5
|-
| 18
| 10L 8s
|-
| 19
| [[5L 14s]]
|-
| 20
| [[20 EDO]]
| equal
|-
| 21
| 16L 5s
| rowspan="4" |5:4
|-
| 22
| 12L 10s
|-
| 23
| 8L 15s
|-
| 24
| 4L 20s
|-
| 25
| [[25 EDO]]
| equal
|-
| 26
| 22L 4s
| rowspan="8" |4:3
|-
| 27
| 19L 8s
|-
| 28
| 16L 12s
|-
| 29
| 13L 16s
|-
| 30
| 10L 20s
|-
| 31
| 7L 24s
|-
| 32
| 4L 28s
|-
| 33
| 1L 32s
|-
| 34
| 32L 2s
| rowspan="16" | 3:2
|-
| 35
| 30L 5s
|-
| 36
| 28L 8s
|-
| 37
| 26L 11s
|-
| 38
| 24L 14s
|-
| 39
| 22L 17s
|-
| 40
| 20L 20s
|-
| 41
| 18L 23s
|-
| 42
| 16L 26s
|-
| 43
| 14L 29s
|-
| 44
| 12L 32s
|-
| 45
| 10L 35s
|-
| 46
| 8L 38s
|-
| 47
| 6L 41s
|-
| 48
| 4L 44s
|-
| 49
| 2L 47s
|-
| 50
| [[50 EDO]]
| equal
|-
| 51
| 49L 2s
| rowspan="29" |2:1
|-
| 52
| 48L 4s
|-
| 53
| 47L 6s
|-
| 54
| 46L 8s
|-
| 55
| 45L 10s
|-
| 56
| 44L 12s
|-
| 57
| 43L 14s
|-
| 58
| 42L 16s
|-
| 59
| 41L 18s
|-
| 60
| 40L 20s
|-
| 61
| 39L 22s
|-
| 62
| 38L 24s
|-
| 63
| 37L 26s
|-
| 64
| 36L 28s
|-
| 65
| 35L 30s
|-
| 66
| 34L 32s
|-
| 67
| 33L 34s
|-
| 68
| 32L 36s
|-
| 69
| 31L 38s
|-
| 70
| 30L 40s
|-
| 71
| 29L 42s
|-
| 72
| 28L 44s
|-
| 73
| 27L 46s
|-
| 74
| 26L 48s
|-
| 75
| 25L 50s
|-
| 76
| 24L 52s
|-
| 77
| 23L 54s
|-
| 78
| 22L 56s
|-
| 79
| 21L 58s
|}

== 100bddd and the 22-note scales ==

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22 EDO]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12 EDO counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12 Edo.

{| class="wikitable"
|-
| Steps of 22-note MODMOS
| Interval name (decatonic)
| Interval name (superpyth diatonic)
| Pure interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
| Dog interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60¢ [12] &lt;br /&gt;5.4545¢
| 48¢ [10] &lt;br /&gt;-6.5455¢
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108¢ [20] &lt;br /&gt;-1.091¢
| 120¢ [2] &lt;br /&gt;10.909¢
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168¢ [14]'' &lt;br /&gt;4.364¢
| ''156¢ [8]'' &lt;br /&gt;-7.636¢
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216¢ [18] &lt;br /&gt;-2.182¢
| 228¢ [4] &lt;br /&gt;9.818¢
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276¢ [16] &lt;br /&gt;3.273¢
| 264¢ [6] &lt;br /&gt;-8.727¢
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324¢ [16]''&lt;br /&gt;-3.273¢
| ''336¢ [6]''&lt;br /&gt;8.727¢
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384¢ [18] &lt;br /&gt;2.182¢
| 372¢ [4] &lt;br /&gt;-9.818¢
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432¢ [14] &lt;br /&gt;-4.364¢
| 444¢ [8] &lt;br /&gt;7.636¢
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492¢ [20] &lt;br /&gt;1.091¢
| 480¢ [2] &lt;br /&gt;-10.909¢
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540¢ [12] &lt;br /&gt;-5.4545¢
| 552¢ [10] &lt;br /&gt;6.5455¢
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600¢ [20]
| 588¢ [1] &lt;br&gt;-12¢ &lt;br&gt; 612¢ [1] &lt;br&gt;12¢
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660¢ [12] &lt;br /&gt;6.5455¢
| 648¢ [10] &lt;br /&gt;-5.4545¢
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708¢ [20] &lt;br /&gt;-1.091¢
| 720¢ [2] &lt;br /&gt;10.909¢
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768¢ [14] &lt;br /&gt;4.364¢
| 756¢ [8] &lt;br /&gt;-7.636¢
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816¢ [18] &lt;br /&gt;-2.182¢
| 828¢ [4] &lt;br /&gt;9.818¢
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876¢ [16]''&lt;br /&gt;3.273¢
| ''864¢ [6]''&lt;br /&gt;-8.727¢
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924¢ [16] &lt;br /&gt;-3.273¢
| 936¢ [6] &lt;br /&gt;8.727¢
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984¢ [18] &lt;br /&gt;2.182¢
| 972¢ [4] &lt;br /&gt;-9.818¢
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032¢ [14]''&lt;br /&gt;-4.364¢
| ''1044¢ [8]''&lt;br /&gt;7.636¢
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092¢ [20] &lt;br /&gt;1.091¢
| 1080¢ [2] &lt;br /&gt;-10.909¢
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140¢ [12] &lt;br /&gt;-5.4545¢
| 1152¢ [10] &lt;br /&gt;6.5455¢
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200¢ [22]
| N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[wikipedia:Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22 EDO for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&amp;[[34edo|34d]] temperaments is also usable. On the other hand, the “rough” alternatives to 22 EDO for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&amp;[[34edo|34d]] temperaments up to 58d is also usable.

[[Category:100edo]]
[[Category:Equal divisions of the octave]]</text>
      <sha1>q1s09eizx7drhfh33mk1fuuys4v560x</sha1>
    </revision>
    <revision>
      <id>92909</id>
      <parentid>71933</parentid>
      <timestamp>2022-07-02T05:50:33Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92909</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8913" sha1="1s01xoky3o36jlq8km92yo6ufor3ydn" xml:space="preserve">&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

'''100 EDO''' is the [[equal division of the octave]] into 100 parts of exact 12 cents each. It is closely related to [[50 EDO]], but the [[patent val]]s differ on the mapping for 7. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50 EDO [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6 EDO]], [[35 EDO]], [[47 EDO]] and [[88 EDO]], 100 EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12 EDO]].

== Scales ==

* [[Greeley8]]
* [[Greeley15]]

Since 100 EDO has a step of 12 cents, it also allows one to use its MOS scales as circulating temperaments.

{| class="wikitable"
|+Circulating temperaments in 100 EDO
! Tones
! Pattern
! L:s
|-
| 5
| [[5 EDO]]
| equal
|-
| 6
| [[4L 2s]]
| 17:16
|-
| 7
| [[2L 5s]]
| 15:14
|-
| 8
| [[4L 4s]]
| 13:12
|-
| 9
| [[1L 8s]]
| 12:11
|-
| 10
| [[10 EDO]]
| equal
|-
| 11
| [[1L 10s]]
| 10:9
|-
| 12
| [[4L 8s]]
| 9:8
|-
| 13
| [[9L 4s]]
| rowspan="2" |8:7
|-
| 14
| [[2L 12s]]
|-
| 15
| [[10L 5s]]
| rowspan="2" |7:6
|-
| 16
| 4L 12s
|-
| 17
| [[15L 2s]]
| rowspan="3" |6:5
|-
| 18
| 10L 8s
|-
| 19
| [[5L 14s]]
|-
| 20
| [[20 EDO]]
| equal
|-
| 21
| 16L 5s
| rowspan="4" |5:4
|-
| 22
| 12L 10s
|-
| 23
| 8L 15s
|-
| 24
| 4L 20s
|-
| 25
| [[25 EDO]]
| equal
|-
| 26
| 22L 4s
| rowspan="8" |4:3
|-
| 27
| 19L 8s
|-
| 28
| 16L 12s
|-
| 29
| 13L 16s
|-
| 30
| 10L 20s
|-
| 31
| 7L 24s
|-
| 32
| 4L 28s
|-
| 33
| 1L 32s
|-
| 34
| 32L 2s
| rowspan="16" | 3:2
|-
| 35
| 30L 5s
|-
| 36
| 28L 8s
|-
| 37
| 26L 11s
|-
| 38
| 24L 14s
|-
| 39
| 22L 17s
|-
| 40
| 20L 20s
|-
| 41
| 18L 23s
|-
| 42
| 16L 26s
|-
| 43
| 14L 29s
|-
| 44
| 12L 32s
|-
| 45
| 10L 35s
|-
| 46
| 8L 38s
|-
| 47
| 6L 41s
|-
| 48
| 4L 44s
|-
| 49
| 2L 47s
|-
| 50
| [[50 EDO]]
| equal
|-
| 51
| 49L 2s
| rowspan="29" |2:1
|-
| 52
| 48L 4s
|-
| 53
| 47L 6s
|-
| 54
| 46L 8s
|-
| 55
| 45L 10s
|-
| 56
| 44L 12s
|-
| 57
| 43L 14s
|-
| 58
| 42L 16s
|-
| 59
| 41L 18s
|-
| 60
| 40L 20s
|-
| 61
| 39L 22s
|-
| 62
| 38L 24s
|-
| 63
| 37L 26s
|-
| 64
| 36L 28s
|-
| 65
| 35L 30s
|-
| 66
| 34L 32s
|-
| 67
| 33L 34s
|-
| 68
| 32L 36s
|-
| 69
| 31L 38s
|-
| 70
| 30L 40s
|-
| 71
| 29L 42s
|-
| 72
| 28L 44s
|-
| 73
| 27L 46s
|-
| 74
| 26L 48s
|-
| 75
| 25L 50s
|-
| 76
| 24L 52s
|-
| 77
| 23L 54s
|-
| 78
| 22L 56s
|-
| 79
| 21L 58s
|}

== 100bddd and the 22-note scales ==

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22 EDO]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12 EDO counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12 Edo.

{| class="wikitable"
|-
| Steps of 22-note MODMOS
| Interval name (decatonic)
| Interval name (superpyth diatonic)
| Pure interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
| Dog interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60¢ [12] &lt;br /&gt;5.4545¢
| 48¢ [10] &lt;br /&gt;-6.5455¢
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108¢ [20] &lt;br /&gt;-1.091¢
| 120¢ [2] &lt;br /&gt;10.909¢
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168¢ [14]'' &lt;br /&gt;4.364¢
| ''156¢ [8]'' &lt;br /&gt;-7.636¢
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216¢ [18] &lt;br /&gt;-2.182¢
| 228¢ [4] &lt;br /&gt;9.818¢
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276¢ [16] &lt;br /&gt;3.273¢
| 264¢ [6] &lt;br /&gt;-8.727¢
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324¢ [16]''&lt;br /&gt;-3.273¢
| ''336¢ [6]''&lt;br /&gt;8.727¢
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384¢ [18] &lt;br /&gt;2.182¢
| 372¢ [4] &lt;br /&gt;-9.818¢
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432¢ [14] &lt;br /&gt;-4.364¢
| 444¢ [8] &lt;br /&gt;7.636¢
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492¢ [20] &lt;br /&gt;1.091¢
| 480¢ [2] &lt;br /&gt;-10.909¢
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540¢ [12] &lt;br /&gt;-5.4545¢
| 552¢ [10] &lt;br /&gt;6.5455¢
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600¢ [20]
| 588¢ [1] &lt;br&gt;-12¢ &lt;br&gt; 612¢ [1] &lt;br&gt;12¢
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660¢ [12] &lt;br /&gt;6.5455¢
| 648¢ [10] &lt;br /&gt;-5.4545¢
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708¢ [20] &lt;br /&gt;-1.091¢
| 720¢ [2] &lt;br /&gt;10.909¢
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768¢ [14] &lt;br /&gt;4.364¢
| 756¢ [8] &lt;br /&gt;-7.636¢
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816¢ [18] &lt;br /&gt;-2.182¢
| 828¢ [4] &lt;br /&gt;9.818¢
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876¢ [16]''&lt;br /&gt;3.273¢
| ''864¢ [6]''&lt;br /&gt;-8.727¢
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924¢ [16] &lt;br /&gt;-3.273¢
| 936¢ [6] &lt;br /&gt;8.727¢
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984¢ [18] &lt;br /&gt;2.182¢
| 972¢ [4] &lt;br /&gt;-9.818¢
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032¢ [14]''&lt;br /&gt;-4.364¢
| ''1044¢ [8]''&lt;br /&gt;7.636¢
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092¢ [20] &lt;br /&gt;1.091¢
| 1080¢ [2] &lt;br /&gt;-10.909¢
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140¢ [12] &lt;br /&gt;-5.4545¢
| 1152¢ [10] &lt;br /&gt;6.5455¢
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200¢ [22]
| N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[wikipedia:Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22 EDO for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&amp;[[34edo|34d]] temperaments is also usable. On the other hand, the “rough” alternatives to 22 EDO for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&amp;[[34edo|34d]] temperaments up to 58d is also usable.

[[Category:100edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>1s01xoky3o36jlq8km92yo6ufor3ydn</sha1>
    </revision>
    <revision>
      <id>96006</id>
      <parentid>92909</parentid>
      <timestamp>2022-10-03T03:18:02Z</timestamp>
      <contributor>
        <username>Royalmilktea</username>
        <id>285</id>
      </contributor>
      <comment>prime harmonics table</comment>
      <origin>96006</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8963" sha1="39y3sgh01o7zlhdbf122sj8uzvi0xto" xml:space="preserve">&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

'''100 EDO''' is the [[equal division of the octave]] into 100 parts of exact 12 cents each. It is closely related to [[50 EDO]], but the [[patent val]]s differ on the mapping for 7. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50 EDO [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6 EDO]], [[35 EDO]], [[47 EDO]] and [[88 EDO]], 100 EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12 EDO]].
{{harmonics in equal|100|start=2|intervals=prime}}
== Scales ==

* [[Greeley8]]
* [[Greeley15]]

Since 100 EDO has a step of 12 cents, it also allows one to use its MOS scales as circulating temperaments.

{| class="wikitable"
|+Circulating temperaments in 100 EDO
! Tones
! Pattern
! L:s
|-
| 5
| [[5 EDO]]
| equal
|-
| 6
| [[4L 2s]]
| 17:16
|-
| 7
| [[2L 5s]]
| 15:14
|-
| 8
| [[4L 4s]]
| 13:12
|-
| 9
| [[1L 8s]]
| 12:11
|-
| 10
| [[10 EDO]]
| equal
|-
| 11
| [[1L 10s]]
| 10:9
|-
| 12
| [[4L 8s]]
| 9:8
|-
| 13
| [[9L 4s]]
| rowspan="2" |8:7
|-
| 14
| [[2L 12s]]
|-
| 15
| [[10L 5s]]
| rowspan="2" |7:6
|-
| 16
| 4L 12s
|-
| 17
| [[15L 2s]]
| rowspan="3" |6:5
|-
| 18
| 10L 8s
|-
| 19
| [[5L 14s]]
|-
| 20
| [[20 EDO]]
| equal
|-
| 21
| 16L 5s
| rowspan="4" |5:4
|-
| 22
| 12L 10s
|-
| 23
| 8L 15s
|-
| 24
| 4L 20s
|-
| 25
| [[25 EDO]]
| equal
|-
| 26
| 22L 4s
| rowspan="8" |4:3
|-
| 27
| 19L 8s
|-
| 28
| 16L 12s
|-
| 29
| 13L 16s
|-
| 30
| 10L 20s
|-
| 31
| 7L 24s
|-
| 32
| 4L 28s
|-
| 33
| 1L 32s
|-
| 34
| 32L 2s
| rowspan="16" | 3:2
|-
| 35
| 30L 5s
|-
| 36
| 28L 8s
|-
| 37
| 26L 11s
|-
| 38
| 24L 14s
|-
| 39
| 22L 17s
|-
| 40
| 20L 20s
|-
| 41
| 18L 23s
|-
| 42
| 16L 26s
|-
| 43
| 14L 29s
|-
| 44
| 12L 32s
|-
| 45
| 10L 35s
|-
| 46
| 8L 38s
|-
| 47
| 6L 41s
|-
| 48
| 4L 44s
|-
| 49
| 2L 47s
|-
| 50
| [[50 EDO]]
| equal
|-
| 51
| 49L 2s
| rowspan="29" |2:1
|-
| 52
| 48L 4s
|-
| 53
| 47L 6s
|-
| 54
| 46L 8s
|-
| 55
| 45L 10s
|-
| 56
| 44L 12s
|-
| 57
| 43L 14s
|-
| 58
| 42L 16s
|-
| 59
| 41L 18s
|-
| 60
| 40L 20s
|-
| 61
| 39L 22s
|-
| 62
| 38L 24s
|-
| 63
| 37L 26s
|-
| 64
| 36L 28s
|-
| 65
| 35L 30s
|-
| 66
| 34L 32s
|-
| 67
| 33L 34s
|-
| 68
| 32L 36s
|-
| 69
| 31L 38s
|-
| 70
| 30L 40s
|-
| 71
| 29L 42s
|-
| 72
| 28L 44s
|-
| 73
| 27L 46s
|-
| 74
| 26L 48s
|-
| 75
| 25L 50s
|-
| 76
| 24L 52s
|-
| 77
| 23L 54s
|-
| 78
| 22L 56s
|-
| 79
| 21L 58s
|}

== 100bddd and the 22-note scales ==

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22 EDO]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12 EDO counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12 Edo.

{| class="wikitable"
|-
| Steps of 22-note MODMOS
| Interval name (decatonic)
| Interval name (superpyth diatonic)
| Pure interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
| Dog interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60¢ [12] &lt;br /&gt;5.4545¢
| 48¢ [10] &lt;br /&gt;-6.5455¢
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108¢ [20] &lt;br /&gt;-1.091¢
| 120¢ [2] &lt;br /&gt;10.909¢
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168¢ [14]'' &lt;br /&gt;4.364¢
| ''156¢ [8]'' &lt;br /&gt;-7.636¢
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216¢ [18] &lt;br /&gt;-2.182¢
| 228¢ [4] &lt;br /&gt;9.818¢
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276¢ [16] &lt;br /&gt;3.273¢
| 264¢ [6] &lt;br /&gt;-8.727¢
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324¢ [16]''&lt;br /&gt;-3.273¢
| ''336¢ [6]''&lt;br /&gt;8.727¢
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384¢ [18] &lt;br /&gt;2.182¢
| 372¢ [4] &lt;br /&gt;-9.818¢
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432¢ [14] &lt;br /&gt;-4.364¢
| 444¢ [8] &lt;br /&gt;7.636¢
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492¢ [20] &lt;br /&gt;1.091¢
| 480¢ [2] &lt;br /&gt;-10.909¢
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540¢ [12] &lt;br /&gt;-5.4545¢
| 552¢ [10] &lt;br /&gt;6.5455¢
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600¢ [20]
| 588¢ [1] &lt;br&gt;-12¢ &lt;br&gt; 612¢ [1] &lt;br&gt;12¢
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660¢ [12] &lt;br /&gt;6.5455¢
| 648¢ [10] &lt;br /&gt;-5.4545¢
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708¢ [20] &lt;br /&gt;-1.091¢
| 720¢ [2] &lt;br /&gt;10.909¢
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768¢ [14] &lt;br /&gt;4.364¢
| 756¢ [8] &lt;br /&gt;-7.636¢
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816¢ [18] &lt;br /&gt;-2.182¢
| 828¢ [4] &lt;br /&gt;9.818¢
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876¢ [16]''&lt;br /&gt;3.273¢
| ''864¢ [6]''&lt;br /&gt;-8.727¢
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924¢ [16] &lt;br /&gt;-3.273¢
| 936¢ [6] &lt;br /&gt;8.727¢
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984¢ [18] &lt;br /&gt;2.182¢
| 972¢ [4] &lt;br /&gt;-9.818¢
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032¢ [14]''&lt;br /&gt;-4.364¢
| ''1044¢ [8]''&lt;br /&gt;7.636¢
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092¢ [20] &lt;br /&gt;1.091¢
| 1080¢ [2] &lt;br /&gt;-10.909¢
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140¢ [12] &lt;br /&gt;-5.4545¢
| 1152¢ [10] &lt;br /&gt;6.5455¢
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200¢ [22]
| N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[wikipedia:Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22 EDO for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&amp;[[34edo|34d]] temperaments is also usable. On the other hand, the “rough” alternatives to 22 EDO for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&amp;[[34edo|34d]] temperaments up to 58d is also usable.

[[Category:100edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>39y3sgh01o7zlhdbf122sj8uzvi0xto</sha1>
    </revision>
    <revision>
      <id>96184</id>
      <parentid>96006</parentid>
      <timestamp>2022-10-04T18:46:23Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96184</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8990" sha1="gtpsmp6xrq0mjy0spnyrzipcw2ekyx6" xml:space="preserve">{{Infobox ET}}
'''100 EDO''' is the [[equal division of the octave]] into 100 parts of exact 12 cents each. It is closely related to [[50 EDO]], but the [[patent val]]s differ on the mapping for 7. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50 EDO [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6 EDO]], [[35 EDO]], [[47 EDO]] and [[88 EDO]], 100 EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12 EDO]].
{{harmonics in equal|100|start=2|intervals=prime}}
== Scales ==

* [[Greeley8]]
* [[Greeley15]]

Since 100 EDO has a step of 12 cents, it also allows one to use its MOS scales as circulating temperaments.

{| class="wikitable"
|+Circulating temperaments in 100 EDO
! Tones
! Pattern
! L:s
|-
| 5
| [[5 EDO]]
| equal
|-
| 6
| [[4L 2s]]
| 17:16
|-
| 7
| [[2L 5s]]
| 15:14
|-
| 8
| [[4L 4s]]
| 13:12
|-
| 9
| [[1L 8s]]
| 12:11
|-
| 10
| [[10 EDO]]
| equal
|-
| 11
| [[1L 10s]]
| 10:9
|-
| 12
| [[4L 8s]]
| 9:8
|-
| 13
| [[9L 4s]]
| rowspan="2" |8:7
|-
| 14
| [[2L 12s]]
|-
| 15
| [[10L 5s]]
| rowspan="2" |7:6
|-
| 16
| 4L 12s
|-
| 17
| [[15L 2s]]
| rowspan="3" |6:5
|-
| 18
| 10L 8s
|-
| 19
| [[5L 14s]]
|-
| 20
| [[20 EDO]]
| equal
|-
| 21
| 16L 5s
| rowspan="4" |5:4
|-
| 22
| 12L 10s
|-
| 23
| 8L 15s
|-
| 24
| 4L 20s
|-
| 25
| [[25 EDO]]
| equal
|-
| 26
| 22L 4s
| rowspan="8" |4:3
|-
| 27
| 19L 8s
|-
| 28
| 16L 12s
|-
| 29
| 13L 16s
|-
| 30
| 10L 20s
|-
| 31
| 7L 24s
|-
| 32
| 4L 28s
|-
| 33
| 1L 32s
|-
| 34
| 32L 2s
| rowspan="16" | 3:2
|-
| 35
| 30L 5s
|-
| 36
| 28L 8s
|-
| 37
| 26L 11s
|-
| 38
| 24L 14s
|-
| 39
| 22L 17s
|-
| 40
| 20L 20s
|-
| 41
| 18L 23s
|-
| 42
| 16L 26s
|-
| 43
| 14L 29s
|-
| 44
| 12L 32s
|-
| 45
| 10L 35s
|-
| 46
| 8L 38s
|-
| 47
| 6L 41s
|-
| 48
| 4L 44s
|-
| 49
| 2L 47s
|-
| 50
| [[50 EDO]]
| equal
|-
| 51
| 49L 2s
| rowspan="29" |2:1
|-
| 52
| 48L 4s
|-
| 53
| 47L 6s
|-
| 54
| 46L 8s
|-
| 55
| 45L 10s
|-
| 56
| 44L 12s
|-
| 57
| 43L 14s
|-
| 58
| 42L 16s
|-
| 59
| 41L 18s
|-
| 60
| 40L 20s
|-
| 61
| 39L 22s
|-
| 62
| 38L 24s
|-
| 63
| 37L 26s
|-
| 64
| 36L 28s
|-
| 65
| 35L 30s
|-
| 66
| 34L 32s
|-
| 67
| 33L 34s
|-
| 68
| 32L 36s
|-
| 69
| 31L 38s
|-
| 70
| 30L 40s
|-
| 71
| 29L 42s
|-
| 72
| 28L 44s
|-
| 73
| 27L 46s
|-
| 74
| 26L 48s
|-
| 75
| 25L 50s
|-
| 76
| 24L 52s
|-
| 77
| 23L 54s
|-
| 78
| 22L 56s
|-
| 79
| 21L 58s
|}

== 100bddd and the 22-note scales ==

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22 EDO]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12 EDO counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12 Edo.

{| class="wikitable"
|-
| Steps of 22-note MODMOS
| Interval name (decatonic)
| Interval name (superpyth diatonic)
| Pure interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
| Dog interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60¢ [12] &lt;br /&gt;5.4545¢
| 48¢ [10] &lt;br /&gt;-6.5455¢
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108¢ [20] &lt;br /&gt;-1.091¢
| 120¢ [2] &lt;br /&gt;10.909¢
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168¢ [14]'' &lt;br /&gt;4.364¢
| ''156¢ [8]'' &lt;br /&gt;-7.636¢
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216¢ [18] &lt;br /&gt;-2.182¢
| 228¢ [4] &lt;br /&gt;9.818¢
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276¢ [16] &lt;br /&gt;3.273¢
| 264¢ [6] &lt;br /&gt;-8.727¢
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324¢ [16]''&lt;br /&gt;-3.273¢
| ''336¢ [6]''&lt;br /&gt;8.727¢
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384¢ [18] &lt;br /&gt;2.182¢
| 372¢ [4] &lt;br /&gt;-9.818¢
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432¢ [14] &lt;br /&gt;-4.364¢
| 444¢ [8] &lt;br /&gt;7.636¢
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492¢ [20] &lt;br /&gt;1.091¢
| 480¢ [2] &lt;br /&gt;-10.909¢
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540¢ [12] &lt;br /&gt;-5.4545¢
| 552¢ [10] &lt;br /&gt;6.5455¢
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600¢ [20]
| 588¢ [1] &lt;br&gt;-12¢ &lt;br&gt; 612¢ [1] &lt;br&gt;12¢
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660¢ [12] &lt;br /&gt;6.5455¢
| 648¢ [10] &lt;br /&gt;-5.4545¢
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708¢ [20] &lt;br /&gt;-1.091¢
| 720¢ [2] &lt;br /&gt;10.909¢
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768¢ [14] &lt;br /&gt;4.364¢
| 756¢ [8] &lt;br /&gt;-7.636¢
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816¢ [18] &lt;br /&gt;-2.182¢
| 828¢ [4] &lt;br /&gt;9.818¢
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876¢ [16]''&lt;br /&gt;3.273¢
| ''864¢ [6]''&lt;br /&gt;-8.727¢
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924¢ [16] &lt;br /&gt;-3.273¢
| 936¢ [6] &lt;br /&gt;8.727¢
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984¢ [18] &lt;br /&gt;2.182¢
| 972¢ [4] &lt;br /&gt;-9.818¢
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032¢ [14]''&lt;br /&gt;-4.364¢
| ''1044¢ [8]''&lt;br /&gt;7.636¢
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092¢ [20] &lt;br /&gt;1.091¢
| 1080¢ [2] &lt;br /&gt;-10.909¢
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140¢ [12] &lt;br /&gt;-5.4545¢
| 1152¢ [10] &lt;br /&gt;6.5455¢
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200¢ [22]
| N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[wikipedia:Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22 EDO for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&amp;[[34edo|34d]] temperaments is also usable. On the other hand, the “rough” alternatives to 22 EDO for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&amp;[[34edo|34d]] temperaments up to 58d is also usable.

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

[[Category:100edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>gtpsmp6xrq0mjy0spnyrzipcw2ekyx6</sha1>
    </revision>
    <revision>
      <id>99374</id>
      <parentid>96184</parentid>
      <timestamp>2022-11-29T16:35:22Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>style, sections, intro</comment>
      <origin>99374</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8951" sha1="k4ao3avmxb9ji2kulj9k3kzi998wp6a" xml:space="preserve">{{Infobox ET}}
{{EDO intro|100}}
==Theory==
It is closely related to [[50 EDO]], but the [[patent val]]s differ on the mapping for 7. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50 EDO [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6 EDO]], [[35 EDO]], [[47 EDO]] and [[88 EDO]], 100 EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12 EDO]].
===Prime harmonics===
{{harmonics in equal|100|start=2|intervals=prime}}

== Scales ==

* [[Greeley8]]
* [[Greeley15]]

Since 100 EDO has a step of 12 cents, it also allows one to use its MOS scales as circulating temperaments.

{| class="wikitable"
|+Circulating temperaments in 100 EDO
! Tones
! Pattern
! L:s
|-
| 5
| [[5 EDO]]
| equal
|-
| 6
| [[4L 2s]]
| 17:16
|-
| 7
| [[2L 5s]]
| 15:14
|-
| 8
| [[4L 4s]]
| 13:12
|-
| 9
| [[1L 8s]]
| 12:11
|-
| 10
| [[10 EDO]]
| equal
|-
| 11
| [[1L 10s]]
| 10:9
|-
| 12
| [[4L 8s]]
| 9:8
|-
| 13
| [[9L 4s]]
| rowspan="2" |8:7
|-
| 14
| [[2L 12s]]
|-
| 15
| [[10L 5s]]
| rowspan="2" |7:6
|-
| 16
| 4L 12s
|-
| 17
| [[15L 2s]]
| rowspan="3" |6:5
|-
| 18
| 10L 8s
|-
| 19
| [[5L 14s]]
|-
| 20
| [[20 EDO]]
| equal
|-
| 21
| 16L 5s
| rowspan="4" |5:4
|-
| 22
| 12L 10s
|-
| 23
| 8L 15s
|-
| 24
| 4L 20s
|-
| 25
| [[25 EDO]]
| equal
|-
| 26
| 22L 4s
| rowspan="8" |4:3
|-
| 27
| 19L 8s
|-
| 28
| 16L 12s
|-
| 29
| 13L 16s
|-
| 30
| 10L 20s
|-
| 31
| 7L 24s
|-
| 32
| 4L 28s
|-
| 33
| 1L 32s
|-
| 34
| 32L 2s
| rowspan="16" | 3:2
|-
| 35
| 30L 5s
|-
| 36
| 28L 8s
|-
| 37
| 26L 11s
|-
| 38
| 24L 14s
|-
| 39
| 22L 17s
|-
| 40
| 20L 20s
|-
| 41
| 18L 23s
|-
| 42
| 16L 26s
|-
| 43
| 14L 29s
|-
| 44
| 12L 32s
|-
| 45
| 10L 35s
|-
| 46
| 8L 38s
|-
| 47
| 6L 41s
|-
| 48
| 4L 44s
|-
| 49
| 2L 47s
|-
| 50
| [[50 EDO]]
| equal
|-
| 51
| 49L 2s
| rowspan="29" |2:1
|-
| 52
| 48L 4s
|-
| 53
| 47L 6s
|-
| 54
| 46L 8s
|-
| 55
| 45L 10s
|-
| 56
| 44L 12s
|-
| 57
| 43L 14s
|-
| 58
| 42L 16s
|-
| 59
| 41L 18s
|-
| 60
| 40L 20s
|-
| 61
| 39L 22s
|-
| 62
| 38L 24s
|-
| 63
| 37L 26s
|-
| 64
| 36L 28s
|-
| 65
| 35L 30s
|-
| 66
| 34L 32s
|-
| 67
| 33L 34s
|-
| 68
| 32L 36s
|-
| 69
| 31L 38s
|-
| 70
| 30L 40s
|-
| 71
| 29L 42s
|-
| 72
| 28L 44s
|-
| 73
| 27L 46s
|-
| 74
| 26L 48s
|-
| 75
| 25L 50s
|-
| 76
| 24L 52s
|-
| 77
| 23L 54s
|-
| 78
| 22L 56s
|-
| 79
| 21L 58s
|}

=== 100bddd and the 22-note scales ===

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22 EDO]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12 EDO counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12 Edo.

{| class="wikitable"
|-
| Steps of 22-note MODMOS
| Interval name (decatonic)
| Interval name (superpyth diatonic)
| Pure interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
| Dog interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60¢ [12] &lt;br /&gt;5.4545¢
| 48¢ [10] &lt;br /&gt;-6.5455¢
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108¢ [20] &lt;br /&gt;-1.091¢
| 120¢ [2] &lt;br /&gt;10.909¢
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168¢ [14]'' &lt;br /&gt;4.364¢
| ''156¢ [8]'' &lt;br /&gt;-7.636¢
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216¢ [18] &lt;br /&gt;-2.182¢
| 228¢ [4] &lt;br /&gt;9.818¢
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276¢ [16] &lt;br /&gt;3.273¢
| 264¢ [6] &lt;br /&gt;-8.727¢
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324¢ [16]''&lt;br /&gt;-3.273¢
| ''336¢ [6]''&lt;br /&gt;8.727¢
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384¢ [18] &lt;br /&gt;2.182¢
| 372¢ [4] &lt;br /&gt;-9.818¢
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432¢ [14] &lt;br /&gt;-4.364¢
| 444¢ [8] &lt;br /&gt;7.636¢
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492¢ [20] &lt;br /&gt;1.091¢
| 480¢ [2] &lt;br /&gt;-10.909¢
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540¢ [12] &lt;br /&gt;-5.4545¢
| 552¢ [10] &lt;br /&gt;6.5455¢
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600¢ [20]
| 588¢ [1] &lt;br&gt;-12¢ &lt;br&gt; 612¢ [1] &lt;br&gt;12¢
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660¢ [12] &lt;br /&gt;6.5455¢
| 648¢ [10] &lt;br /&gt;-5.4545¢
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708¢ [20] &lt;br /&gt;-1.091¢
| 720¢ [2] &lt;br /&gt;10.909¢
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768¢ [14] &lt;br /&gt;4.364¢
| 756¢ [8] &lt;br /&gt;-7.636¢
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816¢ [18] &lt;br /&gt;-2.182¢
| 828¢ [4] &lt;br /&gt;9.818¢
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876¢ [16]''&lt;br /&gt;3.273¢
| ''864¢ [6]''&lt;br /&gt;-8.727¢
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924¢ [16] &lt;br /&gt;-3.273¢
| 936¢ [6] &lt;br /&gt;8.727¢
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984¢ [18] &lt;br /&gt;2.182¢
| 972¢ [4] &lt;br /&gt;-9.818¢
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032¢ [14]''&lt;br /&gt;-4.364¢
| ''1044¢ [8]''&lt;br /&gt;7.636¢
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092¢ [20] &lt;br /&gt;1.091¢
| 1080¢ [2] &lt;br /&gt;-10.909¢
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140¢ [12] &lt;br /&gt;-5.4545¢
| 1152¢ [10] &lt;br /&gt;6.5455¢
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200¢ [22]
| N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[wikipedia:Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22 EDO for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&amp;[[34edo|34d]] temperaments is also usable. On the other hand, the “rough” alternatives to 22 EDO for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&amp;[[34edo|34d]] temperaments up to 58d is also usable.

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

[[Category:100edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>k4ao3avmxb9ji2kulj9k3kzi998wp6a</sha1>
    </revision>
    <revision>
      <id>115186</id>
      <parentid>99374</parentid>
      <timestamp>2023-05-30T21:10:36Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Scales */</comment>
      <origin>115186</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7169" sha1="1j1at6jlnxk3lb1jk4fpc7olr3d29j1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|100}}
==Theory==
It is closely related to [[50 EDO]], but the [[patent val]]s differ on the mapping for 7. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50 EDO [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6 EDO]], [[35 EDO]], [[47 EDO]] and [[88 EDO]], 100 EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12 EDO]].
===Prime harmonics===
{{harmonics in equal|100|start=2|intervals=prime}}

== Scales ==

* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22 EDO]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12 EDO counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12 Edo.

{| class="wikitable"
|-
| Steps of 22-note MODMOS
| Interval name (decatonic)
| Interval name (superpyth diatonic)
| Pure interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
| Dog interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60¢ [12] &lt;br /&gt;5.4545¢
| 48¢ [10] &lt;br /&gt;-6.5455¢
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108¢ [20] &lt;br /&gt;-1.091¢
| 120¢ [2] &lt;br /&gt;10.909¢
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168¢ [14]'' &lt;br /&gt;4.364¢
| ''156¢ [8]'' &lt;br /&gt;-7.636¢
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216¢ [18] &lt;br /&gt;-2.182¢
| 228¢ [4] &lt;br /&gt;9.818¢
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276¢ [16] &lt;br /&gt;3.273¢
| 264¢ [6] &lt;br /&gt;-8.727¢
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324¢ [16]''&lt;br /&gt;-3.273¢
| ''336¢ [6]''&lt;br /&gt;8.727¢
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384¢ [18] &lt;br /&gt;2.182¢
| 372¢ [4] &lt;br /&gt;-9.818¢
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432¢ [14] &lt;br /&gt;-4.364¢
| 444¢ [8] &lt;br /&gt;7.636¢
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492¢ [20] &lt;br /&gt;1.091¢
| 480¢ [2] &lt;br /&gt;-10.909¢
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540¢ [12] &lt;br /&gt;-5.4545¢
| 552¢ [10] &lt;br /&gt;6.5455¢
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600¢ [20]
| 588¢ [1] &lt;br&gt;-12¢ &lt;br&gt; 612¢ [1] &lt;br&gt;12¢
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660¢ [12] &lt;br /&gt;6.5455¢
| 648¢ [10] &lt;br /&gt;-5.4545¢
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708¢ [20] &lt;br /&gt;-1.091¢
| 720¢ [2] &lt;br /&gt;10.909¢
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768¢ [14] &lt;br /&gt;4.364¢
| 756¢ [8] &lt;br /&gt;-7.636¢
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816¢ [18] &lt;br /&gt;-2.182¢
| 828¢ [4] &lt;br /&gt;9.818¢
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876¢ [16]''&lt;br /&gt;3.273¢
| ''864¢ [6]''&lt;br /&gt;-8.727¢
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924¢ [16] &lt;br /&gt;-3.273¢
| 936¢ [6] &lt;br /&gt;8.727¢
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984¢ [18] &lt;br /&gt;2.182¢
| 972¢ [4] &lt;br /&gt;-9.818¢
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032¢ [14]''&lt;br /&gt;-4.364¢
| ''1044¢ [8]''&lt;br /&gt;7.636¢
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092¢ [20] &lt;br /&gt;1.091¢
| 1080¢ [2] &lt;br /&gt;-10.909¢
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140¢ [12] &lt;br /&gt;-5.4545¢
| 1152¢ [10] &lt;br /&gt;6.5455¢
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200¢ [22]
| N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[wikipedia:Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22 EDO for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&amp;[[34edo|34d]] temperaments is also usable. On the other hand, the “rough” alternatives to 22 EDO for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&amp;[[34edo|34d]] temperaments up to 58d is also usable.

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

[[Category:100edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>1j1at6jlnxk3lb1jk4fpc7olr3d29j1</sha1>
    </revision>
    <revision>
      <id>132168</id>
      <parentid>115186</parentid>
      <timestamp>2024-01-20T00:45:18Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added intervals table</comment>
      <origin>132168</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7205" sha1="kexc6edvas393zn13eqbcrq5z650izh" xml:space="preserve">{{Infobox ET}}
{{EDO intro|100}}
==Theory==
It is closely related to [[50 EDO]], but the [[patent val]]s differ on the mapping for 7. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50 EDO [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6 EDO]], [[35 EDO]], [[47 EDO]] and [[88 EDO]], 100 EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12 EDO]].
===Prime harmonics===
{{harmonics in equal|100|start=2|intervals=prime}}

== Intervals ==
{{Interval table}}

== Scales ==

* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22 EDO]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12 EDO counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note MODMOS 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term ''dog'' rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even ''if'' the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to 12 Edo.

{| class="wikitable"
|-
| Steps of 22-note MODMOS
| Interval name (decatonic)
| Interval name (superpyth diatonic)
| Pure interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
| Dog interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60¢ [12] &lt;br /&gt;5.4545¢
| 48¢ [10] &lt;br /&gt;-6.5455¢
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108¢ [20] &lt;br /&gt;-1.091¢
| 120¢ [2] &lt;br /&gt;10.909¢
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168¢ [14]'' &lt;br /&gt;4.364¢
| ''156¢ [8]'' &lt;br /&gt;-7.636¢
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216¢ [18] &lt;br /&gt;-2.182¢
| 228¢ [4] &lt;br /&gt;9.818¢
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276¢ [16] &lt;br /&gt;3.273¢
| 264¢ [6] &lt;br /&gt;-8.727¢
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324¢ [16]''&lt;br /&gt;-3.273¢
| ''336¢ [6]''&lt;br /&gt;8.727¢
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384¢ [18] &lt;br /&gt;2.182¢
| 372¢ [4] &lt;br /&gt;-9.818¢
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432¢ [14] &lt;br /&gt;-4.364¢
| 444¢ [8] &lt;br /&gt;7.636¢
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492¢ [20] &lt;br /&gt;1.091¢
| 480¢ [2] &lt;br /&gt;-10.909¢
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540¢ [12] &lt;br /&gt;-5.4545¢
| 552¢ [10] &lt;br /&gt;6.5455¢
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600¢ [20]
| 588¢ [1] &lt;br&gt;-12¢ &lt;br&gt; 612¢ [1] &lt;br&gt;12¢
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660¢ [12] &lt;br /&gt;6.5455¢
| 648¢ [10] &lt;br /&gt;-5.4545¢
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708¢ [20] &lt;br /&gt;-1.091¢
| 720¢ [2] &lt;br /&gt;10.909¢
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768¢ [14] &lt;br /&gt;4.364¢
| 756¢ [8] &lt;br /&gt;-7.636¢
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816¢ [18] &lt;br /&gt;-2.182¢
| 828¢ [4] &lt;br /&gt;9.818¢
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876¢ [16]''&lt;br /&gt;3.273¢
| ''864¢ [6]''&lt;br /&gt;-8.727¢
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924¢ [16] &lt;br /&gt;-3.273¢
| 936¢ [6] &lt;br /&gt;8.727¢
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984¢ [18] &lt;br /&gt;2.182¢
| 972¢ [4] &lt;br /&gt;-9.818¢
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032¢ [14]''&lt;br /&gt;-4.364¢
| ''1044¢ [8]''&lt;br /&gt;7.636¢
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092¢ [20] &lt;br /&gt;1.091¢
| 1080¢ [2] &lt;br /&gt;-10.909¢
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140¢ [12] &lt;br /&gt;-5.4545¢
| 1152¢ [10] &lt;br /&gt;6.5455¢
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200¢ [22]
| N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[wikipedia:Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22 EDO for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&amp;[[34edo|34d]] temperaments is also usable. On the other hand, the “rough” alternatives to 22 EDO for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&amp;[[34edo|34d]] temperaments up to 58d is also usable.

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

[[Category:100edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>kexc6edvas393zn13eqbcrq5z650izh</sha1>
    </revision>
    <revision>
      <id>135514</id>
      <parentid>132168</parentid>
      <timestamp>2024-02-14T01:33:22Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <comment>italic overload</comment>
      <origin>135514</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7206" sha1="dipy9hu38cpphuqmilxnwp850imcmnm" xml:space="preserve">{{Infobox ET}}
{{EDO intro|100}}
==Theory==
It is closely related to [[50 EDO]], but the [[patent val]]s differ on the mapping for 7. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50 EDO [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43&amp;amp;57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6 EDO]], [[35 EDO]], [[47 EDO]] and [[88 EDO]], 100 EDO possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12 EDO]].
===Prime harmonics===
{{harmonics in equal|100|start=2|intervals=prime}}

== Intervals ==
{{Interval table}}

== Scales ==

* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22 EDO]] for [[pajara]] temperament and for tuning Paul Erlich's decatonic scales, as well as diatonic scales (via superpyth temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12 EDO counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[MODMOS]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this MODMOS still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
| Steps of 22-note MODMOS
| Interval name (decatonic)
| Interval name (superpyth diatonic)
| Pure interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
| Dog interval size [multiplicity] &lt;br /&gt; Difference from 22 EDO
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60¢ [12] &lt;br /&gt;5.4545¢
| 48¢ [10] &lt;br /&gt;-6.5455¢
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108¢ [20] &lt;br /&gt;-1.091¢
| 120¢ [2] &lt;br /&gt;10.909¢
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168¢ [14]'' &lt;br /&gt;4.364¢
| ''156¢ [8]'' &lt;br /&gt;-7.636¢
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216¢ [18] &lt;br /&gt;-2.182¢
| 228¢ [4] &lt;br /&gt;9.818¢
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276¢ [16] &lt;br /&gt;3.273¢
| 264¢ [6] &lt;br /&gt;-8.727¢
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324¢ [16]''&lt;br /&gt;-3.273¢
| ''336¢ [6]''&lt;br /&gt;8.727¢
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384¢ [18] &lt;br /&gt;2.182¢
| 372¢ [4] &lt;br /&gt;-9.818¢
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432¢ [14] &lt;br /&gt;-4.364¢
| 444¢ [8] &lt;br /&gt;7.636¢
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492¢ [20] &lt;br /&gt;1.091¢
| 480¢ [2] &lt;br /&gt;-10.909¢
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540¢ [12] &lt;br /&gt;-5.4545¢
| 552¢ [10] &lt;br /&gt;6.5455¢
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600¢ [20]
| 588¢ [1] &lt;br&gt;-12¢ &lt;br&gt; 612¢ [1] &lt;br&gt;12¢
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660¢ [12] &lt;br /&gt;6.5455¢
| 648¢ [10] &lt;br /&gt;-5.4545¢
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708¢ [20] &lt;br /&gt;-1.091¢
| 720¢ [2] &lt;br /&gt;10.909¢
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768¢ [14] &lt;br /&gt;4.364¢
| 756¢ [8] &lt;br /&gt;-7.636¢
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816¢ [18] &lt;br /&gt;-2.182¢
| 828¢ [4] &lt;br /&gt;9.818¢
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876¢ [16]''&lt;br /&gt;3.273¢
| ''864¢ [6]''&lt;br /&gt;-8.727¢
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924¢ [16] &lt;br /&gt;-3.273¢
| 936¢ [6] &lt;br /&gt;8.727¢
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984¢ [18] &lt;br /&gt;2.182¢
| 972¢ [4] &lt;br /&gt;-9.818¢
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032¢ [14]''&lt;br /&gt;-4.364¢
| ''1044¢ [8]''&lt;br /&gt;7.636¢
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092¢ [20] &lt;br /&gt;1.091¢
| 1080¢ [2] &lt;br /&gt;-10.909¢
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140¢ [12] &lt;br /&gt;-5.4545¢
| 1152¢ [10] &lt;br /&gt;6.5455¢
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200¢ [22]
| N/A
|}

Alternatively, the unmodified, symmetrical 2MOS scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they [[wikipedia:Augmented-fourths_tuning|are tuned in tritones]]. This makes guitar construction much easier compared to other non-equally-tempered scales. The MODMOS would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2MOS is simpler, the MODMOS very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22 EDO for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22&amp;[[34edo|34d]] temperaments is also usable. On the other hand, the “rough” alternatives to 22 EDO for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12&amp;[[34edo|34d]] temperaments up to 58d is also usable.

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

[[Category:100edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>dipy9hu38cpphuqmilxnwp850imcmnm</sha1>
    </revision>
    <revision>
      <id>146050</id>
      <parentid>135514</parentid>
      <timestamp>2024-06-10T17:01:13Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>146050</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7108" sha1="enddx4ftvjqxk38kog2sqdnkl2i02kb" xml:space="preserve">{{Infobox ET}}
{{EDO intro|100}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43 &amp;amp; 57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note Modmos
! Interval Name (Decatonic)
! Interval Name (Superpyth diatonic)
! Pure Interval Size [Multiplicity] &lt;br /&gt; Difference from 22edo
! Dog Interval Size [Multiplicity] &lt;br /&gt; Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60¢ [12] &lt;br /&gt;5.4545¢
| 48¢ [10] &lt;br /&gt;-6.5455¢
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108¢ [20] &lt;br /&gt;-1.091¢
| 120¢ [2] &lt;br /&gt;10.909¢
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168¢ [14]'' &lt;br /&gt;4.364¢
| ''156¢ [8]'' &lt;br /&gt;-7.636¢
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216¢ [18] &lt;br /&gt;-2.182¢
| 228¢ [4] &lt;br /&gt;9.818¢
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276¢ [16] &lt;br /&gt;3.273¢
| 264¢ [6] &lt;br /&gt;-8.727¢
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324¢ [16]''&lt;br /&gt;-3.273¢
| ''336¢ [6]''&lt;br /&gt;8.727¢
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384¢ [18] &lt;br /&gt;2.182¢
| 372¢ [4] &lt;br /&gt;-9.818¢
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432¢ [14] &lt;br /&gt;-4.364¢
| 444¢ [8] &lt;br /&gt;7.636¢
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492¢ [20] &lt;br /&gt;1.091¢
| 480¢ [2] &lt;br /&gt;-10.909¢
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540¢ [12] &lt;br /&gt;-5.4545¢
| 552¢ [10] &lt;br /&gt;6.5455¢
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600¢ [20]
| 588¢ [1] &lt;br&gt;-12¢ &lt;br&gt; 612¢ [1] &lt;br&gt;12¢
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660¢ [12] &lt;br /&gt;6.5455¢
| 648¢ [10] &lt;br /&gt;-5.4545¢
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708¢ [20] &lt;br /&gt;-1.091¢
| 720¢ [2] &lt;br /&gt;10.909¢
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768¢ [14] &lt;br /&gt;4.364¢
| 756¢ [8] &lt;br /&gt;-7.636¢
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816¢ [18] &lt;br /&gt;-2.182¢
| 828¢ [4] &lt;br /&gt;9.818¢
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876¢ [16]''&lt;br /&gt;3.273¢
| ''864¢ [6]''&lt;br /&gt;-8.727¢
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924¢ [16] &lt;br /&gt;-3.273¢
| 936¢ [6] &lt;br /&gt;8.727¢
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984¢ [18] &lt;br /&gt;2.182¢
| 972¢ [4] &lt;br /&gt;-9.818¢
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032¢ [14]''&lt;br /&gt;-4.364¢
| ''1044¢ [8]''&lt;br /&gt;7.636¢
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092¢ [20] &lt;br /&gt;1.091¢
| 1080¢ [2] &lt;br /&gt;-10.909¢
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140¢ [12] &lt;br /&gt;-5.4545¢
| 1152¢ [10] &lt;br /&gt;6.5455¢
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200¢ [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22 &amp; [[34edo|34d]] temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12 &amp; [[34edo|34d]] temperaments up to 58d is also usable.

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;</text>
      <sha1>enddx4ftvjqxk38kog2sqdnkl2i02kb</sha1>
    </revision>
    <revision>
      <id>146096</id>
      <parentid>146050</parentid>
      <timestamp>2024-06-11T11:53:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ +subsets and supersets</comment>
      <origin>146096</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7248" sha1="6vypw8z55wrzc7ntixmfszle5ra4kyc" xml:space="preserve">{{Infobox ET}}
{{EDO intro|100}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43 &amp;amp; 57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===

The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60¢ which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note Modmos
! Interval Name (Decatonic)
! Interval Name (Superpyth diatonic)
! Pure Interval Size [Multiplicity] &lt;br /&gt; Difference from 22edo
! Dog Interval Size [Multiplicity] &lt;br /&gt; Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60¢ [12] &lt;br /&gt;5.4545¢
| 48¢ [10] &lt;br /&gt;-6.5455¢
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108¢ [20] &lt;br /&gt;-1.091¢
| 120¢ [2] &lt;br /&gt;10.909¢
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168¢ [14]'' &lt;br /&gt;4.364¢
| ''156¢ [8]'' &lt;br /&gt;-7.636¢
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216¢ [18] &lt;br /&gt;-2.182¢
| 228¢ [4] &lt;br /&gt;9.818¢
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276¢ [16] &lt;br /&gt;3.273¢
| 264¢ [6] &lt;br /&gt;-8.727¢
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324¢ [16]''&lt;br /&gt;-3.273¢
| ''336¢ [6]''&lt;br /&gt;8.727¢
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384¢ [18] &lt;br /&gt;2.182¢
| 372¢ [4] &lt;br /&gt;-9.818¢
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432¢ [14] &lt;br /&gt;-4.364¢
| 444¢ [8] &lt;br /&gt;7.636¢
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492¢ [20] &lt;br /&gt;1.091¢
| 480¢ [2] &lt;br /&gt;-10.909¢
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540¢ [12] &lt;br /&gt;-5.4545¢
| 552¢ [10] &lt;br /&gt;6.5455¢
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600¢ [20]
| 588¢ [1] &lt;br&gt;-12¢ &lt;br&gt; 612¢ [1] &lt;br&gt;12¢
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660¢ [12] &lt;br /&gt;6.5455¢
| 648¢ [10] &lt;br /&gt;-5.4545¢
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708¢ [20] &lt;br /&gt;-1.091¢
| 720¢ [2] &lt;br /&gt;10.909¢
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768¢ [14] &lt;br /&gt;4.364¢
| 756¢ [8] &lt;br /&gt;-7.636¢
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816¢ [18] &lt;br /&gt;-2.182¢
| 828¢ [4] &lt;br /&gt;9.818¢
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876¢ [16]''&lt;br /&gt;3.273¢
| ''864¢ [6]''&lt;br /&gt;-8.727¢
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924¢ [16] &lt;br /&gt;-3.273¢
| 936¢ [6] &lt;br /&gt;8.727¢
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984¢ [18] &lt;br /&gt;2.182¢
| 972¢ [4] &lt;br /&gt;-9.818¢
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032¢ [14]''&lt;br /&gt;-4.364¢
| ''1044¢ [8]''&lt;br /&gt;7.636¢
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092¢ [20] &lt;br /&gt;1.091¢
| 1080¢ [2] &lt;br /&gt;-10.909¢
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140¢ [12] &lt;br /&gt;-5.4545¢
| 1152¢ [10] &lt;br /&gt;6.5455¢
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200¢ [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600 cents. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of 22 &amp; [[34edo|34d]] temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of 12 &amp; [[34edo|34d]] temperaments up to 58d is also usable.

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;</text>
      <sha1>6vypw8z55wrzc7ntixmfszle5ra4kyc</sha1>
    </revision>
    <revision>
      <id>181301</id>
      <parentid>146096</parentid>
      <timestamp>2025-02-18T16:45:04Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181301</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7570" sha1="maa3qwoq2ijqlwssibhol8xqbdqwf0w" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43 &amp;amp; 57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name (decatonic)
! Interval name (superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;</text>
      <sha1>maa3qwoq2ijqlwssibhol8xqbdqwf0w</sha1>
    </revision>
    <revision>
      <id>181302</id>
      <parentid>181301</parentid>
      <timestamp>2025-02-18T16:45:26Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181302</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7580" sha1="exvx0z5k0jocih6997bdy8djieund9c" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43 &amp;amp; 57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name&lt;br /&gt;(decatonic)
! Interval name&lt;br /&gt;(superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;</text>
      <sha1>exvx0z5k0jocih6997bdy8djieund9c</sha1>
    </revision>
    <revision>
      <id>184830</id>
      <parentid>181302</parentid>
      <timestamp>2025-03-06T13:31:13Z</timestamp>
      <contributor>
        <username>MisterShafXen</username>
        <id>509</id>
      </contributor>
      <minor/>
      <origin>184830</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7633" sha1="lrw72laamwslwgxikc34qkutcpy95ja" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit 43 &amp;amp; 57 temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name&lt;br /&gt;(decatonic)
! Interval name&lt;br /&gt;(superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Instruments ==
[[Skip fretting system 100 5 13]]

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;</text>
      <sha1>lrw72laamwslwgxikc34qkutcpy95ja</sha1>
    </revision>
    <revision>
      <id>185461</id>
      <parentid>184830</parentid>
      <timestamp>2025-03-11T17:50:35Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>185461</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7644" sha1="f10ts07we9fib25johgyeqggtgihq9v" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit {{nowrap|43 &amp;amp; 57}} temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name&lt;br /&gt;(decatonic)
! Interval name&lt;br /&gt;(superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Instruments ==
[[Skip fretting system 100 5 13]]

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;</text>
      <sha1>f10ts07we9fib25johgyeqggtgihq9v</sha1>
    </revision>
    <revision>
      <id>194040</id>
      <parentid>185461</parentid>
      <timestamp>2025-04-26T05:27:32Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>/* Instruments */ Merge into EDO page as voted on XA Discord #wiki channel</comment>
      <origin>194040</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8306" sha1="pmrn32crxxw945jybrex4hb86rrs93x" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit {{nowrap|43 &amp;amp; 57}} temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name&lt;br /&gt;(decatonic)
! Interval name&lt;br /&gt;(superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Instruments ==
=== Skip fretting ===
One way to play 100edo on a [[20edo]] guitar is to tune the strings 13\100 apart, or 156 [[cents]]. All examples on this page are for 7-string guitar.

; Prime harmonics

1/1: string 2 open 

2/1: string 7 fret 7

3/2: string 5 fret 4

5/4: string 1 fret 9

7/4: string 4 fret 11 

11/8: string 4 fret 4 

13/8: string 7 fret 1

17/16: string 5 fret 14 

19/16: string 2 fret 5 

23/16: string 6 open 

29/16: string 4 fret 12 

31/16: string 7 fret 6 

37/32: string 4 fret 19

41/32: string 4 fret 2 

43/32: string 3 fret 6 

47/32: string 4 fret 6

53/32: string 3 fret 12

59/32: string 3 fret 15

61/32: string 3 fret 16

[[Category:Skip fretting]][[Category:100edo]]

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;</text>
      <sha1>pmrn32crxxw945jybrex4hb86rrs93x</sha1>
    </revision>
    <revision>
      <id>194042</id>
      <parentid>194040</parentid>
      <timestamp>2025-04-26T05:28:57Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>194042</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8259" sha1="ggb32usns63hpi6qjwkawdv025ek5so" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit {{nowrap|43 &amp;amp; 57}} temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name&lt;br /&gt;(decatonic)
! Interval name&lt;br /&gt;(superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Instruments ==
=== Skip fretting ===
One way to play 100edo on a [[20edo]] guitar is to tune the strings 13\100 apart, or 156 [[cents]]. All examples on this page are for 7-string guitar.

; Prime harmonics

1/1: string 2 open 

2/1: string 7 fret 7

3/2: string 5 fret 4

5/4: string 1 fret 9

7/4: string 4 fret 11 

11/8: string 4 fret 4 

13/8: string 7 fret 1

17/16: string 5 fret 14 

19/16: string 2 fret 5 

23/16: string 6 open 

29/16: string 4 fret 12 

31/16: string 7 fret 6 

37/32: string 4 fret 19

41/32: string 4 fret 2 

43/32: string 3 fret 6 

47/32: string 4 fret 6

53/32: string 3 fret 12

59/32: string 3 fret 15

61/32: string 3 fret 16

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;</text>
      <sha1>ggb32usns63hpi6qjwkawdv025ek5so</sha1>
    </revision>
    <revision>
      <id>200244</id>
      <parentid>194042</parentid>
      <timestamp>2025-06-21T07:44:32Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Video */ Insert music section after this, starting with Bryan Deister's ''100edo'' (2023)</comment>
      <origin>200244</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8358" sha1="eluhtl5lk1uj8cbuigbf3wbq8fgdm93" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit {{nowrap|43 &amp;amp; 57}} temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name&lt;br /&gt;(decatonic)
! Interval name&lt;br /&gt;(superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Instruments ==
=== Skip fretting ===
One way to play 100edo on a [[20edo]] guitar is to tune the strings 13\100 apart, or 156 [[cents]]. All examples on this page are for 7-string guitar.

; Prime harmonics

1/1: string 2 open 

2/1: string 7 fret 7

3/2: string 5 fret 4

5/4: string 1 fret 9

7/4: string 4 fret 11 

11/8: string 4 fret 4 

13/8: string 7 fret 1

17/16: string 5 fret 14 

19/16: string 2 fret 5 

23/16: string 6 open 

29/16: string 4 fret 12 

31/16: string 7 fret 6 

37/32: string 4 fret 19

41/32: string 4 fret 2 

43/32: string 3 fret 6 

47/32: string 4 fret 6

53/32: string 3 fret 12

59/32: string 3 fret 15

61/32: string 3 fret 16

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

== Music ==

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/37bFvBsKXqo ''100edo''] (2023)</text>
      <sha1>eluhtl5lk1uj8cbuigbf3wbq8fgdm93</sha1>
    </revision>
    <revision>
      <id>200259</id>
      <parentid>200244</parentid>
      <timestamp>2025-06-21T09:55:18Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Music */ Fix year of Bryan Deister's 100edo composition</comment>
      <origin>200259</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8358" sha1="mihtnncci5b8v7q4usldcwipjcfe01k" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit {{nowrap|43 &amp;amp; 57}} temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name&lt;br /&gt;(decatonic)
! Interval name&lt;br /&gt;(superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Instruments ==
=== Skip fretting ===
One way to play 100edo on a [[20edo]] guitar is to tune the strings 13\100 apart, or 156 [[cents]]. All examples on this page are for 7-string guitar.

; Prime harmonics

1/1: string 2 open 

2/1: string 7 fret 7

3/2: string 5 fret 4

5/4: string 1 fret 9

7/4: string 4 fret 11 

11/8: string 4 fret 4 

13/8: string 7 fret 1

17/16: string 5 fret 14 

19/16: string 2 fret 5 

23/16: string 6 open 

29/16: string 4 fret 12 

31/16: string 7 fret 6 

37/32: string 4 fret 19

41/32: string 4 fret 2 

43/32: string 3 fret 6 

47/32: string 4 fret 6

53/32: string 3 fret 12

59/32: string 3 fret 15

61/32: string 3 fret 16

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

== Music ==

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/37bFvBsKXqo ''100edo''] (2022)</text>
      <sha1>mihtnncci5b8v7q4usldcwipjcfe01k</sha1>
    </revision>
    <revision>
      <id>205066</id>
      <parentid>200259</parentid>
      <timestamp>2025-07-24T09:19:47Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Music */ Todo|add lumatone mapping</comment>
      <origin>205066</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8389" sha1="8d6qdm5aurw5dqtcanxh67yvhk6eiao" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit {{nowrap|43 &amp;amp; 57}} temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name&lt;br /&gt;(decatonic)
! Interval name&lt;br /&gt;(superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Instruments ==
=== Skip fretting ===
One way to play 100edo on a [[20edo]] guitar is to tune the strings 13\100 apart, or 156 [[cents]]. All examples on this page are for 7-string guitar.

; Prime harmonics

1/1: string 2 open 

2/1: string 7 fret 7

3/2: string 5 fret 4

5/4: string 1 fret 9

7/4: string 4 fret 11 

11/8: string 4 fret 4 

13/8: string 7 fret 1

17/16: string 5 fret 14 

19/16: string 2 fret 5 

23/16: string 6 open 

29/16: string 4 fret 12 

31/16: string 7 fret 6 

37/32: string 4 fret 19

41/32: string 4 fret 2 

43/32: string 3 fret 6 

47/32: string 4 fret 6

53/32: string 3 fret 12

59/32: string 3 fret 15

61/32: string 3 fret 16

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

== Music ==

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/37bFvBsKXqo ''100edo''] (2022)

{{Todo|add lumatone mapping}}</text>
      <sha1>8d6qdm5aurw5dqtcanxh67yvhk6eiao</sha1>
    </revision>
    <revision>
      <id>206821</id>
      <parentid>205066</parentid>
      <timestamp>2025-08-12T21:12:02Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <comment>Add lumatone mapping link.</comment>
      <origin>206821</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8407" sha1="503onmrjz20cqc2epu0o9rto1n2wsrj" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit {{nowrap|43 &amp;amp; 57}} temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name&lt;br /&gt;(decatonic)
! Interval name&lt;br /&gt;(superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Instruments ==
=== Lumatone ===
[[Lumatone mapping for 100edo]]
=== Skip fretting ===
One way to play 100edo on a [[20edo]] guitar is to tune the strings 13\100 apart, or 156 [[cents]]. All examples on this page are for 7-string guitar.

; Prime harmonics

1/1: string 2 open 

2/1: string 7 fret 7

3/2: string 5 fret 4

5/4: string 1 fret 9

7/4: string 4 fret 11 

11/8: string 4 fret 4 

13/8: string 7 fret 1

17/16: string 5 fret 14 

19/16: string 2 fret 5 

23/16: string 6 open 

29/16: string 4 fret 12 

31/16: string 7 fret 6 

37/32: string 4 fret 19

41/32: string 4 fret 2 

43/32: string 3 fret 6 

47/32: string 4 fret 6

53/32: string 3 fret 12

59/32: string 3 fret 15

61/32: string 3 fret 16

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

== Music ==

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/37bFvBsKXqo ''100edo''] (2022)</text>
      <sha1>503onmrjz20cqc2epu0o9rto1n2wsrj</sha1>
    </revision>
    <revision>
      <id>206822</id>
      <parentid>206821</parentid>
      <timestamp>2025-08-12T22:45:37Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Music */ Add normal link to Iceface H. Wakabayashi's ''Microtonal Piano in 100 tone equal temperament (100EDO) (Microtonal Music)'' (2016) in case the embedded video won't play</comment>
      <origin>206822</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8694" sha1="fkxllhiistvqn8wktvr8bnoprirqsxk" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit {{nowrap|43 &amp;amp; 57}} temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name&lt;br /&gt;(decatonic)
! Interval name&lt;br /&gt;(superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Instruments ==
=== Lumatone ===
[[Lumatone mapping for 100edo]]
=== Skip fretting ===
One way to play 100edo on a [[20edo]] guitar is to tune the strings 13\100 apart, or 156 [[cents]]. All examples on this page are for 7-string guitar.

; Prime harmonics

1/1: string 2 open 

2/1: string 7 fret 7

3/2: string 5 fret 4

5/4: string 1 fret 9

7/4: string 4 fret 11 

11/8: string 4 fret 4 

13/8: string 7 fret 1

17/16: string 5 fret 14 

19/16: string 2 fret 5 

23/16: string 6 open 

29/16: string 4 fret 12 

31/16: string 7 fret 6 

37/32: string 4 fret 19

41/32: string 4 fret 2 

43/32: string 3 fret 6 

47/32: string 4 fret 6

53/32: string 3 fret 12

59/32: string 3 fret 15

61/32: string 3 fret 16

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

== Music ==

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/37bFvBsKXqo ''100edo''] (2022)

; [[Iceface H. Wakabayashi]] (微分音チャンネル)
* [https://www.youtube.com/watch?v=shcrw2vtmJU ''Microtonal Piano in 100 tone equal temperament (100EDO) (Microtonal Music)''] (2016) (this is the same as the video linked above, to use in case the embedded video refuses to play)</text>
      <sha1>fkxllhiistvqn8wktvr8bnoprirqsxk</sha1>
    </revision>
    <revision>
      <id>210389</id>
      <parentid>206822</parentid>
      <timestamp>2025-09-22T06:11:22Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>added subgroup suitable for use</comment>
      <origin>210389</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8760" sha1="c026h3gpglv8yx53fmwvu8fkzfr3h0a" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit {{nowrap|43 &amp;amp; 57}} temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. It is therefore a strong 2.9.5.7.11.13.17.19 system for its size. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name&lt;br /&gt;(decatonic)
! Interval name&lt;br /&gt;(superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Instruments ==
=== Lumatone ===
[[Lumatone mapping for 100edo]]
=== Skip fretting ===
One way to play 100edo on a [[20edo]] guitar is to tune the strings 13\100 apart, or 156 [[cents]]. All examples on this page are for 7-string guitar.

; Prime harmonics

1/1: string 2 open 

2/1: string 7 fret 7

3/2: string 5 fret 4

5/4: string 1 fret 9

7/4: string 4 fret 11 

11/8: string 4 fret 4 

13/8: string 7 fret 1

17/16: string 5 fret 14 

19/16: string 2 fret 5 

23/16: string 6 open 

29/16: string 4 fret 12 

31/16: string 7 fret 6 

37/32: string 4 fret 19

41/32: string 4 fret 2 

43/32: string 3 fret 6 

47/32: string 4 fret 6

53/32: string 3 fret 12

59/32: string 3 fret 15

61/32: string 3 fret 16

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

== Music ==

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/37bFvBsKXqo ''100edo''] (2022)

; [[Iceface H. Wakabayashi]] (微分音チャンネル)
* [https://www.youtube.com/watch?v=shcrw2vtmJU ''Microtonal Piano in 100 tone equal temperament (100EDO) (Microtonal Music)''] (2016) (this is the same as the video linked above, to use in case the embedded video refuses to play)</text>
      <sha1>c026h3gpglv8yx53fmwvu8fkzfr3h0a</sha1>
    </revision>
    <revision>
      <id>223593</id>
      <parentid>210389</parentid>
      <timestamp>2026-02-08T18:44:44Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Subsets and supersets */ + supersets</comment>
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      <text bytes="8983" sha1="19k5jej786jtug5z7tv2rod1qp1mz4o" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
100edo is closely related to [[50edo]], but the [[patent val]]s differ on the mapping for [[7/1|7]]. It tempers out [[6144/6125]] in the 7-limit, [[99/98]] and [[441/440]] in the 11-limit and [[144/143]] in the 13-limit, and like 50edo [[81/80]] in the 5-limit. It provides the [[optimal patent val]] for the 11- and 13- limit {{nowrap|43 &amp;amp; 57}} temperament tempering out 81/80, 99/98, 1350/1331, and in the 13-limit, 144/143.

Like [[6edo|6-]], [[35edo|35-]], [[47edo|47-]] and [[88edo]], 100edo possesses two approximations of the perfect fifth (at 58\100 and 59\100 respectively), each almost exactly six cents from just. It is therefore a strong 2.9.5.7.11.13.17.19 system for its size. One interesting consequence of this property is that one may have a closed circle of twelve good fifths (four wide, eight narrow) that bears little resemblance to [[12edo]].

=== Odd harmonics ===
{{Harmonics in equal|100}}

=== Subsets and supersets ===
Since 100 factors into {{factorization|100}}, 100edo has subset edos {{EDOs| 2, 4, 5, 10, 20, 25, and 50 }}. [[200edo]], which doubles it, corrects the perfect fifth to near-just quality. [[400edo]] further corrects many harmonics, making for a strong 19-limit system. [[1600edo]] and [[2000edo]] do very well in high prime limits.

== Intervals ==
{{Interval table}}

== Scales ==
* [[Greeley8]]
* [[Greeley15]]

=== 100bddd and the 22-note scales ===
The 100bddd val (which maps 3/2 onto 59\100, 5/4 onto its patent value of 32\100, and 7/4 onto 82\100) is of special interest as it provides a good alternative to [[22edo]] for [[pajara]] temperament and for tuning [[Paul Erlich]]'s decatonic scales, as well as diatonic scales (via [[superpyth]] temperament). This alternative tuning prioritizes the 3- and 5-limits over the 7-limit (although the latter is still within striking distance); its pure intervals are also all closer to their 12edo counterparts, and for both reasons it is much less xenharmonic overall. Melodically its properties are superior as well; decatonic scales are more expressive due to the larger difference between step sizes, and the superpyth diatonic scale has a minor second of 60{{c}} which just barely falls within the 60-80 cent range [http://www.anaphoria.com/Secor17puzzle.pdf favored by George Secor] for neomedieval compositions.

The 22-note [[modmos]] 5 4 5 4 5 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 could be used to construct a 22-tone piano; this tuning has two chains of fifths (one with 10 notes in it and one with 12), and thus has two "wolf" fifths. Much like meantone, this tuning has "wolf" intervals but in this case they are only twelve cents away from their pure counterparts, and as such they don't sound nearly as bad. They are xenharmonic but not unpleasant and could easily be used in compositions, which makes this tuning akin to well temperaments as well as to meantone. Because most if not all of the "wolves" are still usable (albeit xenharmonic), it might be better to use the term "dog" rather than wolf for these intervals. Dog intervals frequently provide ''closer'' matches to intervals involving the 7th and 11th harmonics. Even if the dog intervals are completely avoided, this modmos still allows for decatonic music in 12 different keys, and diatonic (superpyth) music in 10 different keys, and thus the freedom of modulation and key choice is still comparable to [[12edo]].

{| class="wikitable"
|-
! Steps of 22-note MODMOS
! Interval name&lt;br /&gt;(decatonic)
! Interval name&lt;br /&gt;(superpyth diatonic)
! Pure interval size [multiplicity]&lt;br /&gt;Difference from 22edo
! Dog interval size [multiplicity]&lt;br /&gt;Difference from 22edo
|-
| 1
| Diminished 2nd&lt;sub&gt;10&lt;/sub&gt;
| Minor second
| 60{{c}} [12] &lt;br /&gt;5.4545{{c}}
| 48{{c}} [10] &lt;br /&gt;−6.5455{{c}}
|-
| 2
| Minor 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented seventh
| 108{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 120{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 3
| Major 2nd&lt;sub&gt;10&lt;/sub&gt;
| Augmented unison
| ''168{{c}} [14]'' &lt;br /&gt;4.364{{c}}
| ''156{{c}} [8]'' &lt;br /&gt;−7.636{{c}}
|-
| 4
| Minor 3rd&lt;sub&gt;10&lt;/sub&gt;
| Major second
| 216{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 228{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 5
| Major 3rd&lt;sub&gt;10&lt;/sub&gt;
| Minor third
| 276{{c}} [16] &lt;br /&gt;3.273{{c}}
| 264{{c}} [6] &lt;br /&gt;−8.727{{c}}
|-
| 6
| Minor 4th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fourth
| ''324{{c}} [16]''&lt;br /&gt;−3.273{{c}}
| ''336{{c}} [6]''&lt;br /&gt;8.727{{c}}
|-
| 7
| Major 4th&lt;sub&gt;10&lt;/sub&gt;
| Augmented second
| 384{{c}} [18] &lt;br /&gt;2.182{{c}}
| 372{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 8
| Augmented 4th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 5th&lt;sub&gt;10&lt;/sub&gt;
| Major third
| 432{{c}} [14] &lt;br /&gt;−4.364{{c}}
| 444{{c}} [8] &lt;br /&gt;7.636{{c}}
|-
| 9
| Perfect 5th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fourth
| 492{{c}} [20] &lt;br /&gt;1.091{{c}}
| 480{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 10
| Augmented 5th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 6th&lt;sub&gt;10&lt;/sub&gt;
| Diminished fifth
| 540{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 552{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 11
| Perfect 6th&lt;sub&gt;10&lt;/sub&gt;
| Augmented third &lt;br&gt; Diminished sixth
| 600{{c}} [20]
| 588{{c}} [1] &lt;br&gt;−12{{c}} &lt;br&gt; 612{{c}} [1] &lt;br&gt;12{{c}}
|-
| 12
| Augmented 6th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 7th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fourth
| 660{{c}} [12] &lt;br /&gt;6.5455{{c}}
| 648{{c}} [10] &lt;br /&gt;−5.4545{{c}}
|-
| 13
| Perfect 7th&lt;sub&gt;10&lt;/sub&gt;
| Perfect fifth
| 708{{c}} [20] &lt;br /&gt;−1.091{{c}}
| 720{{c}} [2] &lt;br /&gt;10.909{{c}}
|-
| 14
| Augmented 7th&lt;sub&gt;10&lt;/sub&gt;

Diminished 8th&lt;sub&gt;10&lt;/sub&gt;
| Minor sixth
| 768{{c}} [14] &lt;br /&gt;4.364{{c}}
| 756{{c}} [8] &lt;br /&gt;−7.636{{c}}
|-
| 15
| Minor 8th&lt;sub&gt;10&lt;/sub&gt;
| Diminished seventh
| 816{{c}} [18] &lt;br /&gt;−2.182{{c}}
| 828{{c}} [4] &lt;br /&gt;9.818{{c}}
|-
| 16
| Major 8th&lt;sub&gt;10&lt;/sub&gt;
| Augmented fifth
| ''876{{c}} [16]''&lt;br /&gt;3.273{{c}}
| ''864{{c}} [6]''&lt;br /&gt;−8.727{{c}}
|-
| 17
| Minor 9th&lt;sub&gt;10&lt;/sub&gt;
| Major sixth
| 924{{c}} [16] &lt;br /&gt;−3.273{{c}}
| 936{{c}} [6] &lt;br /&gt;8.727{{c}}
|-
| 18
| Major 9th&lt;sub&gt;10&lt;/sub&gt;
| Minor seventh
| 984{{c}} [18] &lt;br /&gt;2.182{{c}}
| 972{{c}} [4] &lt;br /&gt;−9.818{{c}}
|-
| 19
| Minor 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished octave
| ''1032{{c}} [14]''&lt;br /&gt;−4.364{{c}}
| ''1044{{c}} [8]''&lt;br /&gt;7.636{{c}}
|-
| 20
| Major 10th&lt;sub&gt;10&lt;/sub&gt;
| Diminished second
| 1092{{c}} [20] &lt;br /&gt;1.091{{c}}
| 1080{{c}} [2] &lt;br /&gt;−10.909{{c}}
|-
| 21
| Augmented 10th&lt;sub&gt;10&lt;/sub&gt; &lt;br&gt; Diminished 11th&lt;sub&gt;10&lt;/sub&gt;
| Major seventh
| 1140{{c}} [12] &lt;br /&gt;−5.4545{{c}}
| 1152{{c}} [10] &lt;br /&gt;6.5455{{c}}
|-
| 22
| 11th&lt;sub&gt;10&lt;/sub&gt;
| Octave
| 1200{{c}} [22]
| N/A
|}

Alternatively, the unmodified, symmetrical [[2mos]] scale 5 4 5 4 5 4 5 4 5 4 5 5 4 5 4 5 4 5 4 5 4 5 could be used instead. This scale is very similar to the modified version except that it lacks dog tritones; every 6th&lt;sub&gt;10&lt;/sub&gt; is exactly 600{{c}}. Because it repeats every half-octave, this scale could be used to construct straight-fretted guitars as long as they are {{w|Augmented-fourths tuning|tuned in tritones}}. This makes guitar construction much easier compared to other non-equally-tempered scales. The modmos would allow ''almost'' all the frets to be straight if the tritones tuning is used; only every eleventh fret would need to be curved. While the 2mos is simpler, the modmos very closely approximates the [[Indian]] sruti system.

Other, "gentle" alternatives to 22edo for pajara include [[78edo|78ddd]] and [[56edo|56d]]. The resulting 22-note scales have large and small steps in ratios of 4:3 or 3:2, respectively, and the rest of the spectrum of {{nowrap|22 &amp;amp; [[34edo|34d]]}} temperaments is also usable. On the other hand, the "rough" alternatives to 22edo for pajara include [[58edo|58d]] and [[46edo|46d]]. The resulting 22-note scales have large and small steps in ratios of 4:1 or 3:1, respectively, and the rest of the spectrum of {{nowrap|12 &amp;amp; [[34edo|34d]]}} temperaments up to 58d is also usable.

== Instruments ==
=== Lumatone ===
[[Lumatone mapping for 100edo]]
=== Skip fretting ===
One way to play 100edo on a [[20edo]] guitar is to tune the strings 13\100 apart, or 156 [[cents]]. All examples on this page are for 7-string guitar.

; Prime harmonics

1/1: string 2 open 

2/1: string 7 fret 7

3/2: string 5 fret 4

5/4: string 1 fret 9

7/4: string 4 fret 11 

11/8: string 4 fret 4 

13/8: string 7 fret 1

17/16: string 5 fret 14 

19/16: string 2 fret 5 

23/16: string 6 open 

29/16: string 4 fret 12 

31/16: string 7 fret 6 

37/32: string 4 fret 19

41/32: string 4 fret 2 

43/32: string 3 fret 6 

47/32: string 4 fret 6

53/32: string 3 fret 12

59/32: string 3 fret 15

61/32: string 3 fret 16

== Video ==
&lt;youtube&gt;shcrw2vtmJU&lt;/youtube&gt;

== Music ==

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/37bFvBsKXqo ''100edo''] (2022)

; [[Iceface H. Wakabayashi]] (微分音チャンネル)
* [https://www.youtube.com/watch?v=shcrw2vtmJU ''Microtonal Piano in 100 tone equal temperament (100EDO) (Microtonal Music)''] (2016) (this is the same as the video linked above, to use in case the embedded video refuses to play)</text>
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      <text bytes="451" sha1="55y356j7ow2sywmln27smhqemr2whil" xml:space="preserve">The 1012 equal division divides the octave into 1012 equal parts of 1.1858 cents each. It is a strong 13-limit system, distinctly consistent through the 15 limit. It is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is 2401/2400, 4096/4095, 6656/6655, 9801/9800 and | 2 6 -1 2 0 4 &amp;gt;. 1012 is divisible by [[22edo|22]], [[46edo|46]] and [[253edo|253]].</text>
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      <comment>**Imported revision 601732944 - Original comment: **</comment>
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-12-08 14:43:05 UTC&lt;/tt&gt;.&lt;br&gt;
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&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 1012 equal division divides the octave into 1012 equal parts of 1.1858 cents each. It is a strong 13-limit system, distinctly consistent through the 15 limit. It is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is 2401/2400, 4096/4095, 6656/6655, 9801/9800 and | 2 6 -1 2 0 4 &amp;gt;. 1012 is divisible by [[22edo|22]], [[46edo|46]] and [[253edo|253]].&lt;/pre&gt;&lt;/div&gt;
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
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      <text bytes="524" sha1="a9aklz7jj6ztj8hhy5uwl1htcbjx3ma" xml:space="preserve">The 1012 equal division divides the octave into 1012 equal parts of 1.1858 cents each. It is a strong 13-limit system, distinctly consistent through the 15 limit. It is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is 2401/2400, 4096/4095, 6656/6655, 9801/9800 and | 2 6 -1 2 0 4 &amp;gt;. 1012 is divisible by [[22edo|22]], [[46edo|46]] and [[253edo|253]].

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
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      <text bytes="539" sha1="2hb23loix8hw4k5nfpx15opqu8uhai3" xml:space="preserve">{{Infobox ET}}
The 1012 equal division divides the octave into 1012 equal parts of 1.1858 cents each. It is a strong 13-limit system, distinctly consistent through the 15 limit. It is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is 2401/2400, 4096/4095, 6656/6655, 9801/9800 and | 2 6 -1 2 0 4 &amp;gt;. 1012 is divisible by [[22edo|22]], [[46edo|46]] and [[253edo|253]].

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
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      <comment>hold on, my browser is lagging</comment>
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      <text bytes="432" sha1="22g0sbjbwu0d4mgh4glojoefxcb649q" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}

== Theory ==
It is a strong 13-limit system, distinctly consistent through the 15 limit. It is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is 2401/2400, 4096/4095, 6656/6655, 9801/9800 and | 2 6 -1 2 0 4 &amp;gt;.

1012edo conta

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="813" sha1="k4j02tnimrrrd9bag5y3r8z1z68px99" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}

== Theory ==
It is a strong 13-limit system, distinctly consistent through the 15 limit. It is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is 2401/2400, 4096/4095, 6656/6655, 9801/9800 and {{monzo|2 6 -1 2 0 4}}.

1012 has divisors {{EDOs|1, 2, 4, 11, 22, 23, 44, 46, 92, 253, 506}}. 

In addition to containing 22edo and 23edo, it contains a [[22L 1s|quartismoid]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with a 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo|18 15 -12 -1  0 -3}}.

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>k4j02tnimrrrd9bag5y3r8z1z68px99</sha1>
    </revision>
    <revision>
      <id>97824</id>
      <parentid>97787</parentid>
      <timestamp>2022-10-22T13:56:29Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>A little rework</comment>
      <origin>97824</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="839" sha1="o4ri82phqsf2cl57pyrp6e667lwa525" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}

== Theory ==
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.

1012 has divisors {{EDOs| 1, 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>o4ri82phqsf2cl57pyrp6e667lwa525</sha1>
    </revision>
    <revision>
      <id>102180</id>
      <parentid>97824</parentid>
      <timestamp>2023-01-31T15:21:06Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup and +prime error table</comment>
      <origin>102180</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="838" sha1="0q4avmyc57m0a5d16g9njn7yokqxxnv" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}

1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Divisors ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

=== Trivia ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.</text>
      <sha1>0q4avmyc57m0a5d16g9njn7yokqxxnv</sha1>
    </revision>
    <revision>
      <id>102280</id>
      <parentid>102180</parentid>
      <timestamp>2023-02-01T01:02:20Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>102280</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="943" sha1="ctno20cysrs4jhzo93a2phzfj3uyy7c" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}

1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Divisors ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

=== Trivia ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.</text>
      <sha1>ctno20cysrs4jhzo93a2phzfj3uyy7c</sha1>
    </revision>
    <revision>
      <id>102389</id>
      <parentid>102280</parentid>
      <timestamp>2023-02-03T11:56:32Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>rank 2 temperaments</comment>
      <origin>102389</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1361" sha1="3qxfk2hok2volflmo3qmrg2ldyytebn" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}
== Theory ==
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Divisors ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

=== Trivia ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}</text>
      <sha1>3qxfk2hok2volflmo3qmrg2ldyytebn</sha1>
    </revision>
    <revision>
      <id>102602</id>
      <parentid>102389</parentid>
      <timestamp>2023-02-10T12:58:37Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Better not to advertise the 2.3.7.11.101 subgroup property</comment>
      <origin>102602</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1374" sha1="rymx9p6n58lsfuwpp9ts51qyzasyesi" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}
== Theory ==
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

=== Trivia ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}</text>
      <sha1>rymx9p6n58lsfuwpp9ts51qyzasyesi</sha1>
    </revision>
    <revision>
      <id>103654</id>
      <parentid>102602</parentid>
      <timestamp>2023-02-23T23:54:24Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Subsets and supersets */</comment>
      <origin>103654</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1480" sha1="hzqil8iy51c5vmlfqlto0kdnuv1sub8" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}
== Theory ==
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

[[2024edo]], which divides the edostep in two, provides a good correction for the 17- and the 19-limit. 

=== Trivia ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}</text>
      <sha1>hzqil8iy51c5vmlfqlto0kdnuv1sub8</sha1>
    </revision>
    <revision>
      <id>103655</id>
      <parentid>103654</parentid>
      <timestamp>2023-02-23T23:54:45Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Prime harmonics */ oops</comment>
      <origin>103655</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1473" sha1="ks991z3gn90aehqaxgbnzuahemnxsct" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}
== Theory ==
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

[[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic. 

=== Trivia ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}</text>
      <sha1>ks991z3gn90aehqaxgbnzuahemnxsct</sha1>
    </revision>
    <revision>
      <id>115935</id>
      <parentid>103655</parentid>
      <timestamp>2023-06-07T13:23:13Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>115935</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1583" sha1="lyul159qsd2pzsjr8o1irvd2754omsw" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}
== Theory ==
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit. 

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

[[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic. 

=== Trivia ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}</text>
      <sha1>lyul159qsd2pzsjr8o1irvd2754omsw</sha1>
    </revision>
    <revision>
      <id>115939</id>
      <parentid>115935</parentid>
      <timestamp>2023-06-07T14:22:49Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>115939</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1905" sha1="p0rrinoth26tx3zr9xm9rvqxpvepot7" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}
== Theory ==
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, it is enfactored, with the same mapping as [[506edo]], providing a tuning for [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it tempers out the [[breedsma]], 2401/2400. Furthermore, it is a [[septiruthenia]]&lt;nowiki/&gt;n system, tempering 64/63 comma to 23 steps, 1/44th of the octave.It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

[[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic. 

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}</text>
      <sha1>p0rrinoth26tx3zr9xm9rvqxpvepot7</sha1>
    </revision>
    <revision>
      <id>115941</id>
      <parentid>115939</parentid>
      <timestamp>2023-06-07T14:27:22Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */ elaborate, "trivia" is too light and simple of a word and i've outlined what "trivia" means on xa discord, "other techniques" is more neutral</comment>
      <origin>115941</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2008" sha1="asdk8c4qh9taea5qztu7fm3qlrd8v1j" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}
== Theory ==
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is enfactored, with the same mapping as [[506edo]], providing a tuning for [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it tempers out the [[breedsma]], 2401/2400, and tunes [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 subgroup, it is a [[septiruthenia]]&lt;nowiki/&gt;n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

[[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic. 

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}</text>
      <sha1>asdk8c4qh9taea5qztu7fm3qlrd8v1j</sha1>
    </revision>
    <revision>
      <id>115942</id>
      <parentid>115941</parentid>
      <timestamp>2023-06-07T14:27:52Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
      <origin>115942</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1998" sha1="a39e78rfh1f9js1vzwsi7rxw9bdn3jg" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}
== Theory ==
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is enfactored, with the same mapping as [[506edo]], providing a tuning for [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it tempers out the [[breedsma]], 2401/2400, and tunes [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 subgroup, it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

[[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}</text>
      <sha1>a39e78rfh1f9js1vzwsi7rxw9bdn3jg</sha1>
    </revision>
    <revision>
      <id>119390</id>
      <parentid>115942</parentid>
      <timestamp>2023-07-09T05:23:23Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119390</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2017" sha1="qpcg7k3zqh87i8gbxkj6lwvyzglenef" xml:space="preserve">{{novelty}}{{stub}}{{Infobox ET}}
{{EDO intro|1012}}
== Theory ==
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is enfactored, with the same mapping as [[506edo]], providing a tuning for [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it tempers out the [[breedsma]], 2401/2400, and tunes [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 subgroup, it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

[[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}</text>
      <sha1>qpcg7k3zqh87i8gbxkj6lwvyzglenef</sha1>
    </revision>
    <revision>
      <id>119598</id>
      <parentid>119390</parentid>
      <timestamp>2023-07-09T06:49:24Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119598</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1998" sha1="a39e78rfh1f9js1vzwsi7rxw9bdn3jg" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}
== Theory ==
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is enfactored, with the same mapping as [[506edo]], providing a tuning for [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it tempers out the [[breedsma]], 2401/2400, and tunes [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 subgroup, it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

[[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}</text>
      <sha1>a39e78rfh1f9js1vzwsi7rxw9bdn3jg</sha1>
    </revision>
    <revision>
      <id>126167</id>
      <parentid>119598</parentid>
      <timestamp>2023-10-19T15:38:24Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Clarify the title row of the rank-2 temp table</comment>
      <origin>126167</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2133" sha1="bfg4n1csvw5asoai59vbc7cwppzj7nv" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}

== Theory ==
1012edo is a strong 13-limit system, distinctly [[consistent]] through the 15-odd-limit. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. A basis for the 13-limit commas is [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is enfactored, with the same mapping as [[506edo]], providing a tuning for [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it tempers out the [[breedsma]], 2401/2400, and tunes [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 subgroup, it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
1012 has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. 

[[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct</text>
      <sha1>bfg4n1csvw5asoai59vbc7cwppzj7nv</sha1>
    </revision>
    <revision>
      <id>126672</id>
      <parentid>126167</parentid>
      <timestamp>2023-10-31T10:59:38Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Adopt template: Factorization; misc. cleanup</comment>
      <origin>126672</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="gtywmnx2pl6i053a6ahkujzmv1r4hbb" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}

== Theory ==
1012edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
Since 1012 factors into {{factorization|1012}}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct</text>
      <sha1>gtywmnx2pl6i053a6ahkujzmv1r4hbb</sha1>
    </revision>
    <revision>
      <id>130884</id>
      <parentid>126672</parentid>
      <timestamp>2024-01-05T22:13:18Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Regular temperament properties */ +major arcana</comment>
      <origin>130884</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2371" sha1="fftgktl2mguep67bny9b1vtik2udak6" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}

== Theory ==
1012edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
Since 1012 factors into {{factorization|1012}}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
|22
|420\1012&lt;br&gt;(6\1012)
|498.023&lt;br&gt;(7.115)
|4/3
({{monzo|16 -13 2}})
|[[Major arcana]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct</text>
      <sha1>fftgktl2mguep67bny9b1vtik2udak6</sha1>
    </revision>
    <revision>
      <id>130885</id>
      <parentid>130884</parentid>
      <timestamp>2024-01-05T22:17:58Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Regular temperament properties */ oops remove that as that is already supported by 506et</comment>
      <origin>130885</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2278" sha1="gtywmnx2pl6i053a6ahkujzmv1r4hbb" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}

== Theory ==
1012edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
Since 1012 factors into {{factorization|1012}}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.023&lt;br&gt;(7.115)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct</text>
      <sha1>gtywmnx2pl6i053a6ahkujzmv1r4hbb</sha1>
    </revision>
    <revision>
      <id>165753</id>
      <parentid>130885</parentid>
      <timestamp>2024-11-16T02:25:36Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>165753</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1996" sha1="fo5gqk5el5fpp0txrzavsta6e0d8dp4" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}

== Theory ==
1012edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
Since 1012 factors into {{factorization|1012}}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{{rank-2 begin}}
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br /&gt;(6\1012)
| 498.023&lt;br /&gt;(7.115)
| 4/3&lt;br /&gt;(18375/18304)
| [[Ruthenium]]
{{rank-2 end}}
{{orf}}</text>
      <sha1>fo5gqk5el5fpp0txrzavsta6e0d8dp4</sha1>
    </revision>
    <revision>
      <id>166272</id>
      <parentid>165753</parentid>
      <timestamp>2024-11-16T13:08:42Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Partial undo</comment>
      <origin>166272</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2309" sha1="ldm4xnujczckvj36t8bhg3pg6fnldey" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}

== Theory ==
1012edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
Since 1012 factors into {{factorization|1012}}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br /&gt;(6\1012)
| 498.023&lt;br /&gt;(7.115)
| 4/3&lt;br /&gt;(18375/18304)
| [[Ruthenium]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct</text>
      <sha1>ldm4xnujczckvj36t8bhg3pg6fnldey</sha1>
    </revision>
    <revision>
      <id>176141</id>
      <parentid>166272</parentid>
      <timestamp>2025-01-15T16:18:00Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176141</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2303" sha1="j91cfwz67t99jf3zzs4m30g6peq3xud" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1012}}

== Theory ==
1012edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
Since 1012 factors into {{factorization|1012}}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br /&gt;(6\1012)
| 498.023&lt;br /&gt;(7.115)
| 4/3&lt;br /&gt;(18375/18304)
| [[Ruthenium]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct</text>
      <sha1>j91cfwz67t99jf3zzs4m30g6peq3xud</sha1>
    </revision>
    <revision>
      <id>181303</id>
      <parentid>176141</parentid>
      <timestamp>2025-02-18T16:45:51Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181303</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2297" sha1="jgcnte12zvaa44holji8498s610kpqj" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1012edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
Since 1012 factors into {{factorization|1012}}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br /&gt;(6\1012)
| 498.023&lt;br /&gt;(7.115)
| 4/3&lt;br /&gt;(18375/18304)
| [[Ruthenium]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct</text>
      <sha1>jgcnte12zvaa44holji8498s610kpqj</sha1>
    </revision>
    <revision>
      <id>225801</id>
      <parentid>181303</parentid>
      <timestamp>2026-03-13T13:32:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct" to "[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct"</comment>
      <origin>225801</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2352" sha1="lrf3z0pmzu2n1qyxhy2fzcq3qypv64w" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1012edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Other techniques ===
In addition to containing 22edo and 23edo, it contains a [[22L 1s]] scale produced by generator of 45\1012 associated with [[33/32]], and is associated with the 45 &amp; 1012 temperament, making it [[concoctic]]. A comma basis for the 13-limit is 2401/2400, 6656/6655, 123201/123200, {{monzo| 18 15 -12 -1  0 -3 }}.

In the 2.3.7.11.101, it tempers out [[7777/7776]] and is a tuning for the [[neutron star]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
Since 1012 factors into {{factorization|1012}}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br /&gt;(6\1012)
| 498.023&lt;br /&gt;(7.115)
| 4/3&lt;br /&gt;(18375/18304)
| [[Ruthenium]]
|}
&lt;nowiki /&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct</text>
      <sha1>lrf3z0pmzu2n1qyxhy2fzcq3qypv64w</sha1>
    </revision>
    <revision>
      <id>233686</id>
      <parentid>225801</parentid>
      <timestamp>2026-07-12T10:26:50Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>XW: NG</comment>
      <origin>233686</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1909" sha1="rbi6ibrmhsgqfgp0ghtdnbyrl15sqrx" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1012edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
Since 1012 factors into {{factorization|1012}}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 361\1012
| 428.066
| 2800/2187 
| [[Osiris]]
|-
| 2
| 491\1012
| 498.023
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br /&gt;(6\1012)
| 498.023&lt;br /&gt;(7.115)
| 4/3&lt;br /&gt;(18375/18304)
| [[Ruthenium]]
|}
&lt;nowiki /&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct</text>
      <sha1>rbi6ibrmhsgqfgp0ghtdnbyrl15sqrx</sha1>
    </revision>
    <revision>
      <id>233687</id>
      <parentid>233686</parentid>
      <timestamp>2026-07-12T10:30:49Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+ semiosiris. Correct math. Style</comment>
      <origin>233687</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1842" sha1="08j3zxq5mu4kj2e7b65h0diz9vso2u3" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1012edo is a strong 13-limit system, [[consistency|distinctly consistent]] through the [[15-odd-limit]]. It is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. A basis for the 13-limit [[comma]]s consists of [[2401/2400]], [[4096/4095]], [[6656/6655]], [[9801/9800]] and {{monzo| 2 6 -1 2 0 4 }}.  

In the 5-limit, 1012edo is [[enfactoring|enfactored]], with the same tuning as [[506edo]], [[support]]ing [[vishnu]], [[monzismic]], and [[lafa]]. In the 7-limit, it [[tempering out|tempers out]] the [[breedsma]], 2401/2400, and tunes the [[osiris]] temperament. Furthermore, noting its exceptional strength in the 2.3.7 [[subgroup]], it is a [[septiruthenia]]n system, tempering 64/63 comma to 1/44th of the octave, that is 23 steps. It provides the [[optimal patent val]] for [[quarvish]] temperament in the 7-limit and also in the 11-limit.  

=== Prime harmonics ===
{{Harmonics in equal|1012}}

=== Subsets and supersets ===
Since 1012 factors into {{nowrap| 2&lt;sup&gt;2&lt;/sup&gt; × 11 × 23 }}, 1012edo has subset edos {{EDOs| 2, 4, 11, 22, 23, 44, 46, 92, 253, 506 }}. [[2024edo]], which divides the edostep in two, provides a good correction for the 17th harmonic.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 361\1012
| 428.06
| 2800/2187
| [[Osiris]]
|-
| 2
| 145\1012
| 171.94
| 243/220
| [[Semiosiris]]
|-
| 2
| 420\1012
| 498.02
| 7/5
| [[Quarvish]]
|-
| 44
| 420\1012&lt;br&gt;(6\1012)
| 498.02&lt;br&gt;(7.11)
| 4/3&lt;br&gt;(18375/18304)
| [[Ruthenium]]
|}
&lt;nowiki/&gt;* In [[normal forms #Minimal-generator form|minimal-generator form]]</text>
      <sha1>08j3zxq5mu4kj2e7b65h0diz9vso2u3</sha1>
    </revision>
  </page>
  <page>
    <title>101edo</title>
    <ns>0</ns>
    <id>18</id>
    <revision>
      <id>18</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>18</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2852" sha1="kjg427oibvwph2br33kt8542cpff50e" xml:space="preserve">'''''101-EDO''''' divides the [[Octave|octave]] into 101 equal parts of 11.881 [[cent|cent]]s each. It can be used to tune the [[Schismatic_family|grackle temperament]]. It is the 26th [[prime_numbers|prime]] edo. The 101cd val provides an excellent tuning for [[Magic_family#Witchcraft|witchcraft temperament]], falling between the 13 and 15 limit least squares tuning.

[[5-limit|5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit|7-limit]] commas: 126/125, 32805/32768, 2430/2401

==&lt;u&gt;Some important MOS scales:&lt;/u&gt;==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable"
|-
| | 25
| | 297.03
|-
| | 38
| | 451.485
|-
| | 63
| | 748.515
|-
| | 88
| | 1045.545
|}
'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable"
|-
| | '''17'''
| | '''201.98'''
|-
| | 34
| | 403.96
|-
| | '''42'''
| | '''499.01'''
|-
| | '''59'''
| | '''700.99'''
|-
| | '''76'''
| | '''902.97'''
|-
| | 93
| | 1104.95
|}
'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable"
|-
| | 13
| | 154.455
|-
| | 26
| | 308.911
|-
| | 39
| | 463.366
|-
| | 52
| | 617.822
|-
| | 65
| | 772.277
|-
| | 78
| | 926.733
|-
| | 91
| | 1081.188
|}
'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable"
|-
| | '''13/101'''
| | '''154.455'''
|-
| | '''26/101'''
| | '''308.911'''
|-
| | 39/101
| | 463.366
|-
| | '''44/101'''
| | '''522.772'''
|-
| | '''57/101'''
| | '''677.228'''
|-
| | '''70/101'''
| | '''831.683'''
|-
| | '''83/101'''
| | '''986.139'''
|-
| | 96/101
| | 1045.545
|}
'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable"
|-
| | '''10'''
| | '''118.812'''
|-
| | 20
| | 237.624
|-
| | '''27'''
| | '''320.792'''
|-
| | '''37'''
| | '''439.604'''
|-
| | '''47'''
| | '''558.416'''
|-
| | 57
| | 677.228
|-
| | '''64'''
| | '''760.396'''
|-
| | '''74'''
| | '''879.218'''
|-
| | '''84'''
| | '''998.03'''
|-
| | 94
| | 1116.842
|}
'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
|-
| | '''7'''
| | '''83.168'''
|-
| | '''14'''
| | '''166.337'''
|-
| | 22
| | 261.386
|-
| | '''29'''
| | '''344.5545'''
|-
| | '''36'''
| | '''427.723'''
|-
| | '''43'''
| | '''510.891'''
|-
| | '''50'''
| | '''594.0595'''
|-
| | 58
| | 689.119
|-
| | '''65'''
| | '''772.287'''
|-
| | '''72'''
| | '''855.4455'''
|-
| | '''79'''
| | '''938.614'''
|-
| | '''86'''
| | '''1021.782'''
|-
| | 93
| | 1104.95
|}

=Links=
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]      [[Category:101-tone]]
[[Category:101edo]]
[[Category:armodue]]
[[Category:edo]]
[[Category:equal]]
[[Category:grackle]]
[[Category:prime_edo]]
[[Category:pythagorean]]
[[Category:scales]]
[[Category:todo:improve_leayout]]
[[Category:todo:unify_precision]]</text>
      <sha1>kjg427oibvwph2br33kt8542cpff50e</sha1>
    </revision>
    <revision>
      <id>4016</id>
      <timestamp>2016-12-29T10:21:49Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602892676 - Original comment: tel-links removed**</comment>
      <origin>4016</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13588" sha1="dgbhfk4cimm9wb9x41c99szz1hl4hkm" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2016-12-29 10:21:49 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602892676&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;tel-links removed&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**//101-EDO//** divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]]. It is the 26th [[prime numbers|prime]] edo. The 101cd val provides an excellent tuning for [[Magic family#Witchcraft|witchcraft temperament]], falling between the 13 and 15 limit least squares tuning.

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**25 13 25 25 13:** //3L2s MOS// (Pentatonic)
|| 25 || 297.03 ||
|| 38 || 451.485 ||
|| 63 || 748.515 ||
|| 88 || 1045.545 ||
**17 17 8 17 17 17 8:** //5L2s MOS// (Diatonic Pythagorean)
|| **17** || **201.98** ||
|| 34 || 403.96 ||
|| **42** || **499.01** ||
|| **59** || **700.99** ||
|| **76** || **902.97** ||
|| 93 || 1104.95 ||
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic)
|| 13 || 154.455 ||
|| 26 || 308.911 ||
|| 39 || 463.366 ||
|| 52 || 617.822 ||
|| 65 || 772.277 ||
|| 78 || 926.733 ||
|| 91 || 1081.188 ||
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Superdiatonic 1/13-tone 13;5 relation)
|| **13/101** || **154.455** ||
|| **26/101** || **308.911** ||
|| 39/101 || 463.366 ||
|| **44/101** || **522.772** ||
|| **57/101** || **677.228** ||
|| **70/101** || **831.683** ||
|| **83/101** || **986.139** ||
|| 96/101 || 1045.545 ||
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11)
|| **10** || **118.812** ||
|| 20 || 237.624 ||
|| **27** || **320.792** ||
|| **37** || **439.604** ||
|| **47** || **558.416** ||
|| 57 || 677.228 ||
|| **64** || **760.396** ||
|| **74** || **879.218** ||
|| **84** || **998.03** ||
|| 94 || 1116.842 ||
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Anti-Ketradektriatoh)
|| **7** || **83.168** ||
|| **14** || **166.337** ||
|| 22 || 261.386 ||
|| **29** || **344.5545** ||
|| **36** || **427.723** ||
|| **43** || **510.891** ||
|| **50** || **594.0595** ||
|| 58 || 689.119 ||
|| **65** || **772.287** ||
|| **72** || **855.4455** ||
|| **79** || **938.614** ||
|| **86** || **1021.782** ||
|| 93 || 1104.95 ||

=Links= 
[[http://tech.groups.yahoo.com/group/tuning-math/message/11157|The Ellis duodene in 101-equal]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;em&amp;gt;101-EDO&amp;lt;/em&amp;gt;&amp;lt;/strong&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;. It is the 26th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo. The 101cd val provides an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family#Witchcraft"&amp;gt;witchcraft temperament&amp;lt;/a&amp;gt;, falling between the 13 and 15 limit least squares tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;25 13 25 25 13:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Pentatonic)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;297.03&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;38&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;451.485&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;748.515&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1045.545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;17&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;201.98&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;403.96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;42&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;499.01&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;59&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;700.99&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;76&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;902.97&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1104.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;154.455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308.911&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;463.366&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;617.822&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;65&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;772.277&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;78&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;926.733&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1081.188&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Superdiatonic 1/13-tone 13;5 relation)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;13/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;154.455&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;26/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;308.911&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;463.366&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;44/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;522.772&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;57/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;677.228&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;70/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;831.683&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;83/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;986.139&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;96/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1045.545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;10&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;118.812&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;237.624&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;27&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;320.792&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;37&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;439.604&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;47&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;558.416&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;677.228&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;64&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;760.396&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;74&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;879.218&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;84&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;998.03&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1116.842&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Anti-Ketradektriatoh)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;83.168&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;14&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;166.337&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;261.386&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;29&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;344.5545&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;36&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;427.723&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;43&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;510.891&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;50&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;594.0595&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;689.119&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;65&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;772.287&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;72&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;855.4455&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;79&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;938.614&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;86&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;1021.782&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1104.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11157" rel="nofollow"&amp;gt;The Ellis duodene in 101-equal&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>dgbhfk4cimm9wb9x41c99szz1hl4hkm</sha1>
    </revision>
    <revision>
      <id>4017</id>
      <timestamp>2016-10-28T08:55:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 597378010 - Original comment: **</comment>
      <origin>4017</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14507" sha1="ky8uebk7t1bw6hc46plfliou0ltz6ur" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-28 08:55:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;597378010&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**//101-EDO//** divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]]. It is the 26th [[prime numbers|prime]] edo. The 101cd val provides an excellent tuning for [[Magic family#Witchcraft|witchcraft temperament]], falling between the 13 and 15 limit least squares tuning.

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**[[tel/25 13 25 25 13|25 13 25 25 13]]:** //3L2s MOS// (Pentatonic)
|| 25 || 297.03 ||
|| 38 || 451.485 ||
|| 63 || 748.515 ||
|| 88 || 1045.545 ||
**[[tel/17 17 8 17 17 17 8|17 17 8 17 17 17 8]]:** //5L2s MOS// (Diatonic Pythagorean)
|| **17** || **201.98** ||
|| 34 || 403.96 ||
|| **42** || **499.01** ||
|| **59** || **700.99** ||
|| **76** || **902.97** ||
|| 93 || 1104.95 ||
**[[tel/13 13 13 13 13 13 13|13 13 13 13 13 13 13]] 10:** //7L1s MOS// (Grumpy Octatonic)
|| 13 || 154.455 ||
|| 26 || 308.911 ||
|| 39 || 463.366 ||
|| 52 || 617.822 ||
|| 65 || 772.277 ||
|| 78 || 926.733 ||
|| 91 || 1081.188 ||
**[[tel/13 13 13 5 13 13 13|13 13 13 5 13 13 13]] 13 5:** //7L2s MOS// (Superdiatonic 1/13-tone 13;5 relation)
|| **13/101** || **154.455** ||
|| **26/101** || **308.911** ||
|| 39/101 || 463.366 ||
|| **44/101** || **522.772** ||
|| **57/101** || **677.228** ||
|| **70/101** || **831.683** ||
|| **83/101** || **986.139** ||
|| 96/101 || 1045.545 ||
**[[tel/10 10 7 10 10 10 7 10|10 10 7 10 10 10 7 10]] 10 10 7:** //8L3s MOS// (Improper Sensi-11)
|| **10** || **118.812** ||
|| 20 || 237.624 ||
|| **27** || **320.792** ||
|| **37** || **439.604** ||
|| **47** || **558.416** ||
|| 57 || 677.228 ||
|| **64** || **760.396** ||
|| **74** || **879.218** ||
|| **84** || **998.03** ||
|| 94 || 1116.842 ||
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Anti-Ketradektriatoh)
|| **7** || **83.168** ||
|| **14** || **166.337** ||
|| 22 || 261.386 ||
|| **29** || **[[tel/344.5545|344.5545]]** ||
|| **36** || **427.723** ||
|| **43** || **510.891** ||
|| **50** || **[[tel/594.0595|594.0595]]** ||
|| 58 || 689.119 ||
|| **65** || **772.287** ||
|| **72** || **[[tel/855.4455|855.4455]]** ||
|| **79** || **938.614** ||
|| **86** || **1021.782** ||
|| 93 || 1104.95 ||

=Links= 
[[http://tech.groups.yahoo.com/group/tuning-math/message/11157|The Ellis duodene in 101-equal]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;em&amp;gt;101-EDO&amp;lt;/em&amp;gt;&amp;lt;/strong&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;. It is the 26th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo. The 101cd val provides an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family#Witchcraft"&amp;gt;witchcraft temperament&amp;lt;/a&amp;gt;, falling between the 13 and 15 limit least squares tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/25%2013%2025%2025%2013"&amp;gt;25 13 25 25 13&amp;lt;/a&amp;gt;:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Pentatonic)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;297.03&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;38&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;451.485&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;748.515&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1045.545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/17%2017%208%2017%2017%2017%208"&amp;gt;17 17 8 17 17 17 8&amp;lt;/a&amp;gt;:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;17&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;201.98&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;403.96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;42&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;499.01&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;59&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;700.99&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;76&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;902.97&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1104.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/13%2013%2013%2013%2013%2013%2013"&amp;gt;13 13 13 13 13 13 13&amp;lt;/a&amp;gt; 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;154.455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308.911&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;463.366&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;617.822&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;65&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;772.277&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;78&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;926.733&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1081.188&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/13%2013%2013%205%2013%2013%2013"&amp;gt;13 13 13 5 13 13 13&amp;lt;/a&amp;gt; 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Superdiatonic 1/13-tone 13;5 relation)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;13/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;154.455&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;26/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;308.911&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;463.366&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;44/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;522.772&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;57/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;677.228&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;70/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;831.683&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;83/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;986.139&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;96/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1045.545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/10%2010%207%2010%2010%2010%207%2010"&amp;gt;10 10 7 10 10 10 7 10&amp;lt;/a&amp;gt; 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;10&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;118.812&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;237.624&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;27&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;320.792&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;37&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;439.604&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;47&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;558.416&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;677.228&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;64&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;760.396&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;74&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;879.218&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;84&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;998.03&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1116.842&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Anti-Ketradektriatoh)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;83.168&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;14&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;166.337&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;261.386&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;29&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/344.5545"&amp;gt;344.5545&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;36&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;427.723&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;43&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;510.891&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;50&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/594.0595"&amp;gt;594.0595&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;689.119&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;65&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;772.287&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;72&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/855.4455"&amp;gt;855.4455&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;79&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;938.614&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;86&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;1021.782&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1104.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11157" rel="nofollow"&amp;gt;The Ellis duodene in 101-equal&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ky8uebk7t1bw6hc46plfliou0ltz6ur</sha1>
    </revision>
    <revision>
      <id>4018</id>
      <timestamp>2016-10-28T08:55:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 597377982 - Original comment: **</comment>
      <origin>4018</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14519" sha1="hnskywx6gegrsftpvgnfpiikon5barj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-28 08:55:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;597377982&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**//101-EDO//** divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]]. It is the 26th [[prime numbers|prime]] edo. The 101cd val provides an excellent tuning for [[Magic family#Witchcraft|witchcraft temperament]], falling between the 13 and 15 limit least squares tuning.

[[5-limit]] [[#|commas]]: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**[[tel:25 13 25 25 13|25 13 25 25 13]]:** //3L2s MOS// (Pentatonic)
|| 25 || 297.03 ||
|| 38 || 451.485 ||
|| 63 || 748.515 ||
|| 88 || 1045.545 ||
**[[tel:17 17 8 17 17 17 8|17 17 8 17 17 17 8]]:** //5L2s MOS// (Diatonic Pythagorean)
|| **17** || **201.98** ||
|| 34 || 403.96 ||
|| **42** || **499.01** ||
|| **59** || **700.99** ||
|| **76** || **902.97** ||
|| 93 || 1104.95 ||
**[[tel:13 13 13 13 13 13 13|13 13 13 13 13 13 13]] 10:** //7L1s MOS// (Grumpy Octatonic)
|| 13 || 154.455 ||
|| 26 || 308.911 ||
|| 39 || 463.366 ||
|| 52 || 617.822 ||
|| 65 || 772.277 ||
|| 78 || 926.733 ||
|| 91 || 1081.188 ||
**[[tel:13 13 13 5 13 13 13|13 13 13 5 13 13 13]] 13 5:** //7L2s MOS// (Superdiatonic 1/13-tone 13;5 relation)
|| **13/101** || **154.455** ||
|| **26/101** || **308.911** ||
|| 39/101 || 463.366 ||
|| **44/101** || **522.772** ||
|| **57/101** || **677.228** ||
|| **70/101** || **831.683** ||
|| **83/101** || **986.139** ||
|| 96/101 || 1045.545 ||
**[[tel:10 10 7 10 10 10 7 10|10 10 7 10 10 10 7 10]] 10 10 7:** //8L3s MOS// (Improper Sensi-11)
|| **10** || **118.812** ||
|| 20 || 237.624 ||
|| **27** || **320.792** ||
|| **37** || **439.604** ||
|| **47** || **558.416** ||
|| 57 || 677.228 ||
|| **64** || **760.396** ||
|| **74** || **879.218** ||
|| **84** || **998.03** ||
|| 94 || 1116.842 ||
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Anti-Ketradektriatoh)
|| **7** || **83.168** ||
|| **14** || **166.337** ||
|| 22 || 261.386 ||
|| **29** || **[[tel:344.5545|344.5545]]** ||
|| **36** || **427.723** ||
|| **43** || **510.891** ||
|| **50** || **[[tel:594.0595|594.0595]]** ||
|| 58 || 689.119 ||
|| **65** || **772.287** ||
|| **72** || **[[tel:855.4455|855.4455]]** ||
|| **79** || **938.614** ||
|| **86** || **1021.782** ||
|| 93 || 1104.95 ||

=Links= 
[[http://tech.groups.yahoo.com/group/tuning-math/message/11157|The Ellis duodene in 101-equal]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;em&amp;gt;101-EDO&amp;lt;/em&amp;gt;&amp;lt;/strong&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;. It is the 26th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo. The 101cd val provides an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family#Witchcraft"&amp;gt;witchcraft temperament&amp;lt;/a&amp;gt;, falling between the 13 and 15 limit least squares tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; [[#|commas]]: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/25%2013%2025%2025%2013"&amp;gt;25 13 25 25 13&amp;lt;/a&amp;gt;:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Pentatonic)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;297.03&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;38&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;451.485&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;748.515&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1045.545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/17%2017%208%2017%2017%2017%208"&amp;gt;17 17 8 17 17 17 8&amp;lt;/a&amp;gt;:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;17&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;201.98&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;403.96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;42&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;499.01&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;59&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;700.99&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;76&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;902.97&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1104.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/13%2013%2013%2013%2013%2013%2013"&amp;gt;13 13 13 13 13 13 13&amp;lt;/a&amp;gt; 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;154.455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308.911&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;463.366&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;617.822&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;65&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;772.277&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;78&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;926.733&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1081.188&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/13%2013%2013%205%2013%2013%2013"&amp;gt;13 13 13 5 13 13 13&amp;lt;/a&amp;gt; 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Superdiatonic 1/13-tone 13;5 relation)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;13/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;154.455&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;26/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;308.911&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;463.366&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;44/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;522.772&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;57/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;677.228&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;70/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;831.683&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;83/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;986.139&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;96/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1045.545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/10%2010%207%2010%2010%2010%207%2010"&amp;gt;10 10 7 10 10 10 7 10&amp;lt;/a&amp;gt; 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;10&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;118.812&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;237.624&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;27&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;320.792&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;37&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;439.604&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;47&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;558.416&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;677.228&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;64&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;760.396&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;74&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;879.218&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;84&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;998.03&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1116.842&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Anti-Ketradektriatoh)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;83.168&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;14&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;166.337&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;261.386&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;29&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/344.5545"&amp;gt;344.5545&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;36&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;427.723&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;43&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;510.891&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;50&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/594.0595"&amp;gt;594.0595&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;689.119&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;65&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;772.287&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;72&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/855.4455"&amp;gt;855.4455&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;79&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;938.614&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;86&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;1021.782&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1104.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11157" rel="nofollow"&amp;gt;The Ellis duodene in 101-equal&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>hnskywx6gegrsftpvgnfpiikon5barj</sha1>
    </revision>
    <revision>
      <id>4019</id>
      <timestamp>2016-08-01T14:12:21Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 588490668 - Original comment: **</comment>
      <origin>4019</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13623" sha1="tq2rvoue6loprfo73ts3jlnljkw4w7k" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-08-01 14:12:21 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;588490668&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**//101-EDO//** divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]]. It is the 26th [[prime numbers|prime]] edo. The 101cd val provides an excellent tuning for [[Magic family#Witchcraft|witchcraft temperament]], falling between the 13 and 15 limit least squares tuning.

[[5-limit]] [[#|commas]]: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**25 13 25 25 13:** //3L2s MOS// (Pentatonic)
|| 25 || 297.03 ||
|| 38 || 451.485 ||
|| 63 || 748.515 ||
|| 88 || 1045.545 ||
**17 17 8 17 17 17 8:** //5L2s MOS// ([[#|Diatonic]] Pythagorean)
|| **17** || **201.98** ||
|| 34 || 403.96 ||
|| **42** || **499.01** ||
|| **59** || **700.99** ||
|| **76** || **902.97** ||
|| 93 || 1104.95 ||
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic)
|| 13 || 154.455 ||
|| 26 || 308.911 ||
|| 39 || 463.366 ||
|| 52 || 617.822 ||
|| 65 || 772.277 ||
|| 78 || 926.733 ||
|| 91 || 1081.188 ||
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Superdiatonic 1/13-tone 13;5 relation)
|| **13/101** || **154.455** ||
|| **26/101** || **308.911** ||
|| 39/101 || 463.366 ||
|| **44/101** || **522.772** ||
|| **57/101** || **677.228** ||
|| **70/101** || **831.683** ||
|| **83/101** || **986.139** ||
|| 96/101 || 1045.545 ||
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11)
|| **10** || **118.812** ||
|| 20 || 237.624 ||
|| **27** || **320.792** ||
|| **37** || **439.604** ||
|| **47** || **558.416** ||
|| 57 || 677.228 ||
|| **64** || **760.396** ||
|| **74** || **879.218** ||
|| **84** || **998.03** ||
|| 94 || 1116.842 ||
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Anti-Ketradektriatoh [[#|form]])
|| **7** || **83.168** ||
|| **14** || **166.337** ||
|| 22 || 261.386 ||
|| **29** || **344.5545** ||
|| **36** || **427.723** ||
|| **43** || **510.891** ||
|| **50** || **594.0595** ||
|| 58 || 689.119 ||
|| **65** || **772.287** ||
|| **72** || **855.4455** ||
|| **79** || **938.614** ||
|| **86** || **1021.782** ||
|| 93 || 1104.95 ||

=Links= 
[[http://tech.groups.yahoo.com/group/tuning-math/message/11157|The Ellis duodene in 101-equal]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;em&amp;gt;101-EDO&amp;lt;/em&amp;gt;&amp;lt;/strong&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;. It is the 26th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo. The 101cd val provides an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family#Witchcraft"&amp;gt;witchcraft temperament&amp;lt;/a&amp;gt;, falling between the 13 and 15 limit least squares tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; [[#|commas]]: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;25 13 25 25 13:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Pentatonic)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;297.03&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;38&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;451.485&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;748.515&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1045.545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; ([[#|Diatonic]] Pythagorean)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;17&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;201.98&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;403.96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;42&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;499.01&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;59&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;700.99&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;76&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;902.97&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1104.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;154.455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308.911&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;463.366&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;617.822&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;65&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;772.277&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;78&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;926.733&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1081.188&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Superdiatonic 1/13-tone 13;5 relation)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;13/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;154.455&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;26/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;308.911&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;463.366&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;44/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;522.772&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;57/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;677.228&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;70/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;831.683&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;83/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;986.139&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;96/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1045.545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;10&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;118.812&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;237.624&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;27&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;320.792&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;37&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;439.604&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;47&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;558.416&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;677.228&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;64&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;760.396&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;74&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;879.218&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;84&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;998.03&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1116.842&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Anti-Ketradektriatoh [[#|form]])&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;83.168&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;14&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;166.337&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;261.386&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;29&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;344.5545&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;36&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;427.723&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;43&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;510.891&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;50&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;594.0595&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;689.119&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;65&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;772.287&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;72&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;855.4455&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;79&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;938.614&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;86&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;1021.782&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1104.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11157" rel="nofollow"&amp;gt;The Ellis duodene in 101-equal&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>tq2rvoue6loprfo73ts3jlnljkw4w7k</sha1>
    </revision>
    <revision>
      <id>4020</id>
      <timestamp>2016-08-01T11:30:27Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 588470276 - Original comment: **</comment>
      <origin>4020</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11190" sha1="bxj5j6bg8ja6g7uqg7jr8yr7b9mmkas" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-08-01 11:30:27 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;588470276&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**//101-EDO//** divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]]. It is the 26th [[prime numbers|prime]] edo. The 101cd val provides an excellent tuning for [[Magic family#Witchcraft|witchcraft temperament]], falling between the 13 and 15 limit least squares tuning.

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**25 13 25 25 13:** //3L2s MOS// (Pentatonic)
|| 25/101 || 297.03 ||
|| 38/101 || 451.485 ||
|| 63/101 || 748.515 ||
|| 88/101 || 1045.545 ||
**17 17 8 17 17 17 8:** //5L2s MOS// (Diatonic Pythagorean)
|| **17/101** || **201.98** ||
|| 34/101 || 403.96 ||
|| **42/101** || **499.01** ||
|| **59/101** || **700.99** ||
|| **76/101** || **902.97** ||
|| 93/101 || 1104.95 ||
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic)
|| 13/101 || 154.455 ||
|| 26/101 || 308.911 ||
|| 39/101 || 463.366 ||
|| 52/101 || 617.822 ||
|| 65/101 || 772.277 ||
|| 78/101 || 926.733 ||
|| 91/101 || 1081.188 ||
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Superdiatonic 1/13-tone 13;5 relation)
|| **13/101** || **154.455** ||
|| **26/101** || **308.911** ||
|| 39/101 || 463.366 ||
|| **44/101** || **522.772** ||
|| **57/101** || **677.228** ||
|| **70/101** || **831.683** ||
|| **83/101** || **986.139** ||
|| 96/101 || 1045.545 ||
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11)
|| **10/101** || **118.812** ||
|| 20/101 || 237.624 ||
|| **27/101** || **320.792** ||
|| **37/101** || **439.604** ||
|| **47/101** || **558.416** ||
|| 57/101 || 677.228 ||
|| **64/101** || **760.396** ||
|| **74/101** || **879.218** ||
|| **84/101** || **998.03** ||
|| 94/101 || 1116.842 ||
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Anti-Ketradektriatoh form)

=Links= 
[[http://tech.groups.yahoo.com/group/tuning-math/message/11157|The Ellis duodene in 101-equal]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;em&amp;gt;101-EDO&amp;lt;/em&amp;gt;&amp;lt;/strong&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;. It is the 26th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo. The 101cd val provides an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family#Witchcraft"&amp;gt;witchcraft temperament&amp;lt;/a&amp;gt;, falling between the 13 and 15 limit least squares tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;25 13 25 25 13:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Pentatonic)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;25/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;297.03&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;38/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;451.485&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;63/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;748.515&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;88/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1045.545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;17/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;201.98&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;34/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;403.96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;42/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;499.01&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;59/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;700.99&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;76/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;902.97&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1104.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;154.455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308.911&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;463.366&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;52/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;617.822&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;65/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;772.277&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;78/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;926.733&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;91/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1081.188&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Superdiatonic 1/13-tone 13;5 relation)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;13/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;154.455&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;26/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;308.911&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;463.366&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;44/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;522.772&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;57/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;677.228&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;70/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;831.683&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;83/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;986.139&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;96/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1045.545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;10/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;118.812&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;237.624&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;27/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;320.792&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;37/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;439.604&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;47/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;558.416&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;57/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;677.228&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;64/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;760.396&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;74/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;879.218&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;84/101&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;998.03&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;94/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1116.842&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Anti-Ketradektriatoh form)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11157" rel="nofollow"&amp;gt;The Ellis duodene in 101-equal&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>bxj5j6bg8ja6g7uqg7jr8yr7b9mmkas</sha1>
    </revision>
    <revision>
      <id>4021</id>
      <timestamp>2013-05-06T04:43:43Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 429131848 - Original comment: **</comment>
      <origin>4021</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4435" sha1="0imzk23tjnqyq0euisw6324pq0s3xgi" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2013-05-06 04:43:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;429131848&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**//101-EDO//** divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]]. It is the 26th [[prime numbers|prime]] edo. The 101cd val provides an excellent tuning for [[Magic family#Witchcraft|witchcraft temperament]], falling between the 13 and 15 limit least squares tuning.

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**25 13 25 25 13:** //3L2s MOS// (Pentatonic)
**17 17 8 17 17 17 8:** //5L2s MOS// (Diatonic Pythagorean)
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic)
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Superdiatonic 1/13-tone 13;5 relation)
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11)
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Anti-Ketradektriatoh form)

=Links= 
[[http://tech.groups.yahoo.com/group/tuning-math/message/11157|The Ellis duodene in 101-equal]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;em&amp;gt;101-EDO&amp;lt;/em&amp;gt;&amp;lt;/strong&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;. It is the 26th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo. The 101cd val provides an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family#Witchcraft"&amp;gt;witchcraft temperament&amp;lt;/a&amp;gt;, falling between the 13 and 15 limit least squares tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;25 13 25 25 13:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Pentatonic)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Superdiatonic 1/13-tone 13;5 relation)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Anti-Ketradektriatoh form)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11157" rel="nofollow"&amp;gt;The Ellis duodene in 101-equal&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>0imzk23tjnqyq0euisw6324pq0s3xgi</sha1>
    </revision>
    <revision>
      <id>4022</id>
      <timestamp>2012-11-17T11:46:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 383551858 - Original comment: **</comment>
      <origin>4022</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4465" sha1="nmtn74vax4q49ne0lsojbem1brezj6b" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-11-17 11:46:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;383551858&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**//101-EDO//** divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]]. It is the 26th [[prime numbers|prime]] edo. The 101cd val provides an excellent tuning for [[Magic family#Witchcraft|witchcraft temperament]], falling between the 13 and 15 limit least squares tuning.

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**25 13 25 25 13:** //3L2s MOS// (Sub-Diatonic)
**17 17 8 17 17 17 8:** //5L2s MOS// (Diatonic Pythagorean)
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic)
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Hornbostel 1/13-tone [13;5 superdiatonic relation])
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11)
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Anti-Ketradektriatoh form)

=Links= 
[[http://tech.groups.yahoo.com/group/tuning-math/message/11157|The Ellis duodene in 101-equal]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;em&amp;gt;101-EDO&amp;lt;/em&amp;gt;&amp;lt;/strong&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;. It is the 26th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo. The 101cd val provides an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family#Witchcraft"&amp;gt;witchcraft temperament&amp;lt;/a&amp;gt;, falling between the 13 and 15 limit least squares tuning.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;25 13 25 25 13:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Sub-Diatonic)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Hornbostel 1/13-tone [13;5 superdiatonic relation])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Anti-Ketradektriatoh form)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11157" rel="nofollow"&amp;gt;The Ellis duodene in 101-equal&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>nmtn74vax4q49ne0lsojbem1brezj6b</sha1>
    </revision>
    <revision>
      <id>4023</id>
      <timestamp>2012-04-30T02:58:26Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 327184110 - Original comment: **</comment>
      <origin>4023</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4108" sha1="lxicr9p0zepv9bxagmjdc3lx68osmud" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-04-30 02:58:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;327184110&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**//101-EDO//** divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]]. It is the 26th [[prime numbers|prime]] edo.

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**25 13 25 25 13:** //3L2s MOS// (Sub-Diatonic)
**17 17 8 17 17 17 8:** //5L2s MOS// (Diatonic Pythagorean)
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic)
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Hornbostel 1/13-tone [13;5 superdiatonic relation])
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11)
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Anti-Ketradektriatoh form)

=Links= 
[[http://tech.groups.yahoo.com/group/tuning-math/message/11157|The Ellis duodene in 101-equal]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;em&amp;gt;101-EDO&amp;lt;/em&amp;gt;&amp;lt;/strong&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;. It is the 26th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;25 13 25 25 13:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Sub-Diatonic)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Hornbostel 1/13-tone [13;5 superdiatonic relation])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Anti-Ketradektriatoh form)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11157" rel="nofollow"&amp;gt;The Ellis duodene in 101-equal&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>lxicr9p0zepv9bxagmjdc3lx68osmud</sha1>
    </revision>
    <revision>
      <id>4024</id>
      <timestamp>2011-12-31T02:09:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 288887241 - Original comment: **</comment>
      <origin>4024</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4306" sha1="i1auckvj5mum28w4rkk75tcfhkl25ue" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-12-31 02:09:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288887241&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**//101-EDO//** divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]]. It is the 26th [[prime numbers|prime]] edo.

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**25 13 25 25 13:** //3L2s MOS// (Sub-Diatonic [Pentadiatonic Mold])
**17 17 8 17 17 17 8:** //5L2s MOS// (Diatonic Pythagorean [Heptadiatonic Mold])
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic [Octamonatonic Mold])
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11 [Undecimal Triatonic Mold])
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Tetradecimal Anti-Triatonic mode)

=Links= 
[[http://tech.groups.yahoo.com/group/tuning-math/message/11157|The Ellis duodene in 101-equal]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;em&amp;gt;101-EDO&amp;lt;/em&amp;gt;&amp;lt;/strong&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;. It is the 26th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;25 13 25 25 13:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Sub-Diatonic [Pentadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean [Heptadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic [Octamonatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11 [Undecimal Triatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Tetradecimal Anti-Triatonic mode)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11157" rel="nofollow"&amp;gt;The Ellis duodene in 101-equal&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>i1auckvj5mum28w4rkk75tcfhkl25ue</sha1>
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    <revision>
      <id>4025</id>
      <timestamp>2011-12-26T06:18:49Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 288481982 - Original comment: **</comment>
      <origin>4025</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4165" sha1="3azyoksz7bm7c5pr6ooytmbjqlwkapw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-12-26 06:18:49 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288481982&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**//101-EDO//** divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]].

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**25 13 25 25 13:** //3L2s MOS// (Sub-Diatonic [Pentadiatonic Mold])
**17 17 8 17 17 17 8:** //5L2s MOS// (Diatonic Pythagorean [Heptadiatonic Mold])
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic [Octamonatonic Mold])
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11 [Undecimal Triatonic Mold])
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Tetradecimal Anti-Triatonic mode)

=Links=
[[http://tech.groups.yahoo.com/group/tuning-math/message/11157|The Ellis duodene in 101-equal]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;&amp;lt;em&amp;gt;101-EDO&amp;lt;/em&amp;gt;&amp;lt;/strong&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;25 13 25 25 13:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Sub-Diatonic [Pentadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean [Heptadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic [Octamonatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11 [Undecimal Triatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Tetradecimal Anti-Triatonic mode)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Links"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Links&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/11157" rel="nofollow"&amp;gt;The Ellis duodene in 101-equal&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>3azyoksz7bm7c5pr6ooytmbjqlwkapw</sha1>
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    <revision>
      <id>4026</id>
      <timestamp>2011-08-18T15:15:12Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;clumma</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 246772567 - Original comment: **</comment>
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      <text bytes="3690" sha1="lrytf1dwh284jp40jzfugyo9zina5je" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:clumma|clumma]] and made on &lt;tt&gt;2011-08-18 15:15:12 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;246772567&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**//101-EDO//** divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]].

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**25 13 25 25 13:** //3L2s MOS// (Sub-Diatonic [Pentadiatonic Mold])
**17 17 8 17 17 17 8:** //5L2s MOS// (Diatonic Pythagorean [Heptadiatonic Mold])
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic [Octamonatonic Mold])
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11 [Undecimal Triatonic Mold])
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Tetradecimal Anti-Triatonic mode)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
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&amp;lt;strong&amp;gt;25 13 25 25 13:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Sub-Diatonic [Pentadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean [Heptadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic [Octamonatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11 [Undecimal Triatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Tetradecimal Anti-Triatonic mode)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4027</id>
      <timestamp>2011-07-12T01:37:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 240944039 - Original comment: **</comment>
      <origin>4027</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4268" sha1="4iivs0v6f7bmybsgy5cdlrv6jdkhyqp" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2011-07-12 01:37:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;240944039&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="background-color: #000000; color: #c97aff; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;101 tone equal temperament&amp;lt;/span&amp;gt;= 

**//101-EDO//** divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]].

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**25 13 25 25 13:** //3L2s MOS// (Sub-Diatonic [Pentadiatonic Mold])
**17 17 8 17 17 17 8:** //5L2s MOS// (Diatonic Pythagorean [Heptadiatonic Mold])
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic [Octamonatonic Mold])
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11 [Undecimal Triatonic Mold])
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Tetradecimal Anti-Triatonic mode)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x101 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="background-color: #000000; color: #c97aff; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;101 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;&amp;lt;em&amp;gt;101-EDO&amp;lt;/em&amp;gt;&amp;lt;/strong&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x101 tone equal temperament-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;25 13 25 25 13:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Sub-Diatonic [Pentadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean [Heptadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic [Octamonatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11 [Undecimal Triatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Tetradecimal Anti-Triatonic mode)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4028</id>
      <timestamp>2011-04-23T17:20:36Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 222441734 - Original comment: **</comment>
      <origin>4028</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4145" sha1="kscrtiahhjf97hy6gmryzszw446jt1i" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2011-04-23 17:20:36 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;222441734&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="background-color: #000000; color: #c97aff; font-size: 109%;"&amp;gt;101 tone equal temperament&amp;lt;/span&amp;gt;= 

//101edo// divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]].

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**25 13 25 25 13:** //3L2s MOS// (Sub-Diatonic [Pentadiatonic Mold])
**17 17 8 17 17 17 8:** //5L2s MOS// (Diatonic Pythagorean [Heptadiatonic Mold])
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic [Octamonatonic Mold])
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11 [Undecimal Triatonic Mold])
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Tetradecimal Anti-Triatonic mode)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x101 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="background-color: #000000; color: #c97aff; font-size: 109%;"&amp;gt;101 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;101edo&amp;lt;/em&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x101 tone equal temperament-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;25 13 25 25 13:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Sub-Diatonic [Pentadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean [Heptadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic [Octamonatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11 [Undecimal Triatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Tetradecimal Anti-Triatonic mode)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>kscrtiahhjf97hy6gmryzszw446jt1i</sha1>
    </revision>
    <revision>
      <id>4029</id>
      <timestamp>2011-04-23T01:29:19Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 222374156 - Original comment: **</comment>
      <origin>4029</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4145" sha1="8536qqtgpfnpdf6y2af6v1yu36iujnx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2011-04-23 01:29:19 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;222374156&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="background-color: #000000; color: #c97aff; font-size: 109%;"&amp;gt;101 tone equal temperament&amp;lt;/span&amp;gt;= 

//101edo// divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]].

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**25 13 25 25 13:** //3L2s MOS// (Sub-Diatonic [Pentadiatonic Mold])
**17 17 8 17 17 17 8:** //5L2s MOS// (Diatonic Pythagorean [Heptadiatonic Mold])
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic [Octamonatonic Mold])
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11 [Undecimal Triatonic Mold])
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Anti-Tetradecimal Triatonic mode)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x101 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="background-color: #000000; color: #c97aff; font-size: 109%;"&amp;gt;101 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;101edo&amp;lt;/em&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x101 tone equal temperament-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;25 13 25 25 13:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L2s MOS&amp;lt;/em&amp;gt; (Sub-Diatonic [Pentadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean [Heptadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic [Octamonatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11 [Undecimal Triatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Anti-Tetradecimal Triatonic mode)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>8536qqtgpfnpdf6y2af6v1yu36iujnx</sha1>
    </revision>
    <revision>
      <id>4030</id>
      <timestamp>2011-04-23T01:26:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 222374016 - Original comment: **</comment>
      <origin>4030</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3953" sha1="78zstr2jqmmcnjqux10jiv7lb2ks0et" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2011-04-23 01:26:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;222374016&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="background-color: #000000; color: #c97aff; font-size: 109%;"&amp;gt;101 tone equal temperament&amp;lt;/span&amp;gt;= 

//101edo// divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]].

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==__Some important MOS scales:__== 

**17 17 8 17 17 17 8:** //5L2s MOS// (Diatonic Pythagorean [Heptadiatonic Mold])
**13 13 13 13 13 13 13 10:** //7L1s MOS// (Grumpy Octatonic [Octamonatonic Mold])
**13 13 13 5 13 13 13 13 5:** //7L2s MOS// (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])
**10 10 7 10 10 10 7 10 10 10 7:** //8L3s MOS// (Improper Sensi-11 [Undecimal Triatonic Mold])
**7 7 7 8 7 7 7 7 8 7 7 7 7 8:** //3L11s MOS// (Anti-Tetradecimal Triatonic mode)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x101 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="background-color: #000000; color: #c97aff; font-size: 109%;"&amp;gt;101 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;101edo&amp;lt;/em&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x101 tone equal temperament-Some important MOS scales:"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;u&amp;gt;Some important MOS scales:&amp;lt;/u&amp;gt;&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 8 17 17 17 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;5L2s MOS&amp;lt;/em&amp;gt; (Diatonic Pythagorean [Heptadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 13 13 13 13 10:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L1s MOS&amp;lt;/em&amp;gt; (Grumpy Octatonic [Octamonatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;7L2s MOS&amp;lt;/em&amp;gt; (Armodue-Hornbostel 1/13-tone [Nonadiatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;10 10 7 10 10 10 7 10 10 10 7:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;8L3s MOS&amp;lt;/em&amp;gt; (Improper Sensi-11 [Undecimal Triatonic Mold])&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;7 7 7 8 7 7 7 7 8 7 7 7 7 8:&amp;lt;/strong&amp;gt; &amp;lt;em&amp;gt;3L11s MOS&amp;lt;/em&amp;gt; (Anti-Tetradecimal Triatonic mode)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>78zstr2jqmmcnjqux10jiv7lb2ks0et</sha1>
    </revision>
    <revision>
      <id>4031</id>
      <timestamp>2011-03-31T03:43:31Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 215740424 - Original comment: **</comment>
      <origin>4031</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2721" sha1="q7qordsgqv7cn3b8sgxbz4iid4o5m72" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-03-31 03:43:31 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;215740424&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=101 tone equal temperament= 
//101edo// divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[Schismatic family|grackle temperament]].

[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

==MOS==
17 17 8 17 17 17 8: 5L2s MOS (improper diatonic)

13 13 13 5 13 13 13 13 5: 7L2s MOS (Armodue)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x101 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;101 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;em&amp;gt;101edo&amp;lt;/em&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 101 equal parts of 11.881 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x101 tone equal temperament-MOS"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;MOS&amp;lt;/h2&amp;gt;
17 17 8 17 17 17 8: 5L2s MOS (improper diatonic)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5: 7L2s MOS (Armodue)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>q7qordsgqv7cn3b8sgxbz4iid4o5m72</sha1>
    </revision>
    <revision>
      <id>4032</id>
      <timestamp>2010-09-22T04:57:46Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 164491287 - Original comment: **</comment>
      <origin>4032</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2497" sha1="5elig7crr5wqf2aaezi8fpquiaflkds" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2010-09-22 04:57:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;164491287&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=101 tone equal temperament= 
//101edo// divides the octave into 101 parts of 11.881 cents each. It can be used to tune the [[Schismatic family|grackle temperament]].

5-limit commas: 32805/32768, &amp;lt;5 13 -11|

7-limit commas: 126/125, 32805/32768, 2430/2401

==MOS==
17 17 8 17 17 17 8: 5L2s MOS (improper diatonic)

13 13 13 5 13 13 13 13 5: 7L2s MOS (Armodue)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 &amp;lt;em&amp;gt;101edo&amp;lt;/em&amp;gt; divides the octave into 101 parts of 11.881 cents each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5-limit commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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17 17 8 17 17 17 8: 5L2s MOS (improper diatonic)&amp;lt;br /&amp;gt;
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13 13 13 5 13 13 13 13 5: 7L2s MOS (Armodue)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 164489787 - Original comment: **</comment>
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      <text bytes="2493" sha1="5nx2mll4s6n9v7hsu59vaklmcjkfqop" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2010-09-22 04:38:22 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;164489787&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=101 tone equal temperament= 
//101edo// divides the octave into 101 parts of 11.881 cents each. It can be used to tune the [[Schismatic family|grackle temperament]].

5-limit commas: 32805/32768, &amp;lt;5 13 -11|

7-limit commas: 126/125, 32805/32768, 2430/2401

==MOS==
17 17 8 17 17 17 8: 5L2s MOS (superpythagoren)

13 13 13 5 13 13 13 13 5: 7L2s MOS (Armodue)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 &amp;lt;em&amp;gt;101edo&amp;lt;/em&amp;gt; divides the octave into 101 parts of 11.881 cents each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5-limit commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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17 17 8 17 17 17 8: 5L2s MOS (superpythagoren)&amp;lt;br /&amp;gt;
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13 13 13 5 13 13 13 13 5: 7L2s MOS (Armodue)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 164489577 - Original comment: **</comment>
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      <text bytes="2299" sha1="sk4xvfx90t77efy3eg6ral0zq2hm1sr" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2010-09-22 04:36:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;164489577&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=101 tone equal temperament= 
//101edo// divides the octave into 101 parts of 11.881 cents each. It can be used to tune the [[Schismatic family|grackle temperament]].

5-limit commas: 32805/32768, &amp;lt;5 13 -11|

7-limit commas: 126/125, 32805/32768, 2430/2401


17 17 8 17 17 17 8: 5L2s MOS (superpythagoren)

13 13 13 5 13 13 13 13 5: 7L2s MOS (Armodue)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 &amp;lt;em&amp;gt;101edo&amp;lt;/em&amp;gt; divides the octave into 101 parts of 11.881 cents each. It can be used to tune the &amp;lt;a class="wiki_link" href="/Schismatic%20family"&amp;gt;grackle temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5-limit commas: 32805/32768, &amp;amp;lt;5 13 -11|&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7-limit commas: 126/125, 32805/32768, 2430/2401&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
17 17 8 17 17 17 8: 5L2s MOS (superpythagoren)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5: 7L2s MOS (Armodue)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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        <id>0</id>
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      <comment>**Imported revision 163861775 - Original comment: **</comment>
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      <text bytes="2069" sha1="pja29wpai2nzexxb32uvzlyxd43sd3f" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2010-09-20 03:14:27 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;163861775&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=101 tone equal temperament= 

17 17 8 17 17 17 8 : [[Pytagorean tuning]] (comparable with [[12edo]])
13 13 13 5 13 13 13 13 5 : [[Armodue tuning]] (comparable with [[39edo]] and [[62edo]])&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;101edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x101 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;101 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
17 17 8 17 17 17 8 : &amp;lt;a class="wiki_link" href="/Pytagorean%20tuning"&amp;gt;Pytagorean tuning&amp;lt;/a&amp;gt; (comparable with &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 : &amp;lt;a class="wiki_link" href="/Armodue%20tuning"&amp;gt;Armodue tuning&amp;lt;/a&amp;gt; (comparable with &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/62edo"&amp;gt;62edo&amp;lt;/a&amp;gt;)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 163685341 - Original comment: **</comment>
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      <text bytes="1781" sha1="k3hpo0tm7r27bnh2yihnwcvt66l9sqt" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2010-09-19 01:05:21 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;163685341&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=101 tone equal temperament= 

17 17 8 17 17 17 8 : Pytagorean tuning (comparable with 12edo)
13 13 13 5 13 13 13 13 5 : Armodue tuning (comparable with 39edo and 62edo)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 &amp;lt;br /&amp;gt;
17 17 8 17 17 17 8 : Pytagorean tuning (comparable with 12edo)&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 : Armodue tuning (comparable with 39edo and 62edo)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>33035</id>
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      <timestamp>2018-09-21T02:36:22Z</timestamp>
      <contributor>
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        <id>16</id>
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      <comment>leayout -&gt; layout</comment>
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      <text bytes="2847" sha1="jcn2b4m2tr779ujdcw334ljeexq34zq" xml:space="preserve">'''''101-EDO''''' divides the [[Octave|octave]] into 101 equal parts of 11.881 [[cent|cent]]s each. It can be used to tune the [[Schismatic_family|grackle temperament]]. It is the 26th [[prime_numbers|prime]] edo. The 101cd val provides an excellent tuning for [[Magic_family#Witchcraft|witchcraft temperament]], falling between the 13 and 15 limit least squares tuning.

[[5-limit|5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit|7-limit]] commas: 126/125, 32805/32768, 2430/2401

==&lt;u&gt;Some important MOS scales:&lt;/u&gt;==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable"
|-
| | 25
| | 297.03
|-
| | 38
| | 451.485
|-
| | 63
| | 748.515
|-
| | 88
| | 1045.545
|}
'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable"
|-
| | '''17'''
| | '''201.98'''
|-
| | 34
| | 403.96
|-
| | '''42'''
| | '''499.01'''
|-
| | '''59'''
| | '''700.99'''
|-
| | '''76'''
| | '''902.97'''
|-
| | 93
| | 1104.95
|}
'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable"
|-
| | 13
| | 154.455
|-
| | 26
| | 308.911
|-
| | 39
| | 463.366
|-
| | 52
| | 617.822
|-
| | 65
| | 772.277
|-
| | 78
| | 926.733
|-
| | 91
| | 1081.188
|}
'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable"
|-
| | '''13/101'''
| | '''154.455'''
|-
| | '''26/101'''
| | '''308.911'''
|-
| | 39/101
| | 463.366
|-
| | '''44/101'''
| | '''522.772'''
|-
| | '''57/101'''
| | '''677.228'''
|-
| | '''70/101'''
| | '''831.683'''
|-
| | '''83/101'''
| | '''986.139'''
|-
| | 96/101
| | 1045.545
|}
'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable"
|-
| | '''10'''
| | '''118.812'''
|-
| | 20
| | 237.624
|-
| | '''27'''
| | '''320.792'''
|-
| | '''37'''
| | '''439.604'''
|-
| | '''47'''
| | '''558.416'''
|-
| | 57
| | 677.228
|-
| | '''64'''
| | '''760.396'''
|-
| | '''74'''
| | '''879.218'''
|-
| | '''84'''
| | '''998.03'''
|-
| | 94
| | 1116.842
|}
'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
|-
| | '''7'''
| | '''83.168'''
|-
| | '''14'''
| | '''166.337'''
|-
| | 22
| | 261.386
|-
| | '''29'''
| | '''344.5545'''
|-
| | '''36'''
| | '''427.723'''
|-
| | '''43'''
| | '''510.891'''
|-
| | '''50'''
| | '''594.0595'''
|-
| | 58
| | 689.119
|-
| | '''65'''
| | '''772.287'''
|-
| | '''72'''
| | '''855.4455'''
|-
| | '''79'''
| | '''938.614'''
|-
| | '''86'''
| | '''1021.782'''
|-
| | 93
| | 1104.95
|}

=Links=
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:101edo]]
[[Category:armodue]]
[[Category:edo]]
[[Category:equal]]
[[Category:grackle]]
[[Category:prime_edo]]
[[Category:pythagorean]]
[[Category:scales]]
[[Category:todo:improve_layout]]
[[Category:todo:unify_precision]]</text>
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        <id>6</id>
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      <minor/>
      <comment>removed cat equal since edo includes equal</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2830" sha1="oodfbypqncr3g9xu36utnz1u723gonr" xml:space="preserve">'''''101-EDO''''' divides the [[Octave|octave]] into 101 equal parts of 11.881 [[cent|cent]]s each. It can be used to tune the [[Schismatic_family|grackle temperament]]. It is the 26th [[prime_numbers|prime]] edo. The 101cd val provides an excellent tuning for [[Magic_family#Witchcraft|witchcraft temperament]], falling between the 13 and 15 limit least squares tuning.

[[5-limit|5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit|7-limit]] commas: 126/125, 32805/32768, 2430/2401

==&lt;u&gt;Some important MOS scales:&lt;/u&gt;==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable"
|-
| | 25
| | 297.03
|-
| | 38
| | 451.485
|-
| | 63
| | 748.515
|-
| | 88
| | 1045.545
|}
'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable"
|-
| | '''17'''
| | '''201.98'''
|-
| | 34
| | 403.96
|-
| | '''42'''
| | '''499.01'''
|-
| | '''59'''
| | '''700.99'''
|-
| | '''76'''
| | '''902.97'''
|-
| | 93
| | 1104.95
|}
'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable"
|-
| | 13
| | 154.455
|-
| | 26
| | 308.911
|-
| | 39
| | 463.366
|-
| | 52
| | 617.822
|-
| | 65
| | 772.277
|-
| | 78
| | 926.733
|-
| | 91
| | 1081.188
|}
'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable"
|-
| | '''13/101'''
| | '''154.455'''
|-
| | '''26/101'''
| | '''308.911'''
|-
| | 39/101
| | 463.366
|-
| | '''44/101'''
| | '''522.772'''
|-
| | '''57/101'''
| | '''677.228'''
|-
| | '''70/101'''
| | '''831.683'''
|-
| | '''83/101'''
| | '''986.139'''
|-
| | 96/101
| | 1045.545
|}
'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable"
|-
| | '''10'''
| | '''118.812'''
|-
| | 20
| | 237.624
|-
| | '''27'''
| | '''320.792'''
|-
| | '''37'''
| | '''439.604'''
|-
| | '''47'''
| | '''558.416'''
|-
| | 57
| | 677.228
|-
| | '''64'''
| | '''760.396'''
|-
| | '''74'''
| | '''879.218'''
|-
| | '''84'''
| | '''998.03'''
|-
| | 94
| | 1116.842
|}
'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
|-
| | '''7'''
| | '''83.168'''
|-
| | '''14'''
| | '''166.337'''
|-
| | 22
| | 261.386
|-
| | '''29'''
| | '''344.5545'''
|-
| | '''36'''
| | '''427.723'''
|-
| | '''43'''
| | '''510.891'''
|-
| | '''50'''
| | '''594.0595'''
|-
| | 58
| | 689.119
|-
| | '''65'''
| | '''772.287'''
|-
| | '''72'''
| | '''855.4455'''
|-
| | '''79'''
| | '''938.614'''
|-
| | '''86'''
| | '''1021.782'''
|-
| | 93
| | 1104.95
|}

=Links=
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:101edo]]
[[Category:Armodue]]
[[Category:Edo]]
[[Category:Grackle]]
[[Category:Prime _edo]]
[[Category:Pythagorean]]
[[Category:Scales]]

[[Category:todo:improve_layout]]
[[Category:todo:unify_precision]]</text>
      <sha1>oodfbypqncr3g9xu36utnz1u723gonr</sha1>
    </revision>
    <revision>
      <id>38084</id>
      <parentid>38075</parentid>
      <timestamp>2018-10-28T15:11:49Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>fixed typo; removed cat</comment>
      <origin>38084</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2809" sha1="ddjfoxjuteuqc66agj77nzsb9s47lre" xml:space="preserve">'''''101-EDO''''' divides the [[Octave|octave]] into 101 equal parts of 11.881 [[cent|cent]]s each. It can be used to tune the [[Schismatic_family|grackle temperament]]. It is the 26th [[prime_numbers|prime]] edo. The 101cd val provides an excellent tuning for [[Magic_family#Witchcraft|witchcraft temperament]], falling between the 13 and 15 limit least squares tuning.

[[5-limit|5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|

[[7-limit|7-limit]] commas: 126/125, 32805/32768, 2430/2401

==&lt;u&gt;Some important MOS scales:&lt;/u&gt;==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable"
|-
| | 25
| | 297.03
|-
| | 38
| | 451.485
|-
| | 63
| | 748.515
|-
| | 88
| | 1045.545
|}
'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable"
|-
| | '''17'''
| | '''201.98'''
|-
| | 34
| | 403.96
|-
| | '''42'''
| | '''499.01'''
|-
| | '''59'''
| | '''700.99'''
|-
| | '''76'''
| | '''902.97'''
|-
| | 93
| | 1104.95
|}
'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable"
|-
| | 13
| | 154.455
|-
| | 26
| | 308.911
|-
| | 39
| | 463.366
|-
| | 52
| | 617.822
|-
| | 65
| | 772.277
|-
| | 78
| | 926.733
|-
| | 91
| | 1081.188
|}
'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable"
|-
| | '''13/101'''
| | '''154.455'''
|-
| | '''26/101'''
| | '''308.911'''
|-
| | 39/101
| | 463.366
|-
| | '''44/101'''
| | '''522.772'''
|-
| | '''57/101'''
| | '''677.228'''
|-
| | '''70/101'''
| | '''831.683'''
|-
| | '''83/101'''
| | '''986.139'''
|-
| | 96/101
| | 1045.545
|}
'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable"
|-
| | '''10'''
| | '''118.812'''
|-
| | 20
| | 237.624
|-
| | '''27'''
| | '''320.792'''
|-
| | '''37'''
| | '''439.604'''
|-
| | '''47'''
| | '''558.416'''
|-
| | 57
| | 677.228
|-
| | '''64'''
| | '''760.396'''
|-
| | '''74'''
| | '''879.218'''
|-
| | '''84'''
| | '''998.03'''
|-
| | 94
| | 1116.842
|}
'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
|-
| | '''7'''
| | '''83.168'''
|-
| | '''14'''
| | '''166.337'''
|-
| | 22
| | 261.386
|-
| | '''29'''
| | '''344.5545'''
|-
| | '''36'''
| | '''427.723'''
|-
| | '''43'''
| | '''510.891'''
|-
| | '''50'''
| | '''594.0595'''
|-
| | 58
| | 689.119
|-
| | '''65'''
| | '''772.287'''
|-
| | '''72'''
| | '''855.4455'''
|-
| | '''79'''
| | '''938.614'''
|-
| | '''86'''
| | '''1021.782'''
|-
| | 93
| | 1104.95
|}

=Links=
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Edo]]
[[Category:Grackle]]
[[Category:Prime edo]]
[[Category:Pythagorean]]
[[Category:Scales]]

[[Category:todo:improve_layout]]
[[Category:todo:unify_precision]]</text>
      <sha1>ddjfoxjuteuqc66agj77nzsb9s47lre</sha1>
    </revision>
    <revision>
      <id>38289</id>
      <parentid>38084</parentid>
      <timestamp>2018-11-02T09:49:01Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>cat rename; simplified links and table; added title to table columns</comment>
      <origin>38289</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2601" sha1="jwydg3el0q09oy6ijf2lf53ytifoj3h" xml:space="preserve">'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

;[[5-limit]] commas: 32805/32768, &amp;lt;5 13 -11|
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable"
! Steps
! Cents
|-
| 25
| 297.03
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable"
! Steps
! Cents
|-
| '''17'''
| '''201.98'''
|-
| 34
| 403.96
|-
| '''42'''
| '''499.01'''
|-
| '''59'''
| '''700.99'''
|-
| '''76'''
| '''902.97'''
|-
| 93
| 1104.95
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable"
! Steps
! Cents
|-
| '''13'''
| '''154.455'''
|-
| '''26'''
| '''308.911'''
|-
| 39
| 463.366
|-
| '''44'''
| '''522.772'''
|-
| '''57'''
| '''677.228'''
|-
| '''70'''
| '''831.683'''
|-
| '''83'''
| '''986.139'''
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable"
! Steps
! Cents
|-
| '''10'''
| '''118.812'''
|-
| 20
| 237.624
|-
| '''27'''
| '''320.792'''
|-
| '''37'''
| '''439.604'''
|-
| '''47'''
| '''558.416'''
|-
| 57
| 677.228
|-
| '''64'''
| '''760.396'''
|-
| '''74'''
| '''879.218'''
|-
| '''84'''
| '''998.03'''
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
! Steps
! Cents
|-
| '''7'''
| '''83.168'''
|-
| '''14'''
| '''166.337'''
|-
| 22
| 261.386
|-
| '''29'''
| '''344.5545'''
|-
| '''36'''
| '''427.723'''
|-
| '''43'''
| '''510.891'''
|-
| '''50'''
| '''594.0595'''
|-
| 58
| 689.119
|-
| '''65'''
| '''772.287'''
|-
| '''72'''
| '''855.4455'''
|-
| '''79'''
| '''938.614'''
|-
| '''86'''
| '''1021.782'''
|-
| 93
| 1104.95
|}

== Links ==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Edo]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]

[[Category:todo:improve_layout]]
[[Category:todo:unify_precision]]</text>
      <sha1>jwydg3el0q09oy6ijf2lf53ytifoj3h</sha1>
    </revision>
    <revision>
      <id>38707</id>
      <parentid>38289</parentid>
      <timestamp>2018-11-30T03:04:24Z</timestamp>
      <contributor>
        <username>Spt3125</username>
        <id>16</id>
      </contributor>
      <minor/>
      <comment>unify precision; minor corrections</comment>
      <origin>38707</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2625" sha1="7j48s3ukl7mnb911q9cdjnytp995wjr" xml:space="preserve">'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

;[[5-limit]] commas: 32805/32768 ( {{Monzo| -15 8 1 }} ), 51018336/48828125 ( {{Monzo| 5 13 -11 }} )
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable"
! Steps
! Cents
|-
| '''17'''
| '''201.980'''
|-
| 34
| 403.960
|-
| '''42'''
| '''499.010'''
|-
| '''59'''
| '''700.990'''
|-
| '''76'''
| '''902.970'''
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable"
! Steps
! Cents
|-
| '''13'''
| '''154.455'''
|-
| '''26'''
| '''308.911'''
|-
| 39
| 463.366
|-
| '''44'''
| '''522.772'''
|-
| '''57'''
| '''677.228'''
|-
| '''70'''
| '''831.683'''
|-
| '''83'''
| '''986.139'''
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable"
! Steps
! Cents
|-
| '''10'''
| '''118.812'''
|-
| 20
| 237.624
|-
| '''27'''
| '''320.792'''
|-
| '''37'''
| '''439.604'''
|-
| '''47'''
| '''558.416'''
|-
| 57
| 677.228
|-
| '''64'''
| '''760.396'''
|-
| '''74'''
| '''879.218'''
|-
| '''84'''
| '''998.020'''
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
! Steps
! Cents
|-
| '''7'''
| '''83.168'''
|-
| '''14'''
| '''166.337'''
|-
| 22
| 261.386
|-
| '''29'''
| '''344.554'''
|-
| '''36'''
| '''427.723'''
|-
| '''43'''
| '''510.891'''
|-
| '''50'''
| '''594.059'''
|-
| 58
| 689.119
|-
| '''65'''
| '''772.287'''
|-
| '''72'''
| '''855.446'''
|-
| '''79'''
| '''938.614'''
|-
| '''86'''
| '''1021.782'''
|-
| 93
| 1104.950
|}

== Links ==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Edo]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]
[[Category:todo:improve_layout]]</text>
      <sha1>7j48s3ukl7mnb911q9cdjnytp995wjr</sha1>
    </revision>
    <revision>
      <id>54945</id>
      <parentid>38707</parentid>
      <timestamp>2020-12-04T23:13:04Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54945</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2651" sha1="fy4xnctvxa7jvk3rdq2mqrdsmuqax3x" xml:space="preserve">'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

;[[5-limit]] commas: 32805/32768 ( {{Monzo| -15 8 1 }} ), 51018336/48828125 ( {{Monzo| 5 13 -11 }} )
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable"
! Steps
! Cents
|-
| '''17'''
| '''201.980'''
|-
| 34
| 403.960
|-
| '''42'''
| '''499.010'''
|-
| '''59'''
| '''700.990'''
|-
| '''76'''
| '''902.970'''
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable"
! Steps
! Cents
|-
| '''13'''
| '''154.455'''
|-
| '''26'''
| '''308.911'''
|-
| 39
| 463.366
|-
| '''44'''
| '''522.772'''
|-
| '''57'''
| '''677.228'''
|-
| '''70'''
| '''831.683'''
|-
| '''83'''
| '''986.139'''
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable"
! Steps
! Cents
|-
| '''10'''
| '''118.812'''
|-
| 20
| 237.624
|-
| '''27'''
| '''320.792'''
|-
| '''37'''
| '''439.604'''
|-
| '''47'''
| '''558.416'''
|-
| 57
| 677.228
|-
| '''64'''
| '''760.396'''
|-
| '''74'''
| '''879.218'''
|-
| '''84'''
| '''998.020'''
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
! Steps
! Cents
|-
| '''7'''
| '''83.168'''
|-
| '''14'''
| '''166.337'''
|-
| 22
| 261.386
|-
| '''29'''
| '''344.554'''
|-
| '''36'''
| '''427.723'''
|-
| '''43'''
| '''510.891'''
|-
| '''50'''
| '''594.059'''
|-
| 58
| 689.119
|-
| '''65'''
| '''772.287'''
|-
| '''72'''
| '''855.446'''
|-
| '''79'''
| '''938.614'''
|-
| '''86'''
| '''1021.782'''
|-
| 93
| 1104.950
|}

== Links ==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]
[[Category:todo:improve_layout]]</text>
      <sha1>fy4xnctvxa7jvk3rdq2mqrdsmuqax3x</sha1>
    </revision>
    <revision>
      <id>59162</id>
      <parentid>54945</parentid>
      <timestamp>2021-01-17T11:39:31Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>lc monzo</comment>
      <origin>59162</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2651" sha1="bfzx4i2x7i0jiqazwnpez289huxidmd" xml:space="preserve">'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

;[[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable"
! Steps
! Cents
|-
| '''17'''
| '''201.980'''
|-
| 34
| 403.960
|-
| '''42'''
| '''499.010'''
|-
| '''59'''
| '''700.990'''
|-
| '''76'''
| '''902.970'''
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable"
! Steps
! Cents
|-
| '''13'''
| '''154.455'''
|-
| '''26'''
| '''308.911'''
|-
| 39
| 463.366
|-
| '''44'''
| '''522.772'''
|-
| '''57'''
| '''677.228'''
|-
| '''70'''
| '''831.683'''
|-
| '''83'''
| '''986.139'''
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable"
! Steps
! Cents
|-
| '''10'''
| '''118.812'''
|-
| 20
| 237.624
|-
| '''27'''
| '''320.792'''
|-
| '''37'''
| '''439.604'''
|-
| '''47'''
| '''558.416'''
|-
| 57
| 677.228
|-
| '''64'''
| '''760.396'''
|-
| '''74'''
| '''879.218'''
|-
| '''84'''
| '''998.020'''
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
! Steps
! Cents
|-
| '''7'''
| '''83.168'''
|-
| '''14'''
| '''166.337'''
|-
| 22
| 261.386
|-
| '''29'''
| '''344.554'''
|-
| '''36'''
| '''427.723'''
|-
| '''43'''
| '''510.891'''
|-
| '''50'''
| '''594.059'''
|-
| 58
| 689.119
|-
| '''65'''
| '''772.287'''
|-
| '''72'''
| '''855.446'''
|-
| '''79'''
| '''938.614'''
|-
| '''86'''
| '''1021.782'''
|-
| 93
| 1104.950
|}

== Links ==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]
[[Category:todo:improve_layout]]</text>
      <sha1>bfzx4i2x7i0jiqazwnpez289huxidmd</sha1>
    </revision>
    <revision>
      <id>67307</id>
      <parentid>59162</parentid>
      <timestamp>2021-04-14T20:08:54Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Some important MOS scales */</comment>
      <origin>67307</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3542" sha1="mxpy8plp1qqmlh1pe8xm00qfrpfzs4y" xml:space="preserve">'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

;[[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''13 12 13 13 12 13 12 13:''' ''5L3s MOS'' (Oneirotonic, an 8-tone circulating temperament)

{| class="wikitable"
! Steps
! Cents
|-
|13
|154.455
|-
| '''25'''
| '''297.030'''
|-
| '''38'''
| '''451.485'''
|-
|51
|605.9405
|-
| '''63'''
| '''748.515'''
|-
|75
|891.089
|-
| '''88'''
| '''1045.545'''
|}

'''9 8 9 8 8 9 8 9 8 9 8 8:''' ''5L7s MOS'' (Diatonic Pythagorean, a 12-tone circulating temperament)

{| class="wikitable"
! Steps
! Cents
|-
|9
|106.059
|-
| '''17'''
| '''201.980'''
|-
|26
|308.911
|-
| '''34'''
| '''403.960'''
|-
| '''42'''
| '''499.010'''
|-
|51
|605.940
|-
| '''59'''
| '''700.990'''
|-
|68
|807.921
|-
| '''76'''
| '''902.970'''
|-
|85
|1009.901
|-
| '''93'''
| '''1104.950'''
|}

'''10 3 10 3 10 3 10 3 10 3 10 3 10 3 10:''' ''8L7s MOS'' (Opossum)

{| class="wikitable"
! Steps
! Cents
|-
|10
|118.812
|-
| '''13'''
| '''154.455'''
|-
|23
|273.267
|-
| '''26'''
| '''308.911'''
|-
|36
|427.723
|-
| '''39'''
| '''463.366'''
|-
|49
|582.178
|-
| '''52'''
| '''617.822'''
|-
|62
|736.337
|-
| '''65'''
| '''772.277'''
|-
|75
|891.089
|-
| '''78'''
| '''926.733'''
|-
|88
|1045.4455
|-
| '''91'''
| '''1081.188'''
|}

'''8 5 8 5 8 5 5 8 5 8 5 8 5 8 5 5:''' ''7L9s MOS'' (Golden Mavila chromatic 1/13-tone)

{| class="wikitable"
! Steps
! Cents
|-
|8
|95.0495
|-
| '''13'''
| '''154.455'''
|-
|21
|249.505
|-
| '''26'''
| '''308.911'''
|-
|34
|403.960
|-
| '''39'''
| '''463.366'''
|-
| '''44'''
| '''522.772'''
|-
|52
|617.822
|-
| '''57'''
| '''677.228'''
|-
|65
|772.277
|-
| '''70'''
| '''831.683'''
|-
|78
|926.733
|-
| '''83'''
| '''986.139'''
|-
|91
|1081.188
|-
| '''96'''
| '''1045.545'''
|}

'''7 3 7 3 7 7 3 7 3 7 3 7 7 3 7 3 7 3 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable"
! Steps
! Cents
|-
|7
|83.168
|-
| '''10'''
| '''118.812'''
|-
|17
|201.980
|-
| '''20'''
| '''237.624'''
|-
| '''27'''
| '''320.792'''
|-
|34
|403.960
|-
| '''37'''
| '''439.604'''
|-
|44
|527.772
|-
| '''47'''
| '''558.416'''
|-
|54
|641.584
|-
| '''57'''
| '''677.228'''
|-
| '''64'''
| '''760.396'''
|-
|71
|843.564
|-
| '''74'''
| '''879.218'''
|-
|81
|962.376
|-
| '''84'''
| '''998.020'''
|-
|91
|1081.188
|-
| '''94'''
| '''1116.842'''
|}

'''7 7 7 7 1 7 7 7 7 7 1 7 7 7 7 7 1:''' ''14L3s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
! Steps
! Cents
|-
| '''7'''
| '''83.168'''
|-
| '''14'''
| '''166.337'''
|-
|21
|249.505
|-
| '''22'''
| '''261.386'''
|-
| '''29'''
| '''344.554'''
|-
| '''36'''
| '''427.723'''
|-
| '''43'''
| '''510.891'''
|-
| '''50'''
| '''594.059'''
|-
|57
|677.278
|-
| '''58'''
| '''689.119'''
|-
| '''65'''
| '''772.287'''
|-
| '''72'''
| '''855.446'''
|-
| '''79'''
| '''938.614'''
|-
| '''86'''
| '''1021.782'''
|-
| '''93'''
| '''1104.950'''
|-
|100
|1188.119
|}

== Links ==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]
[[Category:todo:improve_layout]]</text>
      <sha1>mxpy8plp1qqmlh1pe8xm00qfrpfzs4y</sha1>
    </revision>
    <revision>
      <id>67310</id>
      <parentid>67307</parentid>
      <timestamp>2021-04-14T20:12:10Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Some important MOS scales */</comment>
      <origin>67310</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3543" sha1="lfe62wr86p7hqt0vehkgpltq497sktd" xml:space="preserve">'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

;[[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''13 12 13 13 12 13 12 13:''' ''5L3s MOS'' (Oneirotonic, an 8-tone circulating temperament)

{| class="wikitable"
! Steps
! Cents
|-
|13
|154.455
|-
| '''25'''
| '''297.030'''
|-
| '''38'''
| '''451.485'''
|-
|51
|605.9405
|-
| '''63'''
| '''748.515'''
|-
|75
|891.089
|-
| '''88'''
| '''1045.5445'''
|}

'''9 8 9 8 8 9 8 9 8 9 8 8:''' ''5L7s MOS'' (Diatonic Pythagorean, a 12-tone circulating temperament)

{| class="wikitable"
! Steps
! Cents
|-
|9
|106.059
|-
| '''17'''
| '''201.980'''
|-
|26
|308.911
|-
| '''34'''
| '''403.960'''
|-
| '''42'''
| '''499.010'''
|-
|51
|605.940
|-
| '''59'''
| '''700.990'''
|-
|68
|807.921
|-
| '''76'''
| '''902.970'''
|-
|85
|1009.901
|-
| '''93'''
| '''1104.950'''
|}

'''10 3 10 3 10 3 10 3 10 3 10 3 10 3 10:''' ''8L7s MOS'' (Opossum)

{| class="wikitable"
! Steps
! Cents
|-
|10
|118.812
|-
| '''13'''
| '''154.455'''
|-
|23
|273.267
|-
| '''26'''
| '''308.911'''
|-
|36
|427.723
|-
| '''39'''
| '''463.366'''
|-
|49
|582.178
|-
| '''52'''
| '''617.822'''
|-
|62
|736.337
|-
| '''65'''
| '''772.277'''
|-
|75
|891.089
|-
| '''78'''
| '''926.733'''
|-
|88
|1045.5445
|-
| '''91'''
| '''1081.188'''
|}

'''8 5 8 5 8 5 5 8 5 8 5 8 5 8 5 5:''' ''7L9s MOS'' (Golden Mavila chromatic 1/13-tone)

{| class="wikitable"
! Steps
! Cents
|-
|8
|95.0495
|-
| '''13'''
| '''154.455'''
|-
|21
|249.505
|-
| '''26'''
| '''308.911'''
|-
|34
|403.960
|-
| '''39'''
| '''463.366'''
|-
| '''44'''
| '''522.772'''
|-
|52
|617.822
|-
| '''57'''
| '''677.228'''
|-
|65
|772.277
|-
| '''70'''
| '''831.683'''
|-
|78
|926.733
|-
| '''83'''
| '''986.139'''
|-
|91
|1081.188
|-
| '''96'''
| '''1045.545'''
|}

'''7 3 7 3 7 7 3 7 3 7 3 7 7 3 7 3 7 3 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable"
! Steps
! Cents
|-
|7
|83.168
|-
| '''10'''
| '''118.812'''
|-
|17
|201.980
|-
| '''20'''
| '''237.624'''
|-
| '''27'''
| '''320.792'''
|-
|34
|403.960
|-
| '''37'''
| '''439.604'''
|-
|44
|527.772
|-
| '''47'''
| '''558.416'''
|-
|54
|641.584
|-
| '''57'''
| '''677.228'''
|-
| '''64'''
| '''760.396'''
|-
|71
|843.564
|-
| '''74'''
| '''879.218'''
|-
|81
|962.376
|-
| '''84'''
| '''998.020'''
|-
|91
|1081.188
|-
| '''94'''
| '''1116.842'''
|}

'''7 7 7 7 1 7 7 7 7 7 1 7 7 7 7 7 1:''' ''14L3s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
! Steps
! Cents
|-
| '''7'''
| '''83.168'''
|-
| '''14'''
| '''166.337'''
|-
|21
|249.505
|-
| '''22'''
| '''261.386'''
|-
| '''29'''
| '''344.554'''
|-
| '''36'''
| '''427.723'''
|-
| '''43'''
| '''510.891'''
|-
| '''50'''
| '''594.059'''
|-
|57
|677.278
|-
| '''58'''
| '''689.119'''
|-
| '''65'''
| '''772.287'''
|-
| '''72'''
| '''855.446'''
|-
| '''79'''
| '''938.614'''
|-
| '''86'''
| '''1021.782'''
|-
| '''93'''
| '''1104.950'''
|-
|100
|1188.119
|}

== Links ==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]
[[Category:todo:improve_layout]]</text>
      <sha1>lfe62wr86p7hqt0vehkgpltq497sktd</sha1>
    </revision>
    <revision>
      <id>67404</id>
      <parentid>67310</parentid>
      <timestamp>2021-04-15T09:51:30Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>+todo: Please don't add trailing 5 digits! (rounding doesn't really harm and it's not really hard to calculate the "correct" values for everyone)</comment>
      <origin>67404</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3551" sha1="3irpsxdnndhg1nzk87b1gq8o9z79dpw" xml:space="preserve">'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

;[[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''13 12 13 13 12 13 12 13:''' ''5L3s MOS'' (Oneirotonic, an 8-tone circulating temperament)

{| class="wikitable"
! Steps
! Cents
|-
|13
|154.455
|-
| '''25'''
| '''297.030'''
|-
| '''38'''
| '''451.485'''
|-
|51
|605.9405
|-
| '''63'''
| '''748.515'''
|-
|75
|891.089
|-
| '''88'''
| '''1045.5445'''
|}

'''9 8 9 8 8 9 8 9 8 9 8 8:''' ''5L7s MOS'' (Diatonic Pythagorean, a 12-tone circulating temperament)

{| class="wikitable"
! Steps
! Cents
|-
|9
|106.059
|-
| '''17'''
| '''201.980'''
|-
|26
|308.911
|-
| '''34'''
| '''403.960'''
|-
| '''42'''
| '''499.010'''
|-
|51
|605.940
|-
| '''59'''
| '''700.990'''
|-
|68
|807.921
|-
| '''76'''
| '''902.970'''
|-
|85
|1009.901
|-
| '''93'''
| '''1104.950'''
|}

'''10 3 10 3 10 3 10 3 10 3 10 3 10 3 10:''' ''8L7s MOS'' (Opossum)

{| class="wikitable"
! Steps
! Cents
|-
|10
|118.812
|-
| '''13'''
| '''154.455'''
|-
|23
|273.267
|-
| '''26'''
| '''308.911'''
|-
|36
|427.723
|-
| '''39'''
| '''463.366'''
|-
|49
|582.178
|-
| '''52'''
| '''617.822'''
|-
|62
|736.337
|-
| '''65'''
| '''772.277'''
|-
|75
|891.089
|-
| '''78'''
| '''926.733'''
|-
|88
|1045.5445
|-
| '''91'''
| '''1081.188'''
|}

'''8 5 8 5 8 5 5 8 5 8 5 8 5 8 5 5:''' ''7L9s MOS'' (Golden Mavila chromatic 1/13-tone)

{| class="wikitable"
! Steps
! Cents
|-
|8
|95.0495
|-
| '''13'''
| '''154.455'''
|-
|21
|249.505
|-
| '''26'''
| '''308.911'''
|-
|34
|403.960
|-
| '''39'''
| '''463.366'''
|-
| '''44'''
| '''522.772'''
|-
|52
|617.822
|-
| '''57'''
| '''677.228'''
|-
|65
|772.277
|-
| '''70'''
| '''831.683'''
|-
|78
|926.733
|-
| '''83'''
| '''986.139'''
|-
|91
|1081.188
|-
| '''96'''
| '''1045.545'''
|}

'''7 3 7 3 7 7 3 7 3 7 3 7 7 3 7 3 7 3 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable"
! Steps
! Cents
|-
|7
|83.168
|-
| '''10'''
| '''118.812'''
|-
|17
|201.980
|-
| '''20'''
| '''237.624'''
|-
| '''27'''
| '''320.792'''
|-
|34
|403.960
|-
| '''37'''
| '''439.604'''
|-
|44
|527.772
|-
| '''47'''
| '''558.416'''
|-
|54
|641.584
|-
| '''57'''
| '''677.228'''
|-
| '''64'''
| '''760.396'''
|-
|71
|843.564
|-
| '''74'''
| '''879.218'''
|-
|81
|962.376
|-
| '''84'''
| '''998.020'''
|-
|91
|1081.188
|-
| '''94'''
| '''1116.842'''
|}

'''7 7 7 7 1 7 7 7 7 7 1 7 7 7 7 7 1:''' ''14L3s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
! Steps
! Cents
|-
| '''7'''
| '''83.168'''
|-
| '''14'''
| '''166.337'''
|-
|21
|249.505
|-
| '''22'''
| '''261.386'''
|-
| '''29'''
| '''344.554'''
|-
| '''36'''
| '''427.723'''
|-
| '''43'''
| '''510.891'''
|-
| '''50'''
| '''594.059'''
|-
|57
|677.278
|-
| '''58'''
| '''689.119'''
|-
| '''65'''
| '''772.287'''
|-
| '''72'''
| '''855.446'''
|-
| '''79'''
| '''938.614'''
|-
| '''86'''
| '''1021.782'''
|-
| '''93'''
| '''1104.950'''
|-
|100
|1188.119
|}

== Links ==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]

{{todo|improve_layout|unify precision}}</text>
      <sha1>3irpsxdnndhg1nzk87b1gq8o9z79dpw</sha1>
    </revision>
    <revision>
      <id>67991</id>
      <parentid>67404</parentid>
      <timestamp>2021-04-20T03:09:42Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>67991</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6087" sha1="d7jxe67qcnyt3rp62c8zge62dhxf6ud" xml:space="preserve">'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

;[[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''13 12 13 13 12 13 12 13:''' ''5L3s MOS'' (Oneirotonic, an 8-tone circulating temperament)

{| class="wikitable"
! Steps
! Cents
!Difference from 8edo
|-
|13
|154.455
| +4.455¢
|-
|'''25'''
|'''297.030'''
| -2.97¢
|-
|'''38'''
|'''451.485'''
| +1.485¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''63'''
|'''748.515'''
| -1.485¢
|-
|75
|891.089
| -8.911¢
|-
|'''88'''
| '''1045.5445'''
| -4.5555¢
|}

'''9 8 9 8 8 9 8 9 8 9 8 8:''' ''5L7s MOS'' (Diatonic Pythagorean, a 12-tone circulating temperament)

{| class="wikitable"
!Steps
!Cents
!Difference from 12edo
|-
|9
|106.059
| +6.059¢
|-
|'''17'''
|'''201.980'''
| +1.98¢
|-
|26
| 308.911
| +8.911¢
|-
|'''34'''
|'''403.960'''
| +3.96¢
|-
|'''42'''
|'''499.010'''
| -.99¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''59'''
|'''700.990'''
| +.99¢
|-
|68
|807.921
| +7.921¢
|-
|'''76'''
|'''902.970'''
| +2.97¢
|-
| 85
|1009.901
| +9.901¢
|-
| '''93'''
|'''1104.950'''
| +4.95¢
|}

'''10 3 10 3 10 3 10 3 10 3 10 3 10 3 10:''' ''8L7s MOS'' (Opossum)

{| class="wikitable"
!Steps
!Cents
|-
|10
|118.812
|-
|'''13'''
|'''154.455'''
|-
|23
|273.267
|-
|'''26'''
|'''308.911'''
|-
|36
|427.723
|-
|'''39'''
|'''463.366'''
|-
|49
|582.178
|-
|'''52'''
|'''617.822'''
|-
|62
|736.337
|-
|'''65'''
|'''772.277'''
|-
|75
|891.089
|-
|'''78'''
|'''926.733'''
|-
| 88
| 1045.5445
|-
| '''91'''
|'''1081.188'''
|}

'''8 5 8 5 8 5 5 8 5 8 5 8 5 8 5 5:''' ''7L9s MOS'' (Golden Mavila chromatic 1/13-tone)

{| class="wikitable"
!Steps
!Cents
|-
|8
|95.0495
|-
|'''13'''
|'''154.455'''
|-
|21
|249.505
|-
|'''26'''
|'''308.911'''
|-
|34
| 403.960
|-
|'''39'''
|'''463.366'''
|-
|'''44'''
|'''522.772'''
|-
|52
|617.822
|-
|'''57'''
|'''677.228'''
|-
|65
|772.277
|-
|'''70'''
|'''831.683'''
|-
|78
|926.733
|-
|'''83'''
|'''986.139'''
|-
|91
|1081.188
|-
| '''96'''
|'''1045.545'''
|}

'''4 3 3 4 3 3 4 3 4 3 3 4 3 3 4 3 3 4 3 4 3 3 4 3 3 4 3 3 4 3:''' ''11L19s MOS'' (Improper Sensi-30, a 30-tone circulating temperament)

{| class="wikitable"
!Steps
!Cents
!Difference from 30edo
|-
|4
|47.525
| +7.525¢
|-
|'''7'''
|'''83.168'''
| +3.168¢
|-
|'''10'''
|'''118.812'''
| -1.188¢
|-
|14
|166.336
| +6.336¢
|-
|'''17'''
| '''201.980'''
| +1.98¢
|-
|'''20'''
|'''237.624'''
| +7.624¢
|-
|'''24'''
|'''285.1485'''
| +5.1485¢
|-
|'''27'''
|'''320.792'''
| +.792¢
|-
|31
|368.317
| +8.317¢
|-
|'''34'''
|'''403.960'''
| +3.96¢
|-
|'''37'''
|'''439.604'''
| -.396¢
|-
|41
|487.129
| +7.129¢
|-
|'''44'''
|'''527.772'''
| +7.772¢
|-
|'''47'''
|'''558.416'''
| +8.416¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''54'''
| '''641.584'''
| +1.584¢
|-
|'''57'''
|'''677.228'''
| +7.228¢
|-
|61
|724.7525
| +4.7525¢
|-
|'''64'''
|'''760.396'''
| +.396¢
|-
|68
|807.921
| +7.921¢
|-
|'''71'''
|'''843.564'''
| +3.564¢
|-
|'''74'''
|'''879.218'''
| -.792¢
|-
|78
|926.733
| +6.733¢
|-
|'''81'''
|'''962.376'''
| +2.376¢
|-
|'''84'''
|'''998.020'''
| -1.98¢
|-
|88
|1045.5445
| +5.5445¢
|-
|'''91'''
|'''1081.188'''
| +1.188¢
|-
|'''94'''
|'''1116.842'''
| -3.158¢
|-
|98
|1164.3564
| +4.356¢
|}

'''7 7 7 7 1 7 7 7 7 7 1 7 7 7 7 7 1:''' ''14L3s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
!Steps
!Cents
|-
|
'''7'''
|'''83.168'''
|-
|'''14'''
|'''166.337'''
|-
|21
| 249.505
|-
|'''22'''
|'''261.386'''
|-
|'''29'''
|'''344.554'''
|-
|'''36'''
|'''427.723'''
|-
|'''43'''
|'''510.891'''
|-
|'''50'''
|'''594.059'''
|-
|57
| 677.278
|-
|'''58'''
|'''689.119'''
|-
|'''65'''
|'''772.287'''
|-
|'''72'''
|'''855.446'''
|-
|'''79'''
|'''938.614'''
|-
|'''86'''
|'''1021.782'''
|-
|'''93'''
|'''1104.950'''
|-
|100
|1188.119
|}Since 101edo has a step of 11.881 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which truly allows one to use an 80 tone or larger MOS scale as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 101edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|21:20
|-
|6
|[[5L 1s]]
|17:16
|-
|7
|[[3L 4s]]
|15:14
|-
|8
|[[5L 3s]]
|13:12
|-
|9
|[[2L 7s]]
|12:11
|-
|10
|[[1L 9s]]
|11:10
|-
|11
|2L 9s
|10:9
|-
|12
|[[4L 8s]]
|9:8
|-
|13
|[[9L 4s]]
| rowspan="2" |8:7
|-
|14
|[[2L 12s]]
|-
|15
|[[11L 4s]]
| rowspan="2" |7:6
|-
|16
|[[5L 11s]]
|-
|17
|16L 1s
| rowspan="4" |6:5
|-
|18
|11L 7s
|-
|19
|[[6L 13s]]
|-
|20
|1L 19s
|-
|21
|17L 4s
| rowspan="5" |5:4
|-
|22
|13L 9s
|-
|23
|9L 14s
|-
|24
|[[5L 19s]]
|-
|25
|1L 24s
|-
|26
|23L 3s
| rowspan="8" |4:3
|-
|27
|20L 7s
|-
|28
|17L 11s
|-
|29
|14L 15s
|-
|30
|11L 19s
|-
|31
|8L 23s
|-
|32
|5L 27s
|-
|33
|2L 31s
|-
|34
|33L 1s
| rowspan="17" |3:2
|-
|35
|31L 4s
|-
|36
|29L 7s
|-
|37
|27L 10s
|-
|38
|25L 13s
|-
|39
|23L 16s
|-
|40
|21L 19s
|-
|41
|19L 22s
|-
|42
|17L 25s
|-
|43
|15L 28s
|-
|44
|13L 31s
|-
|45
|11L 34s
|-
|46
|9L 37s
|-
|47
|7L 40s
|-
|48
|5L 43s
|-
|49
|3L 49s
|-
|50
|1L 49s
|-
|51
|50L 1s
| rowspan="30" |2:1
|-
|52
|49L 3s
|-
|53
|48L 5s
|-
|54
|47L 7s
|-
|55
|46L 9s
|-
|56
|45L 11s
|-
|57
|44L 13s
|-
|58
|43L 15s
|-
|59
|42L 17s
|-
|60
|41L 19s
|-
|61
|40L 21s
|-
|62
|39L 23s
|-
|63
|38L 25s
|-
|64
|37L 27s
|-
|65
|36L 29s
|-
|66
|35L 31s
|-
|67
|34L 33s
|-
|68
|33L 35s
|-
|69
|32L 37s
|-
|70
|31L 39s
|-
|71
|30L 41s
|-
|72
|29L 43s
|-
|73
|28L 45s
|-
|74
|27L 47s
|-
|75
|26L 49s
|-
|76
|25L 51s
|-
|77
|24L 53s
|-
|78
|23L 55s
|-
|79
|22L 57s
|-
|80
|21L 59s
|}
==Links==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]

{{todo|improve_layout|unify precision}}</text>
      <sha1>d7jxe67qcnyt3rp62c8zge62dhxf6ud</sha1>
    </revision>
    <revision>
      <id>67994</id>
      <parentid>67991</parentid>
      <timestamp>2021-04-20T04:38:28Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>67994</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6091" sha1="5pmudfagn42vuarh5o4u2xnnefq6zpz" xml:space="preserve">'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

;[[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''13 12 13 13 12 13 12 13:''' ''5L3s MOS'' (Oneirotonic, an 8-tone circulating temperament)

{| class="wikitable"
! Steps
! Cents
!Difference from 8edo
|-
|13
|154.455
| +4.455¢
|-
|'''25'''
|'''297.030'''
| -2.97¢
|-
|'''38'''
|'''451.485'''
| +1.485¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''63'''
|'''748.515'''
| -1.485¢
|-
|75
|891.089
| -8.911¢
|-
|'''88'''
| '''1045.5445'''
| -4.5555¢
|}

'''9 8 9 8 8 9 8 9 8 9 8 8:''' ''5L7s MOS'' (Diatonic Pythagorean, a 12-tone circulating temperament)

{| class="wikitable"
!Steps
!Cents
!Difference from 12edo
|-
|9
|106.059
| +6.059¢
|-
|'''17'''
|'''201.980'''
| +1.98¢
|-
|26
| 308.911
| +8.911¢
|-
|'''34'''
|'''403.960'''
| +3.96¢
|-
|'''42'''
|'''499.010'''
| -.99¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''59'''
|'''700.990'''
| +.99¢
|-
|68
|807.921
| +7.921¢
|-
|'''76'''
|'''902.970'''
| +2.97¢
|-
| 85
|1009.901
| +9.901¢
|-
| '''93'''
|'''1104.950'''
| +4.95¢
|}

'''10 3 10 3 10 3 10 3 10 3 10 3 10 3 10:''' ''8L7s MOS'' (Opossum)

{| class="wikitable"
!Steps
!Cents
|-
|10
|118.812
|-
|'''13'''
|'''154.455'''
|-
|23
|273.267
|-
|'''26'''
|'''308.911'''
|-
|36
|427.723
|-
|'''39'''
|'''463.366'''
|-
|49
|582.178
|-
|'''52'''
|'''617.822'''
|-
|62
|736.337
|-
|'''65'''
|'''772.277'''
|-
|75
|891.089
|-
|'''78'''
|'''926.733'''
|-
| 88
| 1045.5445
|-
| '''91'''
|'''1081.188'''
|}

'''8 5 8 5 8 5 5 8 5 8 5 8 5 8 5 5:''' ''7L9s MOS'' (Golden Mavila chromatic 1/13-tone)

{| class="wikitable"
!Steps
!Cents
|-
|8
|95.0495
|-
|'''13'''
|'''154.455'''
|-
|21
|249.505
|-
|'''26'''
|'''308.911'''
|-
|34
| 403.960
|-
|'''39'''
|'''463.366'''
|-
|'''44'''
|'''522.772'''
|-
|52
|617.822
|-
|'''57'''
|'''677.228'''
|-
|65
|772.277
|-
|'''70'''
|'''831.683'''
|-
|78
|926.733
|-
|'''83'''
|'''986.139'''
|-
|91
|1081.188
|-
| '''96'''
|'''1045.545'''
|}

'''4 3 3 4 3 3 4 3 4 3 3 4 3 3 4 3 3 4 3 4 3 3 4 3 3 4 3 3 4 3:''' ''11L19s MOS'' (Improper Sensi-30, a 30-tone circulating temperament)

{| class="wikitable"
!Steps
!Cents
!Difference from 30edo
|-
|4
|47.525
| +7.525¢
|-
|'''7'''
|'''83.168'''
| +3.168¢
|-
|'''10'''
|'''118.812'''
| -1.188¢
|-
|14
|166.336
| +6.336¢
|-
|'''17'''
| '''201.980'''
| +1.98¢
|-
|'''20'''
|'''237.624'''
| +7.624¢
|-
|'''24'''
|'''285.1485'''
| +5.1485¢
|-
|'''27'''
|'''320.792'''
| +.792¢
|-
|31
|368.317
| +8.317¢
|-
|'''34'''
|'''403.960'''
| +3.96¢
|-
|'''37'''
|'''439.604'''
| -.396¢
|-
|41
|487.129
| +7.129¢
|-
|'''44'''
|'''527.772'''
| +7.772¢
|-
|'''47'''
|'''558.416'''
| +8.416¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''54'''
| '''641.584'''
| +1.584¢
|-
|'''57'''
|'''677.228'''
| +7.228¢
|-
|61
|724.7525
| +4.7525¢
|-
|'''64'''
|'''760.396'''
| +.396¢
|-
|68
|807.921
| +7.921¢
|-
|'''71'''
|'''843.564'''
| +3.564¢
|-
|'''74'''
|'''879.218'''
| -.792¢
|-
|78
|926.733
| +6.733¢
|-
|'''81'''
|'''962.376'''
| +2.376¢
|-
|'''84'''
|'''998.020'''
| -1.98¢
|-
|88
|1045.5445
| +5.5445¢
|-
|'''91'''
|'''1081.188'''
| +1.188¢
|-
|'''94'''
|'''1116.842'''
| -3.158¢
|-
|98
|1164.3564
| +4.356¢
|}

'''7 7 7 7 1 7 7 7 7 7 1 7 7 7 7 7 1:''' ''14L3s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
!Steps
!Cents
|-
|
'''7'''
|'''83.168'''
|-
|'''14'''
|'''166.337'''
|-
|21
| 249.505
|-
|'''22'''
|'''261.386'''
|-
|'''29'''
|'''344.554'''
|-
|'''36'''
|'''427.723'''
|-
|'''43'''
|'''510.891'''
|-
|'''50'''
|'''594.059'''
|-
|57
| 677.278
|-
|'''58'''
|'''689.119'''
|-
|'''65'''
|'''772.287'''
|-
|'''72'''
|'''855.446'''
|-
|'''79'''
|'''938.614'''
|-
|'''86'''
|'''1021.782'''
|-
|'''93'''
|'''1104.950'''
|-
|100
|1188.119
|}Since 101edo has a step of 11.881 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which truly allows one to use an 80 tone or larger MOS scale as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 101edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|21:20
|-
|6
|[[5L 1s]]
|17:16
|-
|7
|[[3L 4s]]
|15:14
|-
|8
|[[5L 3s]]
|13:12
|-
|9
|[[2L 7s]]
|12:11
|-
|10
|[[1L 9s]]
|11:10
|-
|11
|[[2L 9s]]
|10:9
|-
|12
|[[5L 7s]]
|9:8
|-
|13
|[[9L 4s]]
| rowspan="2" |8:7
|-
|14
|[[2L 12s]]
|-
|15
|[[11L 4s]]
| rowspan="2" |7:6
|-
|16
|[[5L 11s]]
|-
|17
|16L 1s
| rowspan="4" |6:5
|-
|18
|11L 7s
|-
|19
|[[6L 13s]]
|-
|20
|1L 19s
|-
|21
|17L 4s
| rowspan="5" |5:4
|-
|22
|13L 9s
|-
|23
|9L 14s
|-
|24
|[[5L 19s]]
|-
|25
|1L 24s
|-
|26
|23L 3s
| rowspan="8" |4:3
|-
|27
|20L 7s
|-
|28
|17L 11s
|-
|29
|14L 15s
|-
|30
|11L 19s
|-
|31
|8L 23s
|-
|32
|5L 27s
|-
|33
|2L 31s
|-
|34
|33L 1s
| rowspan="17" |3:2
|-
|35
|31L 4s
|-
|36
|29L 7s
|-
|37
|27L 10s
|-
|38
|25L 13s
|-
|39
|23L 16s
|-
|40
|21L 19s
|-
|41
|19L 22s
|-
|42
|17L 25s
|-
|43
|15L 28s
|-
|44
|13L 31s
|-
|45
|11L 34s
|-
|46
|9L 37s
|-
|47
|7L 40s
|-
|48
|5L 43s
|-
|49
|3L 49s
|-
|50
|1L 49s
|-
|51
|50L 1s
| rowspan="30" |2:1
|-
|52
|49L 3s
|-
|53
|48L 5s
|-
|54
|47L 7s
|-
|55
|46L 9s
|-
|56
|45L 11s
|-
|57
|44L 13s
|-
|58
|43L 15s
|-
|59
|42L 17s
|-
|60
|41L 19s
|-
|61
|40L 21s
|-
|62
|39L 23s
|-
|63
|38L 25s
|-
|64
|37L 27s
|-
|65
|36L 29s
|-
|66
|35L 31s
|-
|67
|34L 33s
|-
|68
|33L 35s
|-
|69
|32L 37s
|-
|70
|31L 39s
|-
|71
|30L 41s
|-
|72
|29L 43s
|-
|73
|28L 45s
|-
|74
|27L 47s
|-
|75
|26L 49s
|-
|76
|25L 51s
|-
|77
|24L 53s
|-
|78
|23L 55s
|-
|79
|22L 57s
|-
|80
|21L 59s
|}
==Links==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]

{{todo|improve_layout|unify precision}}</text>
      <sha1>5pmudfagn42vuarh5o4u2xnnefq6zpz</sha1>
    </revision>
    <revision>
      <id>68020</id>
      <parentid>67994</parentid>
      <timestamp>2021-04-20T18:51:30Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68020</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6092" sha1="1k2z4ujr7x1gzusyfnvt1bdv63kvqhd" xml:space="preserve">'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

;[[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''13 12 13 13 12 13 12 13:''' ''5L3s MOS'' (Oneirotonic, an 8-tone circulating temperament)

{| class="wikitable"
! Steps
! Cents
!Difference from 8edo
|-
|13
|154.455
| +4.455¢
|-
|'''25'''
|'''297.030'''
| -2.97¢
|-
|'''38'''
|'''451.485'''
| +1.485¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''63'''
|'''748.515'''
| -1.485¢
|-
|75
|891.089
| -8.911¢
|-
|'''88'''
| '''1045.5445'''
| -4.5555¢
|}

'''9 8 9 8 8 9 8 9 8 9 8 8:''' ''5L7s MOS'' (Diatonic Pythagorean, a 12-tone circulating temperament)

{| class="wikitable"
!Steps
!Cents
!Difference from 12edo
|-
|9
|106.059
| +6.059¢
|-
|'''17'''
|'''201.980'''
| +1.98¢
|-
|26
| 308.911
| +8.911¢
|-
|'''34'''
|'''403.960'''
| +3.96¢
|-
|'''42'''
|'''499.010'''
| -.99¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''59'''
|'''700.990'''
| +.99¢
|-
|68
|807.921
| +7.921¢
|-
|'''76'''
|'''902.970'''
| +2.97¢
|-
| 85
|1009.901
| +9.901¢
|-
| '''93'''
|'''1104.950'''
| +4.95¢
|}

'''10 3 10 3 10 3 10 3 10 3 10 3 10 3 10:''' ''8L7s MOS'' (Opossum)

{| class="wikitable"
!Steps
!Cents
|-
|10
|118.812
|-
|'''13'''
|'''154.455'''
|-
|23
|273.267
|-
|'''26'''
|'''308.911'''
|-
|36
|427.723
|-
|'''39'''
|'''463.366'''
|-
|49
|582.178
|-
|'''52'''
|'''617.822'''
|-
|62
|736.337
|-
|'''65'''
|'''772.277'''
|-
|75
|891.089
|-
|'''78'''
|'''926.733'''
|-
| 88
| 1045.5445
|-
| '''91'''
|'''1081.188'''
|}

'''8 5 8 5 8 5 5 8 5 8 5 8 5 8 5 5:''' ''7L9s MOS'' (Golden Mavila chromatic 1/13-tone)

{| class="wikitable"
!Steps
!Cents
|-
|8
|95.0495
|-
|'''13'''
|'''154.455'''
|-
|21
|249.505
|-
|'''26'''
|'''308.911'''
|-
|34
| 403.960
|-
|'''39'''
|'''463.366'''
|-
|'''44'''
|'''522.772'''
|-
|52
|617.822
|-
|'''57'''
|'''677.228'''
|-
|65
|772.277
|-
|'''70'''
|'''831.683'''
|-
|78
|926.733
|-
|'''83'''
|'''986.139'''
|-
|91
|1081.188
|-
| '''96'''
|'''1045.545'''
|}

'''4 3 3 4 3 3 4 3 4 3 3 4 3 3 4 3 3 4 3 4 3 3 4 3 3 4 3 3 4 3:''' ''11L19s MOS'' (Improper Sensi-30, a 30-tone circulating temperament)

{| class="wikitable"
!Steps
!Cents
!Difference from 30edo
|-
|4
|47.525
| +7.525¢
|-
|'''7'''
|'''83.168'''
| +3.168¢
|-
|'''10'''
|'''118.812'''
| -1.188¢
|-
|14
|166.336
| +6.336¢
|-
|'''17'''
| '''201.980'''
| +1.98¢
|-
|'''20'''
|'''237.624'''
| +7.624¢
|-
|'''24'''
|'''285.1485'''
| +5.1485¢
|-
|'''27'''
|'''320.792'''
| +.792¢
|-
|31
|368.317
| +8.317¢
|-
|'''34'''
|'''403.960'''
| +3.96¢
|-
|'''37'''
|'''439.604'''
| -.396¢
|-
|41
|487.129
| +7.129¢
|-
|'''44'''
|'''527.772'''
| +7.772¢
|-
|'''47'''
|'''558.416'''
| +8.416¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''54'''
| '''641.584'''
| +1.584¢
|-
|'''57'''
|'''677.228'''
| +7.228¢
|-
|61
|724.7525
| +4.7525¢
|-
|'''64'''
|'''760.396'''
| +.396¢
|-
|68
|807.921
| +7.921¢
|-
|'''71'''
|'''843.564'''
| +3.564¢
|-
|'''74'''
|'''879.218'''
| -.792¢
|-
|78
|926.733
| +6.733¢
|-
|'''81'''
|'''962.376'''
| +2.376¢
|-
|'''84'''
|'''998.020'''
| -1.98¢
|-
|88
|1045.5445
| +5.5445¢
|-
|'''91'''
|'''1081.188'''
| +1.188¢
|-
|'''94'''
|'''1116.842'''
| -3.158¢
|-
|98
|1164.3564
| +4.356¢
|}

'''7 7 7 7 1 7 7 7 7 7 1 7 7 7 7 7 1:''' ''14L3s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
!Steps
!Cents
|-
|
'''7'''
|'''83.168'''
|-
|'''14'''
|'''166.337'''
|-
|21
| 249.505
|-
|'''22'''
|'''261.386'''
|-
|'''29'''
|'''344.554'''
|-
|'''36'''
|'''427.723'''
|-
|'''43'''
|'''510.891'''
|-
|'''50'''
|'''594.059'''
|-
|57
| 677.278
|-
|'''58'''
|'''689.119'''
|-
|'''65'''
|'''772.287'''
|-
|'''72'''
|'''855.446'''
|-
|'''79'''
|'''938.614'''
|-
|'''86'''
|'''1021.782'''
|-
|'''93'''
|'''1104.950'''
|-
|100
|1188.119
|}Since 101edo has a step of 11.881 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which truly allows one to use an 80 tone or larger MOS scale as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 101edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|21:20
|-
|6
|[[5L 1s]]
|17:16
|-
|7
|[[3L 4s]]
|15:14
|-
|8
|[[5L 3s]]
|13:12
|-
|9
|[[2L 7s]]
|12:11
|-
|10
|[[1L 9s]]
|11:10
|-
|11
|[[2L 9s]]
|10:9
|-
|12
|[[5L 7s]]
|9:8
|-
|13
|[[10L 3s]]
| rowspan="2" |8:7
|-
|14
|[[2L 12s]]
|-
|15
|[[11L 4s]]
| rowspan="2" |7:6
|-
|16
|[[5L 11s]]
|-
|17
|16L 1s
| rowspan="4" |6:5
|-
|18
|11L 7s
|-
|19
|[[6L 13s]]
|-
|20
|1L 19s
|-
|21
|17L 4s
| rowspan="5" |5:4
|-
|22
|13L 9s
|-
|23
|9L 14s
|-
|24
|[[5L 19s]]
|-
|25
|1L 24s
|-
|26
|23L 3s
| rowspan="8" |4:3
|-
|27
|20L 7s
|-
|28
|17L 11s
|-
|29
|14L 15s
|-
|30
|11L 19s
|-
|31
|8L 23s
|-
|32
|5L 27s
|-
|33
|2L 31s
|-
|34
|33L 1s
| rowspan="17" |3:2
|-
|35
|31L 4s
|-
|36
|29L 7s
|-
|37
|27L 10s
|-
|38
|25L 13s
|-
|39
|23L 16s
|-
|40
|21L 19s
|-
|41
|19L 22s
|-
|42
|17L 25s
|-
|43
|15L 28s
|-
|44
|13L 31s
|-
|45
|11L 34s
|-
|46
|9L 37s
|-
|47
|7L 40s
|-
|48
|5L 43s
|-
|49
|3L 49s
|-
|50
|1L 49s
|-
|51
|50L 1s
| rowspan="30" |2:1
|-
|52
|49L 3s
|-
|53
|48L 5s
|-
|54
|47L 7s
|-
|55
|46L 9s
|-
|56
|45L 11s
|-
|57
|44L 13s
|-
|58
|43L 15s
|-
|59
|42L 17s
|-
|60
|41L 19s
|-
|61
|40L 21s
|-
|62
|39L 23s
|-
|63
|38L 25s
|-
|64
|37L 27s
|-
|65
|36L 29s
|-
|66
|35L 31s
|-
|67
|34L 33s
|-
|68
|33L 35s
|-
|69
|32L 37s
|-
|70
|31L 39s
|-
|71
|30L 41s
|-
|72
|29L 43s
|-
|73
|28L 45s
|-
|74
|27L 47s
|-
|75
|26L 49s
|-
|76
|25L 51s
|-
|77
|24L 53s
|-
|78
|23L 55s
|-
|79
|22L 57s
|-
|80
|21L 59s
|}
==Links==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]

{{todo|improve_layout|unify precision}}</text>
      <sha1>1k2z4ujr7x1gzusyfnvt1bdv63kvqhd</sha1>
    </revision>
    <revision>
      <id>68021</id>
      <parentid>68020</parentid>
      <timestamp>2021-04-20T19:19:58Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68021</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6183" sha1="86b3zwynvsegyjxwqxvfqgclaymtorq" xml:space="preserve">'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

;[[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''13 12 13 13 12 13 12 13:''' ''5L3s MOS'' (Oneirotonic, an 8-tone circulating temperament)

{| class="wikitable"
! Steps
! Cents
!Difference from 8edo
|-
|13
|154.455
| +4.455¢
|-
|'''25'''
|'''297.030'''
| -2.97¢
|-
|'''38'''
|'''451.485'''
| +1.485¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''63'''
|'''748.515'''
| -1.485¢
|-
|75
|891.089
| -8.911¢
|-
|'''88'''
| '''1045.5445'''
| -4.5555¢
|}

'''9 8 9 8 8 9 8 9 8 9 8 8:''' ''5L7s MOS'' (Diatonic Pythagorean, a 12-tone circulating temperament)

{| class="wikitable"
!Steps
!Cents
!Difference from 12edo
|-
|9
|106.059
| +6.059¢
|-
|'''17'''
|'''201.980'''
| +1.98¢
|-
|26
| 308.911
| +8.911¢
|-
|'''34'''
|'''403.960'''
| +3.96¢
|-
|'''42'''
|'''499.010'''
| -.99¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''59'''
|'''700.990'''
| +.99¢
|-
|68
|807.921
| +7.921¢
|-
|'''76'''
|'''902.970'''
| +2.97¢
|-
| 85
|1009.901
| +9.901¢
|-
| '''93'''
|'''1104.950'''
| +4.95¢
|}

'''10 3 10 3 10 3 10 3 10 3 10 3 10 3 10:''' ''8L7s MOS'' (Opossum)

{| class="wikitable"
!Steps
!Cents
|-
|10
|118.812
|-
|'''13'''
|'''154.455'''
|-
|23
|273.267
|-
|'''26'''
|'''308.911'''
|-
|36
|427.723
|-
|'''39'''
|'''463.366'''
|-
|49
|582.178
|-
|'''52'''
|'''617.822'''
|-
|62
|736.337
|-
|'''65'''
|'''772.277'''
|-
|75
|891.089
|-
|'''78'''
|'''926.733'''
|-
| 88
| 1045.5445
|-
| '''91'''
|'''1081.188'''
|}

'''8 5 8 5 8 5 5 8 5 8 5 8 5 8 5 5:''' ''7L9s MOS'' (Golden Mavila chromatic 1/13-tone)

{| class="wikitable"
!Steps
!Cents
|-
|8
|95.0495
|-
|'''13'''
|'''154.455'''
|-
|21
|249.505
|-
|'''26'''
|'''308.911'''
|-
|34
| 403.960
|-
|'''39'''
|'''463.366'''
|-
|'''44'''
|'''522.772'''
|-
|52
|617.822
|-
|'''57'''
|'''677.228'''
|-
|65
|772.277
|-
|'''70'''
|'''831.683'''
|-
|78
|926.733
|-
|'''83'''
|'''986.139'''
|-
|91
|1081.188
|-
| '''96'''
|'''1045.545'''
|}

'''4 3 3 4 3 3 4 3 4 3 3 4 3 3 4 3 3 4 3 4 3 3 4 3 3 4 3 3 4 3:''' ''11L19s MOS'' (Improper Sensi-30, a 30-tone circulating temperament)

{| class="wikitable"
!Steps
!Cents
!Difference from 30edo
|-
|4
|47.525
| +7.525¢
|-
|'''7'''
|'''83.168'''
| +3.168¢
|-
|'''10'''
|'''118.812'''
| -1.188¢
|-
|14
|166.336
| +6.336¢
|-
|'''17'''
| '''201.980'''
| +1.98¢
|-
|'''20'''
|'''237.624'''
| +7.624¢
|-
|'''24'''
|'''285.1485'''
| +5.1485¢
|-
|'''27'''
|'''320.792'''
| +.792¢
|-
|31
|368.317
| +8.317¢
|-
|'''34'''
|'''403.960'''
| +3.96¢
|-
|'''37'''
|'''439.604'''
| -.396¢
|-
|41
|487.129
| +7.129¢
|-
|'''44'''
|'''527.772'''
| +7.772¢
|-
|'''47'''
|'''558.416'''
| +8.416¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''54'''
| '''641.584'''
| +1.584¢
|-
|'''57'''
|'''677.228'''
| +7.228¢
|-
|61
|724.7525
| +4.7525¢
|-
|'''64'''
|'''760.396'''
| +.396¢
|-
|68
|807.921
| +7.921¢
|-
|'''71'''
|'''843.564'''
| +3.564¢
|-
|'''74'''
|'''879.218'''
| -.792¢
|-
|78
|926.733
| +6.733¢
|-
|'''81'''
|'''962.376'''
| +2.376¢
|-
|'''84'''
|'''998.020'''
| -1.98¢
|-
|88
|1045.5445
| +5.5445¢
|-
|'''91'''
|'''1081.188'''
| +1.188¢
|-
|'''94'''
|'''1116.842'''
| -3.158¢
|-
|98
|1164.3564
| +4.356¢
|}

'''7 7 7 7 1 7 7 7 7 7 1 7 7 7 7 7 1:''' ''14L3s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
!Steps
!Cents
|-
|
'''7'''
|'''83.168'''
|-
|'''14'''
|'''166.337'''
|-
|21
| 249.505
|-
|'''22'''
|'''261.386'''
|-
|'''29'''
|'''344.554'''
|-
|'''36'''
|'''427.723'''
|-
|'''43'''
|'''510.891'''
|-
|'''50'''
|'''594.059'''
|-
|57
| 677.278
|-
|'''58'''
|'''689.119'''
|-
|'''65'''
|'''772.287'''
|-
|'''72'''
|'''855.446'''
|-
|'''79'''
|'''938.614'''
|-
|'''86'''
|'''1021.782'''
|-
|'''93'''
|'''1104.950'''
|-
|100
|1188.119
|}Since 101edo has a step of 11.881 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which truly allows one to use an 80 tone or larger MOS scale as a circulating temperament or allows one to use an MOS scale with 50 or more large steps as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 101edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|21:20
|-
|6
|[[5L 1s]]
|17:16
|-
|7
|[[3L 4s]]
|15:14
|-
|8
|[[5L 3s]]
|13:12
|-
|9
|[[2L 7s]]
|12:11
|-
|10
|[[1L 9s]]
|11:10
|-
|11
|[[2L 9s]]
|10:9
|-
|12
|[[5L 7s]]
|9:8
|-
|13
|[[10L 3s]]
| rowspan="2" |8:7
|-
|14
|[[2L 12s]]
|-
|15
|[[11L 4s]]
| rowspan="2" |7:6
|-
|16
|[[5L 11s]]
|-
|17
|16L 1s
| rowspan="4" |6:5
|-
|18
|11L 7s
|-
|19
|[[6L 13s]]
|-
|20
|1L 19s
|-
|21
|17L 4s
| rowspan="5" |5:4
|-
|22
|13L 9s
|-
|23
|9L 14s
|-
|24
|[[5L 19s]]
|-
|25
|1L 24s
|-
|26
|23L 3s
| rowspan="8" |4:3
|-
|27
|20L 7s
|-
|28
|17L 11s
|-
|29
|14L 15s
|-
|30
|11L 19s
|-
|31
|8L 23s
|-
|32
|5L 27s
|-
|33
|2L 31s
|-
|34
|33L 1s
| rowspan="17" |3:2
|-
|35
|31L 4s
|-
|36
|29L 7s
|-
|37
|27L 10s
|-
|38
|25L 13s
|-
|39
|23L 16s
|-
|40
|21L 19s
|-
|41
|19L 22s
|-
|42
|17L 25s
|-
|43
|15L 28s
|-
|44
|13L 31s
|-
|45
|11L 34s
|-
|46
|9L 37s
|-
|47
|7L 40s
|-
|48
|5L 43s
|-
|49
|3L 49s
|-
|50
|1L 49s
|-
|51
|50L 1s
| rowspan="30" |2:1
|-
|52
|49L 3s
|-
|53
|48L 5s
|-
|54
|47L 7s
|-
|55
|46L 9s
|-
|56
|45L 11s
|-
|57
|44L 13s
|-
|58
|43L 15s
|-
|59
|42L 17s
|-
|60
|41L 19s
|-
|61
|40L 21s
|-
|62
|39L 23s
|-
|63
|38L 25s
|-
|64
|37L 27s
|-
|65
|36L 29s
|-
|66
|35L 31s
|-
|67
|34L 33s
|-
|68
|33L 35s
|-
|69
|32L 37s
|-
|70
|31L 39s
|-
|71
|30L 41s
|-
|72
|29L 43s
|-
|73
|28L 45s
|-
|74
|27L 47s
|-
|75
|26L 49s
|-
|76
|25L 51s
|-
|77
|24L 53s
|-
|78
|23L 55s
|-
|79
|22L 57s
|-
|80
|21L 59s
|}
==Links==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]

{{todo|improve_layout|unify precision}}</text>
      <sha1>86b3zwynvsegyjxwqxvfqgclaymtorq</sha1>
    </revision>
    <revision>
      <id>68023</id>
      <parentid>68021</parentid>
      <timestamp>2021-04-20T19:41:45Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68023</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6183" sha1="dtqmz1e7nxctq3jgan6nr4mblsvatrq" xml:space="preserve">'''''101-EDO''''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

;[[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
;[[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''13 12 13 13 12 13 12 13:''' ''5L3s MOS'' (Oneirotonic, an 8-tone circulating temperament)

{| class="wikitable"
! Steps
! Cents
!Difference from 8edo
|-
|13
|154.455
| +4.455¢
|-
|'''25'''
|'''297.030'''
| -2.97¢
|-
|'''38'''
|'''451.485'''
| +1.485¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''63'''
|'''748.515'''
| -1.485¢
|-
|75
|891.089
| -8.911¢
|-
|'''88'''
| '''1045.5445'''
| -4.5555¢
|}

'''9 8 9 8 8 9 8 9 8 9 8 8:''' ''5L7s MOS'' (Diatonic Pythagorean, a 12-tone circulating temperament)

{| class="wikitable"
!Steps
!Cents
!Difference from 12edo
|-
|9
|106.059
| +6.059¢
|-
|'''17'''
|'''201.980'''
| +1.98¢
|-
|26
| 308.911
| +8.911¢
|-
|'''34'''
|'''403.960'''
| +3.96¢
|-
|'''42'''
|'''499.010'''
| -.99¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''59'''
|'''700.990'''
| +.99¢
|-
|68
|807.921
| +7.921¢
|-
|'''76'''
|'''902.970'''
| +2.97¢
|-
| 85
|1009.901
| +9.901¢
|-
| '''93'''
|'''1104.950'''
| +4.95¢
|}

'''10 3 10 3 10 3 10 3 10 3 10 3 10 3 10:''' ''8L7s MOS'' (Opossum)

{| class="wikitable"
!Steps
!Cents
|-
|10
|118.812
|-
|'''13'''
|'''154.455'''
|-
|23
|273.267
|-
|'''26'''
|'''308.911'''
|-
|36
|427.723
|-
|'''39'''
|'''463.366'''
|-
|49
|582.178
|-
|'''52'''
|'''617.822'''
|-
|62
|736.337
|-
|'''65'''
|'''772.277'''
|-
|75
|891.089
|-
|'''78'''
|'''926.733'''
|-
| 88
| 1045.5445
|-
| '''91'''
|'''1081.188'''
|}

'''8 5 8 5 8 5 5 8 5 8 5 8 5 8 5 5:''' ''7L9s MOS'' (Golden Mavila chromatic 1/13-tone)

{| class="wikitable"
!Steps
!Cents
|-
|8
|95.0495
|-
|'''13'''
|'''154.455'''
|-
|21
|249.505
|-
|'''26'''
|'''308.911'''
|-
|34
| 403.960
|-
|'''39'''
|'''463.366'''
|-
|'''44'''
|'''522.772'''
|-
|52
|617.822
|-
|'''57'''
|'''677.228'''
|-
|65
|772.277
|-
|'''70'''
|'''831.683'''
|-
|78
|926.733
|-
|'''83'''
|'''986.139'''
|-
|91
|1081.188
|-
| '''96'''
|'''1045.545'''
|}

'''4 3 3 4 3 3 4 3 4 3 3 4 3 3 4 3 3 4 3 4 3 3 4 3 3 4 3 3 4 3:''' ''11L19s MOS'' (Improper Sensi-30, a 30-tone circulating temperament)

{| class="wikitable"
!Steps
!Cents
!Difference from 30edo
|-
|4
|47.525
| +7.525¢
|-
|'''7'''
|'''83.168'''
| +3.168¢
|-
|'''10'''
|'''118.812'''
| -1.188¢
|-
|14
|166.336
| +6.336¢
|-
|'''17'''
| '''201.980'''
| +1.98¢
|-
|'''20'''
|'''237.624'''
| +7.624¢
|-
|'''24'''
|'''285.1485'''
| +5.1485¢
|-
|'''27'''
|'''320.792'''
| +.792¢
|-
|31
|368.317
| +8.317¢
|-
|'''34'''
|'''403.960'''
| +3.96¢
|-
|'''37'''
|'''439.604'''
| -.396¢
|-
|41
|487.129
| +7.129¢
|-
|'''44'''
|'''527.772'''
| +7.772¢
|-
|'''47'''
|'''558.416'''
| +8.416¢
|-
|51
|605.9405
| +5.9405¢
|-
|'''54'''
| '''641.584'''
| +1.584¢
|-
|'''57'''
|'''677.228'''
| +7.228¢
|-
|61
|724.7525
| +4.7525¢
|-
|'''64'''
|'''760.396'''
| +.396¢
|-
|68
|807.921
| +7.921¢
|-
|'''71'''
|'''843.564'''
| +3.564¢
|-
|'''74'''
|'''879.218'''
| -.792¢
|-
|78
|926.733
| +6.733¢
|-
|'''81'''
|'''962.376'''
| +2.376¢
|-
|'''84'''
|'''998.020'''
| -1.98¢
|-
|88
|1045.5445
| +5.5445¢
|-
|'''91'''
|'''1081.188'''
| +1.188¢
|-
|'''94'''
|'''1116.842'''
| -3.158¢
|-
|98
|1164.3564
| +4.356¢
|}

'''7 7 7 7 1 7 7 7 7 7 1 7 7 7 7 7 1:''' ''14L3s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
!Steps
!Cents
|-
|
'''7'''
|'''83.168'''
|-
|'''14'''
|'''166.337'''
|-
|21
| 249.505
|-
|'''22'''
|'''261.386'''
|-
|'''29'''
|'''344.554'''
|-
|'''36'''
|'''427.723'''
|-
|'''43'''
|'''510.891'''
|-
|'''50'''
|'''594.059'''
|-
|57
| 677.278
|-
|'''58'''
|'''689.119'''
|-
|'''65'''
|'''772.287'''
|-
|'''72'''
|'''855.446'''
|-
|'''79'''
|'''938.614'''
|-
|'''86'''
|'''1021.782'''
|-
|'''93'''
|'''1104.950'''
|-
|100
|1188.119
|}Since 101edo has a step of 11.881 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which truly allows one to use an 80 tone or larger MOS scale as a circulating temperament or allows one to use an MOS scale with 50 or more large steps as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 101edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|21:20
|-
|6
|[[5L 1s]]
|17:16
|-
|7
|[[3L 4s]]
|15:14
|-
|8
|[[5L 3s]]
|13:12
|-
|9
|[[2L 7s]]
|12:11
|-
|10
|[[1L 9s]]
|11:10
|-
|11
|[[2L 9s]]
|10:9
|-
|12
|[[5L 7s]]
|9:8
|-
|13
|[[10L 3s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[11L 4s]]
| rowspan="2" |7:6
|-
|16
|[[5L 11s]]
|-
|17
|16L 1s
| rowspan="4" |6:5
|-
|18
|11L 7s
|-
|19
|[[6L 13s]]
|-
|20
|1L 19s
|-
|21
|17L 4s
| rowspan="5" |5:4
|-
|22
|13L 9s
|-
|23
|9L 14s
|-
|24
|[[5L 19s]]
|-
|25
|1L 24s
|-
|26
|23L 3s
| rowspan="8" |4:3
|-
|27
|20L 7s
|-
|28
|17L 11s
|-
|29
|14L 15s
|-
|30
|11L 19s
|-
|31
|8L 23s
|-
|32
|5L 27s
|-
|33
|2L 31s
|-
|34
|33L 1s
| rowspan="17" |3:2
|-
|35
|31L 4s
|-
|36
|29L 7s
|-
|37
|27L 10s
|-
|38
|25L 13s
|-
|39
|23L 16s
|-
|40
|21L 19s
|-
|41
|19L 22s
|-
|42
|17L 25s
|-
|43
|15L 28s
|-
|44
|13L 31s
|-
|45
|11L 34s
|-
|46
|9L 37s
|-
|47
|7L 40s
|-
|48
|5L 43s
|-
|49
|3L 49s
|-
|50
|1L 49s
|-
|51
|50L 1s
| rowspan="30" |2:1
|-
|52
|49L 3s
|-
|53
|48L 5s
|-
|54
|47L 7s
|-
|55
|46L 9s
|-
|56
|45L 11s
|-
|57
|44L 13s
|-
|58
|43L 15s
|-
|59
|42L 17s
|-
|60
|41L 19s
|-
|61
|40L 21s
|-
|62
|39L 23s
|-
|63
|38L 25s
|-
|64
|37L 27s
|-
|65
|36L 29s
|-
|66
|35L 31s
|-
|67
|34L 33s
|-
|68
|33L 35s
|-
|69
|32L 37s
|-
|70
|31L 39s
|-
|71
|30L 41s
|-
|72
|29L 43s
|-
|73
|28L 45s
|-
|74
|27L 47s
|-
|75
|26L 49s
|-
|76
|25L 51s
|-
|77
|24L 53s
|-
|78
|23L 55s
|-
|79
|22L 57s
|-
|80
|21L 59s
|}
==Links==
[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal]

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]

{{todo|improve_layout|unify precision}}</text>
      <sha1>dtqmz1e7nxctq3jgan6nr4mblsvatrq</sha1>
    </revision>
    <revision>
      <id>71938</id>
      <parentid>68023</parentid>
      <timestamp>2021-06-05T14:33:16Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>cleanup, resetting to https://en.xen.wiki/index.php?title=101edo&amp;oldid=59162 which causes reverting additions of [[User:Moremajorthanmajor]] https://en.xen.wiki/index.php?title=101edo&amp;type=revision&amp;diff=68023&amp;oldid=59162</comment>
      <origin>71938</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2317" sha1="935tuaob2jc979csqaf2x6ip62p03vq" xml:space="preserve">'''101 EDO''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]</text>
      <sha1>935tuaob2jc979csqaf2x6ip62p03vq</sha1>
    </revision>
    <revision>
      <id>71939</id>
      <parentid>71938</parentid>
      <timestamp>2021-06-05T14:35:31Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* Some important MOS scales */ right-align table remaining cells</comment>
      <origin>71939</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2367" sha1="cx4uc4zng7wsmah6vs7v304spphta21" xml:space="preserve">'''101 EDO''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]</text>
      <sha1>cx4uc4zng7wsmah6vs7v304spphta21</sha1>
    </revision>
    <revision>
      <id>79731</id>
      <parentid>71939</parentid>
      <timestamp>2021-10-21T06:47:35Z</timestamp>
      <contributor>
        <username>Royalmilktea</username>
        <id>285</id>
      </contributor>
      <comment>added interval table</comment>
      <origin>79731</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3232" sha1="r3fnten22ajkvi7mevr3zxeh8u7ep40" xml:space="preserve">'''101 EDO''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Intervals ==
{| class="wikitable"
!Degree
!Cents
|-
|0
|0.000
|-
|1
|11.881
|-
|2
|23.762
|-
|3
|35.644
|-
|4
|47.525
|-
|5
|59.406
|-
|6
|71.287
|-
|7
|83.168
|-
|8
|95.050
|-
|9
|106.931
|-
|10
|118.812
|-
|11
|130.693
|-
|12
|142.574
|-
|13
|154.455
|-
|14
|166.337
|-
|15
|178.218
|-
|16
|190.099
|-
|17
|201.980
|-
|18
|213.861
|-
|19
|225.743
|-
|20
|237.624
|-
|21
|249.505
|-
|22
|261.386
|-
|23
|273.267
|-
|24
|285.149
|-
|25
|297.030
|-
|26
|308.911
|-
|27
|320.792
|-
|28
|332.673
|-
|29
|344.554
|-
|30
|356.436
|-
|31
|368.317
|-
|32
|380.198
|-
|33
|392.079
|-
|34
|403.960
|-
|35
|415.842
|-
|36
|427.723
|-
|37
|439.604
|-
|38
|451.485
|-
|39
|463.366
|-
|40
|475.248
|-
|41
|487.129
|-
|42
|499.010
|-
|43
|510.891
|-
|44
|522.772
|-
|45
|534.653
|-
|46
|546.535
|-
|47
|558.416
|-
|48
|570.297
|-
|49
|582.178
|-
|50
|594.059
|-
|...
|...
|}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Scales]]</text>
      <sha1>r3fnten22ajkvi7mevr3zxeh8u7ep40</sha1>
    </revision>
    <revision>
      <id>87061</id>
      <parentid>79731</parentid>
      <timestamp>2022-02-25T14:54:41Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87061</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3212" sha1="p5es9ylqteoe4qx70mt6492fzqtfcjm" xml:space="preserve">'''101 EDO''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Intervals ==
{| class="wikitable"
!Degree
!Cents
|-
|0
|0.000
|-
|1
|11.881
|-
|2
|23.762
|-
|3
|35.644
|-
|4
|47.525
|-
|5
|59.406
|-
|6
|71.287
|-
|7
|83.168
|-
|8
|95.050
|-
|9
|106.931
|-
|10
|118.812
|-
|11
|130.693
|-
|12
|142.574
|-
|13
|154.455
|-
|14
|166.337
|-
|15
|178.218
|-
|16
|190.099
|-
|17
|201.980
|-
|18
|213.861
|-
|19
|225.743
|-
|20
|237.624
|-
|21
|249.505
|-
|22
|261.386
|-
|23
|273.267
|-
|24
|285.149
|-
|25
|297.030
|-
|26
|308.911
|-
|27
|320.792
|-
|28
|332.673
|-
|29
|344.554
|-
|30
|356.436
|-
|31
|368.317
|-
|32
|380.198
|-
|33
|392.079
|-
|34
|403.960
|-
|35
|415.842
|-
|36
|427.723
|-
|37
|439.604
|-
|38
|451.485
|-
|39
|463.366
|-
|40
|475.248
|-
|41
|487.129
|-
|42
|499.010
|-
|43
|510.891
|-
|44
|522.772
|-
|45
|534.653
|-
|46
|546.535
|-
|47
|558.416
|-
|48
|570.297
|-
|49
|582.178
|-
|50
|594.059
|-
|...
|...
|}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:101-tone]]
[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]</text>
      <sha1>p5es9ylqteoe4qx70mt6492fzqtfcjm</sha1>
    </revision>
    <revision>
      <id>87563</id>
      <parentid>87061</parentid>
      <timestamp>2022-02-26T18:15:51Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:101-tone]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87563</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3190" sha1="9p1e2eakxja72fv5m7sf7l3d5nn6z6g" xml:space="preserve">'''101 EDO''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Intervals ==
{| class="wikitable"
!Degree
!Cents
|-
|0
|0.000
|-
|1
|11.881
|-
|2
|23.762
|-
|3
|35.644
|-
|4
|47.525
|-
|5
|59.406
|-
|6
|71.287
|-
|7
|83.168
|-
|8
|95.050
|-
|9
|106.931
|-
|10
|118.812
|-
|11
|130.693
|-
|12
|142.574
|-
|13
|154.455
|-
|14
|166.337
|-
|15
|178.218
|-
|16
|190.099
|-
|17
|201.980
|-
|18
|213.861
|-
|19
|225.743
|-
|20
|237.624
|-
|21
|249.505
|-
|22
|261.386
|-
|23
|273.267
|-
|24
|285.149
|-
|25
|297.030
|-
|26
|308.911
|-
|27
|320.792
|-
|28
|332.673
|-
|29
|344.554
|-
|30
|356.436
|-
|31
|368.317
|-
|32
|380.198
|-
|33
|392.079
|-
|34
|403.960
|-
|35
|415.842
|-
|36
|427.723
|-
|37
|439.604
|-
|38
|451.485
|-
|39
|463.366
|-
|40
|475.248
|-
|41
|487.129
|-
|42
|499.010
|-
|43
|510.891
|-
|44
|522.772
|-
|45
|534.653
|-
|46
|546.535
|-
|47
|558.416
|-
|48
|570.297
|-
|49
|582.178
|-
|50
|594.059
|-
|...
|...
|}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]</text>
      <sha1>9p1e2eakxja72fv5m7sf7l3d5nn6z6g</sha1>
    </revision>
    <revision>
      <id>88085</id>
      <parentid>87563</parentid>
      <timestamp>2022-03-01T00:48:36Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Pythagorean]] to [[Category:3-limit]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>88085</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3186" sha1="s5fkoo5acarv7fusez5qv2c3rymmmhu" xml:space="preserve">'''101 EDO''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Intervals ==
{| class="wikitable"
!Degree
!Cents
|-
|0
|0.000
|-
|1
|11.881
|-
|2
|23.762
|-
|3
|35.644
|-
|4
|47.525
|-
|5
|59.406
|-
|6
|71.287
|-
|7
|83.168
|-
|8
|95.050
|-
|9
|106.931
|-
|10
|118.812
|-
|11
|130.693
|-
|12
|142.574
|-
|13
|154.455
|-
|14
|166.337
|-
|15
|178.218
|-
|16
|190.099
|-
|17
|201.980
|-
|18
|213.861
|-
|19
|225.743
|-
|20
|237.624
|-
|21
|249.505
|-
|22
|261.386
|-
|23
|273.267
|-
|24
|285.149
|-
|25
|297.030
|-
|26
|308.911
|-
|27
|320.792
|-
|28
|332.673
|-
|29
|344.554
|-
|30
|356.436
|-
|31
|368.317
|-
|32
|380.198
|-
|33
|392.079
|-
|34
|403.960
|-
|35
|415.842
|-
|36
|427.723
|-
|37
|439.604
|-
|38
|451.485
|-
|39
|463.366
|-
|40
|475.248
|-
|41
|487.129
|-
|42
|499.010
|-
|43
|510.891
|-
|44
|522.772
|-
|45
|534.653
|-
|46
|546.535
|-
|47
|558.416
|-
|48
|570.297
|-
|49
|582.178
|-
|50
|594.059
|-
|...
|...
|}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Equal divisions of the octave]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:3-limit]]</text>
      <sha1>s5fkoo5acarv7fusez5qv2c3rymmmhu</sha1>
    </revision>
    <revision>
      <id>92911</id>
      <parentid>88085</parentid>
      <timestamp>2022-07-02T16:16:05Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92911</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3214" sha1="398t5hmzgpc5q3y6j1z6mlrbnr4d9fl" xml:space="preserve">'''101 EDO''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401

== Intervals ==
{| class="wikitable"
!Degree
!Cents
|-
|0
|0.000
|-
|1
|11.881
|-
|2
|23.762
|-
|3
|35.644
|-
|4
|47.525
|-
|5
|59.406
|-
|6
|71.287
|-
|7
|83.168
|-
|8
|95.050
|-
|9
|106.931
|-
|10
|118.812
|-
|11
|130.693
|-
|12
|142.574
|-
|13
|154.455
|-
|14
|166.337
|-
|15
|178.218
|-
|16
|190.099
|-
|17
|201.980
|-
|18
|213.861
|-
|19
|225.743
|-
|20
|237.624
|-
|21
|249.505
|-
|22
|261.386
|-
|23
|273.267
|-
|24
|285.149
|-
|25
|297.030
|-
|26
|308.911
|-
|27
|320.792
|-
|28
|332.673
|-
|29
|344.554
|-
|30
|356.436
|-
|31
|368.317
|-
|32
|380.198
|-
|33
|392.079
|-
|34
|403.960
|-
|35
|415.842
|-
|36
|427.723
|-
|37
|439.604
|-
|38
|451.485
|-
|39
|463.366
|-
|40
|475.248
|-
|41
|487.129
|-
|42
|499.010
|-
|43
|510.891
|-
|44
|522.772
|-
|45
|534.653
|-
|46
|546.535
|-
|47
|558.416
|-
|48
|570.297
|-
|49
|582.178
|-
|50
|594.059
|-
|...
|...
|}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Armodue]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:3-limit]]</text>
      <sha1>398t5hmzgpc5q3y6j1z6mlrbnr4d9fl</sha1>
    </revision>
    <revision>
      <id>96003</id>
      <parentid>92911</parentid>
      <timestamp>2022-10-03T02:50:06Z</timestamp>
      <contributor>
        <username>Royalmilktea</username>
        <id>285</id>
      </contributor>
      <origin>96003</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3249" sha1="0diykisu7xneb0q7zqfivaewqtae94j" xml:space="preserve">'''101 EDO''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401
{{harmonics in equal|101|start=2}}

== Intervals ==
{| class="wikitable"
!Degree
!Cents
|-
|0
|0.000
|-
|1
|11.881
|-
|2
|23.762
|-
|3
|35.644
|-
|4
|47.525
|-
|5
|59.406
|-
|6
|71.287
|-
|7
|83.168
|-
|8
|95.050
|-
|9
|106.931
|-
|10
|118.812
|-
|11
|130.693
|-
|12
|142.574
|-
|13
|154.455
|-
|14
|166.337
|-
|15
|178.218
|-
|16
|190.099
|-
|17
|201.980
|-
|18
|213.861
|-
|19
|225.743
|-
|20
|237.624
|-
|21
|249.505
|-
|22
|261.386
|-
|23
|273.267
|-
|24
|285.149
|-
|25
|297.030
|-
|26
|308.911
|-
|27
|320.792
|-
|28
|332.673
|-
|29
|344.554
|-
|30
|356.436
|-
|31
|368.317
|-
|32
|380.198
|-
|33
|392.079
|-
|34
|403.960
|-
|35
|415.842
|-
|36
|427.723
|-
|37
|439.604
|-
|38
|451.485
|-
|39
|463.366
|-
|40
|475.248
|-
|41
|487.129
|-
|42
|499.010
|-
|43
|510.891
|-
|44
|522.772
|-
|45
|534.653
|-
|46
|546.535
|-
|47
|558.416
|-
|48
|570.297
|-
|49
|582.178
|-
|50
|594.059
|-
|...
|...
|}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Armodue]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:3-limit]]</text>
      <sha1>0diykisu7xneb0q7zqfivaewqtae94j</sha1>
    </revision>
    <revision>
      <id>96185</id>
      <parentid>96003</parentid>
      <timestamp>2022-10-04T18:46:51Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96185</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3264" sha1="q3utwi5qkyjmhq7qwt6ae3v2d66wjdk" xml:space="preserve">{{Infobox ET}}
'''101 EDO''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401
{{harmonics in equal|101|start=2}}

== Intervals ==
{| class="wikitable"
!Degree
!Cents
|-
|0
|0.000
|-
|1
|11.881
|-
|2
|23.762
|-
|3
|35.644
|-
|4
|47.525
|-
|5
|59.406
|-
|6
|71.287
|-
|7
|83.168
|-
|8
|95.050
|-
|9
|106.931
|-
|10
|118.812
|-
|11
|130.693
|-
|12
|142.574
|-
|13
|154.455
|-
|14
|166.337
|-
|15
|178.218
|-
|16
|190.099
|-
|17
|201.980
|-
|18
|213.861
|-
|19
|225.743
|-
|20
|237.624
|-
|21
|249.505
|-
|22
|261.386
|-
|23
|273.267
|-
|24
|285.149
|-
|25
|297.030
|-
|26
|308.911
|-
|27
|320.792
|-
|28
|332.673
|-
|29
|344.554
|-
|30
|356.436
|-
|31
|368.317
|-
|32
|380.198
|-
|33
|392.079
|-
|34
|403.960
|-
|35
|415.842
|-
|36
|427.723
|-
|37
|439.604
|-
|38
|451.485
|-
|39
|463.366
|-
|40
|475.248
|-
|41
|487.129
|-
|42
|499.010
|-
|43
|510.891
|-
|44
|522.772
|-
|45
|534.653
|-
|46
|546.535
|-
|47
|558.416
|-
|48
|570.297
|-
|49
|582.178
|-
|50
|594.059
|-
|...
|...
|}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Armodue]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:3-limit]]</text>
      <sha1>q3utwi5qkyjmhq7qwt6ae3v2d66wjdk</sha1>
    </revision>
    <revision>
      <id>132169</id>
      <parentid>96185</parentid>
      <timestamp>2024-01-20T00:46:24Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ Replace less detailed manual table with more detailed template table</comment>
      <origin>132169</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2435" sha1="7jbcwxoxf2k3ghcu499qw01gi5e0sc7" xml:space="preserve">{{Infobox ET}}
'''101 EDO''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401
{{harmonics in equal|101|start=2}}

== Intervals ==
{{Interval table}}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Armodue]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:3-limit]]</text>
      <sha1>7jbcwxoxf2k3ghcu499qw01gi5e0sc7</sha1>
    </revision>
    <revision>
      <id>132346</id>
      <parentid>132169</parentid>
      <timestamp>2024-01-21T09:59:33Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Undo revision 132169 by [[Special:Contributions/BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]])</comment>
      <origin>132346</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3264" sha1="q3utwi5qkyjmhq7qwt6ae3v2d66wjdk" xml:space="preserve">{{Infobox ET}}
'''101 EDO''' divides the [[octave]] into 101 equal parts of 11.881 [[cent]]s each. It can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401
{{harmonics in equal|101|start=2}}

== Intervals ==
{| class="wikitable"
!Degree
!Cents
|-
|0
|0.000
|-
|1
|11.881
|-
|2
|23.762
|-
|3
|35.644
|-
|4
|47.525
|-
|5
|59.406
|-
|6
|71.287
|-
|7
|83.168
|-
|8
|95.050
|-
|9
|106.931
|-
|10
|118.812
|-
|11
|130.693
|-
|12
|142.574
|-
|13
|154.455
|-
|14
|166.337
|-
|15
|178.218
|-
|16
|190.099
|-
|17
|201.980
|-
|18
|213.861
|-
|19
|225.743
|-
|20
|237.624
|-
|21
|249.505
|-
|22
|261.386
|-
|23
|273.267
|-
|24
|285.149
|-
|25
|297.030
|-
|26
|308.911
|-
|27
|320.792
|-
|28
|332.673
|-
|29
|344.554
|-
|30
|356.436
|-
|31
|368.317
|-
|32
|380.198
|-
|33
|392.079
|-
|34
|403.960
|-
|35
|415.842
|-
|36
|427.723
|-
|37
|439.604
|-
|38
|451.485
|-
|39
|463.366
|-
|40
|475.248
|-
|41
|487.129
|-
|42
|499.010
|-
|43
|510.891
|-
|44
|522.772
|-
|45
|534.653
|-
|46
|546.535
|-
|47
|558.416
|-
|48
|570.297
|-
|49
|582.178
|-
|50
|594.059
|-
|...
|...
|}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Armodue]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:3-limit]]</text>
      <sha1>q3utwi5qkyjmhq7qwt6ae3v2d66wjdk</sha1>
    </revision>
    <revision>
      <id>132879</id>
      <parentid>132346</parentid>
      <timestamp>2024-01-25T08:56:18Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Add EDO intro template</comment>
      <origin>132879</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3203" sha1="4ve9zb8u944kqju6aewhqipsc5m5z8n" xml:space="preserve">{{Infobox ET}}
{{EDO intro|101}}

101edo can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401
{{harmonics in equal|101|start=2}}

== Intervals ==
{| class="wikitable"
!Degree
!Cents
|-
|0
|0.000
|-
|1
|11.881
|-
|2
|23.762
|-
|3
|35.644
|-
|4
|47.525
|-
|5
|59.406
|-
|6
|71.287
|-
|7
|83.168
|-
|8
|95.050
|-
|9
|106.931
|-
|10
|118.812
|-
|11
|130.693
|-
|12
|142.574
|-
|13
|154.455
|-
|14
|166.337
|-
|15
|178.218
|-
|16
|190.099
|-
|17
|201.980
|-
|18
|213.861
|-
|19
|225.743
|-
|20
|237.624
|-
|21
|249.505
|-
|22
|261.386
|-
|23
|273.267
|-
|24
|285.149
|-
|25
|297.030
|-
|26
|308.911
|-
|27
|320.792
|-
|28
|332.673
|-
|29
|344.554
|-
|30
|356.436
|-
|31
|368.317
|-
|32
|380.198
|-
|33
|392.079
|-
|34
|403.960
|-
|35
|415.842
|-
|36
|427.723
|-
|37
|439.604
|-
|38
|451.485
|-
|39
|463.366
|-
|40
|475.248
|-
|41
|487.129
|-
|42
|499.010
|-
|43
|510.891
|-
|44
|522.772
|-
|45
|534.653
|-
|46
|546.535
|-
|47
|558.416
|-
|48
|570.297
|-
|49
|582.178
|-
|50
|594.059
|-
|...
|...
|}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Armodue]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:3-limit]]</text>
      <sha1>4ve9zb8u944kqju6aewhqipsc5m5z8n</sha1>
    </revision>
    <revision>
      <id>132880</id>
      <parentid>132879</parentid>
      <timestamp>2024-01-25T08:56:56Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added theory subheading</comment>
      <origin>132880</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3216" sha1="rmvk7vfoye1p6uwkj0ym3hvnzimqlh1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|101}}

101edo can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

== Theory ==
; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401
{{harmonics in equal|101|start=2}}

== Intervals ==
{| class="wikitable"
!Degree
!Cents
|-
|0
|0.000
|-
|1
|11.881
|-
|2
|23.762
|-
|3
|35.644
|-
|4
|47.525
|-
|5
|59.406
|-
|6
|71.287
|-
|7
|83.168
|-
|8
|95.050
|-
|9
|106.931
|-
|10
|118.812
|-
|11
|130.693
|-
|12
|142.574
|-
|13
|154.455
|-
|14
|166.337
|-
|15
|178.218
|-
|16
|190.099
|-
|17
|201.980
|-
|18
|213.861
|-
|19
|225.743
|-
|20
|237.624
|-
|21
|249.505
|-
|22
|261.386
|-
|23
|273.267
|-
|24
|285.149
|-
|25
|297.030
|-
|26
|308.911
|-
|27
|320.792
|-
|28
|332.673
|-
|29
|344.554
|-
|30
|356.436
|-
|31
|368.317
|-
|32
|380.198
|-
|33
|392.079
|-
|34
|403.960
|-
|35
|415.842
|-
|36
|427.723
|-
|37
|439.604
|-
|38
|451.485
|-
|39
|463.366
|-
|40
|475.248
|-
|41
|487.129
|-
|42
|499.010
|-
|43
|510.891
|-
|44
|522.772
|-
|45
|534.653
|-
|46
|546.535
|-
|47
|558.416
|-
|48
|570.297
|-
|49
|582.178
|-
|50
|594.059
|-
|...
|...
|}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Armodue]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:3-limit]]</text>
      <sha1>rmvk7vfoye1p6uwkj0ym3hvnzimqlh1</sha1>
    </revision>
    <revision>
      <id>132881</id>
      <parentid>132880</parentid>
      <timestamp>2024-01-25T08:57:28Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ made table collapsible</comment>
      <origin>132881</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3231" sha1="gfmdpitnhh2gxsfammbf5daan5uavdt" xml:space="preserve">{{Infobox ET}}
{{EDO intro|101}}

101edo can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

== Theory ==
; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401
{{harmonics in equal|101|start=2}}

== Intervals ==
{| class="wikitable mw-collapsible"
!Degree
!Cents
|-
|0
|0.000
|-
|1
|11.881
|-
|2
|23.762
|-
|3
|35.644
|-
|4
|47.525
|-
|5
|59.406
|-
|6
|71.287
|-
|7
|83.168
|-
|8
|95.050
|-
|9
|106.931
|-
|10
|118.812
|-
|11
|130.693
|-
|12
|142.574
|-
|13
|154.455
|-
|14
|166.337
|-
|15
|178.218
|-
|16
|190.099
|-
|17
|201.980
|-
|18
|213.861
|-
|19
|225.743
|-
|20
|237.624
|-
|21
|249.505
|-
|22
|261.386
|-
|23
|273.267
|-
|24
|285.149
|-
|25
|297.030
|-
|26
|308.911
|-
|27
|320.792
|-
|28
|332.673
|-
|29
|344.554
|-
|30
|356.436
|-
|31
|368.317
|-
|32
|380.198
|-
|33
|392.079
|-
|34
|403.960
|-
|35
|415.842
|-
|36
|427.723
|-
|37
|439.604
|-
|38
|451.485
|-
|39
|463.366
|-
|40
|475.248
|-
|41
|487.129
|-
|42
|499.010
|-
|43
|510.891
|-
|44
|522.772
|-
|45
|534.653
|-
|46
|546.535
|-
|47
|558.416
|-
|48
|570.297
|-
|49
|582.178
|-
|50
|594.059
|-
|...
|...
|}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Armodue]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:3-limit]]</text>
      <sha1>gfmdpitnhh2gxsfammbf5daan5uavdt</sha1>
    </revision>
    <revision>
      <id>132895</id>
      <parentid>132881</parentid>
      <timestamp>2024-01-25T09:38:08Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Return harmonics in equal start to 1</comment>
      <origin>132895</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3223" sha1="cuyytlesijyyo3yxtwguan04yobya9n" xml:space="preserve">{{Infobox ET}}
{{EDO intro|101}}

101edo can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

== Theory ==
; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401
{{harmonics in equal|101}}

== Intervals ==
{| class="wikitable mw-collapsible"
!Degree
!Cents
|-
|0
|0.000
|-
|1
|11.881
|-
|2
|23.762
|-
|3
|35.644
|-
|4
|47.525
|-
|5
|59.406
|-
|6
|71.287
|-
|7
|83.168
|-
|8
|95.050
|-
|9
|106.931
|-
|10
|118.812
|-
|11
|130.693
|-
|12
|142.574
|-
|13
|154.455
|-
|14
|166.337
|-
|15
|178.218
|-
|16
|190.099
|-
|17
|201.980
|-
|18
|213.861
|-
|19
|225.743
|-
|20
|237.624
|-
|21
|249.505
|-
|22
|261.386
|-
|23
|273.267
|-
|24
|285.149
|-
|25
|297.030
|-
|26
|308.911
|-
|27
|320.792
|-
|28
|332.673
|-
|29
|344.554
|-
|30
|356.436
|-
|31
|368.317
|-
|32
|380.198
|-
|33
|392.079
|-
|34
|403.960
|-
|35
|415.842
|-
|36
|427.723
|-
|37
|439.604
|-
|38
|451.485
|-
|39
|463.366
|-
|40
|475.248
|-
|41
|487.129
|-
|42
|499.010
|-
|43
|510.891
|-
|44
|522.772
|-
|45
|534.653
|-
|46
|546.535
|-
|47
|558.416
|-
|48
|570.297
|-
|49
|582.178
|-
|50
|594.059
|-
|...
|...
|}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Armodue]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:3-limit]]</text>
      <sha1>cuyytlesijyyo3yxtwguan04yobya9n</sha1>
    </revision>
    <revision>
      <id>137185</id>
      <parentid>132895</parentid>
      <timestamp>2024-02-26T03:56:52Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <comment>Replace with the automatic interval table that has ups and downs notation</comment>
      <origin>137185</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2379" sha1="nyyi9h78bmirzj2ynt8xitcv7hmyi79" xml:space="preserve">{{Infobox ET}}
{{EDO intro|101}}

101edo can be used to tune the [[grackle]] temperament. It is the 26th [[prime EDO]]. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13 and 15 limit least squares tuning.

== Theory ==
; [[5-limit]] commas: 32805/32768 ( {{monzo| -15 8 1 }} ), 51018336/48828125 ( {{monzo| 5 13 -11 }} )
; [[7-limit]] commas: 126/125, 32805/32768, 2430/2401
{{harmonics in equal|101}}

== Intervals ==
{{Interval table}}

== Some important MOS scales ==

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== Links ==

[http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Armodue]]
[[Category:Grackle]]
[[Category:Prime EDO]]
[[Category:3-limit]]</text>
      <sha1>nyyi9h78bmirzj2ynt8xitcv7hmyi79</sha1>
    </revision>
    <revision>
      <id>146046</id>
      <parentid>137185</parentid>
      <timestamp>2024-06-10T16:31:36Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>146046</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2412" sha1="djnhr5juggg1gossp2olom81zydc75s" xml:space="preserve">{{Infobox ET}}
{{EDO intro|101}}

== Theory ==
Using the [[patent val]], the equal temperament [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning.

=== Odd harmonics ===
{{Harmonics in equal|101}}

=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]]. 

== Intervals ==
{{Interval table}}

== Scales ==
=== Mos scales ===

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== External links ==
* [http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Grackle]]</text>
      <sha1>djnhr5juggg1gossp2olom81zydc75s</sha1>
    </revision>
    <revision>
      <id>146047</id>
      <parentid>146046</parentid>
      <timestamp>2024-06-10T16:35:26Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ +one-liner overview</comment>
      <origin>146047</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2658" sha1="556qgbpnf4wkikc4walp8o7ttacfnnd" xml:space="preserve">{{Infobox ET}}
{{EDO intro|101}}

== Theory ==
101edo is in[[consistent]] in the [[5-odd-limit]], with [[harmonic]]s [[5/1|5]] and [[7/1|7]] falling about halfway between its steps. As such, {{val| 101 160 '''235''' '''284''' }} ([[patent val]]) and {{val| 101 160 '''234''' '''283''' }} (101cd) are about as viable. Using the patent val, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning.

=== Odd harmonics ===
{{Harmonics in equal|101}}

=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]].

== Intervals ==
{{Interval table}}

== Scales ==
=== Mos scales ===

'''25 13 25 25 13:''' ''3L2s MOS'' (Pentatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 25
| 297.030
|-
| 38
| 451.485
|-
| 63
| 748.515
|-
| 88
| 1045.545
|}

'''17 17 8 17 17 17 8:''' ''5L2s MOS'' (Diatonic Pythagorean)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 17
| 201.980
|-
| 34
| 403.960
|-
| 42
| 499.010
|-
| 59
| 700.990
|-
| 76
| 902.970
|-
| 93
| 1104.950
|}

'''13 13 13 13 13 13 13 10:''' ''7L1s MOS'' (Grumpy Octatonic)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 52
| 617.822
|-
| 65
| 772.277
|-
| 78
| 926.733
|-
| 91
| 1081.188
|}

'''13 13 13 5 13 13 13 13 5:''' ''7L2s MOS'' (Superdiatonic 1/13-tone 13;5 relation)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 13
| 154.455
|-
| 26
| 308.911
|-
| 39
| 463.366
|-
| 44
| 522.772
|-
| 57
| 677.228
|-
| 70
| 831.683
|-
| 83
| 986.139
|-
| 96
| 1045.545
|}

'''10 10 7 10 10 10 7 10 10 10 7:''' ''8L3s MOS'' (Improper Sensi-11)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 10
| 118.812
|-
| 20
| 237.624
|-
| 27
| 320.792
|-
| 37
| 439.604
|-
| 47
| 558.416
|-
| 57
| 677.228
|-
| 64
| 760.396
|-
| 74
| 879.218
|-
| 84
| 998.020
|-
| 94
| 1116.842
|}

'''7 7 7 8 7 7 7 7 8 7 7 7 7 8:''' ''3L11s MOS'' (Anti-Ketradektriatoh)

{| class="wikitable right-all"
! Steps
! Cents
|-
| 7
| 83.168
|-
| 14
| 166.337
|-
| 22
| 261.386
|-
| 29
| 344.554
|-
| 36
| 427.723
|-
| 43
| 510.891
|-
| 50
| 594.059
|-
| 58
| 689.119
|-
| 65
| 772.287
|-
| 72
| 855.446
|-
| 79
| 938.614
|-
| 86
| 1021.782
|-
| 93
| 1104.950
|}

== External links ==
* [http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Grackle]]</text>
      <sha1>556qgbpnf4wkikc4walp8o7ttacfnnd</sha1>
    </revision>
    <revision>
      <id>146049</id>
      <parentid>146047</parentid>
      <timestamp>2024-06-10T16:53:36Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Scales */ save space and request clarification</comment>
      <origin>146049</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1647" sha1="jfegk5amtcrgd4dgz4vbk3so3ondmbq" xml:space="preserve">{{Infobox ET}}
{{EDO intro|101}}

== Theory ==
101edo is in[[consistent]] in the [[5-odd-limit]], with [[harmonic]]s [[5/1|5]] and [[7/1|7]] falling about halfway between its steps. As such, {{val| 101 160 '''235''' '''284''' }} ([[patent val]]) and {{val| 101 160 '''234''' '''283''' }} (101cd) are about as viable. Using the patent val, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning.

=== Odd harmonics ===
{{Harmonics in equal|101}}

=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]].

== Intervals ==
{{Interval table}}

== Scales ==
=== Mos scales ===
* 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101){{clarify}} &lt;!-- why is this significant? --&gt;
* Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
* Grumpy Octatonic 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
* Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
* Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101){{clarify}} &lt;!-- which val? --&gt;
* Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)

== External links ==
* [http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Grackle]]</text>
      <sha1>jfegk5amtcrgd4dgz4vbk3so3ondmbq</sha1>
    </revision>
    <revision>
      <id>181289</id>
      <parentid>146049</parentid>
      <timestamp>2025-02-18T14:14:02Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <origin>181289</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1741" sha1="lirxoczwvcdsl8d9fr8s8pydkqfpvbs" xml:space="preserve">{{Infobox ET}}
{{EDO intro|101}}

== Theory ==
101edo is in[[consistent]] in the [[5-odd-limit]], with [[harmonic]]s [[5/1|5]] and [[7/1|7]] falling about halfway between its steps. As such, {{val| 101 160 '''235''' '''284''' }} ([[patent val]]) and {{val| 101 160 '''234''' '''283''' }} (101cd) are about as viable. Using the patent val, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning.

=== Odd harmonics ===
{{Harmonics in equal|101}}

=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]]. [[202edo]], which doubles it, provides a good correction to the 5th, 7th, and 11th harmonics.

== Intervals ==
{{Interval table}}

== Scales ==
=== Mos scales ===
* 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101){{clarify}} &lt;!-- why is this significant? --&gt;
* Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
* Grumpy Octatonic 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
* Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
* Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101){{clarify}} &lt;!-- which val? --&gt;
* Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)

== External links ==
* [http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Grackle]]</text>
      <sha1>lirxoczwvcdsl8d9fr8s8pydkqfpvbs</sha1>
    </revision>
    <revision>
      <id>181304</id>
      <parentid>181289</parentid>
      <timestamp>2025-02-18T16:46:08Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181304</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1736" sha1="12rkcmoaxbusxfdvhu9vxapkp6ic3sd" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
101edo is in[[consistent]] in the [[5-odd-limit]], with [[harmonic]]s [[5/1|5]] and [[7/1|7]] falling about halfway between its steps. As such, {{val| 101 160 '''235''' '''284''' }} ([[patent val]]) and {{val| 101 160 '''234''' '''283''' }} (101cd) are about as viable. Using the patent val, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning.

=== Odd harmonics ===
{{Harmonics in equal|101}}

=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]]. [[202edo]], which doubles it, provides a good correction to the 5th, 7th, and 11th harmonics.

== Intervals ==
{{Interval table}}

== Scales ==
=== Mos scales ===
* 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101){{clarify}} &lt;!-- why is this significant? --&gt;
* Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
* Grumpy Octatonic 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
* Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
* Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101){{clarify}} &lt;!-- which val? --&gt;
* Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)

== External links ==
* [http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Grackle]]</text>
      <sha1>12rkcmoaxbusxfdvhu9vxapkp6ic3sd</sha1>
    </revision>
    <revision>
      <id>191034</id>
      <parentid>181304</parentid>
      <timestamp>2025-04-11T07:40:40Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ (''See [[regular temperament]] for more about what all this means and how to use it.'')</comment>
      <origin>191034</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1824" sha1="pdg7sv0t7duxxdrhy1u80h4ov4b7qhi" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
101edo is in[[consistent]] in the [[5-odd-limit]], with [[harmonic]]s [[5/1|5]] and [[7/1|7]] falling about halfway between its steps. As such, {{val| 101 160 '''235''' '''284''' }} ([[patent val]]) and {{val| 101 160 '''234''' '''283''' }} (101cd) are about as viable. Using the patent val, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning. (''See [[regular temperament]] for more about what all this means and how to use it.'')

=== Odd harmonics ===
{{Harmonics in equal|101}}

=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]]. [[202edo]], which doubles it, provides a good correction to the 5th, 7th, and 11th harmonics.

== Intervals ==
{{Interval table}}

== Scales ==
=== Mos scales ===
* 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101){{clarify}} &lt;!-- why is this significant? --&gt;
* Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
* Grumpy Octatonic 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
* Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
* Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101){{clarify}} &lt;!-- which val? --&gt;
* Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)

== External links ==
* [http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Grackle]]</text>
      <sha1>pdg7sv0t7duxxdrhy1u80h4ov4b7qhi</sha1>
    </revision>
    <revision>
      <id>191149</id>
      <parentid>191034</parentid>
      <timestamp>2025-04-11T11:16:36Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>No need to remind readers of what a regular temperament is everywhere</comment>
      <origin>191149</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1736" sha1="12rkcmoaxbusxfdvhu9vxapkp6ic3sd" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
101edo is in[[consistent]] in the [[5-odd-limit]], with [[harmonic]]s [[5/1|5]] and [[7/1|7]] falling about halfway between its steps. As such, {{val| 101 160 '''235''' '''284''' }} ([[patent val]]) and {{val| 101 160 '''234''' '''283''' }} (101cd) are about as viable. Using the patent val, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning.

=== Odd harmonics ===
{{Harmonics in equal|101}}

=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]]. [[202edo]], which doubles it, provides a good correction to the 5th, 7th, and 11th harmonics.

== Intervals ==
{{Interval table}}

== Scales ==
=== Mos scales ===
* 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101){{clarify}} &lt;!-- why is this significant? --&gt;
* Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
* Grumpy Octatonic 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
* Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
* Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101){{clarify}} &lt;!-- which val? --&gt;
* Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)

== External links ==
* [http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Grackle]]</text>
      <sha1>12rkcmoaxbusxfdvhu9vxapkp6ic3sd</sha1>
    </revision>
    <revision>
      <id>195018</id>
      <parentid>191149</parentid>
      <timestamp>2025-05-03T14:01:57Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>+music</comment>
      <origin>195018</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1983" sha1="fbfilsa68x9lfue50ytr0eukpg6lt2h" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
101edo is in[[consistent]] in the [[5-odd-limit]], with [[harmonic]]s [[5/1|5]] and [[7/1|7]] falling about halfway between its steps. As such, {{val| 101 160 '''235''' '''284''' }} ([[patent val]]) and {{val| 101 160 '''234''' '''283''' }} (101cd) are about as viable. Using the patent val, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning.

=== Odd harmonics ===
{{Harmonics in equal|101}}

=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]]. [[202edo]], which doubles it, provides a good correction to the 5th, 7th, and 11th harmonics.

== Intervals ==
{{Interval table}}

== Scales ==
=== Mos scales ===
* 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101){{clarify}} &lt;!-- why is this significant? --&gt;
* Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
* Grumpy Octatonic 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
* Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
* Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101){{clarify}} &lt;!-- which val? --&gt;
* Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)

== Music ==
; [[Francium]]
* "Eggclent" from ''Eggs'' (2025) – [https://open.spotify.com/track/4S0BTeb9yDdMUuT1QJy26H Spotify] | [https://francium223.bandcamp.com/track/eggclent Bandcamp] | [https://www.youtube.com/watch?v=FAe4O71Mvj0 YouTube]

== External links ==
* [http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Grackle]]</text>
      <sha1>fbfilsa68x9lfue50ytr0eukpg6lt2h</sha1>
    </revision>
    <revision>
      <id>196150</id>
      <parentid>195018</parentid>
      <timestamp>2025-05-14T05:32:01Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>196150</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1971" sha1="1v3522lwtl99msye9ah1gj5qvlmqyl9" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
101edo is in[[consistent]] in the [[5-odd-limit]], with [[harmonic]]s [[5/1|5]] and [[7/1|7]] falling about halfway between its steps. As such, {{val| 101 160 '''235''' '''284''' }} ([[patent val]]) and {{val| 101 160 '''234''' '''283''' }} (101cd) are about as viable. Using the patent val, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning.

=== Odd harmonics ===
{{Harmonics in equal|101}}

=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]]. [[202edo]], which doubles it, provides a good correction to the 5th, 7th, and 11th harmonics.

== Intervals ==
{{Interval table}}

== Scales ==
=== Mos scales ===
* 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101){{clarify}} &lt;!-- why is this significant? --&gt;
* Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
* Pine 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
* Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
* Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101){{clarify}} &lt;!-- which val? --&gt;
* Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)

== Music ==
; [[Francium]]
* "Eggclent" from ''Eggs'' (2025) – [https://open.spotify.com/track/4S0BTeb9yDdMUuT1QJy26H Spotify] | [https://francium223.bandcamp.com/track/eggclent Bandcamp] | [https://www.youtube.com/watch?v=FAe4O71Mvj0 YouTube]

== External links ==
* [http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Grackle]]</text>
      <sha1>1v3522lwtl99msye9ah1gj5qvlmqyl9</sha1>
    </revision>
    <revision>
      <id>207462</id>
      <parentid>196150</parentid>
      <timestamp>2025-08-18T11:03:36Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Link to lumatone mapping</comment>
      <origin>207462</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2024" sha1="kcqpepi4y2iiub5gstrpx6hwll0o6k5" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
101edo is in[[consistent]] in the [[5-odd-limit]], with [[harmonic]]s [[5/1|5]] and [[7/1|7]] falling about halfway between its steps. As such, {{val| 101 160 '''235''' '''284''' }} ([[patent val]]) and {{val| 101 160 '''234''' '''283''' }} (101cd) are about as viable. Using the patent val, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning.

=== Odd harmonics ===
{{Harmonics in equal|101}}

=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]]. [[202edo]], which doubles it, provides a good correction to the 5th, 7th, and 11th harmonics.

== Intervals ==
{{Interval table}}

== Scales ==
=== Mos scales ===
* 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101){{clarify}} &lt;!-- why is this significant? --&gt;
* Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
* Pine 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
* Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
* Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101){{clarify}} &lt;!-- which val? --&gt;
* Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)

== Instruments ==
* [[Lumatone mapping for 101edo]]

== Music ==
; [[Francium]]
* "Eggclent" from ''Eggs'' (2025) – [https://open.spotify.com/track/4S0BTeb9yDdMUuT1QJy26H Spotify] | [https://francium223.bandcamp.com/track/eggclent Bandcamp] | [https://www.youtube.com/watch?v=FAe4O71Mvj0 YouTube]

== External links ==
* [http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Grackle]]</text>
      <sha1>kcqpepi4y2iiub5gstrpx6hwll0o6k5</sha1>
    </revision>
    <revision>
      <id>216142</id>
      <parentid>207462</parentid>
      <timestamp>2025-11-03T23:57:21Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Music */ Add Bryan Deister's ''101edo improv'' (2025)</comment>
      <origin>216142</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2117" sha1="1lmj144m0uk8w0udlw3mj6nlpkelky9" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
101edo is in[[consistent]] in the [[5-odd-limit]], with [[harmonic]]s [[5/1|5]] and [[7/1|7]] falling about halfway between its steps. As such, {{val| 101 160 '''235''' '''284''' }} ([[patent val]]) and {{val| 101 160 '''234''' '''283''' }} (101cd) are about as viable. Using the patent val, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning.

=== Odd harmonics ===
{{Harmonics in equal|101}}

=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]]. [[202edo]], which doubles it, provides a good correction to the 5th, 7th, and 11th harmonics.

== Intervals ==
{{Interval table}}

== Scales ==
=== Mos scales ===
* 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101){{clarify}} &lt;!-- why is this significant? --&gt;
* Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
* Pine 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
* Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
* Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101){{clarify}} &lt;!-- which val? --&gt;
* Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)

== Instruments ==
* [[Lumatone mapping for 101edo]]

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/qS4fpB_uIyo ''101edo improv''] (2025)

; [[Francium]]
* "Eggclent" from ''Eggs'' (2025) – [https://open.spotify.com/track/4S0BTeb9yDdMUuT1QJy26H Spotify] | [https://francium223.bandcamp.com/track/eggclent Bandcamp] | [https://www.youtube.com/watch?v=FAe4O71Mvj0 YouTube]

== External links ==
* [http://tech.groups.yahoo.com/group/tuning-math/message/11157 The Ellis duodene in 101-equal] {{dead link}}

[[Category:Armodue]]
[[Category:Grackle]]</text>
      <sha1>1lmj144m0uk8w0udlw3mj6nlpkelky9</sha1>
    </revision>
    <revision>
      <id>229590</id>
      <parentid>216142</parentid>
      <timestamp>2026-05-04T01:31:23Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* External links */ fix dead link</comment>
      <origin>229590</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2123" sha1="kk1y48gpwfn62zxynolm03sausk6b5r" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
101edo is in[[consistent]] in the [[5-odd-limit]], with [[harmonic]]s [[5/1|5]] and [[7/1|7]] falling about halfway between its steps. As such, {{val| 101 160 '''235''' '''284''' }} ([[patent val]]) and {{val| 101 160 '''234''' '''283''' }} (101cd) are about as viable. Using the patent val, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and 51018336/48828125 in the 5-limit; [[126/125]] and [[2430/2401]] in the [[7-limit]]. It can be used to tune the [[grackle]] temperament. The 101cd val provides an excellent tuning for [[witchcraft]] temperament, falling between the 13- and 15-odd-limit least squares tuning.

=== Odd harmonics ===
{{Harmonics in equal|101}}

=== Subsets and supersets ===
101edo is the 26th [[prime edo]], following [[97edo]] and before [[103edo]]. [[202edo]], which doubles it, provides a good correction to the 5th, 7th, and 11th harmonics.

== Intervals ==
{{Interval table}}

== Scales ==
=== Mos scales ===
* 3L 2s: 25 13 25 25 13 ((25 38 63 88 101)\101){{clarify}} &lt;!-- why is this significant? --&gt;
* Grackle[7] 5L 2s: 17 17 8 17 17 17 8 ((17 34 42 59 76 93)\101)
* Pine 7L 1s: 13 13 13 13 13 13 13 10 ((13 26 39 52 65 78 91 101)\101)
* Superdiatonic 1/13-tone 13;5 relation: 13 13 13 5 13 13 13 13 5 ((13 26 39 44 57 70 83 96 101)\101)
* Sensi[11] 8L 3s: 10 10 7 10 10 10 7 10 10 10 7 ((10 20 27 37 47 57 64 74 84 94)\101){{clarify}} &lt;!-- which val? --&gt;
* Anti-Ketradektriatoh 3L 11s: 7 7 7 8 7 7 7 7 8 7 7 7 7 8 ((7 14 22 29 36 43 50 58 65 72 79 86 93 101)\101)

== Instruments ==
* [[Lumatone mapping for 101edo]]

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/qS4fpB_uIyo ''101edo improv''] (2025)

; [[Francium]]
* "Eggclent" from ''Eggs'' (2025) – [https://open.spotify.com/track/4S0BTeb9yDdMUuT1QJy26H Spotify] | [https://francium223.bandcamp.com/track/eggclent Bandcamp] | [https://www.youtube.com/watch?v=FAe4O71Mvj0 YouTube]

== External links ==
* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_11157.html The Ellis duodene in 101-equal]

[[Category:Armodue]]
[[Category:Grackle]]</text>
      <sha1>kk1y48gpwfn62zxynolm03sausk6b5r</sha1>
    </revision>
  </page>
  <page>
    <title>102edo</title>
    <ns>0</ns>
    <id>19</id>
    <revision>
      <id>19</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>19</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1290" sha1="ptdabkniyx3mv0qgmyn6sgevikjjzik" xml:space="preserve">'''102edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 102 steps of size 11.765 [[cent|cent]]s each. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] the same [[Comma|comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo|34edo]]. In the [[7-limit|7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit|11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit|13-limit]] 91/90 and 169/168; in the [[17-limit|17-limit]] 136/135 and 154/153; and in the [[19-limit|19-limit]] 133/132 and 190/189. It is the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Diaschismic_family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.

===13-limit Echidnic===

{| class="wikitable"
|-
| | 2
| | 23.529
|-
| | 4
| | 47.059
|-
| | 7
| | 82.353
|-
| | 9
| | 105.882
|-
| | 11
| | 129.412
|-
| | 13
| | 152.941
|-
| | 16
| | 188.235
|-
| | 18
| | 211.765
|-
| | 20
| | 235.294
|-
| | 22
| | 258.824
|-
| | 24
| | 282.353
|-
| | 27
| | 317.647
|-
| | 29
| | 341.176
|-
| | 31
| | 364.706
|-
| | 33
| | 388.235
|-
| | 35
| | 411.765
|-
| | 38
| | 447.059
|-
| | 40
| | 470.588
|-
| | 42
| | 494.117
|-
| | 44
| | 517.647
|-
| | 47
| | 552.941
|-
| | 49
| | 576.471
|}
[[Category:echidnic]]
[[Category:edo]]</text>
      <sha1>ptdabkniyx3mv0qgmyn6sgevikjjzik</sha1>
    </revision>
    <revision>
      <id>4037</id>
      <timestamp>2016-12-29T13:27:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602896474 - Original comment: **</comment>
      <origin>4037</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6609" sha1="habe0tzrr9eumb4uzjmjjym723afyx5" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2016-12-29 13:27:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602896474&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**102edo** is the [[equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit]] it [[tempering out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo]]. In the [[7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit]] 91/90 and 169/168; in the [[17-limit]] 136/135 and 154/153; and in the [[19-limit]] 133/132 and 190/189. It is the [[optimal patent val]] for 13-limit [[Diaschismic family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.
===13-limit Echidnic=== 
|| 2 || 23.529 ||
|| 4 || 47.059 ||
|| 7 || 82.353 ||
|| 9 || 105.882 ||
|| 11 || 129.412 ||
|| 13 || 152.941 ||
|| 16 || 188.235 ||
|| 18 || 211.765 ||
|| 20 || 235.294 ||
|| 22 || 258.824 ||
|| 24 || 282.353 ||
|| 27 || 317.647 ||
|| 29 || 341.176 ||
|| 31 || 364.706 ||
|| 33 || 388.235 ||
|| 35 || 411.765 ||
|| 38 || 447.059 ||
|| 40 || 470.588 ||
|| 42 || 494.117 ||
|| 44 || 517.647 ||
|| 47 || 552.941 ||
|| 49 || 576.471 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;102edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;102edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 102 steps of size 11.765 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the same &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s (2048/2025, 15625/15552, 20000/19683) as &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34edo&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; it tempers out 686/675 and 1029/1024; in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; 385/384, 441/440 and 4000/3993; in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; 91/90 and 169/168; in the &amp;lt;a class="wiki_link" href="/17-limit"&amp;gt;17-limit&amp;lt;/a&amp;gt; 136/135 and 154/153; and in the &amp;lt;a class="wiki_link" href="/19-limit"&amp;gt;19-limit&amp;lt;/a&amp;gt; 133/132 and 190/189. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Diaschismic%20family#Echidnic"&amp;gt;echidnic temperament&amp;lt;/a&amp;gt;, and the rank five temperament tempering out 91/90.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;a name="x--13-limit Echidnic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;13-limit Echidnic&amp;lt;/h3&amp;gt;
 

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        &amp;lt;td&amp;gt;211.765&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;282.353&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;341.176&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;35&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;552.941&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>habe0tzrr9eumb4uzjmjjym723afyx5</sha1>
    </revision>
    <revision>
      <id>4038</id>
      <timestamp>2016-12-07T11:56:31Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 601628380 - Original comment: **</comment>
      <origin>4038</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6621" sha1="7tveuqpldnqlg6zuydwxwdoo96ol8ja" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-12-07 11:56:31 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;601628380&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**102edo** is the [[equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit]] it [[tempering out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo]]. In the [[7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit]] 91/90 and 169/168; in the [[17-limit]] 136/135 and 154/153; and in the [[19-limit]] 133/132 and 190/189. It is the [[optimal patent val]] for 13-limit [[Diaschismic family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.
===13-limit Echidnic=== 
|| 2 || 23.529 ||
|| 4 || 47.059 ||
|| 7 || 82.353 ||
|| 9 || 105.882 ||
|| 11 || 129.412 ||
|| 13 || 152.941 ||
|| 16 || 188.235 ||
|| 18 || 211.765 ||
|| 20 || 235.294 ||
|| 22 || 258.824 ||
|| 24 || 282.353 ||
|| 27 || 317.647 ||
|| 29 || 341.1765/1 ||
|| 31 || 364.706 ||
|| 33 || 388.235 ||
|| 35 || 411.765 ||
|| 38 || 447.059 ||
|| 40 || 470.588 ||
|| 42 || 494.117 ||
|| 44 || 517.647 ||
|| 47 || 552.941 ||
|| 49 || 576.471 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;102edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;102edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 102 steps of size 11.765 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the same &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s (2048/2025, 15625/15552, 20000/19683) as &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34edo&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; it tempers out 686/675 and 1029/1024; in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; 385/384, 441/440 and 4000/3993; in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; 91/90 and 169/168; in the &amp;lt;a class="wiki_link" href="/17-limit"&amp;gt;17-limit&amp;lt;/a&amp;gt; 136/135 and 154/153; and in the &amp;lt;a class="wiki_link" href="/19-limit"&amp;gt;19-limit&amp;lt;/a&amp;gt; 133/132 and 190/189. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Diaschismic%20family#Echidnic"&amp;gt;echidnic temperament&amp;lt;/a&amp;gt;, and the rank five temperament tempering out 91/90.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;a name="x--13-limit Echidnic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;13-limit Echidnic&amp;lt;/h3&amp;gt;
 

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        &amp;lt;td&amp;gt;47.059&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;82.353&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;105.882&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;129.412&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;152.941&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;317.647&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>7tveuqpldnqlg6zuydwxwdoo96ol8ja</sha1>
    </revision>
    <revision>
      <id>4039</id>
      <timestamp>2016-08-27T12:54:32Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 590257298 - Original comment: **</comment>
      <origin>4039</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6713" sha1="h5k9hgnuyk24lrfwbs30qkbfe8b2gt0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-08-27 12:54:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;590257298&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**102edo** is the [[equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit]] it [[tempering out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo]]. In the [[7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit]] 91/90 and 169/168; in the [[17-limit]] 136/135 and 154/153; and in the [[19-limit]] 133/132 and 190/189. It is the [[optimal patent val]] for 13-limit [[Diaschismic family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.
===13-limit Echidnic=== 
|| 2 || 23.529 ||
|| 4 || 47.059 ||
|| 7 || 82.353 ||
|| 9 || 105.882 ||
|| 11 || 129.412 ||
|| 13 || 152.941 ||
|| 16 || 188.235 ||
|| 18 || 211.765 ||
|| 20 || 235.294 ||
|| 22 || 258.824 ||
|| 24 || 282.353 ||
|| 27 || 317.647 ||
|| 29 || [[tel:341.1765|341.1765]] ||
|| 31 || 364.706 ||
|| 33 || 388.235 ||
|| 35 || 411.765 ||
|| 38 || 447.059 ||
|| 40 || 470.588 ||
|| 42 || 494.117 ||
|| 44 || 517.647 ||
|| 47 || 552.941 ||
|| 49 || 576.471 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;102edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;102edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 102 steps of size 11.765 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the same &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s (2048/2025, 15625/15552, 20000/19683) as &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34edo&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; it tempers out 686/675 and 1029/1024; in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; 385/384, 441/440 and 4000/3993; in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; 91/90 and 169/168; in the &amp;lt;a class="wiki_link" href="/17-limit"&amp;gt;17-limit&amp;lt;/a&amp;gt; 136/135 and 154/153; and in the &amp;lt;a class="wiki_link" href="/19-limit"&amp;gt;19-limit&amp;lt;/a&amp;gt; 133/132 and 190/189. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Diaschismic%20family#Echidnic"&amp;gt;echidnic temperament&amp;lt;/a&amp;gt;, and the rank five temperament tempering out 91/90.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;a name="x--13-limit Echidnic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;13-limit Echidnic&amp;lt;/h3&amp;gt;
 

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        &amp;lt;td&amp;gt;129.412&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;317.647&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/341.1765"&amp;gt;341.1765&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;33&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;388.235&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;411.765&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;470.588&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;494.117&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;517.647&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;552.941&amp;lt;br /&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;576.471&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>h5k9hgnuyk24lrfwbs30qkbfe8b2gt0</sha1>
    </revision>
    <revision>
      <id>4040</id>
      <timestamp>2016-08-01T14:52:49Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 588501714 - Original comment: **</comment>
      <origin>4040</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6629" sha1="7anie555abtudlm5gqixl40kvko8mze" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-08-01 14:52:49 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;588501714&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**102edo** is the [[equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit]] it [[tempering out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo]]. In the [[7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit]] 91/90 and 169/168; in the [[17-limit]] 136/135 and 154/153; and in the [[19-limit]] 133/132 and 190/189. It is the [[optimal patent val]] for 13-limit [[Diaschismic family#Echidnic|echidnic temperament]], and the rank five [[#|temperament]] tempering out 91/90.
===13-limit Echidnic=== 
|| 2 || 23.529 ||
|| 4 || 47.059 ||
|| 7 || 82.353 ||
|| 9 || 105.882 ||
|| 11 || 129.412 ||
|| 13 || 152.941 ||
|| 16 || 188.235 ||
|| 18 || 211.765 ||
|| 20 || 235.294 ||
|| 22 || 258.824 ||
|| 24 || 282.353 ||
|| 27 || 317.647 ||
|| 29 || 341.1765 ||
|| 31 || 364.706 ||
|| 33 || 388.235 ||
|| 35 || 411.765 ||
|| 38 || 447.059 ||
|| 40 || 470.588 ||
|| 42 || 494.117 ||
|| 44 || 517.647 ||
|| 47 || 552.941 ||
|| 49 || 576.471 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;102edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;102edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 102 steps of size 11.765 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the same &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s (2048/2025, 15625/15552, 20000/19683) as &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34edo&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; it tempers out 686/675 and 1029/1024; in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; 385/384, 441/440 and 4000/3993; in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; 91/90 and 169/168; in the &amp;lt;a class="wiki_link" href="/17-limit"&amp;gt;17-limit&amp;lt;/a&amp;gt; 136/135 and 154/153; and in the &amp;lt;a class="wiki_link" href="/19-limit"&amp;gt;19-limit&amp;lt;/a&amp;gt; 133/132 and 190/189. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Diaschismic%20family#Echidnic"&amp;gt;echidnic temperament&amp;lt;/a&amp;gt;, and the rank five [[#|temperament]] tempering out 91/90.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;a name="x--13-limit Echidnic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;13-limit Echidnic&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;47.059&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;82.353&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;152.941&amp;lt;br /&amp;gt;
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      <comment>**Imported revision 256133420 - Original comment: **</comment>
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      <text bytes="3130" sha1="6eoop1kii70ln18f0fssg0f4v13yrr8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-09-20 12:15:05 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;256133420&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**102edo** is the [[equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit]] it [[tempering out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo]]. In the [[7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit]] 91/90 and 169/168; in the [[17-limit]] 136/135 and 154/153; and in the [[19-limit]] 133/132 and 190/189. It is the [[optimal patent val]] for 13-limit [[Diaschismic family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;102edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;102edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 102 steps of size 11.765 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the same &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s (2048/2025, 15625/15552, 20000/19683) as &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34edo&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; it tempers out 686/675 and 1029/1024; in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; 385/384, 441/440 and 4000/3993; in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; 91/90 and 169/168; in the &amp;lt;a class="wiki_link" href="/17-limit"&amp;gt;17-limit&amp;lt;/a&amp;gt; 136/135 and 154/153; and in the &amp;lt;a class="wiki_link" href="/19-limit"&amp;gt;19-limit&amp;lt;/a&amp;gt; 133/132 and 190/189. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Diaschismic%20family#Echidnic"&amp;gt;echidnic temperament&amp;lt;/a&amp;gt;, and the rank five temperament tempering out 91/90.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 239785937 - Original comment: **</comment>
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      <text bytes="3112" sha1="6drkokhleu0nlbmciy9r9lirfi5kknl" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-07-02 21:17:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239785937&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**102edo** is the [[equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit]] it [[tempering out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo]]. In the [[7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit]] 91/90 and 169/168; in the [[17-limit]] 136/135 and 154/153; and in the [[19-limit]] 133/132 and 190/189. It is the [[optimal patent val]] for 13-limit [[Diaschismic family|echidnic temperament]], and the rank five temperament tempering out 91/90.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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      <comment>**Imported revision 239303455 - Original comment: **</comment>
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      <text bytes="2998" sha1="d6o6inlzp0arg04tnyz4get7zq4q2i3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 08:28:03 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239303455&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**102edo** is the [[equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit]] it [[tempering out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo]]. In the [[7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit]] 91/90 and 169/168; in the [[17-limit]] 136/135 and 154/153; and in the [[19-limit]] 133/132 and 190/189. It is the [[optimal patent val]] for 13-limit [[Diaschismic family|echidnic temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;102edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;102edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 102 steps of size 11.765 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the same &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s (2048/2025, 15625/15552, 20000/19683) as &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34edo&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; it tempers out 686/675 and 1029/1024; in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; 385/384, 441/440 and 4000/3993; in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; 91/90 and 169/168; in the &amp;lt;a class="wiki_link" href="/17-limit"&amp;gt;17-limit&amp;lt;/a&amp;gt; 136/135 and 154/153; and in the &amp;lt;a class="wiki_link" href="/19-limit"&amp;gt;19-limit&amp;lt;/a&amp;gt; 133/132 and 190/189. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;echidnic temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 214892922 - Original comment: **</comment>
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      <text bytes="2373" sha1="13lxfw7zzen3x695kdo3pgqyqr7u8tq" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-28 21:30:54 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214892922&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 102 equal temperament divides the octave into 102 equal steps of size 11.765 cents each. In the 5-limit it tempers out the same commas (2048/2025, 15625/15552, 20000/19683) as [[34edo]]. In the 7-limit it tempers out 686/675 and 1029/1024; in the 11-limit 385/384, 441/440 and 4000/3993; in the 13-limit 91/90 and 169/168; in the 17-limit 136/135 and 154/153; and in the 19-limit 133/132 and 190/189. It is the [[optimal patent val]] for 13-limit [[Diaschismic family|echidnic temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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      <comment>**Imported revision 213464736 - Original comment: **</comment>
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      <text bytes="2343" sha1="1bzg757wpxgdy2uu5dih8h47msgn574" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-24 00:11:44 UTC&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 102 equal temperament divides the octave into 102 equal steps of size 11.765 cents each. It tempers out the same commas (2048/2025, 15625/15552, 20000/19683) as [[34edo]]. In the 7-limit it tempers out 686/675 and 1029/1024; in the 11-limit 385/384, 441/440 and 4000/3993; in the 13-limit 91/90 and 169/168; in the 17-limit 136/135 and 154/153; and in the 19-limit 133/132 and 190/189. It is the [[optimal patent val]] for 13-limit [[Diaschismic family|echidnic temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;102edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 102 equal temperament divides the octave into 102 equal steps of size 11.765 cents each. It tempers out the same commas (2048/2025, 15625/15552, 20000/19683) as &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34edo&amp;lt;/a&amp;gt;. In the 7-limit it tempers out 686/675 and 1029/1024; in the 11-limit 385/384, 441/440 and 4000/3993; in the 13-limit 91/90 and 169/168; in the 17-limit 136/135 and 154/153; and in the 19-limit 133/132 and 190/189. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;echidnic temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>1bzg757wpxgdy2uu5dih8h47msgn574</sha1>
    </revision>
    <revision>
      <id>54946</id>
      <parentid>19</parentid>
      <timestamp>2020-12-04T23:13:05Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54946</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1316" sha1="1y0f22hpjngh6d9zis1izm5rrymia7g" xml:space="preserve">'''102edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 102 steps of size 11.765 [[cent|cent]]s each. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] the same [[Comma|comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo|34edo]]. In the [[7-limit|7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit|11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit|13-limit]] 91/90 and 169/168; in the [[17-limit|17-limit]] 136/135 and 154/153; and in the [[19-limit|19-limit]] 133/132 and 190/189. It is the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Diaschismic_family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.

===13-limit Echidnic===

{| class="wikitable"
|-
| | 2
| | 23.529
|-
| | 4
| | 47.059
|-
| | 7
| | 82.353
|-
| | 9
| | 105.882
|-
| | 11
| | 129.412
|-
| | 13
| | 152.941
|-
| | 16
| | 188.235
|-
| | 18
| | 211.765
|-
| | 20
| | 235.294
|-
| | 22
| | 258.824
|-
| | 24
| | 282.353
|-
| | 27
| | 317.647
|-
| | 29
| | 341.176
|-
| | 31
| | 364.706
|-
| | 33
| | 388.235
|-
| | 35
| | 411.765
|-
| | 38
| | 447.059
|-
| | 40
| | 470.588
|-
| | 42
| | 494.117
|-
| | 44
| | 517.647
|-
| | 47
| | 552.941
|-
| | 49
| | 576.471
|}
[[Category:echidnic]]
[[Category:Equal divisions of the octave]]</text>
      <sha1>1y0f22hpjngh6d9zis1izm5rrymia7g</sha1>
    </revision>
    <revision>
      <id>68022</id>
      <parentid>54946</parentid>
      <timestamp>2021-04-20T19:29:06Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68022</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3291" sha1="epbps3b13e980t6roxdrjkpsuhwht8l" xml:space="preserve">'''102edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo|34edo]]. In the [[7-limit|7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit|11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit|13-limit]] 91/90 and 169/168; in the [[17-limit|17-limit]] 136/135 and 154/153; and in the [[19-limit|19-limit]] 133/132 and 190/189. It is the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Diaschismic_family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.

===13-limit Echidnic===

{| class="wikitable"
!Degree
!Cents
!Difference from 46edo
|-
| |2
| |23.529
|  -2.5575¢
|-
| |4
| |47.059
| -5.115¢
|-
| | 7
| |82.353
|4.092¢
|-
| |9
| |105.882
|1.5345¢
|-
| |11
| |129.412
| -1.023¢
|-
| |13
| |152.941
|8.184¢
|-
| |16
| |188.235
|5.627¢
|-
| | 18
| |211.765
|3.069¢
|-
| |20
| |235.294
|0.511¢
|-
| |22
| |258.824
| -2.046¢
|-
| | 24
| |282.353
| -4.604¢
|-
| |27
| |317.647
|4.604¢
|-
| |29
| |341.176
|2.046¢
|-
| |31
| |364.706
| -0.5115¢
|-
| |33
| |388.235
|  -3.069¢
|-
| |35
| |411.765
| -5.627¢
|-
| |38
| |447.059
| 3.581¢
|-
| |40
| |470.588
|1.023¢
|-
| |42
| |494.117
| -1.5345¢
|-
| |44
| |517.647
| -4.092¢
|-
| |47
| |552.941
|5.115¢
|-
| |49
| |576.471
|2.5575¢
|}Since 102edo has a step of 11.765 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which truly allows one to use an MOS scale with 60 or more small steps as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 102edo
!Tones
!Pattern
!L:s
|-
|5
|[[2L 3s]]
|21:20
|-
|6
|[[6edo]]
|equal
|-
|7
|[[4L 3s]]
|15:14
|-
|8
|[[6L 2s]]
|13:12
|-
|9
|[[3L 6s]]
|12:11
|-
|10
|[[2L 8s]]
|11:10
|-
|11
|[[3L 8s]]
|10:9
|-
|12
|[[6L 6s]]
|9:8
|-
|13
|[[11L 2s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[12L 3s]]
| rowspan="2" |7:6
|-
|16
|[[6L 10s]]
|-
|17
|[[17edo]]
|equal
|-
|18
|10L 8s
| rowspan="3" |6:5
|-
|19
|[[7L 12s]]
|-
|20
|2L 18s
|-
|21
|18L 3s
| rowspan="5" |5:4
|-
|22
|14L 8s
|-
|23
|10L 13s
|-
|24
|6L 18s
|-
|25
|2L 23s
|-
|26
|24L 2s
| rowspan="8" |4:3
|-
|27
|21L 6s
|-
|28
|18L 10s
|-
|29
|15L 14s
|-
|30
|12L 18s
|-
|31
|9L 22s
|-
|32
|6L 26s
|-
|33
|3L 30s
|-
|34
|[[34edo]]
|equal
|-
|35
|32L 3s
| rowspan="16" |3:2
|-
|36
|30L 6s
|-
|37
|28L 9s
|-
|38
|26L 12s
|-
|39
|24L 15s
|-
|40
|22L 18s
|-
|41
|20L 21s
|-
|42
|18L 24s
|-
|43
|16L 27s
|-
|44
|14L 30s
|-
|45
|12L 33s
|-
|46
|10L 36s
|-
|47
|8L 39s
|-
|48
|6L 42s
|-
|49
|4L 45s
|-
|50
|2L 48s
|-
|51
|[[51edo]]
|equal
|-
|52
|50L 2s
| rowspan="30" |2:1
|-
|53
|49L 4s
|-
|54
|48L 6s
|-
|55
|47L 8s
|-
|56
|46L 10s
|-
|57
|45L 12s
|-
|58
|44L 14s
|-
|59
|43L 16s
|-
|60
|42L 18s
|-
|61
|41L 20s
|-
|62
|40L 22s
|-
|63
|39L 24s
|-
|64
|38L 26s
|-
|65
|37L 28s
|-
|66
|36L 30s
|-
|67
|35L 32s
|-
|68
|34L 34s
|-
|69
|33L 36s
|-
|70
|32L 38s
|-
|71
|31L 40s
|-
|72
|30L 42s
|-
|73
|29L 44s
|-
|74
|28L 46s
|-
|75
|27L 48s
|-
|76
|26L 50s
|-
|77
|25L 52s
|-
|78
|24L 54s
|-
|79
|23L 56s
|-
|80
|22L 58s
|-
|81
|21L 60s
|}
[[Category:echidnic]]
[[Category:Equal divisions of the octave]]</text>
      <sha1>epbps3b13e980t6roxdrjkpsuhwht8l</sha1>
    </revision>
    <revision>
      <id>68024</id>
      <parentid>68022</parentid>
      <timestamp>2021-04-20T19:42:56Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* 13-limit Echidnic */</comment>
      <origin>68024</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3291" sha1="l6ppcrgusgzrbl5v88zsl9t6jckn5g7" xml:space="preserve">'''102edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo|34edo]]. In the [[7-limit|7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit|11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit|13-limit]] 91/90 and 169/168; in the [[17-limit|17-limit]] 136/135 and 154/153; and in the [[19-limit|19-limit]] 133/132 and 190/189. It is the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Diaschismic_family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.

===13-limit Echidnic===

{| class="wikitable"
!Degree
!Cents
!Difference from 46edo
|-
| |2
| |23.529
|  -2.5575¢
|-
| |4
| |47.059
| -5.115¢
|-
| | 7
| |82.353
|4.092¢
|-
| |9
| |105.882
|1.5345¢
|-
| |11
| |129.412
| -1.023¢
|-
| |13
| |152.941
|8.184¢
|-
| |16
| |188.235
|5.627¢
|-
| | 18
| |211.765
|3.069¢
|-
| |20
| |235.294
|0.511¢
|-
| |22
| |258.824
| -2.046¢
|-
| | 24
| |282.353
| -4.604¢
|-
| |27
| |317.647
|4.604¢
|-
| |29
| |341.176
|2.046¢
|-
| |31
| |364.706
| -0.5115¢
|-
| |33
| |388.235
|  -3.069¢
|-
| |35
| |411.765
| -5.627¢
|-
| |38
| |447.059
| 3.581¢
|-
| |40
| |470.588
|1.023¢
|-
| |42
| |494.117
| -1.5345¢
|-
| |44
| |517.647
| -4.092¢
|-
| |47
| |552.941
|5.115¢
|-
| |49
| |576.471
|2.5575¢
|}Since 102edo has a step of 11.765 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which truly allows one to use an MOS scale with 60 or more small steps as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 102edo
!Tones
!Pattern
!L:s
|-
|5
|[[2L 3s]]
|21:20
|-
|6
|[[6edo]]
|equal
|-
|7
|[[4L 3s]]
|15:14
|-
|8
|[[6L 2s]]
|13:12
|-
|9
|[[3L 6s]]
|12:11
|-
|10
|[[2L 8s]]
|11:10
|-
|11
|[[3L 8s]]
|10:9
|-
|12
|[[6L 6s]]
|9:8
|-
|13
|[[11L 2s]]
| rowspan="2" |8:7
|-
|14
|[[4L 10s]]
|-
|15
|[[12L 3s]]
| rowspan="2" |7:6
|-
|16
|[[6L 10s]]
|-
|17
|[[17edo]]
|equal
|-
|18
|10L 8s
| rowspan="3" |6:5
|-
|19
|[[7L 12s]]
|-
|20
|2L 18s
|-
|21
|18L 3s
| rowspan="5" |5:4
|-
|22
|14L 8s
|-
|23
|10L 13s
|-
|24
|6L 18s
|-
|25
|2L 23s
|-
|26
|24L 2s
| rowspan="8" |4:3
|-
|27
|21L 6s
|-
|28
|18L 10s
|-
|29
|15L 14s
|-
|30
|12L 18s
|-
|31
|9L 22s
|-
|32
|6L 26s
|-
|33
|3L 30s
|-
|34
|[[34edo]]
|equal
|-
|35
|32L 3s
| rowspan="16" |3:2
|-
|36
|30L 6s
|-
|37
|28L 9s
|-
|38
|26L 12s
|-
|39
|24L 15s
|-
|40
|22L 18s
|-
|41
|20L 21s
|-
|42
|18L 24s
|-
|43
|16L 27s
|-
|44
|14L 30s
|-
|45
|12L 33s
|-
|46
|10L 36s
|-
|47
|8L 39s
|-
|48
|6L 42s
|-
|49
|4L 45s
|-
|50
|2L 48s
|-
|51
|[[51edo]]
|equal
|-
|52
|50L 2s
| rowspan="30" |2:1
|-
|53
|49L 4s
|-
|54
|48L 6s
|-
|55
|47L 8s
|-
|56
|46L 10s
|-
|57
|45L 12s
|-
|58
|44L 14s
|-
|59
|43L 16s
|-
|60
|42L 18s
|-
|61
|41L 20s
|-
|62
|40L 22s
|-
|63
|39L 24s
|-
|64
|38L 26s
|-
|65
|37L 28s
|-
|66
|36L 30s
|-
|67
|35L 32s
|-
|68
|34L 34s
|-
|69
|33L 36s
|-
|70
|32L 38s
|-
|71
|31L 40s
|-
|72
|30L 42s
|-
|73
|29L 44s
|-
|74
|28L 46s
|-
|75
|27L 48s
|-
|76
|26L 50s
|-
|77
|25L 52s
|-
|78
|24L 54s
|-
|79
|23L 56s
|-
|80
|22L 58s
|-
|81
|21L 60s
|}
[[Category:echidnic]]
[[Category:Equal divisions of the octave]]</text>
      <sha1>l6ppcrgusgzrbl5v88zsl9t6jckn5g7</sha1>
    </revision>
    <revision>
      <id>92912</id>
      <parentid>68024</parentid>
      <timestamp>2022-07-02T16:16:19Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92912</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3320" sha1="fps9a8l3utpi9ai1un5m0279eqaaizh" xml:space="preserve">'''102edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo|34edo]]. In the [[7-limit|7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit|11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit|13-limit]] 91/90 and 169/168; in the [[17-limit|17-limit]] 136/135 and 154/153; and in the [[19-limit|19-limit]] 133/132 and 190/189. It is the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Diaschismic_family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.

===13-limit Echidnic===

{| class="wikitable"
!Degree
!Cents
!Difference from 46edo
|-
| |2
| |23.529
|  -2.5575¢
|-
| |4
| |47.059
| -5.115¢
|-
| | 7
| |82.353
|4.092¢
|-
| |9
| |105.882
|1.5345¢
|-
| |11
| |129.412
| -1.023¢
|-
| |13
| |152.941
|8.184¢
|-
| |16
| |188.235
|5.627¢
|-
| | 18
| |211.765
|3.069¢
|-
| |20
| |235.294
|0.511¢
|-
| |22
| |258.824
| -2.046¢
|-
| | 24
| |282.353
| -4.604¢
|-
| |27
| |317.647
|4.604¢
|-
| |29
| |341.176
|2.046¢
|-
| |31
| |364.706
| -0.5115¢
|-
| |33
| |388.235
|  -3.069¢
|-
| |35
| |411.765
| -5.627¢
|-
| |38
| |447.059
| 3.581¢
|-
| |40
| |470.588
|1.023¢
|-
| |42
| |494.117
| -1.5345¢
|-
| |44
| |517.647
| -4.092¢
|-
| |47
| |552.941
|5.115¢
|-
| |49
| |576.471
|2.5575¢
|}Since 102edo has a step of 11.765 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which truly allows one to use an MOS scale with 60 or more small steps as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 102edo
!Tones
!Pattern
!L:s
|-
|5
|[[2L 3s]]
|21:20
|-
|6
|[[6edo]]
|equal
|-
|7
|[[4L 3s]]
|15:14
|-
|8
|[[6L 2s]]
|13:12
|-
|9
|[[3L 6s]]
|12:11
|-
|10
|[[2L 8s]]
|11:10
|-
|11
|[[3L 8s]]
|10:9
|-
|12
|[[6L 6s]]
|9:8
|-
|13
|[[11L 2s]]
| rowspan="2" |8:7
|-
|14
|[[4L 10s]]
|-
|15
|[[12L 3s]]
| rowspan="2" |7:6
|-
|16
|[[6L 10s]]
|-
|17
|[[17edo]]
|equal
|-
|18
|10L 8s
| rowspan="3" |6:5
|-
|19
|[[7L 12s]]
|-
|20
|2L 18s
|-
|21
|18L 3s
| rowspan="5" |5:4
|-
|22
|14L 8s
|-
|23
|10L 13s
|-
|24
|6L 18s
|-
|25
|2L 23s
|-
|26
|24L 2s
| rowspan="8" |4:3
|-
|27
|21L 6s
|-
|28
|18L 10s
|-
|29
|15L 14s
|-
|30
|12L 18s
|-
|31
|9L 22s
|-
|32
|6L 26s
|-
|33
|3L 30s
|-
|34
|[[34edo]]
|equal
|-
|35
|32L 3s
| rowspan="16" |3:2
|-
|36
|30L 6s
|-
|37
|28L 9s
|-
|38
|26L 12s
|-
|39
|24L 15s
|-
|40
|22L 18s
|-
|41
|20L 21s
|-
|42
|18L 24s
|-
|43
|16L 27s
|-
|44
|14L 30s
|-
|45
|12L 33s
|-
|46
|10L 36s
|-
|47
|8L 39s
|-
|48
|6L 42s
|-
|49
|4L 45s
|-
|50
|2L 48s
|-
|51
|[[51edo]]
|equal
|-
|52
|50L 2s
| rowspan="30" |2:1
|-
|53
|49L 4s
|-
|54
|48L 6s
|-
|55
|47L 8s
|-
|56
|46L 10s
|-
|57
|45L 12s
|-
|58
|44L 14s
|-
|59
|43L 16s
|-
|60
|42L 18s
|-
|61
|41L 20s
|-
|62
|40L 22s
|-
|63
|39L 24s
|-
|64
|38L 26s
|-
|65
|37L 28s
|-
|66
|36L 30s
|-
|67
|35L 32s
|-
|68
|34L 34s
|-
|69
|33L 36s
|-
|70
|32L 38s
|-
|71
|31L 40s
|-
|72
|30L 42s
|-
|73
|29L 44s
|-
|74
|28L 46s
|-
|75
|27L 48s
|-
|76
|26L 50s
|-
|77
|25L 52s
|-
|78
|24L 54s
|-
|79
|23L 56s
|-
|80
|22L 58s
|-
|81
|21L 60s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Echidnic]]</text>
      <sha1>fps9a8l3utpi9ai1un5m0279eqaaizh</sha1>
    </revision>
    <revision>
      <id>96186</id>
      <parentid>92912</parentid>
      <timestamp>2022-10-04T18:48:05Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96186</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3335" sha1="sc087v0caa017x0ue6r9gqx3s5xr1e0" xml:space="preserve">{{Infobox ET}}
'''102edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo|34edo]]. In the [[7-limit|7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit|11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit|13-limit]] 91/90 and 169/168; in the [[17-limit|17-limit]] 136/135 and 154/153; and in the [[19-limit|19-limit]] 133/132 and 190/189. It is the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Diaschismic_family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.

===13-limit Echidnic===

{| class="wikitable"
!Degree
!Cents
!Difference from 46edo
|-
| |2
| |23.529
|  -2.5575¢
|-
| |4
| |47.059
| -5.115¢
|-
| | 7
| |82.353
|4.092¢
|-
| |9
| |105.882
|1.5345¢
|-
| |11
| |129.412
| -1.023¢
|-
| |13
| |152.941
|8.184¢
|-
| |16
| |188.235
|5.627¢
|-
| | 18
| |211.765
|3.069¢
|-
| |20
| |235.294
|0.511¢
|-
| |22
| |258.824
| -2.046¢
|-
| | 24
| |282.353
| -4.604¢
|-
| |27
| |317.647
|4.604¢
|-
| |29
| |341.176
|2.046¢
|-
| |31
| |364.706
| -0.5115¢
|-
| |33
| |388.235
|  -3.069¢
|-
| |35
| |411.765
| -5.627¢
|-
| |38
| |447.059
| 3.581¢
|-
| |40
| |470.588
|1.023¢
|-
| |42
| |494.117
| -1.5345¢
|-
| |44
| |517.647
| -4.092¢
|-
| |47
| |552.941
|5.115¢
|-
| |49
| |576.471
|2.5575¢
|}Since 102edo has a step of 11.765 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which truly allows one to use an MOS scale with 60 or more small steps as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 102edo
!Tones
!Pattern
!L:s
|-
|5
|[[2L 3s]]
|21:20
|-
|6
|[[6edo]]
|equal
|-
|7
|[[4L 3s]]
|15:14
|-
|8
|[[6L 2s]]
|13:12
|-
|9
|[[3L 6s]]
|12:11
|-
|10
|[[2L 8s]]
|11:10
|-
|11
|[[3L 8s]]
|10:9
|-
|12
|[[6L 6s]]
|9:8
|-
|13
|[[11L 2s]]
| rowspan="2" |8:7
|-
|14
|[[4L 10s]]
|-
|15
|[[12L 3s]]
| rowspan="2" |7:6
|-
|16
|[[6L 10s]]
|-
|17
|[[17edo]]
|equal
|-
|18
|10L 8s
| rowspan="3" |6:5
|-
|19
|[[7L 12s]]
|-
|20
|2L 18s
|-
|21
|18L 3s
| rowspan="5" |5:4
|-
|22
|14L 8s
|-
|23
|10L 13s
|-
|24
|6L 18s
|-
|25
|2L 23s
|-
|26
|24L 2s
| rowspan="8" |4:3
|-
|27
|21L 6s
|-
|28
|18L 10s
|-
|29
|15L 14s
|-
|30
|12L 18s
|-
|31
|9L 22s
|-
|32
|6L 26s
|-
|33
|3L 30s
|-
|34
|[[34edo]]
|equal
|-
|35
|32L 3s
| rowspan="16" |3:2
|-
|36
|30L 6s
|-
|37
|28L 9s
|-
|38
|26L 12s
|-
|39
|24L 15s
|-
|40
|22L 18s
|-
|41
|20L 21s
|-
|42
|18L 24s
|-
|43
|16L 27s
|-
|44
|14L 30s
|-
|45
|12L 33s
|-
|46
|10L 36s
|-
|47
|8L 39s
|-
|48
|6L 42s
|-
|49
|4L 45s
|-
|50
|2L 48s
|-
|51
|[[51edo]]
|equal
|-
|52
|50L 2s
| rowspan="30" |2:1
|-
|53
|49L 4s
|-
|54
|48L 6s
|-
|55
|47L 8s
|-
|56
|46L 10s
|-
|57
|45L 12s
|-
|58
|44L 14s
|-
|59
|43L 16s
|-
|60
|42L 18s
|-
|61
|41L 20s
|-
|62
|40L 22s
|-
|63
|39L 24s
|-
|64
|38L 26s
|-
|65
|37L 28s
|-
|66
|36L 30s
|-
|67
|35L 32s
|-
|68
|34L 34s
|-
|69
|33L 36s
|-
|70
|32L 38s
|-
|71
|31L 40s
|-
|72
|30L 42s
|-
|73
|29L 44s
|-
|74
|28L 46s
|-
|75
|27L 48s
|-
|76
|26L 50s
|-
|77
|25L 52s
|-
|78
|24L 54s
|-
|79
|23L 56s
|-
|80
|22L 58s
|-
|81
|21L 60s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Echidnic]]</text>
      <sha1>sc087v0caa017x0ue6r9gqx3s5xr1e0</sha1>
    </revision>
    <revision>
      <id>106363</id>
      <parentid>96186</parentid>
      <timestamp>2023-03-30T09:15:10Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <minor/>
      <comment>Primes table</comment>
      <origin>106363</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3382" sha1="4wgo7gv0n31oo21jzsd7yzy7137f2hv" xml:space="preserve">{{Infobox ET}}
'''102edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo|34edo]]. In the [[7-limit|7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit|11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit|13-limit]] 91/90 and 169/168; in the [[17-limit|17-limit]] 136/135 and 154/153; and in the [[19-limit|19-limit]] 133/132 and 190/189. It is the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Diaschismic_family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.

=== Prime harmonics ===
{{Primes in edo|102}}

===13-limit Echidnic===

{| class="wikitable"
!Degree
!Cents
!Difference from 46edo
|-
| |2
| |23.529
|  -2.5575¢
|-
| |4
| |47.059
| -5.115¢
|-
| | 7
| |82.353
|4.092¢
|-
| |9
| |105.882
|1.5345¢
|-
| |11
| |129.412
| -1.023¢
|-
| |13
| |152.941
|8.184¢
|-
| |16
| |188.235
|5.627¢
|-
| | 18
| |211.765
|3.069¢
|-
| |20
| |235.294
|0.511¢
|-
| |22
| |258.824
| -2.046¢
|-
| | 24
| |282.353
| -4.604¢
|-
| |27
| |317.647
|4.604¢
|-
| |29
| |341.176
|2.046¢
|-
| |31
| |364.706
| -0.5115¢
|-
| |33
| |388.235
|  -3.069¢
|-
| |35
| |411.765
| -5.627¢
|-
| |38
| |447.059
| 3.581¢
|-
| |40
| |470.588
|1.023¢
|-
| |42
| |494.117
| -1.5345¢
|-
| |44
| |517.647
| -4.092¢
|-
| |47
| |552.941
|5.115¢
|-
| |49
| |576.471
|2.5575¢
|}Since 102edo has a step of 11.765 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which truly allows one to use an MOS scale with 60 or more small steps as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 102edo
!Tones
!Pattern
!L:s
|-
|5
|[[2L 3s]]
|21:20
|-
|6
|[[6edo]]
|equal
|-
|7
|[[4L 3s]]
|15:14
|-
|8
|[[6L 2s]]
|13:12
|-
|9
|[[3L 6s]]
|12:11
|-
|10
|[[2L 8s]]
|11:10
|-
|11
|[[3L 8s]]
|10:9
|-
|12
|[[6L 6s]]
|9:8
|-
|13
|[[11L 2s]]
| rowspan="2" |8:7
|-
|14
|[[4L 10s]]
|-
|15
|[[12L 3s]]
| rowspan="2" |7:6
|-
|16
|[[6L 10s]]
|-
|17
|[[17edo]]
|equal
|-
|18
|10L 8s
| rowspan="3" |6:5
|-
|19
|[[7L 12s]]
|-
|20
|2L 18s
|-
|21
|18L 3s
| rowspan="5" |5:4
|-
|22
|14L 8s
|-
|23
|10L 13s
|-
|24
|6L 18s
|-
|25
|2L 23s
|-
|26
|24L 2s
| rowspan="8" |4:3
|-
|27
|21L 6s
|-
|28
|18L 10s
|-
|29
|15L 14s
|-
|30
|12L 18s
|-
|31
|9L 22s
|-
|32
|6L 26s
|-
|33
|3L 30s
|-
|34
|[[34edo]]
|equal
|-
|35
|32L 3s
| rowspan="16" |3:2
|-
|36
|30L 6s
|-
|37
|28L 9s
|-
|38
|26L 12s
|-
|39
|24L 15s
|-
|40
|22L 18s
|-
|41
|20L 21s
|-
|42
|18L 24s
|-
|43
|16L 27s
|-
|44
|14L 30s
|-
|45
|12L 33s
|-
|46
|10L 36s
|-
|47
|8L 39s
|-
|48
|6L 42s
|-
|49
|4L 45s
|-
|50
|2L 48s
|-
|51
|[[51edo]]
|equal
|-
|52
|50L 2s
| rowspan="30" |2:1
|-
|53
|49L 4s
|-
|54
|48L 6s
|-
|55
|47L 8s
|-
|56
|46L 10s
|-
|57
|45L 12s
|-
|58
|44L 14s
|-
|59
|43L 16s
|-
|60
|42L 18s
|-
|61
|41L 20s
|-
|62
|40L 22s
|-
|63
|39L 24s
|-
|64
|38L 26s
|-
|65
|37L 28s
|-
|66
|36L 30s
|-
|67
|35L 32s
|-
|68
|34L 34s
|-
|69
|33L 36s
|-
|70
|32L 38s
|-
|71
|31L 40s
|-
|72
|30L 42s
|-
|73
|29L 44s
|-
|74
|28L 46s
|-
|75
|27L 48s
|-
|76
|26L 50s
|-
|77
|25L 52s
|-
|78
|24L 54s
|-
|79
|23L 56s
|-
|80
|22L 58s
|-
|81
|21L 60s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Echidnic]]</text>
      <sha1>4wgo7gv0n31oo21jzsd7yzy7137f2hv</sha1>
    </revision>
    <revision>
      <id>115187</id>
      <parentid>106363</parentid>
      <timestamp>2023-05-30T21:11:22Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>115187</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1618" sha1="j3cnrph5h5zt30y6mharbm4qeaq9k6h" xml:space="preserve">{{Infobox ET}}
'''102edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo|34edo]]. In the [[7-limit|7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit|11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit|13-limit]] 91/90 and 169/168; in the [[17-limit|17-limit]] 136/135 and 154/153; and in the [[19-limit|19-limit]] 133/132 and 190/189. It is the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Diaschismic_family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.

=== Prime harmonics ===
{{Primes in edo|102}}

===13-limit Echidnic===

{| class="wikitable"
!Degree
!Cents
!Difference from 46edo
|-
| |2
| |23.529
|  -2.5575¢
|-
| |4
| |47.059
| -5.115¢
|-
| | 7
| |82.353
|4.092¢
|-
| |9
| |105.882
|1.5345¢
|-
| |11
| |129.412
| -1.023¢
|-
| |13
| |152.941
|8.184¢
|-
| |16
| |188.235
|5.627¢
|-
| | 18
| |211.765
|3.069¢
|-
| |20
| |235.294
|0.511¢
|-
| |22
| |258.824
| -2.046¢
|-
| | 24
| |282.353
| -4.604¢
|-
| |27
| |317.647
|4.604¢
|-
| |29
| |341.176
|2.046¢
|-
| |31
| |364.706
| -0.5115¢
|-
| |33
| |388.235
|  -3.069¢
|-
| |35
| |411.765
| -5.627¢
|-
| |38
| |447.059
| 3.581¢
|-
| |40
| |470.588
|1.023¢
|-
| |42
| |494.117
| -1.5345¢
|-
| |44
| |517.647
| -4.092¢
|-
| |47
| |552.941
|5.115¢
|-
| |49
| |576.471
|2.5575¢
|}
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Echidnic]]</text>
      <sha1>j3cnrph5h5zt30y6mharbm4qeaq9k6h</sha1>
    </revision>
    <revision>
      <id>132170</id>
      <parentid>115187</parentid>
      <timestamp>2024-01-20T00:50:23Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Made heading style consistent with other edo pages, replaced old harmonics template with new one, added interval table</comment>
      <origin>132170</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1651" sha1="mjvlo95gbmzdfsryoglmrehigk9o774" xml:space="preserve">{{Infobox ET}}
'''102edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 102 steps of size 11.765 [[cent]]s each. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo|34edo]]. In the [[7-limit|7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit|11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit|13-limit]] 91/90 and 169/168; in the [[17-limit|17-limit]] 136/135 and 154/153; and in the [[19-limit|19-limit]] 133/132 and 190/189. It is the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Diaschismic_family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.

== Harmonics ==
{{Harmonics in equal|102}}

== Intervals ==
{{Interval table}}

== 13-limit Echidnic ==

{| class="wikitable"
!Degree
!Cents
!Difference from 46edo
|-
| |2
| |23.529
|  -2.5575¢
|-
| |4
| |47.059
| -5.115¢
|-
| | 7
| |82.353
|4.092¢
|-
| |9
| |105.882
|1.5345¢
|-
| |11
| |129.412
| -1.023¢
|-
| |13
| |152.941
|8.184¢
|-
| |16
| |188.235
|5.627¢
|-
| | 18
| |211.765
|3.069¢
|-
| |20
| |235.294
|0.511¢
|-
| |22
| |258.824
| -2.046¢
|-
| | 24
| |282.353
| -4.604¢
|-
| |27
| |317.647
|4.604¢
|-
| |29
| |341.176
|2.046¢
|-
| |31
| |364.706
| -0.5115¢
|-
| |33
| |388.235
|  -3.069¢
|-
| |35
| |411.765
| -5.627¢
|-
| |38
| |447.059
| 3.581¢
|-
| |40
| |470.588
|1.023¢
|-
| |42
| |494.117
| -1.5345¢
|-
| |44
| |517.647
| -4.092¢
|-
| |47
| |552.941
|5.115¢
|-
| |49
| |576.471
|2.5575¢
|}
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Echidnic]]</text>
      <sha1>mjvlo95gbmzdfsryoglmrehigk9o774</sha1>
    </revision>
    <revision>
      <id>132882</id>
      <parentid>132170</parentid>
      <timestamp>2024-01-25T08:59:12Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Add Edo intro</comment>
      <origin>132882</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1542" sha1="hmppzv6421h95yzrejdy01oyq2kdrep" xml:space="preserve">{{Infobox ET}}
{{EDO intro|102}}

In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] the same [[comma]]s (2048/2025, 15625/15552, 20000/19683) as [[34edo|34edo]]. In the [[7-limit|7-limit]] it tempers out 686/675 and 1029/1024; in the [[11-limit|11-limit]] 385/384, 441/440 and 4000/3993; in the [[13-limit|13-limit]] 91/90 and 169/168; in the [[17-limit|17-limit]] 136/135 and 154/153; and in the [[19-limit|19-limit]] 133/132 and 190/189. It is the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Diaschismic_family#Echidnic|echidnic temperament]], and the rank five temperament tempering out 91/90.

== Harmonics ==
{{Harmonics in equal|102}}

== Intervals ==
{{Interval table}}

== 13-limit Echidnic ==

{| class="wikitable"
!Degree
!Cents
!Difference from 46edo
|-
| |2
| |23.529
|  -2.5575¢
|-
| |4
| |47.059
| -5.115¢
|-
| | 7
| |82.353
|4.092¢
|-
| |9
| |105.882
|1.5345¢
|-
| |11
| |129.412
| -1.023¢
|-
| |13
| |152.941
|8.184¢
|-
| |16
| |188.235
|5.627¢
|-
| | 18
| |211.765
|3.069¢
|-
| |20
| |235.294
|0.511¢
|-
| |22
| |258.824
| -2.046¢
|-
| | 24
| |282.353
| -4.604¢
|-
| |27
| |317.647
|4.604¢
|-
| |29
| |341.176
|2.046¢
|-
| |31
| |364.706
| -0.5115¢
|-
| |33
| |388.235
|  -3.069¢
|-
| |35
| |411.765
| -5.627¢
|-
| |38
| |447.059
| 3.581¢
|-
| |40
| |470.588
|1.023¢
|-
| |42
| |494.117
| -1.5345¢
|-
| |44
| |517.647
| -4.092¢
|-
| |47
| |552.941
|5.115¢
|-
| |49
| |576.471
|2.5575¢
|}
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Echidnic]]</text>
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    </revision>
    <revision>
      <id>146045</id>
      <parentid>132882</parentid>
      <timestamp>2024-06-10T16:24:59Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>146045</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="762" sha1="tgufrwagpjhc51a841j21hh3ukk09td" xml:space="preserve">{{Infobox ET}}
{{EDO intro|102}}

== Theory ==
102edo is [[enfactoring|enfactored]] in the [[5-limit]], where it [[tempering out|tempers out]] the same [[comma]]s ([[2048/2025]], [[15625/15552]], [[20000/19683]]) as [[34edo]]. In the [[7-limit]] it tempers out [[686/675]] and [[1029/1024]]; in the [[11-limit]] [[385/384]], [[441/440]] and [[4000/3993]]; in the [[13-limit]] [[91/90]] and [[169/168]]; in the [[17-limit]] [[136/135]] and [[154/153]]; and in the [[19-limit]] [[133/132]] and [[190/189]]. It is the [[optimal patent val]] for 13-limit [[Diaschismic family #Echidnic|echidnic]] temperament, and the rank-5 temperament tempering out 91/90.

=== Odd harmonics ===
{{Harmonics in equal|102}}

== Intervals ==
{{Interval table}}

[[Category:Echidnic]]</text>
      <sha1>tgufrwagpjhc51a841j21hh3ukk09td</sha1>
    </revision>
    <revision>
      <id>181305</id>
      <parentid>146045</parentid>
      <timestamp>2025-02-18T16:46:22Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181305</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="757" sha1="hdhhlurdgyo45xkc8hbrgqcseekyqfg" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
102edo is [[enfactoring|enfactored]] in the [[5-limit]], where it [[tempering out|tempers out]] the same [[comma]]s ([[2048/2025]], [[15625/15552]], [[20000/19683]]) as [[34edo]]. In the [[7-limit]] it tempers out [[686/675]] and [[1029/1024]]; in the [[11-limit]] [[385/384]], [[441/440]] and [[4000/3993]]; in the [[13-limit]] [[91/90]] and [[169/168]]; in the [[17-limit]] [[136/135]] and [[154/153]]; and in the [[19-limit]] [[133/132]] and [[190/189]]. It is the [[optimal patent val]] for 13-limit [[Diaschismic family #Echidnic|echidnic]] temperament, and the rank-5 temperament tempering out 91/90.

=== Odd harmonics ===
{{Harmonics in equal|102}}

== Intervals ==
{{Interval table}}

[[Category:Echidnic]]</text>
      <sha1>hdhhlurdgyo45xkc8hbrgqcseekyqfg</sha1>
    </revision>
  </page>
  <page>
    <title>103edo</title>
    <ns>0</ns>
    <id>20</id>
    <revision>
      <id>20</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>20</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6193" sha1="7nkt61icc0wny4oiwsun4b5hqhmriv8" xml:space="preserve">'''103edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 103 steps of size 11.6505 [[cent|cent]]s each. It is a good [[Gamelismic_clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limi|13-]] limits and [[Gamelismic_clan|benediction]] and [[Gamelismic_clan|hemisecordite]], two of the [[13-limit|13-limit]] extensions of miracle. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit|5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit|11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel_family#Marvel|marvel temperament]] as well as [[Gamelismic_clan#Miracle|benediction]] and [[Gamelismic_clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime_numbers|prime]] edo.

===13-limit temperaments===

{| class="wikitable"
|-
| | Degree
| | cents
| | temperament
|-
| | 1
| | 11.650
| | Marvel and Benediction
|-
| | 2
| | 23.301
| | Hemisecordite
|-
| | 3
| | 34.951
| | Marvel and Benediction, Hemisecordite
|-
| | 4
| | 46.602
| | Marvel and Benediction
|-
| | 5
| | 58.252
| | Hemisecordite
|-
| | 6
| | 69.903
| | Marvel and Benediction
|-
| | 7
| | 81.553
| | Marvel and Benediction, Hemisecordite
|-
| | 8
| | 93.204
| | Hemisecordite
|-
| | 9
| | 104.854
| | Marvel and Benediction
|-
| | 10
| | 116.505
| | Marvel and Benediction, Hemisecordite
|-
| | 11
| | 128.155
| | Marvel and Benediction
|-
| | 12
| | 139.806
| | Hemisecordite
|-
| | 13
| | 151.456
| | Marvel and Benediction, Hemisecordite
|-
| | 14
| | 163.107
| | Marvel and Benediction
|-
| | 15
| | 174.757
| | Hemisecordite
|-
| | 16
| | 186.408
| | Marvel and Benediction
|-
| | 17
| | 198.058
| | Marvel and Benediction, Hemisecordite
|-
| | 18
| | 209.708
| | Hemisecordite
|-
| | 19
| | 221.359
| | Marvel and Benediction
|-
| | 20
| | 233.010
| | Marvel and Benediction, Hemisecordite
|-
| | 21
| | 244.660
| | Marvel and Benediction
|-
| | 22
| | 256.311
| | Hemisecordite
|-
| | 23
| | 267.961
| | Marvel and Benediction, Hemisecordite
|-
| | 24
| | 279.712
| | Marvel and Benediction
|-
| | 25
| | 291.262
| | Hemisecordite
|-
| | 26
| | 303.013
| | Marvel and Benediction
|-
| | 27
| | 314.563
| | Marvel and Benediction, Hemisecordite
|-
| | 28
| | 326.214
| | Hemisecordite
|-
| | 29
| | 337.864
| | Marvel and Benediction
|-
| | 30
| | 349.615
| | Marvel and Benediction, Hemisecordite
|-
| | 31
| | 361.165
| | Marvel and Benediction
|-
| | 32
| | 372.816
| | Hemisecordite
|-
| | 33
| | 384.466
| | Marvel and Benediction, Hemisecordite
|-
| | 34
| | 396.117
| | Marvel and Benediction
|-
| | 35
| | 407.767
| | Hemisecordite
|-
| | 36
| | 419.4175
| | Marvel and Benediction
|-
| | 37
| | 431.068
| | Marvel and Benediction, Hemisecordite
|-
| | 38
| | 442.708
| | Hemisecordite
|-
| | 39
| | 454.369
| | Marvel and Benediction
|-
| | 40
| | 466.019
| | Marvel and Benediction, Hemisecordite
|-
| | 41
| | 477.670
| | Marvel and Benediction
|-
| | 42
| | 489.320
| | Hemisecordite
|-
| | 43
| | 500.971
| | Marvel and Benediction, Hemisecordite
|-
| | 44
| | 512.621
| | Marvel and Benediction
|-
| | 45
| | 524.272
| | Hemisecordite
|-
| | 46
| | 535.922
| | Marvel and Benediction
|-
| | 47
| | 547.573
| | Marevel and Benediction, Hemisecordite
|-
| | 48
| | 559.223
| | Hemisecordite
|-
| | 49
| | 570.874
| | Marvel and Benediction
|-
| | 50
| | 582.524
| | Marvel and Benediction, Hemisecordite
|-
| | 52
| | 605.825
| | Marvel and Benediction, Hemisecordite
|-
| | 53
| | 617.476
| | Marvel and Benediction, Hemisecordite
|-
| | 54
| | 629.126
| | Marvel and Benediction
|-
| | 55
| | 640.777
| | Hemisecordite
|-
| | 56
| | 652.427
| | Marvel and Benediction, Hemisecordite
|-
| | 57
| | 664.078
| | Marvel and Benediction
|-
| | 58
| | 675.728
| | Hemisecordite
|-
| | 59
| | 687.379
| | Marvel and Benediction
|-
| | 60
| | 699.029
| | Marvel and Benediction, Hemisecordite
|-
| | 61
| | 710.677
| | Hemisecordite
|-
| | 62
| | 722.330
| | Marvel and Benediction
|-
| | 63
| | 733.981
| | Marvel and Benediction, Hemisecordite
|-
| | 64
| | 745.631
| | Marvel and Benediction
|-
| | 65
| | 757.282
| | Hemisecordite
|-
| | 66
| | 768.932
| | Marvel and Benediction, Hemisecordite
|-
| | 67
| | 780.583
| | Marvel and Benediction
|-
| | 68
| | 792.233
| | Hemisecordite
|-
| | 69
| | 803.883
| | Marvel and Benediction
|-
| | 70
| | 815.534
| | Marvel and Benediction, Hemisecordite
|-
| | 71
| | 827.185
| | Hemisecordite
|-
| | 72
| | 838.835
| | Marvel and Benediction
|-
| | 73
| | 850.385
| | Marvel and Benediction. Hemisecordite
|-
| | 74
| | 862.156
| | Marvel and Benediction
|-
| | 75
| | 873.786
| | Hemisecordite
|-
| | 76
| | 885.537
| | Marvel and Benediction, Hemisecordite
|-
| | 77
| | 896.987
| | Marvel and Benediction
|-
| | 78
| | 908.638
| | Hemisecordite
|-
| | 79
| | 920.388
| | Marvel and Benediction, Hemisecordite
|-
| | 80
| | 932.039
| | Marvel and Benediction
|-
| | 81
| | 943.689
| | Hemisecordite
|-
| | 82
| | 955.34
| | Marvel and Benediction
|-
| | 83
| | 966.99
| | Marvel and Benediction, Hemisecordite
|-
| | 84
| | 978.641
| | Marvel and Benediction
|-
| | 85
| | 990.291
| | Hemisecordite
|-
| | 86
| | 1001.942
| | Marvel and Benediction, Hemisecordite
|-
| | 87
| | 1013.592
| | Marvel and Benediction
|-
| | 88
| | 1025.243
| | Hemisecordite
|-
| | 89
| | 1036.893
| | Marvel and Benediction
|-
| | 90
| | 1048.544
| | Marvel and Benediction, Hemisecordite
|-
| | 91
| | 1060.194
| | Hemisecordite
|-
| | 92
| | 1071.845
| | Marvel and Benediction
|-
| | 93
| | 1083.495
| | Marvel and Benediction, Hemisecordite
|-
| | 94
| | 1095.146
| | Marvel and Benediction
|-
| | 95
| | 1106.796
| | Hemisecordite
|-
| | 96
| | 1118.447
| | Marvel and Benediction, Hemisecordite
|-
| | 97
| | 1130.097
| | Marvel and Benediction
|-
| | 98
| | 1141.748
| | Hemisecordite
|-
| | 99
| | 1153.398
| | Marvel and Benediction
|-
| | 100
| | 1165.0485
| | Marvel and Benediction, Hemisecordite
|-
| | 101
| | 1176.609
| | Hemisecordite
|-
| | 102
| | 1188.3495
| | Marvel and Benediction
|}
[[Category:benediction]]
[[Category:edo]]
[[Category:miracle]]
[[Category:prime_edo]]
[[Category:theory]]</text>
      <sha1>7nkt61icc0wny4oiwsun4b5hqhmriv8</sha1>
    </revision>
    <revision>
      <id>4046</id>
      <timestamp>2016-12-29T14:47:57Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602898192 - Original comment: **</comment>
      <origin>4046</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28390" sha1="cecwd4hup56ed2bywqggsk126o3tkkk" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2016-12-29 14:47:57 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602898192&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limi|13-]] limits and [[Gamelismic clan|benediction]] and [[Gamelismic clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel family#Marvel|marvel temperament]] as well as [[Gamelismic clan#Miracle|benediction]] and [[Gamelismic clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime numbers|prime]] edo.
===13-limit temperaments=== 
|| Degree || cents || temperament ||
|| 1 || 11.650 || Marvel and Benediction ||
|| 2 || 23.301 || Hemisecordite ||
|| 3 || 34.951 || Marvel and Benediction, Hemisecordite ||
|| 4 || 46.602 || Marvel and Benediction ||
|| 5 || 58.252 || Hemisecordite ||
|| 6 || 69.903 || Marvel and Benediction ||
|| 7 || 81.553 || Marvel and Benediction, Hemisecordite ||
|| 8 || 93.204 || Hemisecordite ||
|| 9 || 104.854 || Marvel and Benediction ||
|| 10 || 116.505 || Marvel and Benediction, Hemisecordite ||
|| 11 || 128.155 || Marvel and Benediction ||
|| 12 || 139.806 || Hemisecordite ||
|| 13 || 151.456 || Marvel and Benediction, Hemisecordite ||
|| 14 || 163.107 || Marvel and Benediction ||
|| 15 || 174.757 || Hemisecordite ||
|| 16 || 186.408 || Marvel and Benediction ||
|| 17 || 198.058 || Marvel and Benediction, Hemisecordite ||
|| 18 || 209.708 || Hemisecordite ||
|| 19 || 221.359 || Marvel and Benediction ||
|| 20 || 233.010 || Marvel and Benediction, Hemisecordite ||
|| 21 || 244.660 || Marvel and Benediction ||
|| 22 || 256.311 || Hemisecordite ||
|| 23 || 267.961 || Marvel and Benediction, Hemisecordite ||
|| 24 || 279.712 || Marvel and Benediction ||
|| 25 || 291.262 || Hemisecordite ||
|| 26 || 303.013 || Marvel and Benediction ||
|| 27 || 314.563 || Marvel and Benediction, Hemisecordite ||
|| 28 || 326.214 || Hemisecordite ||
|| 29 || 337.864 || Marvel and Benediction ||
|| 30 || 349.615 || Marvel and Benediction, Hemisecordite ||
|| 31 || 361.165 || Marvel and Benediction ||
|| 32 || 372.816 || Hemisecordite ||
|| 33 || 384.466 || Marvel and Benediction, Hemisecordite ||
|| 34 || 396.117 || Marvel and Benediction ||
|| 35 || 407.767 || Hemisecordite ||
|| 36 || 419.4175 || Marvel and Benediction ||
|| 37 || 431.068 || Marvel and Benediction, Hemisecordite ||
|| 38 || 442.708 || Hemisecordite ||
|| 39 || 454.369 || Marvel and Benediction ||
|| 40 || 466.019 || Marvel and Benediction, Hemisecordite ||
|| 41 || 477.670 || Marvel and Benediction ||
|| 42 || 489.320 || Hemisecordite ||
|| 43 || 500.971 || Marvel and Benediction, Hemisecordite ||
|| 44 || 512.621 || Marvel and Benediction ||
|| 45 || 524.272 || Hemisecordite ||
|| 46 || 535.922 || Marvel and Benediction ||
|| 47 || 547.573 || Marevel and Benediction, Hemisecordite ||
|| 48 || 559.223 || Hemisecordite ||
|| 49 || 570.874 || Marvel and Benediction ||
|| 50 || 582.524 || Marvel and Benediction, Hemisecordite ||
|| 52 || 605.825 || Marvel and Benediction, Hemisecordite ||
|| 53 || 617.476 || Marvel and Benediction, Hemisecordite ||
|| 54 || 629.126 || Marvel and Benediction ||
|| 55 || 640.777 || Hemisecordite ||
|| 56 || 652.427 || Marvel and Benediction, Hemisecordite ||
|| 57 || 664.078 || Marvel and Benediction ||
|| 58 || 675.728 || Hemisecordite ||
|| 59 || 687.379 || Marvel and Benediction ||
|| 60 || 699.029 || Marvel and Benediction, Hemisecordite ||
|| 61 || 710.677 || Hemisecordite ||
|| 62 || 722.330 || Marvel and Benediction ||
|| 63 || 733.981 || Marvel and Benediction, Hemisecordite ||
|| 64 || 745.631 || Marvel and Benediction ||
|| 65 || 757.282 || Hemisecordite ||
|| 66 || 768.932 || Marvel and Benediction, Hemisecordite ||
|| 67 || 780.583 || Marvel and Benediction ||
|| 68 || 792.233 || Hemisecordite ||
|| 69 || 803.883 || Marvel and Benediction ||
|| 70 || 815.534 || Marvel and Benediction, Hemisecordite ||
|| 71 || 827.185 || Hemisecordite ||
|| 72 || 838.835 || Marvel and Benediction ||
|| 73 || 850.385 || Marvel and Benediction. Hemisecordite ||
|| 74 || 862.156 || Marvel and Benediction ||
|| 75 || 873.786 || Hemisecordite ||
|| 76 || 885.537 || Marvel and Benediction, Hemisecordite ||
|| 77 || 896.987 || Marvel and Benediction ||
|| 78 || 908.638 || Hemisecordite ||
|| 79 || 920.388 || Marvel and Benediction, Hemisecordite ||
|| 80 || 932.039 || Marvel and Benediction ||
|| 81 || 943.689 || Hemisecordite ||
|| 82 || 955.34 || Marvel and Benediction ||
|| 83 || 966.99 || Marvel and Benediction, Hemisecordite ||
|| 84 || 978.641 || Marvel and Benediction ||
|| 85 || 990.291 || Hemisecordite ||
|| 86 || 1001.942 || Marvel and Benediction, Hemisecordite ||
|| 87 || 1013.592 || Marvel and Benediction ||
|| 88 || 1025.243 || Hemisecordite ||
|| 89 || 1036.893 || Marvel and Benediction ||
|| 90 || 1048.544 || Marvel and Benediction, Hemisecordite ||
|| 91 || 1060.194 || Hemisecordite ||
|| 92 || 1071.845 || Marvel and Benediction ||
|| 93 || 1083.495 || Marvel and Benediction, Hemisecordite ||
|| 94 || 1095.146 || Marvel and Benediction ||
|| 95 || 1106.796 || Hemisecordite ||
|| 96 || 1118.447 || Marvel and Benediction, Hemisecordite ||
|| 97 || 1130.097 || Marvel and Benediction ||
|| 98 || 1141.748 || Hemisecordite ||
|| 99 || 1153.398 || Marvel and Benediction ||
|| 100 || 1165.0485 || Marvel and Benediction, Hemisecordite ||
|| 101 || 1176.609 || Hemisecordite ||
|| 102 || 1188.3495 || Marvel and Benediction ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;103edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;103edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 103 steps of size 11.6505 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is a good &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;miracle&amp;lt;/a&amp;gt; tuning, especially for the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/13-limi"&amp;gt;13-&amp;lt;/a&amp;gt; limits and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;benediction&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;hemisecordite&amp;lt;/a&amp;gt;, two of the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; extensions of miracle. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for &amp;lt;a class="wiki_link" href="/Marvel%20family#Marvel"&amp;gt;marvel temperament&amp;lt;/a&amp;gt; as well as &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;benediction&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Hemisecordite"&amp;gt;hemisecordite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
103edo is the 27th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;a name="x--13-limit temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;13-limit temperaments&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;temperament&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11.650&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23.301&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34.951&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46.602&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;58.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;69.903&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;81.553&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.854&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;128.155&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;139.806&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;151.456&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;163.107&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;174.757&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;186.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;198.058&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;209.708&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;221.359&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;233.010&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;244.660&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256.311&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;267.961&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;279.712&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;291.262&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;303.013&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;314.563&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;28&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;326.214&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;337.864&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;30&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;349.615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;361.165&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;32&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;372.816&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;384.466&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;396.117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;407.767&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;419.4175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.068&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;38&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;442.708&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;454.369&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;466.019&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;477.670&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;42&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;489.320&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;500.971&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;44&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;512.621&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;524.272&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;535.922&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.573&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marevel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;559.223&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;570.874&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;582.524&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;605.825&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;617.476&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;629.126&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;640.777&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;652.427&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;664.078&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;675.728&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;687.379&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;699.029&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;710.677&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;62&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;722.330&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;733.981&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;745.631&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;65&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;757.282&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;66&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;768.932&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;780.583&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;68&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;792.233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;803.883&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;815.534&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;827.185&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;838.835&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;850.385&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction. Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;74&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;862.156&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;873.786&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;885.537&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;896.987&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;78&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;908.638&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;920.388&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;932.039&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;943.689&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;955.34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;966.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;84&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;978.641&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;85&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;990.291&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1001.942&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;87&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1013.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1025.243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;89&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1036.893&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1048.544&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1060.194&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;92&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1071.845&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1083.495&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1095.146&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1106.796&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1118.447&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;97&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1130.097&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1141.748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1153.398&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1165.0485&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1176.609&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;102&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1188.3495&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>cecwd4hup56ed2bywqggsk126o3tkkk</sha1>
    </revision>
    <revision>
      <id>4047</id>
      <timestamp>2016-12-07T11:59:10Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 601628680 - Original comment: **</comment>
      <origin>4047</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28412" sha1="32c37373yzlcvold66sq8b6lo5zpwag" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-12-07 11:59:10 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;601628680&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limi|13-]] limits and [[Gamelismic clan|benediction]] and [[Gamelismic clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel family#Marvel|marvel temperament]] as well as [[Gamelismic clan#Miracle|benediction]] and [[Gamelismic clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime numbers|prime]] edo.
===13-limit temperaments=== 
|| Degree || cents || temperament ||
|| 1 || 11.6505 || Marvel and Benediction ||
|| 2 || 23.301 || Hemisecordite ||
|| 3 || 34.9515 || Marvel and Benediction, Hemisecordite ||
|| 4 || 46.602 || Marvel and Benediction ||
|| 5 || 58.252 || Hemisecordite ||
|| 6 || 69.903 || Marvel and Benediction ||
|| 7 || 81.553 || Marvel and Benediction, Hemisecordite ||
|| 8 || 93.204 || Hemisecordite ||
|| 9 || 104.854 || Marvel and Benediction ||
|| 10 || 116.505 || Marvel and Benediction, Hemisecordite ||
|| 11 || 128.155 || Marvel and Benediction ||
|| 12 || 139.806 || Hemisecordite ||
|| 13 || 151.456 || Marvel and Benediction, Hemisecordite ||
|| 14 || 163.107 || Marvel and Benediction ||
|| 15 || 174.757 || Hemisecordite ||
|| 16 || 186.408 || Marvel and Benediction ||
|| 17 || 198.058 || Marvel and Benediction, Hemisecordite ||
|| 18 || 209.708 || Hemisecordite ||
|| 19 || 221.359 || Marvel and Benediction ||
|| 20 || 233.01 || Marvel and Benediction, Hemisecordite ||
|| 21 || 244.66 || Marvel and Benediction ||
|| 22 || 256.311 || Hemisecordite ||
|| 23 || 267.961 || Marvel and Benediction, Hemisecordite ||
|| 24 || 279.712 || Marvel and Benediction ||
|| 25 || 291.262 || Hemisecordite ||
|| 26 || 303.013 || Marvel and Benediction ||
|| 27 || 314.563 || Marvel and Benediction, Hemisecordite ||
|| 28 || 326.214 || Hemisecordite ||
|| 29 || 337.864 || Marvel and Benediction ||
|| 30 || 349.615 || Marvel and Benediction, Hemisecordite ||
|| 31 || 361.165 || Marvel and Benediction ||
|| 32 || 372.816 || Hemisecordite ||
|| 33 || 384.466 || Marvel and Benediction, Hemisecordite ||
|| 34 || 396.1165/1 || Marvel and Benediction ||
|| 35 || 407.767 || Hemisecordite ||
|| 36 || 419.4175/1 || Marvel and Benediction ||
|| 37 || 431.068 || Marvel and Benediction, Hemisecordite ||
|| 38 || 442.708 || Hemisecordite ||
|| 39 || 454.369 || Marvel and Benediction ||
|| 40 || 466.019 || Marvel and Benediction, Hemisecordite ||
|| 41 || 477.67 || Marvel and Benediction ||
|| 42 || 489.32 || Hemisecordite ||
|| 43 || 500.971 || Marvel and Benediction, Hemisecordite ||
|| 44 || 512.621 || Marvel and Benediction ||
|| 45 || 524.272 || Hemisecordite ||
|| 46 || 535.922 || Marvel and Benediction ||
|| 47 || 547.573 || Marevel and Benediction, Hemisecordite ||
|| 48 || 559.223 || Hemisecordite ||
|| 49 || 570.874 || Marvel and Benediction ||
|| 50 || 582.524 || Marvel and Benediction, Hemisecordite ||
|| 52 || 605.825 || Marvel and Benediction, Hemisecordite ||
|| 53 || 617.476 || Marvel and Benediction, Hemisecordite ||
|| 54 || 629.126 || Marvel and Benediction ||
|| 55 || 640.777 || Hemisecordite ||
|| 56 || 652.427 || Marvel and Benediction, Hemisecordite ||
|| 57 || 664.078 || Marvel and Benediction ||
|| 58 || 675.728 || Hemisecordite ||
|| 59 || 687.379 || Marvel and Benediction ||
|| 60 || 699.029 || Marvel and Benediction, Hemisecordite ||
|| 61 || 710.677 || Hemisecordite ||
|| 62 || 722.33 || Marvel and Benediction ||
|| 63 || 733.981 || Marvel and Benediction, Hemisecordite ||
|| 64 || 745.631 || Marvel and Benediction ||
|| 65 || 757.282 || Hemisecordite ||
|| 66 || 768.932 || Marvel and Benediction, Hemisecordite ||
|| 67 || 780.5825/1 || Marvel and Benediction ||
|| 68 || 792.233 || Hemisecordite ||
|| 69 || 803.8865/1 || Marvel and Benediction ||
|| 70 || 815.534 || Marvel and Benediction, Hemisecordite ||
|| 71 || 827.185 || Hemisecordite ||
|| 72 || 838.835 || Marvel and Benediction ||
|| 73 || 850.385 || Marvel and Benediction. Hemisecordite ||
|| 74 || 862.156 || Marvel and Benediction ||
|| 75 || 873.786 || Hemisecordite ||
|| 76 || 885.537 || Marvel and Benediction, Hemisecordite ||
|| 77 || 896.987 || Marvel and Benediction ||
|| 78 || 908.638 || Hemisecordite ||
|| 79 || 920.388 || Marvel and Benediction, Hemisecordite ||
|| 80 || 932.039 || Marvel and Benediction ||
|| 81 || 943.689 || Hemisecordite ||
|| 82 || 955.34 || Marvel and Benediction ||
|| 83 || 966.99 || Marvel and Benediction, Hemisecordite ||
|| 84 || 978.641 || Marvel and Benediction ||
|| 85 || 990.291 || Hemisecordite ||
|| 86 || 1001.942 || Marvel and Benediction, Hemisecordite ||
|| 87 || 1013.592 || Marvel and Benediction ||
|| 88 || 1025.243 || Hemisecordite ||
|| 89 || 1036.893 || Marvel and Benediction ||
|| 90 || 1048.544 || Marvel and Benediction, Hemisecordite ||
|| 91 || 1060.194 || Hemisecordite ||
|| 92 || 1071.845 || Marvel and Benediction ||
|| 93 || 1083.495 || Marvel and Benediction, Hemisecordite ||
|| 94 || 1095.146 || Marvel and Benediction ||
|| 95 || 1106.796 || Hemisecordite ||
|| 96 || 1118.447 || Marvel and Benediction, Hemisecordite ||
|| 97 || 1130.097 || Marvel and Benediction ||
|| 98 || 1141.748 || Hemisecordite ||
|| 99 || 1153.398 || Marvel and Benediction ||
|| 100 || 1165.0485 || Marvel and Benediction, Hemisecordite ||
|| 101 || 1176.609 || Hemisecordite ||
|| 102 || 1188.3495 || Marvel and Benediction ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;103edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;103edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 103 steps of size 11.6505 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is a good &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;miracle&amp;lt;/a&amp;gt; tuning, especially for the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/13-limi"&amp;gt;13-&amp;lt;/a&amp;gt; limits and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;benediction&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;hemisecordite&amp;lt;/a&amp;gt;, two of the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; extensions of miracle. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for &amp;lt;a class="wiki_link" href="/Marvel%20family#Marvel"&amp;gt;marvel temperament&amp;lt;/a&amp;gt; as well as &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;benediction&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Hemisecordite"&amp;gt;hemisecordite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
103edo is the 27th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;a name="x--13-limit temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;13-limit temperaments&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;temperament&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11.6505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23.301&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34.9515&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46.602&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;58.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;69.903&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;81.553&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.854&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;128.155&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;139.806&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;151.456&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;163.107&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;174.757&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;186.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;198.058&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;209.708&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;221.359&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;233.01&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;244.66&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256.311&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;267.961&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;279.712&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;291.262&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;303.013&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;314.563&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;28&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;326.214&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;337.864&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;30&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;349.615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;361.165&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;32&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;372.816&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;384.466&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;396.1165/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;407.767&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;419.4175/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.068&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;38&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;442.708&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;454.369&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;466.019&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;477.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;42&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;489.32&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;500.971&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;44&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;512.621&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;524.272&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;535.922&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.573&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marevel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;559.223&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;570.874&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;582.524&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;605.825&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;617.476&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;629.126&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;640.777&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;652.427&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;664.078&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;675.728&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;687.379&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;699.029&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;710.677&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;62&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;722.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;733.981&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;745.631&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;65&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;757.282&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;66&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;768.932&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;780.5825/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;68&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;792.233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;803.8865/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;815.534&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;827.185&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;838.835&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;850.385&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction. Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;74&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;862.156&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;873.786&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;885.537&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;896.987&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;78&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;908.638&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;920.388&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;932.039&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;943.689&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;955.34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;966.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;84&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;978.641&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;85&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;990.291&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1001.942&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;87&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1013.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1025.243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;89&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1036.893&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1048.544&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1060.194&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;92&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1071.845&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1083.495&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1095.146&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1106.796&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1118.447&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;97&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1130.097&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1141.748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1153.398&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1165.0485&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1176.609&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;102&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1188.3495&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>32c37373yzlcvold66sq8b6lo5zpwag</sha1>
    </revision>
    <revision>
      <id>4048</id>
      <timestamp>2016-10-28T08:57:36Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 597378124 - Original comment: **</comment>
      <origin>4048</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28780" sha1="mzbea14sauj6ry5irmnkzbhiq0ywvm7" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-28 08:57:36 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;597378124&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limi|13-]] limits and [[Gamelismic clan|benediction]] and [[Gamelismic clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel family#Marvel|marvel temperament]] as well as [[Gamelismic clan#Miracle|benediction]] and [[Gamelismic clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime numbers|prime]] edo.
===13-limit temperaments=== 
|| Degree || cents || temperament ||
|| 1 || 11.6505 || Marvel and Benediction ||
|| 2 || 23.301 || Hemisecordite ||
|| 3 || 34.9515 || Marvel and Benediction, Hemisecordite ||
|| 4 || 46.602 || Marvel and Benediction ||
|| 5 || 58.252 || Hemisecordite ||
|| 6 || 69.903 || Marvel and Benediction ||
|| 7 || 81.553 || Marvel and Benediction, Hemisecordite ||
|| 8 || 93.204 || Hemisecordite ||
|| 9 || 104.854 || Marvel and Benediction ||
|| 10 || 116.505 || Marvel and Benediction, Hemisecordite ||
|| 11 || 128.155 || Marvel and Benediction ||
|| 12 || 139.806 || Hemisecordite ||
|| 13 || 151.456 || Marvel and Benediction, Hemisecordite ||
|| 14 || 163.107 || Marvel and Benediction ||
|| 15 || 174.757 || Hemisecordite ||
|| 16 || 186.408 || Marvel and Benediction ||
|| 17 || 198.058 || Marvel and Benediction, Hemisecordite ||
|| 18 || 209.708 || Hemisecordite ||
|| 19 || 221.359 || Marvel and Benediction ||
|| 20 || 233.01 || Marvel and Benediction, Hemisecordite ||
|| 21 || 244.66 || Marvel and Benediction ||
|| 22 || 256.311 || Hemisecordite ||
|| 23 || 267.961 || Marvel and Benediction, Hemisecordite ||
|| 24 || 279.712 || Marvel and Benediction ||
|| 25 || 291.262 || Hemisecordite ||
|| 26 || 303.013 || Marvel and Benediction ||
|| 27 || 314.563 || Marvel and Benediction, Hemisecordite ||
|| 28 || 326.214 || Hemisecordite ||
|| 29 || 337.864 || Marvel and Benediction ||
|| 30 || 349.615 || Marvel and Benediction, Hemisecordite ||
|| 31 || 361.165 || Marvel and Benediction ||
|| 32 || 372.816 || Hemisecordite ||
|| 33 || 384.466 || Marvel and Benediction, Hemisecordite ||
|| 34 || [[tel/396.1165|396.1165]] || Marvel and Benediction ||
|| 35 || 407.767 || Hemisecordite ||
|| 36 || [[tel/419.4175|419.4175]] || Marvel and Benediction ||
|| 37 || 431.068 || Marvel and Benediction, Hemisecordite ||
|| 38 || 442.708 || Hemisecordite ||
|| 39 || 454.369 || Marvel and Benediction ||
|| 40 || 466.019 || Marvel and Benediction, Hemisecordite ||
|| 41 || 477.67 || Marvel and Benediction ||
|| 42 || 489.32 || Hemisecordite ||
|| 43 || 500.971 || Marvel and Benediction, Hemisecordite ||
|| 44 || 512.621 || Marvel and Benediction ||
|| 45 || 524.272 || Hemisecordite ||
|| 46 || 535.922 || Marvel and Benediction ||
|| 47 || 547.573 || Marevel and Benediction, Hemisecordite ||
|| 48 || 559.223 || Hemisecordite ||
|| 49 || 570.874 || Marvel and Benediction ||
|| 50 || 582.524 || Marvel and Benediction, Hemisecordite ||
|| 52 || 605.825 || Marvel and Benediction, Hemisecordite ||
|| 53 || 617.476 || Marvel and Benediction, Hemisecordite ||
|| 54 || 629.126 || Marvel and Benediction ||
|| 55 || 640.777 || Hemisecordite ||
|| 56 || 652.427 || Marvel and Benediction, Hemisecordite ||
|| 57 || 664.078 || Marvel and Benediction ||
|| 58 || 675.728 || Hemisecordite ||
|| 59 || 687.379 || Marvel and Benediction ||
|| 60 || 699.029 || Marvel and Benediction, Hemisecordite ||
|| 61 || 710.677 || Hemisecordite ||
|| 62 || 722.33 || Marvel and Benediction ||
|| 63 || 733.981 || Marvel and Benediction, Hemisecordite ||
|| 64 || 745.631 || Marvel and Benediction ||
|| 65 || 757.282 || Hemisecordite ||
|| 66 || 768.932 || Marvel and Benediction, Hemisecordite ||
|| 67 || [[tel/780.5825|780.5825]] || Marvel and Benediction ||
|| 68 || 792.233 || Hemisecordite ||
|| 69 || [[tel/803.8865|803.8865]] || Marvel and Benediction ||
|| 70 || 815.534 || Marvel and Benediction, Hemisecordite ||
|| 71 || 827.185 || Hemisecordite ||
|| 72 || 838.835 || Marvel and Benediction ||
|| 73 || 850.385 || Marvel and Benediction. Hemisecordite ||
|| 74 || 862.156 || Marvel and Benediction ||
|| 75 || 873.786 || Hemisecordite ||
|| 76 || 885.537 || Marvel and Benediction, Hemisecordite ||
|| 77 || 896.987 || Marvel and Benediction ||
|| 78 || 908.638 || Hemisecordite ||
|| 79 || 920.388 || Marvel and Benediction, Hemisecordite ||
|| 80 || 932.039 || Marvel and Benediction ||
|| 81 || 943.689 || Hemisecordite ||
|| 82 || 955.34 || Marvel and Benediction ||
|| 83 || 966.99 || Marvel and Benediction, Hemisecordite ||
|| 84 || 978.641 || Marvel and Benediction ||
|| 85 || 990.291 || Hemisecordite ||
|| 86 || 1001.942 || Marvel and Benediction, Hemisecordite ||
|| 87 || 1013.592 || Marvel and Benediction ||
|| 88 || 1025.243 || Hemisecordite ||
|| 89 || 1036.893 || Marvel and Benediction ||
|| 90 || 1048.544 || Marvel and Benediction, Hemisecordite ||
|| 91 || 1060.194 || Hemisecordite ||
|| 92 || 1071.845 || Marvel and Benediction ||
|| 93 || 1083.495 || Marvel and Benediction, Hemisecordite ||
|| 94 || 1095.146 || Marvel and Benediction ||
|| 95 || 1106.796 || Hemisecordite ||
|| 96 || 1118.447 || Marvel and Benediction, Hemisecordite ||
|| 97 || 1130.097 || Marvel and Benediction ||
|| 98 || 1141.748 || Hemisecordite ||
|| 99 || 1153.398 || Marvel and Benediction ||
|| 100 || 1165.0485 || Marvel and Benediction, Hemisecordite ||
|| 101 || 1176.609 || Hemisecordite ||
|| 102 || 1188.3495 || Marvel and Benediction ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;103edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;103edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 103 steps of size 11.6505 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is a good &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;miracle&amp;lt;/a&amp;gt; tuning, especially for the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/13-limi"&amp;gt;13-&amp;lt;/a&amp;gt; limits and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;benediction&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;hemisecordite&amp;lt;/a&amp;gt;, two of the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; extensions of miracle. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for &amp;lt;a class="wiki_link" href="/Marvel%20family#Marvel"&amp;gt;marvel temperament&amp;lt;/a&amp;gt; as well as &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;benediction&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Hemisecordite"&amp;gt;hemisecordite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
103edo is the 27th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;a name="x--13-limit temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;13-limit temperaments&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;temperament&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11.6505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23.301&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34.9515&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46.602&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;58.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;69.903&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;81.553&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.854&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;128.155&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;139.806&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;151.456&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;163.107&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;174.757&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;186.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;198.058&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;209.708&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;221.359&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;233.01&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;244.66&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256.311&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;267.961&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;279.712&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;291.262&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;303.013&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;314.563&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;28&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;326.214&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;337.864&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;30&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;349.615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;361.165&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;32&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;372.816&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;384.466&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/396.1165"&amp;gt;396.1165&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;407.767&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/419.4175"&amp;gt;419.4175&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.068&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;38&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;442.708&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;454.369&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;466.019&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;477.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;42&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;489.32&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;500.971&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;44&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;512.621&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;524.272&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;535.922&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.573&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marevel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;559.223&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;570.874&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;582.524&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;605.825&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;617.476&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;629.126&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;640.777&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;652.427&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;664.078&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;675.728&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;687.379&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;699.029&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;710.677&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;722.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;733.981&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;757.282&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;66&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;768.932&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/780.5825"&amp;gt;780.5825&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;68&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;792.233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/803.8865"&amp;gt;803.8865&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;815.534&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;838.835&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;896.987&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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        &amp;lt;td&amp;gt;908.638&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;920.388&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;932.039&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;943.689&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;955.34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;83&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;966.99&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;84&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;978.641&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;990.291&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1001.942&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;87&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1013.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1025.243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;89&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1036.893&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1048.544&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1060.194&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;92&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1071.845&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1083.495&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
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    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1095.146&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1106.796&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1118.447&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;97&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1130.097&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1141.748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1153.398&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;1165.0485&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1176.609&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;102&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1188.3495&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>mzbea14sauj6ry5irmnkzbhiq0ywvm7</sha1>
    </revision>
    <revision>
      <id>4049</id>
      <timestamp>2016-08-27T12:19:33Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 590256274 - Original comment: **</comment>
      <origin>4049</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28790" sha1="21rwotio2sw1rh1speorvztr1dbtt8s" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-08-27 12:19:33 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;590256274&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limi|13-]] limits and [[Gamelismic clan|benediction]] and [[Gamelismic clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel family#Marvel|marvel temperament]] as well as [[Gamelismic clan#Miracle|benediction]] and [[Gamelismic clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime numbers|prime]] edo.
===13-limit temperaments=== 
|| Degree || cents || temperament ||
|| 1 || 11.6505 || Marvel and Benediction ||
|| 2 || 23.301 || Hemisecordite ||
|| 3 || 34.9515 || Marvel and Benediction, Hemisecordite ||
|| 4 || 46.602 || Marvel and Benediction ||
|| 5 || 58.252 || Hemisecordite ||
|| 6 || 69.903 || Marvel and Benediction ||
|| 7 || 81.553 || Marvel and Benediction, Hemisecordite ||
|| 8 || 93.204 || Hemisecordite ||
|| 9 || 104.854 || Marvel and Benediction ||
|| 10 || 116.505 || Marvel and Benediction, Hemisecordite ||
|| 11 || 128.155 || Marvel and Benediction ||
|| 12 || 139.806 || Hemisecordite ||
|| 13 || 151.456 || Marvel and Benediction, Hemisecordite ||
|| 14 || 163.107 || Marvel and Benediction ||
|| 15 || 174.757 || Hemisecordite ||
|| 16 || 186.408 || Marvel and Benediction ||
|| 17 || 198.058 || Marvel and Benediction, Hemisecordite ||
|| 18 || 209.708 || Hemisecordite ||
|| 19 || 221.359 || Marvel and Benediction ||
|| 20 || 233.01 || Marvel and Benediction, Hemisecordite ||
|| 21 || 244.66 || Marvel and Benediction ||
|| 22 || 256.311 || Hemisecordite ||
|| 23 || 267.961 || Marvel and Benediction, Hemisecordite ||
|| 24 || 279.712 || Marvel and Benediction ||
|| 25 || 291.262 || Hemisecordite ||
|| 26 || 303.013 || Marvel and Benediction ||
|| 27 || 314.563 || Marvel and Benediction, Hemisecordite ||
|| 28 || 326.214 || Hemisecordite ||
|| 29 || 337.864 || Marvel and Benediction ||
|| 30 || 349.615 || Marvel and Benediction, Hemisecordite ||
|| 31 || 361.165 || Marvel and Benediction ||
|| 32 || 372.816 || Hemisecordite ||
|| 33 || 384.466 || Marvel and Benediction, Hemisecordite ||
|| 34 || [[tel:396.1165|396.1165]] || Marvel and Benediction ||
|| 35 || 407.767 || Hemisecordite ||
|| 36 || [[tel:419.4175|419.4175]] || Marvel and Benediction ||
|| 37 || 431.068 || Marvel and Benediction, Hemisecordite ||
|| 38 || 442.708 || Hemisecordite ||
|| 39 || 454.369 || Marvel and Benediction ||
|| 40 || 466.019 || Marvel and Benediction, Hemisecordite ||
|| 41 || 477.67 || Marvel and Benediction ||
|| 42 || 489.32 || Hemisecordite ||
|| 43 || 500.971 || Marvel and Benediction, Hemisecordite ||
|| 44 || 512.621 || Marvel and Benediction ||
|| 45 || 524.272 || Hemisecordite ||
|| 46 || 535.922 || Marvel and Benediction ||
|| 47 || 547.573 || Marvel and Benediction,
Hemisecordite ||
|| 48 || 559.223 || Hemisecordite ||
|| 49 || 570.874 || Marvel and Benediction ||
|| 50 || 582.524 || Marvel and Benediction, Hemisecordite ||
|| 52 || 605.825 || Marvel and Benediction, Hemisecordite ||
|| 53 || 617.476 || Marvel and Benediction, Hemisecordite ||
|| 54 || 629.126 || Marvel and Benediction ||
|| 55 || 640.777 || Hemisecordite ||
|| 56 || 652.427 || Marvel and Benediction, Hemisecordite ||
|| 57 || 664.078 || Marvel and Benediction ||
|| 58 || 675.728 || Hemisecordite ||
|| 59 || 687.379 || Marvel and Benediction ||
|| 60 || 699.029 || Marvel and Benediction, Hemisecordite ||
|| 61 || 710.677 || Hemisecordite ||
|| 62 || 722.33 || Marvel and Benediction ||
|| 63 || 733.981 || Marvel and Benediction, Hemisecordite ||
|| 64 || 745.631 || Marvel and Benediction ||
|| 65 || 757.282 || Hemisecordite ||
|| 66 || 768.932 || Marvel and Benediction, Hemisecordite ||
|| 67 || [[tel:780.5825|780.5825]] || Marvel and Benediction ||
|| 68 || 792.233 || Hemisecordite ||
|| 69 || [[tel:803.8865|803.8865]] || Marvel and Benediction ||
|| 70 || 815.534 || Marvel and Benediction, Hemisecordite ||
|| 71 || 827.185 || Hemisecordite ||
|| 72 || 838.835 || Marvel and Benediction ||
|| 73 || 850.385 || Marvel and Benediction. Hemisecordite ||
|| 74 || 862.156 || Marvel and Benediction ||
|| 75 || 873.786 || Hemisecordite ||
|| 76 || 885.537 || Marvel and Benediction, Hemisecordite ||
|| 77 || 896.987 || Marvel and Benediction ||
|| 78 || 908.638 || Hemisecordite ||
|| 79 || 920.388 || Marvel and Benediction, Hemisecordite ||
|| 80 || 932.039 || Marvel and Benediction ||
|| 81 || 943.689 || Hemisecordite ||
|| 82 || 955.34 || Marvel and Benediction ||
|| 83 || 966.99 || Marvel and Benediction, Hemisecordite ||
|| 84 || 978.641 || Marvel and Benediction ||
|| 85 || 990.291 || Hemisecordite ||
|| 86 || 1001.942 || Marvel and Benediction, Hemisecordite ||
|| 87 || 1013.592 || Marvel and Benediction ||
|| 88 || 1025.243 || Hemisecordite ||
|| 89 || 1036.893 || Marvel and Benediction ||
|| 90 || 1048.544 || Marvel and Benediction, Hemisecordite ||
|| 91 || 1060.194 || Hemisecordite ||
|| 92 || 1071.845 || Marvel and Benediction ||
|| 93 || 1083.495 || Marvel and Benediction, Hemisecordite ||
|| 94 || 1095.146 || Marvel and Benediction ||
|| 95 || 1106.796 || Hemisecordite ||
|| 96 || 1118.447 || Marvel and Benediction, Hemisecordite ||
|| 97 || 1130.097 || Marvel and Benediction ||
|| 98 || 1141.748 || Hemisecordite ||
|| 99 || 1153.398 || Marvel and Benediction ||
|| 100 || 1165.0485 || Marvel and Benediction, Hemisecordite ||
|| 101 || 1176.609 || Hemisecordite ||
|| 102 || 1188.3495 || Marvel and Benediction ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;103edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;103edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 103 steps of size 11.6505 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is a good &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;miracle&amp;lt;/a&amp;gt; tuning, especially for the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/13-limi"&amp;gt;13-&amp;lt;/a&amp;gt; limits and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;benediction&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;hemisecordite&amp;lt;/a&amp;gt;, two of the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; extensions of miracle. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for &amp;lt;a class="wiki_link" href="/Marvel%20family#Marvel"&amp;gt;marvel temperament&amp;lt;/a&amp;gt; as well as &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;benediction&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Hemisecordite"&amp;gt;hemisecordite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
103edo is the 27th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;a name="x--13-limit temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;13-limit temperaments&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;temperament&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11.6505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23.301&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34.9515&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46.602&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;58.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;69.903&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;81.553&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.854&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;128.155&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;139.806&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;151.456&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;163.107&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;174.757&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;186.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;198.058&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;209.708&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;221.359&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;233.01&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;244.66&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256.311&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;267.961&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;279.712&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;291.262&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;303.013&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;314.563&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;28&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;326.214&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;337.864&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;30&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;349.615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;361.165&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;32&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;372.816&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;384.466&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/396.1165"&amp;gt;396.1165&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;407.767&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/419.4175"&amp;gt;419.4175&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.068&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;38&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;442.708&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;454.369&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;466.019&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;477.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;42&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;489.32&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;500.971&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;44&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;512.621&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;524.272&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;535.922&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.573&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction,&amp;lt;br /&amp;gt;
Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;559.223&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;570.874&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;582.524&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;605.825&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;617.476&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;629.126&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;640.777&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;652.427&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;664.078&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;675.728&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;687.379&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;699.029&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;710.677&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;62&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;722.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;733.981&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;745.631&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;65&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;757.282&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;66&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;768.932&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/780.5825"&amp;gt;780.5825&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;68&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;792.233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/803.8865"&amp;gt;803.8865&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;815.534&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;827.185&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;838.835&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;850.385&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction. Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;74&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;862.156&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;873.786&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;885.537&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;896.987&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;78&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;908.638&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;920.388&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;932.039&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;943.689&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;955.34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;966.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;84&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;978.641&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;85&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;990.291&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1001.942&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;87&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1013.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1025.243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;89&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1036.893&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1048.544&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1060.194&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;92&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1071.845&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1083.495&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1095.146&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1106.796&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1118.447&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;97&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1130.097&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1141.748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1153.398&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1165.0485&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1176.609&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;102&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1188.3495&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>21rwotio2sw1rh1speorvztr1dbtt8s</sha1>
    </revision>
    <revision>
      <id>4050</id>
      <timestamp>2016-08-01T18:33:23Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 588563488 - Original comment: **</comment>
      <origin>4050</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28420" sha1="h0k0l07kuorbd9fd3a67i3orn7drinr" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-08-01 18:33:23 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;588563488&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limi|13-]] limits and [[Gamelismic clan|benediction]] and [[Gamelismic clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal [[#|patent]] val for [[Marvel family#Marvel|marvel temperament]] as well as [[Gamelismic clan#Miracle|benediction]] and [[Gamelismic clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime numbers|prime]] edo.
===13-limit temperaments=== 
|| Degree || cents || temperament ||
|| 1 || 11.6505 || Marvel and Benediction ||
|| 2 || 23.301 || Hemisecordite ||
|| 3 || 34.9515 || Marvel and Benediction, Hemisecordite ||
|| 4 || 46.602 || Marvel and Benediction ||
|| 5 || 58.252 || Hemisecordite ||
|| 6 || 69.903 || Marvel and Benediction ||
|| 7 || 81.553 || Marvel and Benediction, Hemisecordite ||
|| 8 || 93.204 || Hemisecordite ||
|| 9 || 104.854 || Marvel and Benediction ||
|| 10 || 116.505 || Marvel and Benediction, Hemisecordite ||
|| 11 || 128.155 || Marvel and Benediction ||
|| 12 || 139.806 || Hemisecordite ||
|| 13 || 151.456 || Marvel and Benediction, Hemisecordite ||
|| 14 || 163.107 || Marvel and Benediction ||
|| 15 || 174.757 || Hemisecordite ||
|| 16 || 186.408 || Marvel and Benediction ||
|| 17 || 198.058 || Marvel and Benediction, Hemisecordite ||
|| 18 || 209.708 || Hemisecordite ||
|| 19 || 221.359 || Marvel and Benediction ||
|| 20 || 233.01 || Marvel and Benediction, Hemisecordite ||
|| 21 || 244.66 || Marvel and Benediction ||
|| 22 || 256.311 || Hemisecordite ||
|| 23 || 267.961 || Marvel and Benediction, Hemisecordite ||
|| 24 || 279.712 || Marvel and Benediction ||
|| 25 || 291.262 || Hemisecordite ||
|| 26 || 303.013 || Marvel and Benediction ||
|| 27 || 314.563 || Marvel and Benediction, Hemisecordite ||
|| 28 || 326.214 || Hemisecordite ||
|| 29 || 337.864 || Marvel and Benediction ||
|| 30 || 349.615 || Marvel and Benediction, Hemisecordite ||
|| 31 || 361.165 || Marvel and Benediction ||
|| 32 || 372.816 || Hemisecordite ||
|| 33 || 384.466 || Marvel and Benediction, Hemisecordite ||
|| 34 || 396.1165 || Marvel and Benediction ||
|| 35 || 407.767 || Hemisecordite ||
|| 36 || 419.4175 || Marvel and Benediction ||
|| 37 || 431.068 || Marvel and Benediction, Hemisecordite ||
|| 38 || 442.708 || Hemisecordite ||
|| 39 || 454.369 || Marvel and Benediction ||
|| 40 || 466.019 || Marvel and Benediction, Hemisecordite ||
|| 41 || 477.67 || Marvel and Benediction ||
|| 42 || 489.32 || Hemisecordite ||
|| 43 || 500.971 || Marvel and Benediction, Hemisecordite ||
|| 44 || 512.621 || Marvel and Benediction ||
|| 45 || 524.272 || Hemisecordite ||
|| 46 || 535.922 || Marvel and Benediction ||
|| 47 || 547.573 || Marvel and Benediction, 
Hemisecordite ||
|| 48 || 559.223 || Hemisecordite ||
|| 49 || 570.874 || Marvel and Benediction ||
|| 50 || 582.524 || Marvel and Benediction, Hemisecordite ||
|| 52 || 605.825 || Marvel and Benediction, Hemisecordite ||
|| 53 || 617.476 || Marvel and Benediction, Hemisecordite ||
|| 54 || 629.126 || Marvel and Benediction ||
|| 55 || 640.777 || Hemisecordite ||
|| 56 || 652.427 || Marvel and Benediction, Hemisecordite ||
|| 57 || 664.078 || Marvel and Benediction ||
|| 58 || 675.728 || Hemisecordite ||
|| 59 || 687.379 || Marvel and Benediction ||
|| 60 || 699.029 || Marvel and Benediction, Hemisecordite ||
|| 61 || 710.677 || Hemisecordite ||
|| 62 || 722.33 || Marvel and Benediction ||
|| 63 || 733.981 || Marvel and Benediction, Hemisecordite ||
|| 64 || 745.631 || Marvel and Benediction ||
|| 65 || 757.282 || Hemisecordite ||
|| 66 || 768.932 || Marvel and Benediction, Hemisecordite ||
|| 67 || 780.5825 || Marvel and Benediction ||
|| 68 || 792.233 || Hemisecordite ||
|| 69 || 803.8865 || Marvel and Benediction ||
|| 70 || 815.534 || Marvel and Benediction, Hemisecordite ||
|| 71 || 827.185 || Hemisecordite ||
|| 72 || 838.835 || Marvel and Benediction ||
|| 73 || 850.385 || Marvel and Benediction. Hemisecordite ||
|| 74 || 862.156 || Marvel and Benediction ||
|| 75 || 873.786 || Hemisecordite ||
|| 76 || 885.537 || Marvel and Benediction, Hemisecordite ||
|| 77 || 896.987 || Marvel and Benediction ||
|| 78 || 908.638 || Hemisecordite ||
|| 79 || 920.388 || Marvel and Benediction, Hemisecordite ||
|| 80 || 932.039 || Marvel and Benediction ||
|| 81 || 943.689 || Hemisecordite ||
|| 82 || 955.34 || Marvel and Benediction ||
|| 83 || 966.99 || Marvel and Benediction, Hemisecordite ||
|| 84 || 978.641 || Marvel and Benediction ||
|| 85 || 990.291 || Hemisecordite ||
|| 86 || 1001.942 || Marvel and Benediction, Hemisecordite ||
|| 87 || 1013.592 || Marvel and Benediction ||
|| 88 || 1025.243 || Hemisecordite ||
|| 89 || 1036.893 || Marvel and Benediction ||
|| 90 || 1048.544 || Marvel and Benediction, Hemisecordite ||
|| 91 || 1060.194 || Hemisecordite ||
|| 92 || 1071.845 || Marvel and Benediction ||
|| 93 || 1083.495 || Marvel and Benediction, Hemisecordite ||
|| 94 || 1095.146 || Marvel and Benediction ||
|| 95 || 1106.796 || Hemisecordite ||
|| 96 || 1118.447 || Marvel and Benediction, Hemisecordite ||
|| 97 || 1130.097 || Marvel and Benediction ||
|| 98 || 1141.748 || Hemisecordite ||
|| 99 || 1153.398 || Marvel and Benediction ||
|| 100 || 1165.0485 || Marvel and Benediction, Hemisecordite ||
|| 101 || 1176.609 || Hemisecordite ||
|| 102 || 1188.3495 || Marvel and Benediction ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;103edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;103edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 103 steps of size 11.6505 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is a good &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;miracle&amp;lt;/a&amp;gt; tuning, especially for the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/13-limi"&amp;gt;13-&amp;lt;/a&amp;gt; limits and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;benediction&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;hemisecordite&amp;lt;/a&amp;gt;, two of the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; extensions of miracle. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal [[#|patent]] val for &amp;lt;a class="wiki_link" href="/Marvel%20family#Marvel"&amp;gt;marvel temperament&amp;lt;/a&amp;gt; as well as &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;benediction&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Hemisecordite"&amp;gt;hemisecordite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
103edo is the 27th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;a name="x--13-limit temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;13-limit temperaments&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;temperament&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11.6505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23.301&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34.9515&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46.602&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;58.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;69.903&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;81.553&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.854&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;128.155&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;139.806&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;151.456&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;163.107&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;174.757&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;186.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;198.058&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;209.708&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;221.359&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;233.01&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;244.66&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256.311&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;267.961&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;279.712&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;291.262&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;303.013&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;314.563&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;28&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;326.214&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;337.864&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;30&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;349.615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;361.165&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;32&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;372.816&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;384.466&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;396.1165&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;407.767&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;419.4175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.068&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;38&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;442.708&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;454.369&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;466.019&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;477.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;42&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;489.32&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;500.971&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;44&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;512.621&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;524.272&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;535.922&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.573&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, &amp;lt;br /&amp;gt;
Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;559.223&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;570.874&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;582.524&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;605.825&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;617.476&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;629.126&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;640.777&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;652.427&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;664.078&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;675.728&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;687.379&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;699.029&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;710.677&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;62&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;722.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;733.981&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;745.631&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;65&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;757.282&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;66&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;768.932&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction, Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;780.5825&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;68&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;792.233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemisecordite&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;803.8865&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel and Benediction&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;70&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4051</id>
      <timestamp>2015-01-03T13:34:01Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 536135726 - Original comment: **</comment>
      <origin>4051</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3541" sha1="a99d87fsha961r4mu7fdy87iqy503u6" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-01-03 13:34:01 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;536135726&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limi|13-]] limits and [[Gamelismic clan|benediction]] and [[Gamelismic clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel family#Marvel|marvel temperament]] as well as [[Gamelismic clan#Miracle|benediction]] and [[Gamelismic clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime numbers|prime]] edo.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;103edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;103edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 103 steps of size 11.6505 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is a good &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;miracle&amp;lt;/a&amp;gt; tuning, especially for the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/13-limi"&amp;gt;13-&amp;lt;/a&amp;gt; limits and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;benediction&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;hemisecordite&amp;lt;/a&amp;gt;, two of the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; extensions of miracle. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for &amp;lt;a class="wiki_link" href="/Marvel%20family#Marvel"&amp;gt;marvel temperament&amp;lt;/a&amp;gt; as well as &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;benediction&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Hemisecordite"&amp;gt;hemisecordite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
103edo is the 27th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4052</id>
      <timestamp>2015-01-03T13:33:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
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      <comment>**Imported revision 536135690 - Original comment: **</comment>
      <origin>4052</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3533" sha1="rbn5p0k9akys03ihbb69ymqchf1yg98" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-01-03 13:33:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;536135690&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit|7-]] [[13-limi|13-]] limits and [[Gamelismic clan|benediction]] and [[Gamelismic clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel family#Marvel|marvel temperament]] as well as [[Gamelismic clan#Miracle|benediction]] and [[Gamelismic clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime numbers|prime]] edo.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
103edo is the 27th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 536135620 - Original comment: **</comment>
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      <text bytes="3448" sha1="7tv3g0j736a30950ujqezfqpe4wo9iu" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-01-03 13:31:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;536135620&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit]] and [[Gamelismic clan|benediction]] and [[Gamelismic clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel family#Marvel|marvel temperament]] as well as [[Gamelismic clan#Miracle|benediction]] and [[Gamelismic clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime numbers|prime]] edo.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
103edo is the 27th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 536135442 - Original comment: **</comment>
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      <text bytes="3277" sha1="r7g0w5d2g1b46g4b95yhwt5x2wi0hat" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-01-03 13:27:34 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;536135442&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit]] and [[Gamelismic clan|benediction]] and [[Gamelismic clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel family#Marvel|marvel temperament]] as well as [[Gamelismic clan#Miracle|benediction]].

103edo is the 27th [[prime numbers|prime]] edo.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
103edo is the 27th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 288887265 - Original comment: **</comment>
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      <text bytes="3166" sha1="jfn1trbbkmpr7ktm21gwk7rs3x5cdof" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-12-31 02:10:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288887265&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit]] and [[Gamelismic clan|benediction]], one of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel family#Marvel|marvel temperament]] as well as [[Gamelismic clan#Miracle|benediction]].

103edo is the 27th [[prime numbers|prime]] edo.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
103edo is the 27th &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 266243836 - Original comment: **</comment>
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      <text bytes="2993" sha1="mz2x2a0l2a9pa2thntiwv24qzbcccyq" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-10-18 23:33:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;266243836&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit]] and [[Gamelismic clan|benediction]], one of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel family#Marvel|marvel temperament]] as well as [[Gamelismic clan#Miracle|benediction]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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      <comment>**Imported revision 239611327 - Original comment: **</comment>
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      <text bytes="2605" sha1="5gfoq3e7oco78wmpz2nverz4reeo37a" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2011-07-01 00:44:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239611327&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit]] and [[Gamelismic clan|benediction]], one of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 in the 13-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;103edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;103edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 103 steps of size 11.6505 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is a good &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;miracle&amp;lt;/a&amp;gt; tuning, especially for the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;benediction&amp;lt;/a&amp;gt;, one of the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; extensions of miracle. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; 351/350 in the 13-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4058</id>
      <timestamp>2011-06-29T08:28:31Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 239303495 - Original comment: **</comment>
      <origin>4058</origin>
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      <text bytes="2591" sha1="7f02u1wpno0e03gc8fjhecjfcz9o4y7" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 08:28:31 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239303495&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of size 11.650 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit]] and [[Gamelismic clan|benediction]], one of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 in the 13-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;103edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;103edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 103 steps of size 11.650 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is a good &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;miracle&amp;lt;/a&amp;gt; tuning, especially for the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;benediction&amp;lt;/a&amp;gt;, one of the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; extensions of miracle. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; 351/350 in the 13-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4059</id>
      <timestamp>2011-06-29T08:25:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
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      <comment>**Imported revision 239303251 - Original comment: **</comment>
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      <text bytes="2581" sha1="24gsmwaib994hhyv4v0xqyq9i7a4bah" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 08:25:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239303251&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo** is the [[equal division of the octave]] into 103 steps of 11.650 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit]] and [[Gamelismic clan|benediction]], one of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 in the 13-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;103edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;103edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 103 steps of 11.650 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is a good &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;miracle&amp;lt;/a&amp;gt; tuning, especially for the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;benediction&amp;lt;/a&amp;gt;, one of the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; extensions of miracle. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; 351/350 in the 13-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2011-06-29T08:13:14Z</timestamp>
      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 239302093 - Original comment: **</comment>
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      <text bytes="2602" sha1="bcnkzkbfro0dcjqpqcn5fzxu8smxcto" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 08:13:14 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239302093&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**103edo**, the //103 equal division// divides the [[octave]] into 103 equal parts of 11.650 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit]] and [[Gamelismic clan|benediction]], one of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 in the 13-limit.&lt;/pre&gt;&lt;/div&gt;
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&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;103edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;103edo&amp;lt;/strong&amp;gt;, the &amp;lt;em&amp;gt;103 equal division&amp;lt;/em&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 103 equal parts of 11.650 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is a good &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;miracle&amp;lt;/a&amp;gt; tuning, especially for the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;benediction&amp;lt;/a&amp;gt;, one of the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; extensions of miracle. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; 351/350 in the 13-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2011-03-28T03:12:09Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
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      <comment>**Imported revision 214557916 - Original comment: **</comment>
      <origin>4061</origin>
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      <text bytes="2474" sha1="m1wh6j5la457mz2vpf7gi2gzoimiz4l" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-03-28 03:12:09 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214557916&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //103 equal division// divides the [[octave]] into 103 equal parts of 11.650 [[cent]]s each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the [[7-limit]] and [[Gamelismic clan|benediction]], one of the [[13-limit]] extensions of miracle. It tempers out 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 in the 13-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;103edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;103 equal division&amp;lt;/em&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 103 equal parts of 11.650 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is a good &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;miracle&amp;lt;/a&amp;gt; tuning, especially for the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;benediction&amp;lt;/a&amp;gt;, one of the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; extensions of miracle. It tempers out 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; 351/350 in the 13-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>m1wh6j5la457mz2vpf7gi2gzoimiz4l</sha1>
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    <revision>
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      <timestamp>2011-03-25T14:55:42Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 214054878 - Original comment: **</comment>
      <origin>4062</origin>
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      <text bytes="2146" sha1="8199jjop2mpwwt3g5ig562hiwk9t81b" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-25 14:55:42 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214054878&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //103 equal division// divides the octave into 103 equal parts of 11.650 cents each. It is a good [[Gamelismic clan|miracle]] tuning, especially for the 7-limit and [[Gamelismic clan|benediction]], one of the 13-limit extensions of miracle. It tempers out 78732/78125 in the 5-limit; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the 11-limit; 351/350 in the 13-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;103edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;103 equal division&amp;lt;/em&amp;gt; divides the octave into 103 equal parts of 11.650 cents each. It is a good &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;miracle&amp;lt;/a&amp;gt; tuning, especially for the 7-limit and &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;benediction&amp;lt;/a&amp;gt;, one of the 13-limit extensions of miracle. It tempers out 78732/78125 in the 5-limit; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the 11-limit; 351/350 in the 13-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>38290</id>
      <parentid>20</parentid>
      <timestamp>2018-11-02T09:50:00Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
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      <minor/>
      <comment>cat rename</comment>
      <origin>38290</origin>
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      <text bytes="6194" sha1="8hr6hx16hvnn5sc11igkmvopq4f9fck" xml:space="preserve">'''103edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 103 steps of size 11.6505 [[cent|cent]]s each. It is a good [[Gamelismic_clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limi|13-]] limits and [[Gamelismic_clan|benediction]] and [[Gamelismic_clan|hemisecordite]], two of the [[13-limit|13-limit]] extensions of miracle. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit|5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit|11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel_family#Marvel|marvel temperament]] as well as [[Gamelismic_clan#Miracle|benediction]] and [[Gamelismic_clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime_numbers|prime]] edo.

===13-limit temperaments===

{| class="wikitable"
|-
| | Degree
| | cents
| | temperament
|-
| | 1
| | 11.650
| | Marvel and Benediction
|-
| | 2
| | 23.301
| | Hemisecordite
|-
| | 3
| | 34.951
| | Marvel and Benediction, Hemisecordite
|-
| | 4
| | 46.602
| | Marvel and Benediction
|-
| | 5
| | 58.252
| | Hemisecordite
|-
| | 6
| | 69.903
| | Marvel and Benediction
|-
| | 7
| | 81.553
| | Marvel and Benediction, Hemisecordite
|-
| | 8
| | 93.204
| | Hemisecordite
|-
| | 9
| | 104.854
| | Marvel and Benediction
|-
| | 10
| | 116.505
| | Marvel and Benediction, Hemisecordite
|-
| | 11
| | 128.155
| | Marvel and Benediction
|-
| | 12
| | 139.806
| | Hemisecordite
|-
| | 13
| | 151.456
| | Marvel and Benediction, Hemisecordite
|-
| | 14
| | 163.107
| | Marvel and Benediction
|-
| | 15
| | 174.757
| | Hemisecordite
|-
| | 16
| | 186.408
| | Marvel and Benediction
|-
| | 17
| | 198.058
| | Marvel and Benediction, Hemisecordite
|-
| | 18
| | 209.708
| | Hemisecordite
|-
| | 19
| | 221.359
| | Marvel and Benediction
|-
| | 20
| | 233.010
| | Marvel and Benediction, Hemisecordite
|-
| | 21
| | 244.660
| | Marvel and Benediction
|-
| | 22
| | 256.311
| | Hemisecordite
|-
| | 23
| | 267.961
| | Marvel and Benediction, Hemisecordite
|-
| | 24
| | 279.712
| | Marvel and Benediction
|-
| | 25
| | 291.262
| | Hemisecordite
|-
| | 26
| | 303.013
| | Marvel and Benediction
|-
| | 27
| | 314.563
| | Marvel and Benediction, Hemisecordite
|-
| | 28
| | 326.214
| | Hemisecordite
|-
| | 29
| | 337.864
| | Marvel and Benediction
|-
| | 30
| | 349.615
| | Marvel and Benediction, Hemisecordite
|-
| | 31
| | 361.165
| | Marvel and Benediction
|-
| | 32
| | 372.816
| | Hemisecordite
|-
| | 33
| | 384.466
| | Marvel and Benediction, Hemisecordite
|-
| | 34
| | 396.117
| | Marvel and Benediction
|-
| | 35
| | 407.767
| | Hemisecordite
|-
| | 36
| | 419.4175
| | Marvel and Benediction
|-
| | 37
| | 431.068
| | Marvel and Benediction, Hemisecordite
|-
| | 38
| | 442.708
| | Hemisecordite
|-
| | 39
| | 454.369
| | Marvel and Benediction
|-
| | 40
| | 466.019
| | Marvel and Benediction, Hemisecordite
|-
| | 41
| | 477.670
| | Marvel and Benediction
|-
| | 42
| | 489.320
| | Hemisecordite
|-
| | 43
| | 500.971
| | Marvel and Benediction, Hemisecordite
|-
| | 44
| | 512.621
| | Marvel and Benediction
|-
| | 45
| | 524.272
| | Hemisecordite
|-
| | 46
| | 535.922
| | Marvel and Benediction
|-
| | 47
| | 547.573
| | Marevel and Benediction, Hemisecordite
|-
| | 48
| | 559.223
| | Hemisecordite
|-
| | 49
| | 570.874
| | Marvel and Benediction
|-
| | 50
| | 582.524
| | Marvel and Benediction, Hemisecordite
|-
| | 52
| | 605.825
| | Marvel and Benediction, Hemisecordite
|-
| | 53
| | 617.476
| | Marvel and Benediction, Hemisecordite
|-
| | 54
| | 629.126
| | Marvel and Benediction
|-
| | 55
| | 640.777
| | Hemisecordite
|-
| | 56
| | 652.427
| | Marvel and Benediction, Hemisecordite
|-
| | 57
| | 664.078
| | Marvel and Benediction
|-
| | 58
| | 675.728
| | Hemisecordite
|-
| | 59
| | 687.379
| | Marvel and Benediction
|-
| | 60
| | 699.029
| | Marvel and Benediction, Hemisecordite
|-
| | 61
| | 710.677
| | Hemisecordite
|-
| | 62
| | 722.330
| | Marvel and Benediction
|-
| | 63
| | 733.981
| | Marvel and Benediction, Hemisecordite
|-
| | 64
| | 745.631
| | Marvel and Benediction
|-
| | 65
| | 757.282
| | Hemisecordite
|-
| | 66
| | 768.932
| | Marvel and Benediction, Hemisecordite
|-
| | 67
| | 780.583
| | Marvel and Benediction
|-
| | 68
| | 792.233
| | Hemisecordite
|-
| | 69
| | 803.883
| | Marvel and Benediction
|-
| | 70
| | 815.534
| | Marvel and Benediction, Hemisecordite
|-
| | 71
| | 827.185
| | Hemisecordite
|-
| | 72
| | 838.835
| | Marvel and Benediction
|-
| | 73
| | 850.385
| | Marvel and Benediction. Hemisecordite
|-
| | 74
| | 862.156
| | Marvel and Benediction
|-
| | 75
| | 873.786
| | Hemisecordite
|-
| | 76
| | 885.537
| | Marvel and Benediction, Hemisecordite
|-
| | 77
| | 896.987
| | Marvel and Benediction
|-
| | 78
| | 908.638
| | Hemisecordite
|-
| | 79
| | 920.388
| | Marvel and Benediction, Hemisecordite
|-
| | 80
| | 932.039
| | Marvel and Benediction
|-
| | 81
| | 943.689
| | Hemisecordite
|-
| | 82
| | 955.34
| | Marvel and Benediction
|-
| | 83
| | 966.99
| | Marvel and Benediction, Hemisecordite
|-
| | 84
| | 978.641
| | Marvel and Benediction
|-
| | 85
| | 990.291
| | Hemisecordite
|-
| | 86
| | 1001.942
| | Marvel and Benediction, Hemisecordite
|-
| | 87
| | 1013.592
| | Marvel and Benediction
|-
| | 88
| | 1025.243
| | Hemisecordite
|-
| | 89
| | 1036.893
| | Marvel and Benediction
|-
| | 90
| | 1048.544
| | Marvel and Benediction, Hemisecordite
|-
| | 91
| | 1060.194
| | Hemisecordite
|-
| | 92
| | 1071.845
| | Marvel and Benediction
|-
| | 93
| | 1083.495
| | Marvel and Benediction, Hemisecordite
|-
| | 94
| | 1095.146
| | Marvel and Benediction
|-
| | 95
| | 1106.796
| | Hemisecordite
|-
| | 96
| | 1118.447
| | Marvel and Benediction, Hemisecordite
|-
| | 97
| | 1130.097
| | Marvel and Benediction
|-
| | 98
| | 1141.748
| | Hemisecordite
|-
| | 99
| | 1153.398
| | Marvel and Benediction
|-
| | 100
| | 1165.0485
| | Marvel and Benediction, Hemisecordite
|-
| | 101
| | 1176.609
| | Hemisecordite
|-
| | 102
| | 1188.3495
| | Marvel and Benediction
|}

[[Category:Benediction]]
[[Category:Edo]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>8hr6hx16hvnn5sc11igkmvopq4f9fck</sha1>
    </revision>
    <revision>
      <id>39147</id>
      <parentid>38290</parentid>
      <timestamp>2018-12-25T07:24:34Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>39147</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6139" sha1="seo6l52lzxrah0g7dzaincuv7zbpqp4" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic_clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]] limits and [[Gamelismic_clan|benediction]] and [[Gamelismic_clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel_family#Marvel|marvel temperament]] as well as [[Gamelismic_clan#Miracle|benediction]] and [[Gamelismic_clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime_numbers|prime]] edo.

===13-limit temperaments===

{| class="wikitable"
|-
| | Degree
| | cents
| | temperament
|-
| | 1
| | 11.650
| | Marvel and Benediction
|-
| | 2
| | 23.301
| | Hemisecordite
|-
| | 3
| | 34.951
| | Marvel and Benediction, Hemisecordite
|-
| | 4
| | 46.602
| | Marvel and Benediction
|-
| | 5
| | 58.252
| | Hemisecordite
|-
| | 6
| | 69.903
| | Marvel and Benediction
|-
| | 7
| | 81.553
| | Marvel and Benediction, Hemisecordite
|-
| | 8
| | 93.204
| | Hemisecordite
|-
| | 9
| | 104.854
| | Marvel and Benediction
|-
| | 10
| | 116.505
| | Marvel and Benediction, Hemisecordite
|-
| | 11
| | 128.155
| | Marvel and Benediction
|-
| | 12
| | 139.806
| | Hemisecordite
|-
| | 13
| | 151.456
| | Marvel and Benediction, Hemisecordite
|-
| | 14
| | 163.107
| | Marvel and Benediction
|-
| | 15
| | 174.757
| | Hemisecordite
|-
| | 16
| | 186.408
| | Marvel and Benediction
|-
| | 17
| | 198.058
| | Marvel and Benediction, Hemisecordite
|-
| | 18
| | 209.708
| | Hemisecordite
|-
| | 19
| | 221.359
| | Marvel and Benediction
|-
| | 20
| | 233.010
| | Marvel and Benediction, Hemisecordite
|-
| | 21
| | 244.660
| | Marvel and Benediction
|-
| | 22
| | 256.311
| | Hemisecordite
|-
| | 23
| | 267.961
| | Marvel and Benediction, Hemisecordite
|-
| | 24
| | 279.712
| | Marvel and Benediction
|-
| | 25
| | 291.262
| | Hemisecordite
|-
| | 26
| | 303.013
| | Marvel and Benediction
|-
| | 27
| | 314.563
| | Marvel and Benediction, Hemisecordite
|-
| | 28
| | 326.214
| | Hemisecordite
|-
| | 29
| | 337.864
| | Marvel and Benediction
|-
| | 30
| | 349.615
| | Marvel and Benediction, Hemisecordite
|-
| | 31
| | 361.165
| | Marvel and Benediction
|-
| | 32
| | 372.816
| | Hemisecordite
|-
| | 33
| | 384.466
| | Marvel and Benediction, Hemisecordite
|-
| | 34
| | 396.117
| | Marvel and Benediction
|-
| | 35
| | 407.767
| | Hemisecordite
|-
| | 36
| | 419.4175
| | Marvel and Benediction
|-
| | 37
| | 431.068
| | Marvel and Benediction, Hemisecordite
|-
| | 38
| | 442.708
| | Hemisecordite
|-
| | 39
| | 454.369
| | Marvel and Benediction
|-
| | 40
| | 466.019
| | Marvel and Benediction, Hemisecordite
|-
| | 41
| | 477.670
| | Marvel and Benediction
|-
| | 42
| | 489.320
| | Hemisecordite
|-
| | 43
| | 500.971
| | Marvel and Benediction, Hemisecordite
|-
| | 44
| | 512.621
| | Marvel and Benediction
|-
| | 45
| | 524.272
| | Hemisecordite
|-
| | 46
| | 535.922
| | Marvel and Benediction
|-
| | 47
| | 547.573
| | Marevel and Benediction, Hemisecordite
|-
| | 48
| | 559.223
| | Hemisecordite
|-
| | 49
| | 570.874
| | Marvel and Benediction
|-
| | 50
| | 582.524
| | Marvel and Benediction, Hemisecordite
|-
| | 52
| | 605.825
| | Marvel and Benediction, Hemisecordite
|-
| | 53
| | 617.476
| | Marvel and Benediction, Hemisecordite
|-
| | 54
| | 629.126
| | Marvel and Benediction
|-
| | 55
| | 640.777
| | Hemisecordite
|-
| | 56
| | 652.427
| | Marvel and Benediction, Hemisecordite
|-
| | 57
| | 664.078
| | Marvel and Benediction
|-
| | 58
| | 675.728
| | Hemisecordite
|-
| | 59
| | 687.379
| | Marvel and Benediction
|-
| | 60
| | 699.029
| | Marvel and Benediction, Hemisecordite
|-
| | 61
| | 710.677
| | Hemisecordite
|-
| | 62
| | 722.330
| | Marvel and Benediction
|-
| | 63
| | 733.981
| | Marvel and Benediction, Hemisecordite
|-
| | 64
| | 745.631
| | Marvel and Benediction
|-
| | 65
| | 757.282
| | Hemisecordite
|-
| | 66
| | 768.932
| | Marvel and Benediction, Hemisecordite
|-
| | 67
| | 780.583
| | Marvel and Benediction
|-
| | 68
| | 792.233
| | Hemisecordite
|-
| | 69
| | 803.883
| | Marvel and Benediction
|-
| | 70
| | 815.534
| | Marvel and Benediction, Hemisecordite
|-
| | 71
| | 827.185
| | Hemisecordite
|-
| | 72
| | 838.835
| | Marvel and Benediction
|-
| | 73
| | 850.385
| | Marvel and Benediction. Hemisecordite
|-
| | 74
| | 862.156
| | Marvel and Benediction
|-
| | 75
| | 873.786
| | Hemisecordite
|-
| | 76
| | 885.537
| | Marvel and Benediction, Hemisecordite
|-
| | 77
| | 896.987
| | Marvel and Benediction
|-
| | 78
| | 908.638
| | Hemisecordite
|-
| | 79
| | 920.388
| | Marvel and Benediction, Hemisecordite
|-
| | 80
| | 932.039
| | Marvel and Benediction
|-
| | 81
| | 943.689
| | Hemisecordite
|-
| | 82
| | 955.34
| | Marvel and Benediction
|-
| | 83
| | 966.99
| | Marvel and Benediction, Hemisecordite
|-
| | 84
| | 978.641
| | Marvel and Benediction
|-
| | 85
| | 990.291
| | Hemisecordite
|-
| | 86
| | 1001.942
| | Marvel and Benediction, Hemisecordite
|-
| | 87
| | 1013.592
| | Marvel and Benediction
|-
| | 88
| | 1025.243
| | Hemisecordite
|-
| | 89
| | 1036.893
| | Marvel and Benediction
|-
| | 90
| | 1048.544
| | Marvel and Benediction, Hemisecordite
|-
| | 91
| | 1060.194
| | Hemisecordite
|-
| | 92
| | 1071.845
| | Marvel and Benediction
|-
| | 93
| | 1083.495
| | Marvel and Benediction, Hemisecordite
|-
| | 94
| | 1095.146
| | Marvel and Benediction
|-
| | 95
| | 1106.796
| | Hemisecordite
|-
| | 96
| | 1118.447
| | Marvel and Benediction, Hemisecordite
|-
| | 97
| | 1130.097
| | Marvel and Benediction
|-
| | 98
| | 1141.748
| | Hemisecordite
|-
| | 99
| | 1153.398
| | Marvel and Benediction
|-
| | 100
| | 1165.0485
| | Marvel and Benediction, Hemisecordite
|-
| | 101
| | 1176.609
| | Hemisecordite
|-
| | 102
| | 1188.3495
| | Marvel and Benediction
|}

[[Category:Benediction]]
[[Category:Edo]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>seo6l52lzxrah0g7dzaincuv7zbpqp4</sha1>
    </revision>
    <revision>
      <id>54947</id>
      <parentid>39147</parentid>
      <timestamp>2020-12-04T23:13:05Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54947</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6165" sha1="iwvasl3o8cjzi9gqt0ousn8sc8ymarc" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic_clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]] limits and [[Gamelismic_clan|benediction]] and [[Gamelismic_clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel_family#Marvel|marvel temperament]] as well as [[Gamelismic_clan#Miracle|benediction]] and [[Gamelismic_clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime_numbers|prime]] edo.

===13-limit temperaments===

{| class="wikitable"
|-
| | Degree
| | cents
| | temperament
|-
| | 1
| | 11.650
| | Marvel and Benediction
|-
| | 2
| | 23.301
| | Hemisecordite
|-
| | 3
| | 34.951
| | Marvel and Benediction, Hemisecordite
|-
| | 4
| | 46.602
| | Marvel and Benediction
|-
| | 5
| | 58.252
| | Hemisecordite
|-
| | 6
| | 69.903
| | Marvel and Benediction
|-
| | 7
| | 81.553
| | Marvel and Benediction, Hemisecordite
|-
| | 8
| | 93.204
| | Hemisecordite
|-
| | 9
| | 104.854
| | Marvel and Benediction
|-
| | 10
| | 116.505
| | Marvel and Benediction, Hemisecordite
|-
| | 11
| | 128.155
| | Marvel and Benediction
|-
| | 12
| | 139.806
| | Hemisecordite
|-
| | 13
| | 151.456
| | Marvel and Benediction, Hemisecordite
|-
| | 14
| | 163.107
| | Marvel and Benediction
|-
| | 15
| | 174.757
| | Hemisecordite
|-
| | 16
| | 186.408
| | Marvel and Benediction
|-
| | 17
| | 198.058
| | Marvel and Benediction, Hemisecordite
|-
| | 18
| | 209.708
| | Hemisecordite
|-
| | 19
| | 221.359
| | Marvel and Benediction
|-
| | 20
| | 233.010
| | Marvel and Benediction, Hemisecordite
|-
| | 21
| | 244.660
| | Marvel and Benediction
|-
| | 22
| | 256.311
| | Hemisecordite
|-
| | 23
| | 267.961
| | Marvel and Benediction, Hemisecordite
|-
| | 24
| | 279.712
| | Marvel and Benediction
|-
| | 25
| | 291.262
| | Hemisecordite
|-
| | 26
| | 303.013
| | Marvel and Benediction
|-
| | 27
| | 314.563
| | Marvel and Benediction, Hemisecordite
|-
| | 28
| | 326.214
| | Hemisecordite
|-
| | 29
| | 337.864
| | Marvel and Benediction
|-
| | 30
| | 349.615
| | Marvel and Benediction, Hemisecordite
|-
| | 31
| | 361.165
| | Marvel and Benediction
|-
| | 32
| | 372.816
| | Hemisecordite
|-
| | 33
| | 384.466
| | Marvel and Benediction, Hemisecordite
|-
| | 34
| | 396.117
| | Marvel and Benediction
|-
| | 35
| | 407.767
| | Hemisecordite
|-
| | 36
| | 419.4175
| | Marvel and Benediction
|-
| | 37
| | 431.068
| | Marvel and Benediction, Hemisecordite
|-
| | 38
| | 442.708
| | Hemisecordite
|-
| | 39
| | 454.369
| | Marvel and Benediction
|-
| | 40
| | 466.019
| | Marvel and Benediction, Hemisecordite
|-
| | 41
| | 477.670
| | Marvel and Benediction
|-
| | 42
| | 489.320
| | Hemisecordite
|-
| | 43
| | 500.971
| | Marvel and Benediction, Hemisecordite
|-
| | 44
| | 512.621
| | Marvel and Benediction
|-
| | 45
| | 524.272
| | Hemisecordite
|-
| | 46
| | 535.922
| | Marvel and Benediction
|-
| | 47
| | 547.573
| | Marevel and Benediction, Hemisecordite
|-
| | 48
| | 559.223
| | Hemisecordite
|-
| | 49
| | 570.874
| | Marvel and Benediction
|-
| | 50
| | 582.524
| | Marvel and Benediction, Hemisecordite
|-
| | 52
| | 605.825
| | Marvel and Benediction, Hemisecordite
|-
| | 53
| | 617.476
| | Marvel and Benediction, Hemisecordite
|-
| | 54
| | 629.126
| | Marvel and Benediction
|-
| | 55
| | 640.777
| | Hemisecordite
|-
| | 56
| | 652.427
| | Marvel and Benediction, Hemisecordite
|-
| | 57
| | 664.078
| | Marvel and Benediction
|-
| | 58
| | 675.728
| | Hemisecordite
|-
| | 59
| | 687.379
| | Marvel and Benediction
|-
| | 60
| | 699.029
| | Marvel and Benediction, Hemisecordite
|-
| | 61
| | 710.677
| | Hemisecordite
|-
| | 62
| | 722.330
| | Marvel and Benediction
|-
| | 63
| | 733.981
| | Marvel and Benediction, Hemisecordite
|-
| | 64
| | 745.631
| | Marvel and Benediction
|-
| | 65
| | 757.282
| | Hemisecordite
|-
| | 66
| | 768.932
| | Marvel and Benediction, Hemisecordite
|-
| | 67
| | 780.583
| | Marvel and Benediction
|-
| | 68
| | 792.233
| | Hemisecordite
|-
| | 69
| | 803.883
| | Marvel and Benediction
|-
| | 70
| | 815.534
| | Marvel and Benediction, Hemisecordite
|-
| | 71
| | 827.185
| | Hemisecordite
|-
| | 72
| | 838.835
| | Marvel and Benediction
|-
| | 73
| | 850.385
| | Marvel and Benediction. Hemisecordite
|-
| | 74
| | 862.156
| | Marvel and Benediction
|-
| | 75
| | 873.786
| | Hemisecordite
|-
| | 76
| | 885.537
| | Marvel and Benediction, Hemisecordite
|-
| | 77
| | 896.987
| | Marvel and Benediction
|-
| | 78
| | 908.638
| | Hemisecordite
|-
| | 79
| | 920.388
| | Marvel and Benediction, Hemisecordite
|-
| | 80
| | 932.039
| | Marvel and Benediction
|-
| | 81
| | 943.689
| | Hemisecordite
|-
| | 82
| | 955.34
| | Marvel and Benediction
|-
| | 83
| | 966.99
| | Marvel and Benediction, Hemisecordite
|-
| | 84
| | 978.641
| | Marvel and Benediction
|-
| | 85
| | 990.291
| | Hemisecordite
|-
| | 86
| | 1001.942
| | Marvel and Benediction, Hemisecordite
|-
| | 87
| | 1013.592
| | Marvel and Benediction
|-
| | 88
| | 1025.243
| | Hemisecordite
|-
| | 89
| | 1036.893
| | Marvel and Benediction
|-
| | 90
| | 1048.544
| | Marvel and Benediction, Hemisecordite
|-
| | 91
| | 1060.194
| | Hemisecordite
|-
| | 92
| | 1071.845
| | Marvel and Benediction
|-
| | 93
| | 1083.495
| | Marvel and Benediction, Hemisecordite
|-
| | 94
| | 1095.146
| | Marvel and Benediction
|-
| | 95
| | 1106.796
| | Hemisecordite
|-
| | 96
| | 1118.447
| | Marvel and Benediction, Hemisecordite
|-
| | 97
| | 1130.097
| | Marvel and Benediction
|-
| | 98
| | 1141.748
| | Hemisecordite
|-
| | 99
| | 1153.398
| | Marvel and Benediction
|-
| | 100
| | 1165.0485
| | Marvel and Benediction, Hemisecordite
|-
| | 101
| | 1176.609
| | Hemisecordite
|-
| | 102
| | 1188.3495
| | Marvel and Benediction
|}

[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>iwvasl3o8cjzi9gqt0ousn8sc8ymarc</sha1>
    </revision>
    <revision>
      <id>58699</id>
      <parentid>54947</parentid>
      <timestamp>2021-01-11T21:25:27Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>+prime intervals</comment>
      <origin>58699</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6187" sha1="rbccxf7eo7i1unl8yl6ngebh972d1hu" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic_clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]] limits and [[Gamelismic_clan|benediction]] and [[Gamelismic_clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel_family#Marvel|marvel temperament]] as well as [[Gamelismic_clan#Miracle|benediction]] and [[Gamelismic_clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime_numbers|prime]] edo.
{{Primes in edo|103}}

===13-limit temperaments===

{| class="wikitable"
|-
| | Degree
| | cents
| | temperament
|-
| | 1
| | 11.650
| | Marvel and Benediction
|-
| | 2
| | 23.301
| | Hemisecordite
|-
| | 3
| | 34.951
| | Marvel and Benediction, Hemisecordite
|-
| | 4
| | 46.602
| | Marvel and Benediction
|-
| | 5
| | 58.252
| | Hemisecordite
|-
| | 6
| | 69.903
| | Marvel and Benediction
|-
| | 7
| | 81.553
| | Marvel and Benediction, Hemisecordite
|-
| | 8
| | 93.204
| | Hemisecordite
|-
| | 9
| | 104.854
| | Marvel and Benediction
|-
| | 10
| | 116.505
| | Marvel and Benediction, Hemisecordite
|-
| | 11
| | 128.155
| | Marvel and Benediction
|-
| | 12
| | 139.806
| | Hemisecordite
|-
| | 13
| | 151.456
| | Marvel and Benediction, Hemisecordite
|-
| | 14
| | 163.107
| | Marvel and Benediction
|-
| | 15
| | 174.757
| | Hemisecordite
|-
| | 16
| | 186.408
| | Marvel and Benediction
|-
| | 17
| | 198.058
| | Marvel and Benediction, Hemisecordite
|-
| | 18
| | 209.708
| | Hemisecordite
|-
| | 19
| | 221.359
| | Marvel and Benediction
|-
| | 20
| | 233.010
| | Marvel and Benediction, Hemisecordite
|-
| | 21
| | 244.660
| | Marvel and Benediction
|-
| | 22
| | 256.311
| | Hemisecordite
|-
| | 23
| | 267.961
| | Marvel and Benediction, Hemisecordite
|-
| | 24
| | 279.712
| | Marvel and Benediction
|-
| | 25
| | 291.262
| | Hemisecordite
|-
| | 26
| | 303.013
| | Marvel and Benediction
|-
| | 27
| | 314.563
| | Marvel and Benediction, Hemisecordite
|-
| | 28
| | 326.214
| | Hemisecordite
|-
| | 29
| | 337.864
| | Marvel and Benediction
|-
| | 30
| | 349.615
| | Marvel and Benediction, Hemisecordite
|-
| | 31
| | 361.165
| | Marvel and Benediction
|-
| | 32
| | 372.816
| | Hemisecordite
|-
| | 33
| | 384.466
| | Marvel and Benediction, Hemisecordite
|-
| | 34
| | 396.117
| | Marvel and Benediction
|-
| | 35
| | 407.767
| | Hemisecordite
|-
| | 36
| | 419.4175
| | Marvel and Benediction
|-
| | 37
| | 431.068
| | Marvel and Benediction, Hemisecordite
|-
| | 38
| | 442.708
| | Hemisecordite
|-
| | 39
| | 454.369
| | Marvel and Benediction
|-
| | 40
| | 466.019
| | Marvel and Benediction, Hemisecordite
|-
| | 41
| | 477.670
| | Marvel and Benediction
|-
| | 42
| | 489.320
| | Hemisecordite
|-
| | 43
| | 500.971
| | Marvel and Benediction, Hemisecordite
|-
| | 44
| | 512.621
| | Marvel and Benediction
|-
| | 45
| | 524.272
| | Hemisecordite
|-
| | 46
| | 535.922
| | Marvel and Benediction
|-
| | 47
| | 547.573
| | Marevel and Benediction, Hemisecordite
|-
| | 48
| | 559.223
| | Hemisecordite
|-
| | 49
| | 570.874
| | Marvel and Benediction
|-
| | 50
| | 582.524
| | Marvel and Benediction, Hemisecordite
|-
| | 52
| | 605.825
| | Marvel and Benediction, Hemisecordite
|-
| | 53
| | 617.476
| | Marvel and Benediction, Hemisecordite
|-
| | 54
| | 629.126
| | Marvel and Benediction
|-
| | 55
| | 640.777
| | Hemisecordite
|-
| | 56
| | 652.427
| | Marvel and Benediction, Hemisecordite
|-
| | 57
| | 664.078
| | Marvel and Benediction
|-
| | 58
| | 675.728
| | Hemisecordite
|-
| | 59
| | 687.379
| | Marvel and Benediction
|-
| | 60
| | 699.029
| | Marvel and Benediction, Hemisecordite
|-
| | 61
| | 710.677
| | Hemisecordite
|-
| | 62
| | 722.330
| | Marvel and Benediction
|-
| | 63
| | 733.981
| | Marvel and Benediction, Hemisecordite
|-
| | 64
| | 745.631
| | Marvel and Benediction
|-
| | 65
| | 757.282
| | Hemisecordite
|-
| | 66
| | 768.932
| | Marvel and Benediction, Hemisecordite
|-
| | 67
| | 780.583
| | Marvel and Benediction
|-
| | 68
| | 792.233
| | Hemisecordite
|-
| | 69
| | 803.883
| | Marvel and Benediction
|-
| | 70
| | 815.534
| | Marvel and Benediction, Hemisecordite
|-
| | 71
| | 827.185
| | Hemisecordite
|-
| | 72
| | 838.835
| | Marvel and Benediction
|-
| | 73
| | 850.385
| | Marvel and Benediction. Hemisecordite
|-
| | 74
| | 862.156
| | Marvel and Benediction
|-
| | 75
| | 873.786
| | Hemisecordite
|-
| | 76
| | 885.537
| | Marvel and Benediction, Hemisecordite
|-
| | 77
| | 896.987
| | Marvel and Benediction
|-
| | 78
| | 908.638
| | Hemisecordite
|-
| | 79
| | 920.388
| | Marvel and Benediction, Hemisecordite
|-
| | 80
| | 932.039
| | Marvel and Benediction
|-
| | 81
| | 943.689
| | Hemisecordite
|-
| | 82
| | 955.34
| | Marvel and Benediction
|-
| | 83
| | 966.99
| | Marvel and Benediction, Hemisecordite
|-
| | 84
| | 978.641
| | Marvel and Benediction
|-
| | 85
| | 990.291
| | Hemisecordite
|-
| | 86
| | 1001.942
| | Marvel and Benediction, Hemisecordite
|-
| | 87
| | 1013.592
| | Marvel and Benediction
|-
| | 88
| | 1025.243
| | Hemisecordite
|-
| | 89
| | 1036.893
| | Marvel and Benediction
|-
| | 90
| | 1048.544
| | Marvel and Benediction, Hemisecordite
|-
| | 91
| | 1060.194
| | Hemisecordite
|-
| | 92
| | 1071.845
| | Marvel and Benediction
|-
| | 93
| | 1083.495
| | Marvel and Benediction, Hemisecordite
|-
| | 94
| | 1095.146
| | Marvel and Benediction
|-
| | 95
| | 1106.796
| | Hemisecordite
|-
| | 96
| | 1118.447
| | Marvel and Benediction, Hemisecordite
|-
| | 97
| | 1130.097
| | Marvel and Benediction
|-
| | 98
| | 1141.748
| | Hemisecordite
|-
| | 99
| | 1153.398
| | Marvel and Benediction
|-
| | 100
| | 1165.0485
| | Marvel and Benediction, Hemisecordite
|-
| | 101
| | 1176.609
| | Hemisecordite
|-
| | 102
| | 1188.3495
| | Marvel and Benediction
|}

[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>rbccxf7eo7i1unl8yl6ngebh972d1hu</sha1>
    </revision>
    <revision>
      <id>68026</id>
      <parentid>58699</parentid>
      <timestamp>2021-04-20T20:12:19Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68026</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7834" sha1="6njvtvp2j80v4si6c0qu1b4afwbqa0n" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic_clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]] limits and [[Gamelismic_clan|benediction]] and [[Gamelismic_clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel_family#Marvel|marvel temperament]] as well as [[Gamelismic_clan#Miracle|benediction]] and [[Gamelismic_clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime_numbers|prime]] edo.
{{Primes in edo|103}}

===13-limit temperaments===

{| class="wikitable"
|-
| | Degree
| | cents
| | temperament
|-
| | 1
| | 11.650
| | Marvel and Benediction
|-
| | 2
| | 23.301
| | Hemisecordite
|-
| | 3
| | 34.951
| | Marvel and Benediction, Hemisecordite
|-
| | 4
| | 46.602
| | Marvel and Benediction
|-
| | 5
| | 58.252
| | Hemisecordite
|-
| | 6
| | 69.903
| | Marvel and Benediction
|-
| | 7
| | 81.553
| | Marvel and Benediction, Hemisecordite
|-
| | 8
| | 93.204
| | Hemisecordite
|-
| | 9
| | 104.854
| | Marvel and Benediction
|-
| | 10
| | 116.505
| | Marvel and Benediction, Hemisecordite
|-
| | 11
| | 128.155
| | Marvel and Benediction
|-
| | 12
| | 139.806
| | Hemisecordite
|-
| | 13
| | 151.456
| | Marvel and Benediction, Hemisecordite
|-
| | 14
| | 163.107
| | Marvel and Benediction
|-
| | 15
| | 174.757
| | Hemisecordite
|-
| | 16
| | 186.408
| | Marvel and Benediction
|-
| | 17
| | 198.058
| | Marvel and Benediction, Hemisecordite
|-
| | 18
| | 209.708
| | Hemisecordite
|-
| | 19
| | 221.359
| | Marvel and Benediction
|-
| | 20
| | 233.010
| | Marvel and Benediction, Hemisecordite
|-
| | 21
| | 244.660
| | Marvel and Benediction
|-
| | 22
| | 256.311
| | Hemisecordite
|-
| | 23
| | 267.961
| | Marvel and Benediction, Hemisecordite
|-
| | 24
| | 279.712
| | Marvel and Benediction
|-
| | 25
| | 291.262
| | Hemisecordite
|-
| | 26
| | 303.013
| | Marvel and Benediction
|-
| | 27
| | 314.563
| | Marvel and Benediction, Hemisecordite
|-
| | 28
| | 326.214
| | Hemisecordite
|-
| | 29
| | 337.864
| | Marvel and Benediction
|-
| | 30
| | 349.615
| | Marvel and Benediction, Hemisecordite
|-
| | 31
| | 361.165
| | Marvel and Benediction
|-
| | 32
| | 372.816
| | Hemisecordite
|-
| | 33
| | 384.466
| | Marvel and Benediction, Hemisecordite
|-
| | 34
| | 396.117
| | Marvel and Benediction
|-
| | 35
| | 407.767
| | Hemisecordite
|-
| | 36
| | 419.4175
| | Marvel and Benediction
|-
| | 37
| | 431.068
| | Marvel and Benediction, Hemisecordite
|-
| | 38
| | 442.708
| | Hemisecordite
|-
| | 39
| | 454.369
| | Marvel and Benediction
|-
| | 40
| | 466.019
| | Marvel and Benediction, Hemisecordite
|-
| | 41
| | 477.670
| | Marvel and Benediction
|-
| | 42
| | 489.320
| | Hemisecordite
|-
| | 43
| | 500.971
| | Marvel and Benediction, Hemisecordite
|-
| | 44
| | 512.621
| | Marvel and Benediction
|-
| | 45
| | 524.272
| | Hemisecordite
|-
| | 46
| | 535.922
| | Marvel and Benediction
|-
| | 47
| | 547.573
| | Marevel and Benediction, Hemisecordite
|-
| | 48
| | 559.223
| | Hemisecordite
|-
| | 49
| | 570.874
| | Marvel and Benediction
|-
| | 50
| | 582.524
| | Marvel and Benediction, Hemisecordite
|-
| | 52
| | 605.825
| | Marvel and Benediction, Hemisecordite
|-
| | 53
| | 617.476
| | Marvel and Benediction, Hemisecordite
|-
| | 54
| | 629.126
| | Marvel and Benediction
|-
| | 55
| | 640.777
| | Hemisecordite
|-
| | 56
| | 652.427
| | Marvel and Benediction, Hemisecordite
|-
| | 57
| | 664.078
| | Marvel and Benediction
|-
| | 58
| | 675.728
| | Hemisecordite
|-
| | 59
| | 687.379
| | Marvel and Benediction
|-
| | 60
| | 699.029
| | Marvel and Benediction, Hemisecordite
|-
| | 61
| | 710.677
| | Hemisecordite
|-
| | 62
| | 722.330
| | Marvel and Benediction
|-
| | 63
| | 733.981
| | Marvel and Benediction, Hemisecordite
|-
| | 64
| | 745.631
| | Marvel and Benediction
|-
| | 65
| | 757.282
| | Hemisecordite
|-
| | 66
| | 768.932
| | Marvel and Benediction, Hemisecordite
|-
| | 67
| | 780.583
| | Marvel and Benediction
|-
| | 68
| | 792.233
| | Hemisecordite
|-
| | 69
| | 803.883
| | Marvel and Benediction
|-
| | 70
| | 815.534
| | Marvel and Benediction, Hemisecordite
|-
| | 71
| | 827.185
| | Hemisecordite
|-
| | 72
| | 838.835
| | Marvel and Benediction
|-
| | 73
| | 850.385
| | Marvel and Benediction. Hemisecordite
|-
| | 74
| | 862.156
| | Marvel and Benediction
|-
| | 75
| | 873.786
| | Hemisecordite
|-
| | 76
| | 885.537
| | Marvel and Benediction, Hemisecordite
|-
| | 77
| | 896.987
| | Marvel and Benediction
|-
| | 78
| | 908.638
| | Hemisecordite
|-
| | 79
| | 920.388
| | Marvel and Benediction, Hemisecordite
|-
| | 80
| | 932.039
| | Marvel and Benediction
|-
| | 81
| | 943.689
| | Hemisecordite
|-
| | 82
| | 955.34
| | Marvel and Benediction
|-
| | 83
| | 966.99
| | Marvel and Benediction, Hemisecordite
|-
| | 84
| | 978.641
| | Marvel and Benediction
|-
| | 85
| | 990.291
| | Hemisecordite
|-
| | 86
| | 1001.942
| | Marvel and Benediction, Hemisecordite
|-
| | 87
| | 1013.592
| | Marvel and Benediction
|-
| | 88
| | 1025.243
| | Hemisecordite
|-
| | 89
| | 1036.893
| | Marvel and Benediction
|-
| | 90
| | 1048.544
| | Marvel and Benediction, Hemisecordite
|-
| | 91
| | 1060.194
| | Hemisecordite
|-
| | 92
| | 1071.845
| | Marvel and Benediction
|-
| | 93
| | 1083.495
| | Marvel and Benediction, Hemisecordite
|-
| | 94
| | 1095.146
| | Marvel and Benediction
|-
| | 95
| | 1106.796
| | Hemisecordite
|-
| | 96
| | 1118.447
| | Marvel and Benediction, Hemisecordite
|-
| | 97
| | 1130.097
| | Marvel and Benediction
|-
| | 98
| | 1141.748
| | Hemisecordite
|-
| | 99
| | 1153.398
| | Marvel and Benediction
|-
| | 100
| | 1165.0485
| | Marvel and Benediction, Hemisecordite
|-
| | 101
| | 1176.609
| | Hemisecordite
|-
| | 102
| | 1188.3495
| | Marvel and Benediction
|}
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="2" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|equal
|-
|18
|11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
|19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|14L 9s
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
|25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
|19L 9s
|-
|29
|16L 13s
|-
|30
|13L 17s
|-
|31
|10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
|33L 2s
| rowspan="17" |3:2
|-
|36
|31L 5s
|-
|37
|29L 8s
|-
|38
|27L 11s
|-
|39
|25L 14s
|-
|40
|23L 17s
|-
|41
|21L 20s
|-
|42
|19L 23s 
|-
|43
|17L 26L
|-
|44
|15L 29s
|-
|45
|13L 32s
|-
|46
|11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
|51L 1s
| rowspan="31" |2:1
|-
|53
|50L 3s
|-
|54
|49L 5s
|-
|55
|48L 7s
|-
|56
|47L 9s
|-
|57
|46L 11s
|-
|58
|45L 13s
|-
|59
|44L 15s
|-
|60
|43L 17s
|-
|61
|42L 19s
|-
|62
|41L 21s
|-
|63
|40L 23s
|-
|64
|39L 25s
|-
|65
|38L 27s
|-
|66
|37L 29s
|-
|67
|36L 31s
|-
|68
|35L 33s
|-
|69
|34L 35s
|-
|70
|33L 37s
|-
|71
|32L 39s
|-
|72
|31L 41s
|-
|73
|30L 43s
|-
|74
|29L 45s
|-
|75
|28L 47s
|-
|76
|27L 49s
|-
|77
|26L 51s
|-
|78
|25L 53s
|-
|79
|24L 55s
|-
|80
|23L 57s
|-
|81
|22L 59s
|-
|82
|21L 61s
|}
[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>6njvtvp2j80v4si6c0qu1b4afwbqa0n</sha1>
    </revision>
    <revision>
      <id>68027</id>
      <parentid>68026</parentid>
      <timestamp>2021-04-20T20:22:08Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68027</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7827" sha1="q1c8eg8vw9odfva2fjp6lnbds385drr" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic_clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]] limits and [[Gamelismic_clan|benediction]] and [[Gamelismic_clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel_family#Marvel|marvel temperament]] as well as [[Gamelismic_clan#Miracle|benediction]] and [[Gamelismic_clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime_numbers|prime]] edo.
{{Primes in edo|103}}

===13-limit temperaments===

{| class="wikitable"
|-
| | Degree
| | cents
| | temperament
|-
| | 1
| | 11.650
| | Marvel and Benediction
|-
| | 2
| | 23.301
| | Hemisecordite
|-
| | 3
| | 34.951
| | Marvel and Benediction, Hemisecordite
|-
| | 4
| | 46.602
| | Marvel and Benediction
|-
| | 5
| | 58.252
| | Hemisecordite
|-
| | 6
| | 69.903
| | Marvel and Benediction
|-
| | 7
| | 81.553
| | Marvel and Benediction, Hemisecordite
|-
| | 8
| | 93.204
| | Hemisecordite
|-
| | 9
| | 104.854
| | Marvel and Benediction
|-
| | 10
| | 116.505
| | Marvel and Benediction, Hemisecordite
|-
| | 11
| | 128.155
| | Marvel and Benediction
|-
| | 12
| | 139.806
| | Hemisecordite
|-
| | 13
| | 151.456
| | Marvel and Benediction, Hemisecordite
|-
| | 14
| | 163.107
| | Marvel and Benediction
|-
| | 15
| | 174.757
| | Hemisecordite
|-
| | 16
| | 186.408
| | Marvel and Benediction
|-
| | 17
| | 198.058
| | Marvel and Benediction, Hemisecordite
|-
| | 18
| | 209.708
| | Hemisecordite
|-
| | 19
| | 221.359
| | Marvel and Benediction
|-
| | 20
| | 233.010
| | Marvel and Benediction, Hemisecordite
|-
| | 21
| | 244.660
| | Marvel and Benediction
|-
| | 22
| | 256.311
| | Hemisecordite
|-
| | 23
| | 267.961
| | Marvel and Benediction, Hemisecordite
|-
| | 24
| | 279.712
| | Marvel and Benediction
|-
| | 25
| | 291.262
| | Hemisecordite
|-
| | 26
| | 303.013
| | Marvel and Benediction
|-
| | 27
| | 314.563
| | Marvel and Benediction, Hemisecordite
|-
| | 28
| | 326.214
| | Hemisecordite
|-
| | 29
| | 337.864
| | Marvel and Benediction
|-
| | 30
| | 349.615
| | Marvel and Benediction, Hemisecordite
|-
| | 31
| | 361.165
| | Marvel and Benediction
|-
| | 32
| | 372.816
| | Hemisecordite
|-
| | 33
| | 384.466
| | Marvel and Benediction, Hemisecordite
|-
| | 34
| | 396.117
| | Marvel and Benediction
|-
| | 35
| | 407.767
| | Hemisecordite
|-
| | 36
| | 419.4175
| | Marvel and Benediction
|-
| | 37
| | 431.068
| | Marvel and Benediction, Hemisecordite
|-
| | 38
| | 442.708
| | Hemisecordite
|-
| | 39
| | 454.369
| | Marvel and Benediction
|-
| | 40
| | 466.019
| | Marvel and Benediction, Hemisecordite
|-
| | 41
| | 477.670
| | Marvel and Benediction
|-
| | 42
| | 489.320
| | Hemisecordite
|-
| | 43
| | 500.971
| | Marvel and Benediction, Hemisecordite
|-
| | 44
| | 512.621
| | Marvel and Benediction
|-
| | 45
| | 524.272
| | Hemisecordite
|-
| | 46
| | 535.922
| | Marvel and Benediction
|-
| | 47
| | 547.573
| | Marevel and Benediction, Hemisecordite
|-
| | 48
| | 559.223
| | Hemisecordite
|-
| | 49
| | 570.874
| | Marvel and Benediction
|-
| | 50
| | 582.524
| | Marvel and Benediction, Hemisecordite
|-
| | 52
| | 605.825
| | Marvel and Benediction, Hemisecordite
|-
| | 53
| | 617.476
| | Marvel and Benediction, Hemisecordite
|-
| | 54
| | 629.126
| | Marvel and Benediction
|-
| | 55
| | 640.777
| | Hemisecordite
|-
| | 56
| | 652.427
| | Marvel and Benediction, Hemisecordite
|-
| | 57
| | 664.078
| | Marvel and Benediction
|-
| | 58
| | 675.728
| | Hemisecordite
|-
| | 59
| | 687.379
| | Marvel and Benediction
|-
| | 60
| | 699.029
| | Marvel and Benediction, Hemisecordite
|-
| | 61
| | 710.677
| | Hemisecordite
|-
| | 62
| | 722.330
| | Marvel and Benediction
|-
| | 63
| | 733.981
| | Marvel and Benediction, Hemisecordite
|-
| | 64
| | 745.631
| | Marvel and Benediction
|-
| | 65
| | 757.282
| | Hemisecordite
|-
| | 66
| | 768.932
| | Marvel and Benediction, Hemisecordite
|-
| | 67
| | 780.583
| | Marvel and Benediction
|-
| | 68
| | 792.233
| | Hemisecordite
|-
| | 69
| | 803.883
| | Marvel and Benediction
|-
| | 70
| | 815.534
| | Marvel and Benediction, Hemisecordite
|-
| | 71
| | 827.185
| | Hemisecordite
|-
| | 72
| | 838.835
| | Marvel and Benediction
|-
| | 73
| | 850.385
| | Marvel and Benediction. Hemisecordite
|-
| | 74
| | 862.156
| | Marvel and Benediction
|-
| | 75
| | 873.786
| | Hemisecordite
|-
| | 76
| | 885.537
| | Marvel and Benediction, Hemisecordite
|-
| | 77
| | 896.987
| | Marvel and Benediction
|-
| | 78
| | 908.638
| | Hemisecordite
|-
| | 79
| | 920.388
| | Marvel and Benediction, Hemisecordite
|-
| | 80
| | 932.039
| | Marvel and Benediction
|-
| | 81
| | 943.689
| | Hemisecordite
|-
| | 82
| | 955.34
| | Marvel and Benediction
|-
| | 83
| | 966.99
| | Marvel and Benediction, Hemisecordite
|-
| | 84
| | 978.641
| | Marvel and Benediction
|-
| | 85
| | 990.291
| | Hemisecordite
|-
| | 86
| | 1001.942
| | Marvel and Benediction, Hemisecordite
|-
| | 87
| | 1013.592
| | Marvel and Benediction
|-
| | 88
| | 1025.243
| | Hemisecordite
|-
| | 89
| | 1036.893
| | Marvel and Benediction
|-
| | 90
| | 1048.544
| | Marvel and Benediction, Hemisecordite
|-
| | 91
| | 1060.194
| | Hemisecordite
|-
| | 92
| | 1071.845
| | Marvel and Benediction
|-
| | 93
| | 1083.495
| | Marvel and Benediction, Hemisecordite
|-
| | 94
| | 1095.146
| | Marvel and Benediction
|-
| | 95
| | 1106.796
| | Hemisecordite
|-
| | 96
| | 1118.447
| | Marvel and Benediction, Hemisecordite
|-
| | 97
| | 1130.097
| | Marvel and Benediction
|-
| | 98
| | 1141.748
| | Hemisecordite
|-
| | 99
| | 1153.398
| | Marvel and Benediction
|-
| | 100
| | 1165.0485
| | Marvel and Benediction, Hemisecordite
|-
| | 101
| | 1176.609
| | Hemisecordite
|-
| | 102
| | 1188.3495
| | Marvel and Benediction
|}
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
|11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
|19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|14L 9s
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
|25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
|19L 9s
|-
|29
|16L 13s
|-
|30
|13L 17s
|-
|31
|10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
|33L 2s
| rowspan="17" |3:2
|-
|36
|31L 5s
|-
|37
|29L 8s
|-
|38
|27L 11s
|-
|39
|25L 14s
|-
|40
|23L 17s
|-
|41
|21L 20s
|-
|42
|19L 23s 
|-
|43
|17L 26L
|-
|44
|15L 29s
|-
|45
|13L 32s
|-
|46
|11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
|51L 1s
| rowspan="31" |2:1
|-
|53
|50L 3s
|-
|54
|49L 5s
|-
|55
|48L 7s
|-
|56
|47L 9s
|-
|57
|46L 11s
|-
|58
|45L 13s
|-
|59
|44L 15s
|-
|60
|43L 17s
|-
|61
|42L 19s
|-
|62
|41L 21s
|-
|63
|40L 23s
|-
|64
|39L 25s
|-
|65
|38L 27s
|-
|66
|37L 29s
|-
|67
|36L 31s
|-
|68
|35L 33s
|-
|69
|34L 35s
|-
|70
|33L 37s
|-
|71
|32L 39s
|-
|72
|31L 41s
|-
|73
|30L 43s
|-
|74
|29L 45s
|-
|75
|28L 47s
|-
|76
|27L 49s
|-
|77
|26L 51s
|-
|78
|25L 53s
|-
|79
|24L 55s
|-
|80
|23L 57s
|-
|81
|22L 59s
|-
|82
|21L 61s
|}
[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>q1c8eg8vw9odfva2fjp6lnbds385drr</sha1>
    </revision>
    <revision>
      <id>68028</id>
      <parentid>68027</parentid>
      <timestamp>2021-04-20T21:22:40Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68028</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7832" sha1="0wr7i0i64boeig85shuo59tyyr92had" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic_clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]] limits and [[Gamelismic_clan|benediction]] and [[Gamelismic_clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel_family#Marvel|marvel temperament]] as well as [[Gamelismic_clan#Miracle|benediction]] and [[Gamelismic_clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime_numbers|prime]] edo.
{{Primes in edo|103}}

===13-limit temperaments===

{| class="wikitable"
|-
| | Degree
| | cents
| | temperament
|-
| | 1
| | 11.650
| | Marvel and Benediction
|-
| | 2
| | 23.301
| | Hemisecordite
|-
| | 3
| | 34.951
| | Marvel and Benediction, Hemisecordite
|-
| | 4
| | 46.602
| | Marvel and Benediction
|-
| | 5
| | 58.252
| | Hemisecordite
|-
| | 6
| | 69.903
| | Marvel and Benediction
|-
| | 7
| | 81.553
| | Marvel and Benediction, Hemisecordite
|-
| | 8
| | 93.204
| | Hemisecordite
|-
| | 9
| | 104.854
| | Marvel and Benediction
|-
| | 10
| | 116.505
| | Marvel and Benediction, Hemisecordite
|-
| | 11
| | 128.155
| | Marvel and Benediction
|-
| | 12
| | 139.806
| | Hemisecordite
|-
| | 13
| | 151.456
| | Marvel and Benediction, Hemisecordite
|-
| | 14
| | 163.107
| | Marvel and Benediction
|-
| | 15
| | 174.757
| | Hemisecordite
|-
| | 16
| | 186.408
| | Marvel and Benediction
|-
| | 17
| | 198.058
| | Marvel and Benediction, Hemisecordite
|-
| | 18
| | 209.708
| | Hemisecordite
|-
| | 19
| | 221.359
| | Marvel and Benediction
|-
| | 20
| | 233.010
| | Marvel and Benediction, Hemisecordite
|-
| | 21
| | 244.660
| | Marvel and Benediction
|-
| | 22
| | 256.311
| | Hemisecordite
|-
| | 23
| | 267.961
| | Marvel and Benediction, Hemisecordite
|-
| | 24
| | 279.712
| | Marvel and Benediction
|-
| | 25
| | 291.262
| | Hemisecordite
|-
| | 26
| | 303.013
| | Marvel and Benediction
|-
| | 27
| | 314.563
| | Marvel and Benediction, Hemisecordite
|-
| | 28
| | 326.214
| | Hemisecordite
|-
| | 29
| | 337.864
| | Marvel and Benediction
|-
| | 30
| | 349.615
| | Marvel and Benediction, Hemisecordite
|-
| | 31
| | 361.165
| | Marvel and Benediction
|-
| | 32
| | 372.816
| | Hemisecordite
|-
| | 33
| | 384.466
| | Marvel and Benediction, Hemisecordite
|-
| | 34
| | 396.117
| | Marvel and Benediction
|-
| | 35
| | 407.767
| | Hemisecordite
|-
| | 36
| | 419.4175
| | Marvel and Benediction
|-
| | 37
| | 431.068
| | Marvel and Benediction, Hemisecordite
|-
| | 38
| | 442.708
| | Hemisecordite
|-
| | 39
| | 454.369
| | Marvel and Benediction
|-
| | 40
| | 466.019
| | Marvel and Benediction, Hemisecordite
|-
| | 41
| | 477.670
| | Marvel and Benediction
|-
| | 42
| | 489.320
| | Hemisecordite
|-
| | 43
| | 500.971
| | Marvel and Benediction, Hemisecordite
|-
| | 44
| | 512.621
| | Marvel and Benediction
|-
| | 45
| | 524.272
| | Hemisecordite
|-
| | 46
| | 535.922
| | Marvel and Benediction
|-
| | 47
| | 547.573
| | Marevel and Benediction, Hemisecordite
|-
| | 48
| | 559.223
| | Hemisecordite
|-
| | 49
| | 570.874
| | Marvel and Benediction
|-
| | 50
| | 582.524
| | Marvel and Benediction, Hemisecordite
|-
| | 52
| | 605.825
| | Marvel and Benediction, Hemisecordite
|-
| | 53
| | 617.476
| | Marvel and Benediction, Hemisecordite
|-
| | 54
| | 629.126
| | Marvel and Benediction
|-
| | 55
| | 640.777
| | Hemisecordite
|-
| | 56
| | 652.427
| | Marvel and Benediction, Hemisecordite
|-
| | 57
| | 664.078
| | Marvel and Benediction
|-
| | 58
| | 675.728
| | Hemisecordite
|-
| | 59
| | 687.379
| | Marvel and Benediction
|-
| | 60
| | 699.029
| | Marvel and Benediction, Hemisecordite
|-
| | 61
| | 710.677
| | Hemisecordite
|-
| | 62
| | 722.330
| | Marvel and Benediction
|-
| | 63
| | 733.981
| | Marvel and Benediction, Hemisecordite
|-
| | 64
| | 745.631
| | Marvel and Benediction
|-
| | 65
| | 757.282
| | Hemisecordite
|-
| | 66
| | 768.932
| | Marvel and Benediction, Hemisecordite
|-
| | 67
| | 780.583
| | Marvel and Benediction
|-
| | 68
| | 792.233
| | Hemisecordite
|-
| | 69
| | 803.883
| | Marvel and Benediction
|-
| | 70
| | 815.534
| | Marvel and Benediction, Hemisecordite
|-
| | 71
| | 827.185
| | Hemisecordite
|-
| | 72
| | 838.835
| | Marvel and Benediction
|-
| | 73
| | 850.385
| | Marvel and Benediction. Hemisecordite
|-
| | 74
| | 862.156
| | Marvel and Benediction
|-
| | 75
| | 873.786
| | Hemisecordite
|-
| | 76
| | 885.537
| | Marvel and Benediction, Hemisecordite
|-
| | 77
| | 896.987
| | Marvel and Benediction
|-
| | 78
| | 908.638
| | Hemisecordite
|-
| | 79
| | 920.388
| | Marvel and Benediction, Hemisecordite
|-
| | 80
| | 932.039
| | Marvel and Benediction
|-
| | 81
| | 943.689
| | Hemisecordite
|-
| | 82
| | 955.34
| | Marvel and Benediction
|-
| | 83
| | 966.99
| | Marvel and Benediction, Hemisecordite
|-
| | 84
| | 978.641
| | Marvel and Benediction
|-
| | 85
| | 990.291
| | Hemisecordite
|-
| | 86
| | 1001.942
| | Marvel and Benediction, Hemisecordite
|-
| | 87
| | 1013.592
| | Marvel and Benediction
|-
| | 88
| | 1025.243
| | Hemisecordite
|-
| | 89
| | 1036.893
| | Marvel and Benediction
|-
| | 90
| | 1048.544
| | Marvel and Benediction, Hemisecordite
|-
| | 91
| | 1060.194
| | Hemisecordite
|-
| | 92
| | 1071.845
| | Marvel and Benediction
|-
| | 93
| | 1083.495
| | Marvel and Benediction, Hemisecordite
|-
| | 94
| | 1095.146
| | Marvel and Benediction
|-
| | 95
| | 1106.796
| | Hemisecordite
|-
| | 96
| | 1118.447
| | Marvel and Benediction, Hemisecordite
|-
| | 97
| | 1130.097
| | Marvel and Benediction
|-
| | 98
| | 1141.748
| | Hemisecordite
|-
| | 99
| | 1153.398
| | Marvel and Benediction
|-
| | 100
| | 1165.0485
| | Marvel and Benediction, Hemisecordite
|-
| | 101
| | 1176.609
| | Hemisecordite
|-
| | 102
| | 1188.3495
| | Marvel and Benediction
|}
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
|11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
|19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
|25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
|19L 9s
|-
|29
|16L 13s
|-
|30
|13L 17s
|-
|31
|10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
|33L 2s
| rowspan="17" |3:2
|-
|36
|31L 5s
|-
|37
|29L 8s
|-
|38
|27L 11s
|-
|39
|25L 14s
|-
|40
|23L 17s
|-
|41
|21L 20s
|-
|42
|19L 23s 
|-
|43
|17L 26L
|-
|44
|15L 29s
|-
|45
|13L 32s
|-
|46
|11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
|51L 1s
| rowspan="31" |2:1
|-
|53
|50L 3s
|-
|54
|49L 5s
|-
|55
|48L 7s
|-
|56
|47L 9s
|-
|57
|46L 11s
|-
|58
|45L 13s
|-
|59
|44L 15s
|-
|60
|43L 17s
|-
|61
|42L 19s
|-
|62
|41L 21s
|-
|63
|40L 23s
|-
|64
|39L 25s
|-
|65
|38L 27s
|-
|66
|37L 29s
|-
|67
|36L 31s
|-
|68
|35L 33s
|-
|69
|34L 35s
|-
|70
|33L 37s
|-
|71
|32L 39s
|-
|72
|31L 41s
|-
|73
|30L 43s
|-
|74
|29L 45s
|-
|75
|28L 47s
|-
|76
|27L 49s
|-
|77
|26L 51s
|-
|78
|25L 53s
|-
|79
|24L 55s
|-
|80
|23L 57s
|-
|81
|22L 59s
|-
|82
|21L 61s
|}
[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>0wr7i0i64boeig85shuo59tyyr92had</sha1>
    </revision>
    <revision>
      <id>68192</id>
      <parentid>68028</parentid>
      <timestamp>2021-04-22T03:51:52Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68192</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11132" sha1="kqthrp0mk8hlb8ufs88ho1vhtwstyac" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic_clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]] limits and [[Gamelismic_clan|benediction]] and [[Gamelismic_clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel_family#Marvel|marvel temperament]] as well as [[Gamelismic_clan#Miracle|benediction]] and [[Gamelismic_clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime_numbers|prime]] edo.
{{Primes in edo|103}}

===13-limit temperaments===

{| class="wikitable"
|-
| | Degree
| | cents
| | temperament
|-
| | 1
| | 11.6505
| |Marvel and Benediction
|-
| |2
| |23.301
| |Hemisecordite
|-
| |3
| |34.9515
| |Marvel and Benediction, Hemisecordite
|-
| |4
| |46.602
| |Marvel and Benediction
|-
| |5
| |58.252
| |Hemisecordite
|-
| |6
| |69.903
| |Marvel and Benediction
|-
| |7
| |81.553
| |Marvel and Benediction, Hemisecordite
|-
| |8
| |93.204
| |Hemisecordite
|-
| |9
| |104.854
| |Marvel and Benediction
|-
| |10
| |116.5045
| |Marvel and Benediction, Hemisecordite
|-
| |11
| |128.155
| |Marvel and Benediction
|-
| |12
| |139.806
| |Hemisecordite
|-
| |13
| |151.456
| |Marvel and Benediction, Hemisecordite
|-
| |14
| |163.107
| |Marvel and Benediction
|-
| |15
| |174.757
| |Hemisecordite
|-
| |16
| |186.408
| |Marvel and Benediction
|-
| |17
| |198.058
| |Marvel and Benediction, Hemisecordite
|-
| |18
| |209.708
| |Hemisecordite
|-
| |19
| |221.359
| |Marvel and Benediction
|-
| |20
| |233.010
| |Marvel and Benediction, Hemisecordite
|-
| |21
| |244.660
| |Marvel and Benediction
|-
| |22
| |256.311
| |Hemisecordite
|-
| |23
| |267.961
| |Marvel and Benediction, Hemisecordite
|-
| |24
| |279.712
| |Marvel and Benediction
|-
| |25
| |291.262
| |Hemisecordite
|-
| |26
| |303.013
| |Marvel and Benediction
|-
| |27
| |314.563
| |Marvel and Benediction, Hemisecordite
|-
| |28
| |326.214
| |Hemisecordite
|-
| |29
| |337.864
| |Marvel and Benediction
|-
| |30
| |349.615
| |Marvel and Benediction, Hemisecordite
|-
| |31
| |361.165
| |Marvel and Benediction
|-
| |32
| |372.816
| |Hemisecordite
|-
| |33
| |384.466
| |Marvel and Benediction, Hemisecordite
|-
| |34
| |396.117
| |Marvel and Benediction
|-
| |35
| |407.767
| |Hemisecordite
|-
| |36
| |419.4175
| |Marvel and Benediction
|-
| |37
| |431.068
| |Marvel and Benediction, Hemisecordite
|-
| |38
| |442.708
| |Hemisecordite
|-
| |39
| |454.369
| |Marvel and Benediction
|-
| |40
| |466.019
| |Marvel and Benediction, Hemisecordite
|-
| |41
| |477.670
| |Marvel and Benediction
|-
| |42
| |489.320
| |Hemisecordite
|-
| |43
| |500.971
| |Marvel and Benediction, Hemisecordite
|-
| |44
| |512.621
| |Marvel and Benediction
|-
| |45
| |524.272
| |Hemisecordite
|-
| |46
| |535.922
| |Marvel and Benediction
|-
| |47
| |547.573
| |Marevel and Benediction, Hemisecordite
|-
| |48
| |559.223
| |Hemisecordite
|-
| |49
| |570.874
| |Marvel and Benediction
|-
| |50
| |582.524
| |Marvel and Benediction, Hemisecordite
|-
| |52
| |605.825
| |Marvel and Benediction, Hemisecordite
|-
| |53
| |617.476
| |Marvel and Benediction, Hemisecordite
|-
| |54
| |629.126
| |Marvel and Benediction
|-
| |55
| |640.777
| |Hemisecordite
|-
| |56
| |652.427
| |Marvel and Benediction, Hemisecordite
|-
| |57
| |664.078
| |Marvel and Benediction
|-
| |58
| |675.728
| |Hemisecordite
|-
| |59
| |687.379
| |Marvel and Benediction
|-
| |60
| |699.029
| |Marvel and Benediction, Hemisecordite
|-
| |61
| |710.677
| |Hemisecordite
|-
| |62
| |722.330
| |Marvel and Benediction
|-
| |63
| |733.981
| |Marvel and Benediction, Hemisecordite
|-
| |64
| |745.631
| |Marvel and Benediction
|-
| |65
| |757.282
| |Hemisecordite
|-
| |66
| |768.932
| |Marvel and Benediction, Hemisecordite
|-
| |67
| |780.583
| |Marvel and Benediction
|-
| |68
| |792.233
| |Hemisecordite
|-
| |69
| |803.883
| |Marvel and Benediction
|-
| |70
| |815.534
| |Marvel and Benediction, Hemisecordite
|-
| |71
| |827.185
| |Hemisecordite
|-
| |72
| |838.835
| |Marvel and Benediction
|-
| |73
| |850.385
| |Marvel and Benediction. Hemisecordite
|-
| |74
| |862.156
| |Marvel and Benediction
|-
| |75
| |873.786
| |Hemisecordite
|-
| |76
| |885.537
| |Marvel and Benediction, Hemisecordite
|-
| |77
| |896.987
| |Marvel and Benediction
|-
| |78
| |908.638
| |Hemisecordite
|-
| |79
| |920.388
| |Marvel and Benediction, Hemisecordite
|-
| |80
| |932.039
| |Marvel and Benediction
|-
| |81
| |943.689
| |Hemisecordite
|-
| |82
| |955.34
| |Marvel and Benediction
|-
| |83
| |966.99
| |Marvel and Benediction, Hemisecordite
|-
| |84
| |978.641
| |Marvel and Benediction
|-
| |85
| |990.291
| |Hemisecordite
|-
| |86
| |1001.942
| |Marvel and Benediction, Hemisecordite
|-
| |87
| |1013.592
| |Marvel and Benediction
|-
| |88
| |1025.243
| |Hemisecordite
|-
| |89
| |1036.893
| |Marvel and Benediction
|-
| |90
| |1048.544
| |Marvel and Benediction, Hemisecordite
|-
| |91
| |1060.194
| |Hemisecordite
|-
| |92
| |1071.845
| |Marvel and Benediction
|-
| |93
| |1083.495
| |Marvel and Benediction, Hemisecordite
|-
| |94
| |1095.146
| |Marvel and Benediction
|-
| |95
| |1106.796
| |Hemisecordite
|-
| |96
| |1118.447
| |Marvel and Benediction, Hemisecordite
|-
| |97
| |1130.097
| |Marvel and Benediction
|-
| |98
| |1141.748
| |Hemisecordite
|-
| |99
| |1153.398
| |Marvel and Benediction
|-
| |100
| |1165.0485
| |Marvel and Benediction, Hemisecordite
|-
| |101
| |1176.609
| |Hemisecordite
|-
| |102
| |1188.3495
| |Marvel and Benediction
|}
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

===13-limit temperaments===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.30097087
|3.946132164
|-
|3
|34.9515
|1.618¢
|3
|34.95145631
| -3.758221109
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}
[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>kqthrp0mk8hlb8ufs88ho1vhtwstyac</sha1>
    </revision>
    <revision>
      <id>68193</id>
      <parentid>68192</parentid>
      <timestamp>2021-04-22T03:52:54Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* 13-limit temperaments */</comment>
      <origin>68193</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11115" sha1="okp0kpv38u4jpxez377uz4snllokz96" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. It is a good [[Gamelismic_clan|miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]] limits and [[Gamelismic_clan|benediction]] and [[Gamelismic_clan|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224, 1029/1024 and 2401/2400 in the 7-limit; 243/242, 441/440 and 540/539 in the [[11-limit]]; 351/350 and 847/845 in the 13-limit. In the 13-limit it provides the optimal patent val for [[Marvel_family#Marvel|marvel temperament]] as well as [[Gamelismic_clan#Miracle|benediction]] and [[Gamelismic_clan#Hemisecordite|hemisecordite]].

103edo is the 27th [[prime_numbers|prime]] edo.
{{Primes in edo|103}}

===13-limit temperaments===

{| class="wikitable"
|-
| | Degree
| | cents
| | temperament
|-
| | 1
| | 11.6505
| |Marvel and Benediction
|-
| |2
| |23.301
| |Hemisecordite
|-
| |3
| |34.9515
| |Marvel and Benediction, Hemisecordite
|-
| |4
| |46.602
| |Marvel and Benediction
|-
| |5
| |58.252
| |Hemisecordite
|-
| |6
| |69.903
| |Marvel and Benediction
|-
| |7
| |81.553
| |Marvel and Benediction, Hemisecordite
|-
| |8
| |93.204
| |Hemisecordite
|-
| |9
| |104.854
| |Marvel and Benediction
|-
| |10
| |116.5045
| |Marvel and Benediction, Hemisecordite
|-
| |11
| |128.155
| |Marvel and Benediction
|-
| |12
| |139.806
| |Hemisecordite
|-
| |13
| |151.456
| |Marvel and Benediction, Hemisecordite
|-
| |14
| |163.107
| |Marvel and Benediction
|-
| |15
| |174.757
| |Hemisecordite
|-
| |16
| |186.408
| |Marvel and Benediction
|-
| |17
| |198.058
| |Marvel and Benediction, Hemisecordite
|-
| |18
| |209.708
| |Hemisecordite
|-
| |19
| |221.359
| |Marvel and Benediction
|-
| |20
| |233.010
| |Marvel and Benediction, Hemisecordite
|-
| |21
| |244.660
| |Marvel and Benediction
|-
| |22
| |256.311
| |Hemisecordite
|-
| |23
| |267.961
| |Marvel and Benediction, Hemisecordite
|-
| |24
| |279.712
| |Marvel and Benediction
|-
| |25
| |291.262
| |Hemisecordite
|-
| |26
| |303.013
| |Marvel and Benediction
|-
| |27
| |314.563
| |Marvel and Benediction, Hemisecordite
|-
| |28
| |326.214
| |Hemisecordite
|-
| |29
| |337.864
| |Marvel and Benediction
|-
| |30
| |349.615
| |Marvel and Benediction, Hemisecordite
|-
| |31
| |361.165
| |Marvel and Benediction
|-
| |32
| |372.816
| |Hemisecordite
|-
| |33
| |384.466
| |Marvel and Benediction, Hemisecordite
|-
| |34
| |396.117
| |Marvel and Benediction
|-
| |35
| |407.767
| |Hemisecordite
|-
| |36
| |419.4175
| |Marvel and Benediction
|-
| |37
| |431.068
| |Marvel and Benediction, Hemisecordite
|-
| |38
| |442.708
| |Hemisecordite
|-
| |39
| |454.369
| |Marvel and Benediction
|-
| |40
| |466.019
| |Marvel and Benediction, Hemisecordite
|-
| |41
| |477.670
| |Marvel and Benediction
|-
| |42
| |489.320
| |Hemisecordite
|-
| |43
| |500.971
| |Marvel and Benediction, Hemisecordite
|-
| |44
| |512.621
| |Marvel and Benediction
|-
| |45
| |524.272
| |Hemisecordite
|-
| |46
| |535.922
| |Marvel and Benediction
|-
| |47
| |547.573
| |Marevel and Benediction, Hemisecordite
|-
| |48
| |559.223
| |Hemisecordite
|-
| |49
| |570.874
| |Marvel and Benediction
|-
| |50
| |582.524
| |Marvel and Benediction, Hemisecordite
|-
| |52
| |605.825
| |Marvel and Benediction, Hemisecordite
|-
| |53
| |617.476
| |Marvel and Benediction, Hemisecordite
|-
| |54
| |629.126
| |Marvel and Benediction
|-
| |55
| |640.777
| |Hemisecordite
|-
| |56
| |652.427
| |Marvel and Benediction, Hemisecordite
|-
| |57
| |664.078
| |Marvel and Benediction
|-
| |58
| |675.728
| |Hemisecordite
|-
| |59
| |687.379
| |Marvel and Benediction
|-
| |60
| |699.029
| |Marvel and Benediction, Hemisecordite
|-
| |61
| |710.677
| |Hemisecordite
|-
| |62
| |722.330
| |Marvel and Benediction
|-
| |63
| |733.981
| |Marvel and Benediction, Hemisecordite
|-
| |64
| |745.631
| |Marvel and Benediction
|-
| |65
| |757.282
| |Hemisecordite
|-
| |66
| |768.932
| |Marvel and Benediction, Hemisecordite
|-
| |67
| |780.583
| |Marvel and Benediction
|-
| |68
| |792.233
| |Hemisecordite
|-
| |69
| |803.883
| |Marvel and Benediction
|-
| |70
| |815.534
| |Marvel and Benediction, Hemisecordite
|-
| |71
| |827.185
| |Hemisecordite
|-
| |72
| |838.835
| |Marvel and Benediction
|-
| |73
| |850.385
| |Marvel and Benediction. Hemisecordite
|-
| |74
| |862.156
| |Marvel and Benediction
|-
| |75
| |873.786
| |Hemisecordite
|-
| |76
| |885.537
| |Marvel and Benediction, Hemisecordite
|-
| |77
| |896.987
| |Marvel and Benediction
|-
| |78
| |908.638
| |Hemisecordite
|-
| |79
| |920.388
| |Marvel and Benediction, Hemisecordite
|-
| |80
| |932.039
| |Marvel and Benediction
|-
| |81
| |943.689
| |Hemisecordite
|-
| |82
| |955.34
| |Marvel and Benediction
|-
| |83
| |966.99
| |Marvel and Benediction, Hemisecordite
|-
| |84
| |978.641
| |Marvel and Benediction
|-
| |85
| |990.291
| |Hemisecordite
|-
| |86
| |1001.942
| |Marvel and Benediction, Hemisecordite
|-
| |87
| |1013.592
| |Marvel and Benediction
|-
| |88
| |1025.243
| |Hemisecordite
|-
| |89
| |1036.893
| |Marvel and Benediction
|-
| |90
| |1048.544
| |Marvel and Benediction, Hemisecordite
|-
| |91
| |1060.194
| |Hemisecordite
|-
| |92
| |1071.845
| |Marvel and Benediction
|-
| |93
| |1083.495
| |Marvel and Benediction, Hemisecordite
|-
| |94
| |1095.146
| |Marvel and Benediction
|-
| |95
| |1106.796
| |Hemisecordite
|-
| |96
| |1118.447
| |Marvel and Benediction, Hemisecordite
|-
| |97
| |1130.097
| |Marvel and Benediction
|-
| |98
| |1141.748
| |Hemisecordite
|-
| |99
| |1153.398
| |Marvel and Benediction
|-
| |100
| |1165.0485
| |Marvel and Benediction, Hemisecordite
|-
| |101
| |1176.609
| |Hemisecordite
|-
| |102
| |1188.3495
| |Marvel and Benediction
|}
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

===13-limit temperaments===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}
[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>okp0kpv38u4jpxez377uz4snllokz96</sha1>
    </revision>
    <revision>
      <id>73604</id>
      <parentid>68193</parentid>
      <timestamp>2021-07-03T14:45:16Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Cleanup</comment>
      <origin>73604</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8541" sha1="fsw31bo9h8212n7fdkkrnbsdsdcm2v0" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]]limit and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Primes in edo|103}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Temperament
|-
| 1
| 11.6505
| Marvel and Benediction
|-
| 2
| 23.301
| Hemisecordite
|-
| 3
| 34.9515
| Marvel and Benediction, Hemisecordite
|-
| 4
| 46.602
| Marvel and Benediction
|-
| 5
| 58.252
| Hemisecordite
|-
| 6
| 69.903
| Marvel and Benediction
|-
| 7
| 81.553
| Marvel and Benediction, Hemisecordite
|-
| 8
| 93.204
| Hemisecordite
|-
| 9
| 104.854
| Marvel and Benediction
|-
| 10
| 116.5045
| Marvel and Benediction, Hemisecordite
|-
| 11
| 128.155
| Marvel and Benediction
|-
| 12
| 139.806
| Hemisecordite
|-
| 13
| 151.456
| Marvel and Benediction, Hemisecordite
|-
| 14
| 163.107
| Marvel and Benediction
|-
| 15
| 174.757
| Hemisecordite
|-
| 16
| 186.408
| Marvel and Benediction
|-
| 17
| 198.058
| Marvel and Benediction, Hemisecordite
|-
| 18
| 209.708
| Hemisecordite
|-
| 19
| 221.359
| Marvel and Benediction
|-
| 20
| 233.010
| Marvel and Benediction, Hemisecordite
|-
| 21
| 244.660
| Marvel and Benediction
|-
| 22
| 256.311
| Hemisecordite
|-
| 23
| 267.961
| Marvel and Benediction, Hemisecordite
|-
| 24
| 279.712
| Marvel and Benediction
|-
| 25
| 291.262
| Hemisecordite
|-
| 26
| 303.013
| Marvel and Benediction
|-
| 27
| 314.563
| Marvel and Benediction, Hemisecordite
|-
| 28
| 326.214
| Hemisecordite
|-
| 29
| 337.864
| Marvel and Benediction
|-
| 30
| 349.615
| Marvel and Benediction, Hemisecordite
|-
| 31
| 361.165
| Marvel and Benediction
|-
| 32
| 372.816
| Hemisecordite
|-
| 33
| 384.466
| Marvel and Benediction, Hemisecordite
|-
| 34
| 396.117
| Marvel and Benediction
|-
| 35
| 407.767
| Hemisecordite
|-
| 36
| 419.4175
| Marvel and Benediction
|-
| 37
| 431.068
| Marvel and Benediction, Hemisecordite
|-
| 38
| 442.708
| Hemisecordite
|-
| 39
| 454.369
| Marvel and Benediction
|-
| 40
| 466.019
| Marvel and Benediction, Hemisecordite
|-
| 41
| 477.670
| Marvel and Benediction
|-
| 42
| 489.320
| Hemisecordite
|-
| 43
| 500.971
| Marvel and Benediction, Hemisecordite
|-
| 44
| 512.621
| Marvel and Benediction
|-
| 45
| 524.272
| Hemisecordite
|-
| 46
| 535.922
| Marvel and Benediction
|-
| 47
| 547.573
| Marevel and Benediction, Hemisecordite
|-
| 48
| 559.223
| Hemisecordite
|-
| 49
| 570.874
| Marvel and Benediction
|-
| 50
| 582.524
| Marvel and Benediction, Hemisecordite
|-
| …
| …
| …
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>fsw31bo9h8212n7fdkkrnbsdsdcm2v0</sha1>
    </revision>
    <revision>
      <id>73605</id>
      <parentid>73604</parentid>
      <timestamp>2021-07-03T14:56:22Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ unify precision and add the 51th step that was missing in the original table</comment>
      <origin>73605</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8558" sha1="na2lc2nv6g4xie2cygvhkrvliyzhnje" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]]limit and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Primes in edo|103}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Temperament
|-
| 1
| 11.650
| Marvel and Benediction
|-
| 2
| 23.301
| Hemisecordite
|-
| 3
| 34.951
| Marvel and Benediction, Hemisecordite
|-
| 4
| 46.602
| Marvel and Benediction
|-
| 5
| 58.252
| Hemisecordite
|-
| 6
| 69.903
| Marvel and Benediction
|-
| 7
| 81.553
| Marvel and Benediction, Hemisecordite
|-
| 8
| 93.204
| Hemisecordite
|-
| 9
| 104.854
| Marvel and Benediction
|-
| 10
| 116.505
| Marvel and Benediction, Hemisecordite
|-
| 11
| 128.155
| Marvel and Benediction
|-
| 12
| 139.806
| Hemisecordite
|-
| 13
| 151.456
| Marvel and Benediction, Hemisecordite
|-
| 14
| 163.107
| Marvel and Benediction
|-
| 15
| 174.757
| Hemisecordite
|-
| 16
| 186.408
| Marvel and Benediction
|-
| 17
| 198.058
| Marvel and Benediction, Hemisecordite
|-
| 18
| 209.708
| Hemisecordite
|-
| 19
| 221.359
| Marvel and Benediction
|-
| 20
| 233.010
| Marvel and Benediction, Hemisecordite
|-
| 21
| 244.660
| Marvel and Benediction
|-
| 22
| 256.311
| Hemisecordite
|-
| 23
| 267.961
| Marvel and Benediction, Hemisecordite
|-
| 24
| 279.712
| Marvel and Benediction
|-
| 25
| 291.262
| Hemisecordite
|-
| 26
| 303.013
| Marvel and Benediction
|-
| 27
| 314.563
| Marvel and Benediction, Hemisecordite
|-
| 28
| 326.214
| Hemisecordite
|-
| 29
| 337.864
| Marvel and Benediction
|-
| 30
| 349.615
| Marvel and Benediction, Hemisecordite
|-
| 31
| 361.165
| Marvel and Benediction
|-
| 32
| 372.816
| Hemisecordite
|-
| 33
| 384.466
| Marvel and Benediction, Hemisecordite
|-
| 34
| 396.117
| Marvel and Benediction
|-
| 35
| 407.767
| Hemisecordite
|-
| 36
| 419.417
| Marvel and Benediction
|-
| 37
| 431.068
| Marvel and Benediction, Hemisecordite
|-
| 38
| 442.708
| Hemisecordite
|-
| 39
| 454.369
| Marvel and Benediction
|-
| 40
| 466.019
| Marvel and Benediction, Hemisecordite
|-
| 41
| 477.670
| Marvel and Benediction
|-
| 42
| 489.320
| Hemisecordite
|-
| 43
| 500.971
| Marvel and Benediction, Hemisecordite
|-
| 44
| 512.621
| Marvel and Benediction
|-
| 45
| 524.272
| Hemisecordite
|-
| 46
| 535.922
| Marvel and Benediction
|-
| 47
| 547.573
| Marevel and Benediction, Hemisecordite
|-
| 48
| 559.223
| Hemisecordite
|-
| 49
| 570.874
| Marvel and Benediction
|-
| 50
| 582.524
| Marvel and Benediction, Hemisecordite
|-
| 51
| 594.175
| 
|-
| …
| …
| …
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>na2lc2nv6g4xie2cygvhkrvliyzhnje</sha1>
    </revision>
    <revision>
      <id>73670</id>
      <parentid>73605</parentid>
      <timestamp>2021-07-04T13:55:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+rank-2 temperaments</comment>
      <origin>73670</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9555" sha1="drj279wtzxze4wfrqck2h0n65umfwhp" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]]limit and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Primes in edo|103}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Temperament
|-
| 1
| 11.650
| Marvel and Benediction
|-
| 2
| 23.301
| Hemisecordite
|-
| 3
| 34.951
| Marvel and Benediction, Hemisecordite
|-
| 4
| 46.602
| Marvel and Benediction
|-
| 5
| 58.252
| Hemisecordite
|-
| 6
| 69.903
| Marvel and Benediction
|-
| 7
| 81.553
| Marvel and Benediction, Hemisecordite
|-
| 8
| 93.204
| Hemisecordite
|-
| 9
| 104.854
| Marvel and Benediction
|-
| 10
| 116.505
| Marvel and Benediction, Hemisecordite
|-
| 11
| 128.155
| Marvel and Benediction
|-
| 12
| 139.806
| Hemisecordite
|-
| 13
| 151.456
| Marvel and Benediction, Hemisecordite
|-
| 14
| 163.107
| Marvel and Benediction
|-
| 15
| 174.757
| Hemisecordite
|-
| 16
| 186.408
| Marvel and Benediction
|-
| 17
| 198.058
| Marvel and Benediction, Hemisecordite
|-
| 18
| 209.708
| Hemisecordite
|-
| 19
| 221.359
| Marvel and Benediction
|-
| 20
| 233.010
| Marvel and Benediction, Hemisecordite
|-
| 21
| 244.660
| Marvel and Benediction
|-
| 22
| 256.311
| Hemisecordite
|-
| 23
| 267.961
| Marvel and Benediction, Hemisecordite
|-
| 24
| 279.712
| Marvel and Benediction
|-
| 25
| 291.262
| Hemisecordite
|-
| 26
| 303.013
| Marvel and Benediction
|-
| 27
| 314.563
| Marvel and Benediction, Hemisecordite
|-
| 28
| 326.214
| Hemisecordite
|-
| 29
| 337.864
| Marvel and Benediction
|-
| 30
| 349.615
| Marvel and Benediction, Hemisecordite
|-
| 31
| 361.165
| Marvel and Benediction
|-
| 32
| 372.816
| Hemisecordite
|-
| 33
| 384.466
| Marvel and Benediction, Hemisecordite
|-
| 34
| 396.117
| Marvel and Benediction
|-
| 35
| 407.767
| Hemisecordite
|-
| 36
| 419.417
| Marvel and Benediction
|-
| 37
| 431.068
| Marvel and Benediction, Hemisecordite
|-
| 38
| 442.708
| Hemisecordite
|-
| 39
| 454.369
| Marvel and Benediction
|-
| 40
| 466.019
| Marvel and Benediction, Hemisecordite
|-
| 41
| 477.670
| Marvel and Benediction
|-
| 42
| 489.320
| Hemisecordite
|-
| 43
| 500.971
| Marvel and Benediction, Hemisecordite
|-
| 44
| 512.621
| Marvel and Benediction
|-
| 45
| 524.272
| Hemisecordite
|-
| 46
| 535.922
| Marvel and Benediction
|-
| 47
| 547.573
| Marevel and Benediction, Hemisecordite
|-
| 48
| 559.223
| Hemisecordite
|-
| 49
| 570.874
| Marvel and Benediction
|-
| 50
| 582.524
| Marvel and Benediction, Hemisecordite
|-
| 51
| 594.175
| 
|-
| …
| …
| …
|}

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>drj279wtzxze4wfrqck2h0n65umfwhp</sha1>
    </revision>
    <revision>
      <id>73672</id>
      <parentid>73670</parentid>
      <timestamp>2021-07-04T14:17:37Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ replace that (unuseful) temperament column with ratios</comment>
      <origin>73672</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8727" sha1="7uf2mdmmkms72jhvgxfxii3qlf5trbl" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]]limit and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Primes in edo|103}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>7uf2mdmmkms72jhvgxfxii3qlf5trbl</sha1>
    </revision>
    <revision>
      <id>78454</id>
      <parentid>73672</parentid>
      <timestamp>2021-09-23T20:01:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ +RTT table</comment>
      <origin>78454</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9663" sha1="qc23hd0yd3pcxakk0suzxh8rsmo5xes" xml:space="preserve">'''103edo''' is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]]limit and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Primes in edo|103}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>qc23hd0yd3pcxakk0suzxh8rsmo5xes</sha1>
    </revision>
    <revision>
      <id>78456</id>
      <parentid>78454</parentid>
      <timestamp>2021-09-23T20:04:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Improve intro</comment>
      <origin>78456</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9836" sha1="ct8m4rhcv5572yydn8bdxziap9wwskn" xml:space="preserve">The '''103 equal divisions of the octave''' ('''103edo'''), or the '''103(-tone) equal temperament''' ('''103tet''', '''103et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]]limit and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Primes in edo|103}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>ct8m4rhcv5572yydn8bdxziap9wwskn</sha1>
    </revision>
    <revision>
      <id>78459</id>
      <parentid>78456</parentid>
      <timestamp>2021-09-23T21:23:40Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ +septendesemi, +subsemifourth, +hemiseptisix</comment>
      <origin>78459</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9995" sha1="ih2eml0qokyrlkc7v7oqan4sur8g95f" xml:space="preserve">The '''103 equal divisions of the octave''' ('''103edo'''), or the '''103(-tone) equal temperament''' ('''103tet''', '''103et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]]limit and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Primes in edo|103}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>ih2eml0qokyrlkc7v7oqan4sur8g95f</sha1>
    </revision>
    <revision>
      <id>87437</id>
      <parentid>78459</parentid>
      <timestamp>2022-02-26T16:48:49Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87437</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9975" sha1="7ktfrq47bd5b02s6c6uk82gkb9ewkiy" xml:space="preserve">The '''103 equal divisions of the octave''' ('''103edo'''), or the '''103(-tone) equal temperament''' ('''103tet''', '''103et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 103 steps of size 11.6505 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]]limit and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Primes in edo|103}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]</text>
      <sha1>7ktfrq47bd5b02s6c6uk82gkb9ewkiy</sha1>
    </revision>
    <revision>
      <id>89464</id>
      <parentid>87437</parentid>
      <timestamp>2022-03-12T20:50:42Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+infobox and adopt new template</comment>
      <origin>89464</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10152" sha1="jo0lmfrfhfyobxxifork1c7zf2oqrcb" xml:space="preserve">{{Infobox ET
| Prime factorization = 103 (prime)
| Step size = 11.6505¢
| Fifth = 60\103 (699.0¢)
| Semitones = 8:9 (93.2¢ : 104.9¢)
| Consistency = 7
}}
The '''103 equal divisions of the octave''' ('''103edo'''), or the '''103(-tone) equal temperament''' ('''103tet''', '''103et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 103 steps of about 11.7 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]]limit and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

[[Category:Benediction]]
[[Category:Equal divisions of the octave]]
[[Category:Miracle]]
[[Category:Prime EDO]]</text>
      <sha1>jo0lmfrfhfyobxxifork1c7zf2oqrcb</sha1>
    </revision>
    <revision>
      <id>92913</id>
      <parentid>89464</parentid>
      <timestamp>2022-07-02T16:16:33Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92913</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10180" sha1="35987kf7lvi813llm6k9yhmd2zntthn" xml:space="preserve">{{Infobox ET
| Prime factorization = 103 (prime)
| Step size = 11.6505¢
| Fifth = 60\103 (699.0¢)
| Semitones = 8:9 (93.2¢ : 104.9¢)
| Consistency = 7
}}
The '''103 equal divisions of the octave''' ('''103edo'''), or the '''103(-tone) equal temperament''' ('''103tet''', '''103et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 103 steps of about 11.7 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit|13-]]limit and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Benediction]]
[[Category:Miracle]]
[[Category:Prime EDO]]</text>
      <sha1>35987kf7lvi813llm6k9yhmd2zntthn</sha1>
    </revision>
    <revision>
      <id>95225</id>
      <parentid>92913</parentid>
      <timestamp>2022-09-02T17:43:33Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>95225</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10215" sha1="10o3astnszwb1fcpybh04zr0jtfuxjo" xml:space="preserve">{{Infobox ET
| Prime factorization = 103 (prime)
| Step size = 11.6505¢
| Fifth = 60\103 (699.0¢)
| Semitones = 8:9 (93.2¢ : 104.9¢)
| Consistency = 7
}}
The '''103 equal divisions of the octave''' ('''103edo'''), or the '''103(-tone) equal temperament''' ('''103tet''', '''103et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 103 steps of about 11.7 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit]] and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
|1
|37\103
|431.06
|77/60
|[[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Benediction]]
[[Category:Miracle]]
[[Category:Prime EDO]]</text>
      <sha1>10o3astnszwb1fcpybh04zr0jtfuxjo</sha1>
    </revision>
    <revision>
      <id>96079</id>
      <parentid>95225</parentid>
      <timestamp>2022-10-04T16:00:21Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET now computes most parameters automatically</comment>
      <origin>96079</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10072" sha1="55h93nia5dso5f63ihtk90vvaew26om" xml:space="preserve">{{Infobox ET}}
The '''103 equal divisions of the octave''' ('''103edo'''), or the '''103(-tone) equal temperament''' ('''103tet''', '''103et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 103 steps of about 11.7 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit]] and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
|1
|37\103
|431.06
|77/60
|[[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Benediction]]
[[Category:Miracle]]
[[Category:Prime EDO]]</text>
      <sha1>55h93nia5dso5f63ihtk90vvaew26om</sha1>
    </revision>
    <revision>
      <id>112104</id>
      <parentid>96079</parentid>
      <timestamp>2023-05-05T16:49:02Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>Added music section</comment>
      <origin>112104</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10182" sha1="s5gn38saljm8yvxd85x4dhsqal8bxuo" xml:space="preserve">{{Infobox ET}}
The '''103 equal divisions of the octave''' ('''103edo'''), or the '''103(-tone) equal temperament''' ('''103tet''', '''103et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 103 steps of about 11.7 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit]] and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
|1
|37\103
|431.06
|77/60
|[[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
Since 103edo has a step of 11.6505 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 103edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|21:20
|-
|6
|[[1L 5s]]
|18:17
|-
|7
|[[5L 2s]]
|15:14
|-
|8
|[[7L 1s]]
|13:12
|-
|9
|[[4L 5s]]
|12:11
|-
|10
|[[3L 7s]]
|11:10
|-
|11
|[[4L 7s]]
|10:9
|-
|12
|[[7L 5s]]
|9:8
|-
|13
|[[12L 1s]]
| rowspan="2" |8:7
|-
|14
|[[3L 11s]]
|-
|15
|[[13L 2s]]
| rowspan="3" |7:6
|-
|16
|[[7L 9s]]
|-
|17
|1L 16s
|-
|18
| 11L 7s
| rowspan="3" |6:5
|-
|19
|8L 11s
|-
|20
|[[3L 17s]]
|-
|21
| 19L 2s
| rowspan="5" |5:4
|-
|22
|[[15L 7s]]
|-
|23
|[[11L 12s]]
|-
|24
|[[7L 17s]]
|-
|25
|3L 22s
|-
|26
| 25L 1s
| rowspan="9" |4:3
|-
|27
|[[22L 5s]]
|-
|28
| 19L 9s
|-
|29
| 16L 13s
|-
|30
| 13L 17s
|-
|31
| 10L 21s
|-
|32
|7L 25s
|-
|33
|4L 29s
|-
|34
|1L 33s
|-
|35
| 33L 2s
| rowspan="17" |3:2
|-
|36
| 31L 5s
|-
|37
| 29L 8s
|-
|38
| 27L 11s
|-
|39
| 25L 14s
|-
|40
| 23L 17s
|-
|41
| 21L 20s
|-
|42
| 19L 23s
|-
|43
| 17L 26L
|-
|44
| 15L 29s
|-
|45
| 13L 32s
|-
|46
| 11L 35s
|-
|47
|9L 38s
|-
|48
|7L 41s
|-
|49
|5L 44s
|-
|50
|3L 47s
|-
|51
|1L 50s
|-
|52
| 51L 1s
| rowspan="31" |2:1
|-
|53
| 50L 3s
|-
|54
| 49L 5s
|-
|55
| 48L 7s
|-
|56
| 47L 9s
|-
|57
| 46L 11s
|-
|58
| 45L 13s
|-
|59
| 44L 15s
|-
|60
| 43L 17s
|-
|61
| 42L 19s
|-
|62
| 41L 21s
|-
|63
| 40L 23s
|-
|64
| 39L 25s
|-
|65
| 38L 27s
|-
|66
| 37L 29s
|-
|67
| 36L 31s
|-
|68
| 35L 33s
|-
|69
| 34L 35s
|-
|70
| 33L 37s
|-
|71
| 32L 39s
|-
|72
| 31L 41s
|-
|73
| 30L 43s
|-
|74
| 29L 45s
|-
|75
| 28L 47s
|-
|76
| 27L 49s
|-
|77
| 26L 51s
|-
|78
| 25L 53s
|-
|79
| 24L 55s
|-
|80
| 23L 57s
|-
|81
| 22L 59s
|-
|82
| 21L 61s
|}

=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

== Music ==
* [https://www.youtube.com/watch?v=5pbnmzXAFcM Forest Tribe Dance] by [[User:Francium|Francium]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Benediction]]
[[Category:Miracle]]
[[Category:Prime EDO]]</text>
      <sha1>s5gn38saljm8yvxd85x4dhsqal8bxuo</sha1>
    </revision>
    <revision>
      <id>115188</id>
      <parentid>112104</parentid>
      <timestamp>2023-05-30T21:11:46Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Scales */</comment>
      <origin>115188</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8475" sha1="s64r59abhi9s8o4luurszo81ejbu91c" xml:space="preserve">{{Infobox ET}}
The '''103 equal divisions of the octave''' ('''103edo'''), or the '''103(-tone) equal temperament''' ('''103tet''', '''103et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 103 steps of about 11.7 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit]] and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
|1
|37\103
|431.06
|77/60
|[[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

== Music ==
* [https://www.youtube.com/watch?v=5pbnmzXAFcM Forest Tribe Dance] by [[User:Francium|Francium]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Benediction]]
[[Category:Miracle]]
[[Category:Prime EDO]]</text>
      <sha1>s64r59abhi9s8o4luurszo81ejbu91c</sha1>
    </revision>
    <revision>
      <id>122965</id>
      <parentid>115188</parentid>
      <timestamp>2023-08-28T14:43:41Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <origin>122965</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8614" sha1="dn15ki4vaddpbgrpehbhnurf2kgql1w" xml:space="preserve">{{Infobox ET}}
The '''103 equal divisions of the octave''' ('''103edo'''), or the '''103(-tone) equal temperament''' ('''103tet''', '''103et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 103 steps of about 11.7 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit]] and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

In 103edo, all intervals in the 17-odd-limit are consistent with only the exception of 9/8 and its octave complement which barely misses.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
|1
|37\103
|431.06
|77/60
|[[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

== Music ==
* [https://www.youtube.com/watch?v=5pbnmzXAFcM Forest Tribe Dance] by [[User:Francium|Francium]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Benediction]]
[[Category:Miracle]]
[[Category:Prime EDO]]</text>
      <sha1>dn15ki4vaddpbgrpehbhnurf2kgql1w</sha1>
    </revision>
    <revision>
      <id>122966</id>
      <parentid>122965</parentid>
      <timestamp>2023-08-28T14:44:10Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <origin>122966</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8619" sha1="b5eemug4hx7o742trm9rs8un76v5y4u" xml:space="preserve">{{Infobox ET}}
The '''103 equal divisions of the octave''' ('''103edo'''), or the '''103(-tone) equal temperament''' ('''103tet''', '''103et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 103 steps of about 11.7 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit]] and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

In 103edo, all intervals in the 17-odd-limit are consistent with only the exception of 9/8 and its octave complement 16/9 which barely misses.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
|1
|37\103
|431.06
|77/60
|[[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

== Music ==
* [https://www.youtube.com/watch?v=5pbnmzXAFcM Forest Tribe Dance] by [[User:Francium|Francium]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Benediction]]
[[Category:Miracle]]
[[Category:Prime EDO]]</text>
      <sha1>b5eemug4hx7o742trm9rs8un76v5y4u</sha1>
    </revision>
    <revision>
      <id>122967</id>
      <parentid>122966</parentid>
      <timestamp>2023-08-28T14:45:36Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>122967</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8624" sha1="kis753u8zx0veya94j0fhhfhzcynhol" xml:space="preserve">{{Infobox ET}}
The '''103 equal divisions of the octave''' ('''103edo'''), or the '''103(-tone) equal temperament''' ('''103tet''', '''103et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 103 steps of about 11.7 [[cent]]s each. 

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]] and [[13-limit]] and [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

103edo is the 27th [[prime numbers|prime]] edo.

In 103-EDO, all intervals within the 17-odd-limit are consistent, with the sole exception of 9/8 and its octave complement 16/9, which barely miss.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
|1
|37\103
|431.06
|77/60
|[[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensipent]] / [[sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" |Marvel and Benediction
! colspan="3" |Hemisecordite
|-
!Degree
!cents
!Difference from 72edo
!Degree
!cents
!Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

== Music ==
* [https://www.youtube.com/watch?v=5pbnmzXAFcM Forest Tribe Dance] by [[User:Francium|Francium]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Benediction]]
[[Category:Miracle]]
[[Category:Prime EDO]]</text>
      <sha1>kis753u8zx0veya94j0fhhfhzcynhol</sha1>
    </revision>
    <revision>
      <id>122970</id>
      <parentid>122967</parentid>
      <timestamp>2023-08-28T15:11:15Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Adopt template: EDO intro; cleanup; -redundant categories</comment>
      <origin>122970</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8292" sha1="kcydt0myfq63lsigknsl0cqrpg8k5bm" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

In 103edo, all intervals within the 17-odd-limit are [[consistent]], with the sole exception of 9/8 and its octave complement 16/9, which barely miss.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]].

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" | Marvel and Benediction
! colspan="3" | Hemisecordite
|-
! Degree
! cents
! Difference from 72edo
! Degree
! cents
! Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

== Music ==
* [https://www.youtube.com/watch?v=5pbnmzXAFcM ''Forest Tribe Dance''] by [[User:Francium|Francium]]

[[Category:Benediction]]
[[Category:Miracle]]</text>
      <sha1>kcydt0myfq63lsigknsl0cqrpg8k5bm</sha1>
    </revision>
    <revision>
      <id>122974</id>
      <parentid>122970</parentid>
      <timestamp>2023-08-28T15:43:04Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <origin>122974</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8315" sha1="8gphg5hxcuvpjeceej6dqtth9hxbo7k" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

In 103edo, all intervals within the 17-odd-limit are [[consistent]], with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]].

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" | Marvel and Benediction
! colspan="3" | Hemisecordite
|-
! Degree
! cents
! Difference from 72edo
! Degree
! cents
! Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

== Music ==
* [https://www.youtube.com/watch?v=5pbnmzXAFcM ''Forest Tribe Dance''] by [[User:Francium|Francium]]

[[Category:Benediction]]
[[Category:Miracle]]</text>
      <sha1>8gphg5hxcuvpjeceej6dqtth9hxbo7k</sha1>
    </revision>
    <revision>
      <id>127667</id>
      <parentid>122974</parentid>
      <timestamp>2023-11-20T06:14:40Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Music */ +links</comment>
      <origin>127667</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8462" sha1="78fkrssghsgjb2x068geg9rhom2p66p" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

In 103edo, all intervals within the 17-odd-limit are [[consistent]], with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]].

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" | Marvel and Benediction
! colspan="3" | Hemisecordite
|-
! Degree
! cents
! Difference from 72edo
! Degree
! cents
! Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Miracle]]</text>
      <sha1>78fkrssghsgjb2x068geg9rhom2p66p</sha1>
    </revision>
    <revision>
      <id>127668</id>
      <parentid>127667</parentid>
      <timestamp>2023-11-20T06:14:59Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <minor/>
      <comment>/* Music */</comment>
      <origin>127668</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8469" sha1="dn6deoxp7mvimuvpwmwb2zf4b79g1up" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

In 103edo, all intervals within the 17-odd-limit are [[consistent]], with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]].

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" | Marvel and Benediction
! colspan="3" | Hemisecordite
|-
! Degree
! cents
! Difference from 72edo
! Degree
! cents
! Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Miracle]]</text>
      <sha1>dn6deoxp7mvimuvpwmwb2zf4b79g1up</sha1>
    </revision>
    <revision>
      <id>132171</id>
      <parentid>127668</parentid>
      <timestamp>2024-01-20T00:53:01Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ Replaced incomplete manual table with template table</comment>
      <origin>132171</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6974" sha1="ndw83p64zr00cf2gmxygdrzuskfybs2" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

In 103edo, all intervals within the 17-odd-limit are [[consistent]], with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]].

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" | Marvel and Benediction
! colspan="3" | Hemisecordite
|-
! Degree
! cents
! Difference from 72edo
! Degree
! cents
! Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Miracle]]</text>
      <sha1>ndw83p64zr00cf2gmxygdrzuskfybs2</sha1>
    </revision>
    <revision>
      <id>132328</id>
      <parentid>132171</parentid>
      <timestamp>2024-01-21T08:45:58Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Revert to handpicked interval table</comment>
      <origin>132328</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8469" sha1="dn6deoxp7mvimuvpwmwb2zf4b79g1up" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

In 103edo, all intervals within the 17-odd-limit are [[consistent]], with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]].

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! Degree
! Cents
! Approximate Ratios
|-
| 1
| 11.650
| 81/80, 126/125
|-
| 2
| 23.301
| 65/64, 66/65, 78/77
|-
| 3
| 34.951
| 49/48, 50/49, 64/63
|-
| 4
| 46.602
| 33/32, 35/34, 36/35
|-
| 5
| 58.252
| 27/26, 34/33
|-
| 6
| 69.903
| 25/24, 26/25, 28/27
|-
| 7
| 81.553
| 21/20, 22/21
|-
| 8
| 93.204
| 18/17
|-
| 9
| 104.854
| 17/16
|-
| 10
| 116.505
| 15/14, 16/15
|-
| 11
| 128.155
| 14/13
|-
| 12
| 139.806
| 13/12
|-
| 13
| 151.456
| 12/11
|-
| 14
| 163.107
| 11/10
|-
| 15
| 174.757
| 72/65
|-
| 16
| 186.408
| 10/9
|-
| 17
| 198.058
| 9/8
|-
| 18
| 209.708
| 
|-
| 19
| 221.359
| 17/15, 25/22
|-
| 20
| 233.010
| 8/7
|-
| 21
| 244.660
| 15/13
|-
| 22
| 256.311
| 
|-
| 23
| 267.961
| 7/6
|-
| 24
| 279.712
| 20/17
|-
| 25
| 291.262
| 13/11
|-
| 26
| 303.013
| 25/21
|-
| 27
| 314.563
| 6/5
|-
| 28
| 326.214
| 63/52, 65/54
|-
| 29
| 337.864
| 17/14, 39/32
|-
| 30
| 349.615
| 11/9, 27/22
|-
| 31
| 361.165
| 16/13, 21/17
|-
| 32
| 372.816
| 26/21, 81/65
|-
| 33
| 384.466
| 5/4
|-
| 34
| 396.117
| 44/35
|-
| 35
| 407.767
| 33/26
|-
| 36
| 419.417
| 14/11
|-
| 37
| 431.068
| 9/7
|-
| 38
| 442.708
| 22/17
|-
| 39
| 454.369
| 13/10
|-
| 40
| 466.019
| 17/13, 21/16
|-
| 41
| 477.670
| 
|-
| 42
| 489.320
| 65/49
|-
| 43
| 500.971
| 4/3
|-
| 44
| 512.621
| 27/20
|-
| 45
| 524.272
| 65/48
|-
| 46
| 535.922
| 15/11
|-
| 47
| 547.573
| 11/8
|-
| 48
| 559.223
| 18/13
|-
| 49
| 570.874
| 25/18
|-
| 50
| 582.524
| 7/5
|-
| 51
| 594.175
| 24/17
|-
| …
| …
| …
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" | Marvel and Benediction
! colspan="3" | Hemisecordite
|-
! Degree
! cents
! Difference from 72edo
! Degree
! cents
! Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Miracle]]</text>
      <sha1>dn6deoxp7mvimuvpwmwb2zf4b79g1up</sha1>
    </revision>
    <revision>
      <id>132971</id>
      <parentid>132328</parentid>
      <timestamp>2024-01-26T09:51:36Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Intervals */ moved content</comment>
      <origin>132971</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6990" sha1="kf51iag9wpll4rdz9r06nuz5kyeys8m" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit|7-]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

In 103edo, all intervals within the 17-odd-limit are [[consistent]], with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Scales ==
=== 13-limit temperaments ===
{| class="wikitable"
|+
! colspan="3" | Marvel and Benediction
! colspan="3" | Hemisecordite
|-
! Degree
! cents
! Difference from 72edo
! Degree
! cents
! Difference from 62edo
|-
|1
|11.6505
| -5.016¢
|2
|23.301
|3.946¢
|-
|3
|34.9515
|1.618¢
|3
|34.9515
| -3.758¢
|-
|4
|46.602
| -3.398¢
|5
|58.252
|0.188¢
|-
|6
|69.903
|3.236¢
|7
|81.553
|4.134¢
|-
|7
|81.553
| -1.78¢
|8
|93.204
| -3.57¢
|-
|9
|104.854
|4.854¢
|10
|116.505
|0.376¢
|-
|10
|116.5045
| -0.162¢
|12
|139.806
|4.322¢
|-
|11
|128.155
| -5.178¢
|13
|151.456
| -3.382¢
|-
|13
|151.456¢
|1.456¢
|15
|174.757
|0.563¢
|-
|14
|163.107¢
| -3.56¢
|17
|198.058
|4.51¢
|-
|16
|186.408
|3.074¢
|18
|209.709
| -3.1945¢
|-
|17
|198.058
| -1.942¢
|20
|233.01
|0.751¢
|-
|19
|221.359
|4.693¢
|22
|256.311
|4.698¢
|-
|20
|233.01
| -0.324¢
|23
|267.961
| -3.007¢
|-
|21
|244.66
| -5.34¢
|25
|291.262
|0.94¢
|-
|23
|267.961
|1.2945¢
|27
|314.563
|4.886¢
|-
|24
|279.612
| -3.722¢
|28
|326.214
| -2.819¢
|-
|26
|302.913
|2.913¢
|30
|349.515
|1.1275¢
|-
|27
|314.563
| -2.104¢
|32
|372.8155
|5.074¢
|-
|29
|337.864
|4.531¢
|33
|384.466
| -2.631¢
|-
|30
|349.515
| -0.485¢
|35
|407.767
|1.315¢
|-
|31
|361.165
| -5.502¢
|37
|431.068
|5.2615¢
|-
|33
|384.466
|1.133¢
|38
|442.718
| -2.443¢
|-
|34
|396.1165
| -3.8835¢
|40
|466.0190
|1.503¢
|-
|36
|419.4175
|2.751¢
|42
|489.32
|5.449¢
|-
|37
|431.068
| -2.265¢
|43
|500.971
| -2.255¢
|-
|39
|454.369
|4.369¢
|45
|524.272
|1.691¢
|-
|40
|466.019
| -0.647¢
|47
|547.573
|5.637¢
|-
|41
|477.67
| -5.663¢
|48
|559.223
| -2.067¢
|-
|43
|500.971
|0.971¢
|50
|582.524
|1.879¢
|-
|44
|512.621¢
| -4.045¢
|52
|605.825
|5.825¢
|-
|46
|535.922¢
|2.589¢
|53
|617.476
| -1.879¢
|-
|47
|547.573¢
| -2.427¢
|55
|640.777¢
|2.067
|-
|49
|570.874¢
|4.207¢
|56
|652.427
| -5.637¢
|-
|50
|582.524
| -0.809¢
|58
|675.728
| -1.691
|-
|52
|605.825
|5.825¢
|60
|699.029
|2.255¢
|-
|53
|617.475
|0.809¢
|61
|710.68
| -5.449¢
|-
|54
|629.126¢
| -4.207¢
|63
|733.981
| -1.503¢
|-
|56
|652.427¢
|2.427¢
|65
|757.282
|2.443¢
|-
|57
|664.078
| -2.589¢
|66
|768.932
| -5.2615¢
|-
|59
|687.379
|4.045¢
|68
|792.233
| -1.315¢
|-
|60
|699.029
| -0.971¢
|70
|815.534
|2.631¢
|-
|62
|722.33
|5.663¢
|71
|827.1845
| -5.074¢
|-
|63
|733.981
|0.647¢
|73
|850.485
| -1.1275¢
|-
|64
|745.631
| -4.369¢
|75
|873.786
|2.819¢
|-
|66
|768.932
|2.265¢
|76
|885.437
| -4.886¢
|-
|67
|780.5825
| -2.751¢
|78
|908.738
| -0.94¢
|-
|69
|803.8835
|3.8835¢
|80
|932.039
|3.007¢
|-
|70
|815.534
| -1.133¢
|81
|943.689
| -4.698¢
|-
|72
|838.835
|5.501¢
|83
|966.99
| -0.752¢
|-
|73
|850.485
|0.485¢
|85
|990.291
|3.1945¢
|-
|74
|862.136
| -4.531¢
|86
|1001.942
| -4.51¢
|-
|76
|885.439
|2.104¢
|88
|1025.243
| -0.564¢
|-
|77
|897.087
| -2.913¢
|90
|1048.544
|3.382¢
|-
|79
|920.388
|3.722¢
|91
|1060.194
| -4.322¢
|-
|80
|932.039
| -1.2945¢
|93
|1083.495
| -0.376¢
|-
|82
|955.34
|5.34¢
|95
|1106.796
|3.57¢
|-
|83
|966.99
|0.324¢
|96
|1118.447¢
| -4.134¢
|-
|84
|978.641
| -4.693¢
|98
|1141.748
| -0.188¢
|-
|86
|1001.942
|1.942¢
|100
|1165.0485
|3.758¢
|-
|87
|1013.592
| -3.074¢
|101
|1176.699
| -3.946¢
|-
|89
|1036.893
|3.56¢
|
|
|
|-
|90
|1048.544
| -1.456¢
|
|
|
|-
|92
|1071.845
|5.178¢
|
|
|
|-
|93
|1083.495
|0.162¢
|
|
|
|-
|94
|1095.146
| -4.854¢
|
|
|
|-
|96
|1118.447
|1.78¢
|
|
|
|-
|97
|1130.097
| -3.236¢
|
|
|
|-
|99
|1153.398
|3.398¢
|
|
|
|-
|100
|1165.0485
| -1.618¢
|
|
|
|-
|102
|1188.3495
|5.016¢
|
|
|
|}

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Miracle]]</text>
      <sha1>kf51iag9wpll4rdz9r06nuz5kyeys8m</sha1>
    </revision>
    <revision>
      <id>133438</id>
      <parentid>132971</parentid>
      <timestamp>2024-01-29T04:34:40Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Fix typo in theory; -useless scale section</comment>
      <origin>133438</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3436" sha1="qwqzacnly72nyd8hfip1i89yv7ae9ko" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

In 103edo, all intervals within the 17-odd-limit are [[consistent]], with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Miracle]]</text>
      <sha1>qwqzacnly72nyd8hfip1i89yv7ae9ko</sha1>
    </revision>
    <revision>
      <id>135443</id>
      <parentid>133438</parentid>
      <timestamp>2024-02-13T17:43:41Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <minor/>
      <comment>+categories</comment>
      <origin>135443</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3456" sha1="k80nwq9xk0rb75vrx3asvmpyxoy2iuk" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

In 103edo, all intervals within the 17-odd-limit are [[consistent]], with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Miracle]]
[[Category:Listen]]</text>
      <sha1>k80nwq9xk0rb75vrx3asvmpyxoy2iuk</sha1>
    </revision>
    <revision>
      <id>140796</id>
      <parentid>135443</parentid>
      <timestamp>2024-04-07T17:42:47Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <origin>140796</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3626" sha1="rdbfvbwcdox296ioihc9xvaukj5v0gm" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

In 103edo, all intervals within the 17-odd-limit are [[consistent]], with the sole exception of 9/8 and its octave complement 16/9, which barely miss (relative error 50.2%).

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| [{{val| 103 166 }}]
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| [{{val| 103 166 239 }}]
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| [{{val| 103 166 239 289 }}]
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| [{{val| 103 166 239 289 356 }}]
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| [{{val| 103 166 239 289 356 381 }}]
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| [{{val| 103 166 239 289 356 381 421 }}]
| +0.694
| 0.595
| 5.10
|}
Its closest [[zeta peak index]], [[596zpi]], stretches the octave by +0.739 cents. This expansion is both consistent and uniquely consistent within the 15-integer limit.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammic]] (5-limit)
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Miracle]]
[[Category:Listen]]</text>
      <sha1>rdbfvbwcdox296ioihc9xvaukj5v0gm</sha1>
    </revision>
    <revision>
      <id>145901</id>
      <parentid>140796</parentid>
      <timestamp>2024-06-08T09:04:36Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup and consolidate tuning information</comment>
      <origin>145901</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3790" sha1="t90r40e79aphow79q5ebyuwis4a2odu" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Miracle]]
[[Category:Listen]]</text>
      <sha1>t90r40e79aphow79q5ebyuwis4a2odu</sha1>
    </revision>
    <revision>
      <id>146689</id>
      <parentid>145901</parentid>
      <timestamp>2024-06-27T09:55:45Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Subsets and supersets */</comment>
      <origin>146689</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3834" sha1="92cf4rd5gdbn0og6iaclna30x2uxy4i" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Miracle]]
[[Category:Listen]]</text>
      <sha1>92cf4rd5gdbn0og6iaclna30x2uxy4i</sha1>
    </revision>
    <revision>
      <id>165920</id>
      <parentid>146689</parentid>
      <timestamp>2024-11-16T04:50:20Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>165920</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3303" sha1="hjbx5dnhseukf61noqnpjx4vr7l3wyd" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Regular temperament properties ==
{{comma basis begin}}
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
{{comma basis end}}

=== Rank-2 temperaments ===
{{rank-2 begin}}
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
{{rank-2 end}}
{{orf}}

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) &amp;ndash; [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]</text>
      <sha1>hjbx5dnhseukf61noqnpjx4vr7l3wyd</sha1>
    </revision>
    <revision>
      <id>166159</id>
      <parentid>165920</parentid>
      <timestamp>2024-11-16T12:23:11Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Partial undo</comment>
      <origin>166159</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3860" sha1="0l4m8np0bmycdr2kdd15j46bf6d78os" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) &amp;ndash; [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]</text>
      <sha1>0l4m8np0bmycdr2kdd15j46bf6d78os</sha1>
    </revision>
    <revision>
      <id>176465</id>
      <parentid>166159</parentid>
      <timestamp>2025-01-16T14:19:58Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176465</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3850" sha1="i7923rin125l0qxub06qzjzfbzlng9q" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]</text>
      <sha1>i7923rin125l0qxub06qzjzfbzlng9q</sha1>
    </revision>
    <revision>
      <id>176975</id>
      <parentid>176465</parentid>
      <timestamp>2025-01-18T23:10:59Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <origin>176975</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4247" sha1="jaiin4i8lrsa9bfhog4ca5a4qozxjj5" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]] and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]] and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Zeta properties ==
===Zeta peak index===
{| class="wikitable"
! colspan="3" |Tuning
! colspan="3" |Strength
! colspan="2" |Closest EDO
! colspan="2" |Integer limit
|-
!ZPI
!Steps per octave
!Step size (cents)
!Height
!Integral
!Gap
!EDO
!Octave (cents)
!Consistent
!Distinct
|-
|[[596zpi]]
|102.936629522070
|11.6576577800491
|8.543510
|1.340775
|18.270998
|103edo
|1200.73875134506
|15
|15
|}
== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]</text>
      <sha1>jaiin4i8lrsa9bfhog4ca5a4qozxjj5</sha1>
    </revision>
    <revision>
      <id>177058</id>
      <parentid>176975</parentid>
      <timestamp>2025-01-19T02:43:22Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Formatting</comment>
      <origin>177058</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4279" sha1="dcss61ur9892vxolvy4de78hrgeq5qz" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]], and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Zeta properties ==
=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[596zpi]]
| 102.936629522070
| 11.6576577800491
| 8.543510
| 1.340775
| 18.270998
| 103edo
| 1200.73875134506
| 15
| 15
|}

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]</text>
      <sha1>dcss61ur9892vxolvy4de78hrgeq5qz</sha1>
    </revision>
    <revision>
      <id>177134</id>
      <parentid>177058</parentid>
      <timestamp>2025-01-19T10:10:40Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Consolidate sections</comment>
      <origin>177134</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4276" sha1="e1un6q4nf4nlspzcth4sqzn7gvyyusq" xml:space="preserve">{{Infobox ET}}
{{EDO intro|103}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]], and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[596zpi]]
| 102.936629522070
| 11.6576577800491
| 8.543510
| 1.340775
| 18.270998
| 103edo
| 1200.73875134506
| 15
| 15
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]</text>
      <sha1>e1un6q4nf4nlspzcth4sqzn7gvyyusq</sha1>
    </revision>
    <revision>
      <id>181306</id>
      <parentid>177134</parentid>
      <timestamp>2025-02-18T16:46:37Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181306</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4271" sha1="ipc99jfx07qio9ws3p45se9uasag8tg" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]], and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[596zpi]]
| 102.936629522070
| 11.6576577800491
| 8.543510
| 1.340775
| 18.270998
| 103edo
| 1200.73875134506
| 15
| 15
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]</text>
      <sha1>ipc99jfx07qio9ws3p45se9uasag8tg</sha1>
    </revision>
    <revision>
      <id>187300</id>
      <parentid>181306</parentid>
      <timestamp>2025-03-20T14:44:43Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Approximation to JI */</comment>
      <origin>187300</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4179" sha1="h6os11oepzhy7tlfwp981xlois42xpo" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]], and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 596
| steps = 102.936629522070
| step size = 11.6576577800491
| tempered height = 8.543510
| pure height = 5.620365
| integral = 1.340775
| gap = 18.270998
| octave = 1200.73875134506
| consistent = 15
| distinct = 15
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]</text>
      <sha1>h6os11oepzhy7tlfwp981xlois42xpo</sha1>
    </revision>
    <revision>
      <id>221280</id>
      <parentid>187300</parentid>
      <timestamp>2026-01-11T02:23:31Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Regular temperament properties */ note absolute error</comment>
      <origin>221280</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4370" sha1="4oqes2zh21wewlmj668rynxo0f0d3ii" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]], and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 596
| steps = 102.936629522070
| step size = 11.6576577800491
| tempered height = 8.543510
| pure height = 5.620365
| integral = 1.340775
| gap = 18.270998
| octave = 1200.73875134506
| consistent = 15
| distinct = 15
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}

* 103et (103h val) has lower absolute errors than any smaller equal temperaments in the [[13-limit|13-]], [[17-limit|17-]], and [[19-limit]]s, being beaten by [[111edo|111]] in all of them.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]</text>
      <sha1>4oqes2zh21wewlmj668rynxo0f0d3ii</sha1>
    </revision>
    <revision>
      <id>225887</id>
      <parentid>221280</parentid>
      <timestamp>2026-03-13T13:32:45Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct" to "[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct"</comment>
      <origin>225887</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4425" sha1="hfew65iql6pai6yfwl8owvc5dl4a18d" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]], and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 596
| steps = 102.936629522070
| step size = 11.6576577800491
| tempered height = 8.543510
| pure height = 5.620365
| integral = 1.340775
| gap = 18.270998
| octave = 1200.73875134506
| consistent = 15
| distinct = 15
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}

* 103et (103h val) has lower absolute errors than any smaller equal temperaments in the [[13-limit|13-]], [[17-limit|17-]], and [[19-limit]]s, being beaten by [[111edo|111]] in all of them.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]</text>
      <sha1>hfew65iql6pai6yfwl8owvc5dl4a18d</sha1>
    </revision>
    <revision>
      <id>233362</id>
      <parentid>225887</parentid>
      <timestamp>2026-07-08T04:00:44Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Update link</comment>
      <origin>233362</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4447" sha1="siuv424108sqecfseyvn8ckx1vxbu0f" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
In 103edo, all intervals within the [[17-odd-limit]] are [[consistent]], with the sole exception of [[9/8]] and its octave complement [[16/9]], which barely miss (relative error 50.2%). Its closest [[zeta peak index]], [[596zpi]], [[stretched and compressed tuning|stretches the octave]] by +0.739 cents. This expansion is uniquely consistent within the 15-integer-limit.

103edo is a good [[miracle]] tuning, especially for the [[7-limit]], and for [[Gamelismic clan #Miracle|benediction]] and [[Gamelismic clan #Miracle|hemisecordite]], two of the [[13-limit]] extensions of miracle. It [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]]; [[225/224]], [[1029/1024]], and [[2401/2400]] in the 7-limit; [[243/242]], [[441/440]], and [[540/539]] in the [[11-limit]]; [[351/350]] and [[847/845]] in the 13-limit. In the 13-limit it provides the [[optimal patent val]] for [[marvel]] temperament as well as benediction and hemisecordite.

=== Prime harmonics ===
{{Harmonics in equal|103|intervals=prime}}

=== Subsets and supersets ===
103edo is the 27th [[prime edo]], following [[101edo]] and before [[107edo]].

== Intervals ==
{{Main|Table of 103edo intervals}}

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 596
| steps = 102.936629522070
| step size = 11.6576577800491
| tempered height = 8.543510
| pure height = 5.620365
| integral = 1.340775
| gap = 18.270998
| octave = 1200.73875134506
| consistent = 15
| distinct = 15
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -163 103 }}
| {{mapping| 103 166 }}
| +0.923
| 0.924
| 7.92
|-
| 2.3.5
| 78732/78125, 34171875/33554432
| {{mapping| 103 166 239 }}
| +0.881
| 0.757
| 6.49
|-
| 2.3.5.7
| 225/224, 1029/1024, 78732/78125
| {{mapping| 103 166 239 289 }}
| +0.824
| 0.663
| 5.68
|-
| 2.3.5.7.11
| 225/224, 243/242, 385/384, 43923/43750
| {{mapping| 103 166 239 289 356 }}
| +0.876
| 0.602
| 5.16
|-
| 2.3.5.7.11.13
| 225/224, 243/242, 351/350, 385/384, 847/845
| {{mapping| 103 166 239 289 356 381 }}
| +0.806
| 0.571
| 4.90
|-
| 2.3.5.7.11.13.17
| 225/224, 243/242, 273/272, 351/350, 375/374, 847/845
| {{mapping| 103 166 239 289 356 381 421 }}
| +0.694
| 0.595
| 5.10
|}

* 103et (103h val) has lower absolute errors than any smaller equal temperaments in the [[13-limit|13-]], [[17-limit|17-]], and [[19-limit]]s, being beaten by [[111edo|111]] in all of them.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 3\103
| 34.951
| 1990656/1953125
| [[Gammy]]
|-
| 1
| 5\103
| 58.252
| 27/26
| [[Hemisecordite]]
|-
| 1
| 9\103
| 104.854
| 17/16
| [[Septendesemi]]
|-
| 1
| 10\103
| 116.505
| 15/14~16/15
| [[Miracle]] / [[benediction]]
|-
| 1
| 16\103
| 186.408
| 10/9
| [[Mintone]]
|-
| 1
| 20\103
| 233.010
| 8/7
| [[Slendric]]
|-
| 1
| 21\103
| 244.660
| 15/13
| [[Subsemifourth]]
|-
| 1
| 26\103
| 303.013
| 25/21
| [[Quinmite]]
|-
| 1
| 31\103
| 361.165
| 16/13
| [[Phicordial]]
|-
| 1
| 37\103
| 431.06
| 77/60
| [[Lockerbie]]
|-
| 1
| 38\103
| 442.708
| 162/125
| [[Sensei]]
|-
| 1
| 39\103
| 454.369
| 13/10
| [[Fibo]]
|-
| 1
| 40\103
| 466.019
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 42\103
| 489.320
| 65/49
| [[Catafourth]]
|-
| 1
| 45\103
| 524.272
| 65/48
| [[Widefourth]]
|-
| 1
| 47\103
| 547.573
| 11/8
| [[Heinz]]
|-
| 1
| 48\103
| 559.223
| 242/175
| [[Tritriple]]
|-
| 1
| 50\103
| 582.524
| 7/5
| [[Neptune (temperament)|Neptune]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Music ==
; [[Francium]]
* "Forest Tribe Dance" from ''Mysteries'' (2023) – [https://open.spotify.com/track/0lPUfgduKoJliGbU3kcow0 Spotify] | [https://francium223.bandcamp.com/track/forest-tribe-dance Bandcamp] | [https://www.youtube.com/watch?v=5pbnmzXAFcM YouTube]

[[Category:Benediction]]
[[Category:Listen]]
[[Category:Miracle]]</text>
      <sha1>siuv424108sqecfseyvn8ckx1vxbu0f</sha1>
    </revision>
  </page>
  <page>
    <title>104edo</title>
    <ns>0</ns>
    <id>21</id>
    <revision>
      <id>21</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>21</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4031" sha1="80e40tn9q73nc93ptnn5bwp5ib0gu17" xml:space="preserve">''104edo'' divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit [[val|val]]s, and both are useful. The flat major third val, &amp;lt;104 165 241|, tempers out 3125/3072, and supports [[Magic_family|magic temperament]]. The sharp major third val, &amp;lt;104 165 242|, tempers out 2048/2025 and supports [[Diaschismic_family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for [[Magic_family|magic temperament]], providing the [[Optimal_patent_val|optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic_family#Magic-13-limit-Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the [[Optimal_patent_val|optimal patent val]].

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

'''17-limit Regular Temperaments'''

{| class="wikitable"
|-
! | Degree
! | Cents
|-
| | '''2'''
| | '''23.08'''
|-
| | 3
| | 34.615
|-
| | 4
| | 46.15
|-
| | '''5'''
| | '''57.69'''
|-
| | '''7'''
| | '''80.77'''
|-
| | 8
| | 92.31
|-
| | 9
| | 103.85
|-
| | 10
| | 115.385
|-
| | 11
| | 126.92
|-
| | 12
| | 138.46
|-
| | '''13'''
| | '''150'''
|-
| | 14
| | 161.54
|-
| | 15
| | 173.08
|-
| | 16
| | 184.615
|-
| | 17
| | 196.15
|-
| | '''18'''
| | '''207.69'''
|-
| | '''20'''
| | '''230.77'''
|-
| | 21
| | 242.31
|-
| | 22
| | 253.85
|-
| | '''23'''
| | '''265.385'''
|-
| | '''25'''
| | '''288.46'''
|-
| | 26
| | 300
|-
| | 27
| | 311.54
|-
| | 28
| | 323.08
|-
| | 29
| | 334.615
|-
| | '''30'''
| | '''346.15'''
|-
| | 31
| | 357.69
|-
| | 32
| | 369.23
|-
| | 33
| | 380.77
|-
| | 34
| | 392.31
|-
| | 35
| | 403.85
|-
| | 36
| | 415.385
|-
| | 38
| | 438.46
|-
| | 39
| | 450
|-
| | 40
| | 461.54
|-
| | '''41'''
| | '''473.08'''
|-
| | '''43'''
| | '''496.15'''
|-
| | '''45'''
| | '''519.23'''
|-
| | 46
| | 530.77
|-
| | 47
| | 542.31
|-
| | '''48'''
| | '''553.85'''
|-
| | 50
| | 576.92
|-
| | 51
| | 588.45
|-
| | 52
| | 600
|-
| | 53
| | 611.54
|-
| | '''54'''
| | '''623.08'''
|-
| | 56
| | 646.15
|-
| | 57
| | 657.69
|-
| | 58
| | 669.23
|-
| | 59
| | 680.77
|-
| | '''61'''
| | '''703.85'''
|-
| | 63
| | 726.92
|-
| | 64
| | 738.46
|-
| | 65
| | 750
|-
| | '''66'''
| | '''761.54'''
|-
| | 67
| | 773.08
|-
| | '''68'''
| | '''784.615'''
|-
| | 69
| | 796.15
|-
| | 70
| | 807.69
|-
| | 71
| | 819.23
|-
| | 72
| | 830.77
|-
| | 73
| | 842.31
|-
| | '''74'''
| | '''853.85'''
|-
| | 75
| | 865.385
|-
| | 76
| | 876.92
|-
| | 77
| | 888.46
|-
| | 78
| | 900
|-
| | 79
| | 911.54
|-
| | '''81'''
| | '''934.615'''
|-
| | 82
| | 946.15
|-
| | 83
| | 957.69
|-
| | '''84'''
| | '''969.23'''
|-
| | 86
| | 992.31
|-
| | 87
| | 1003.85
|-
| | 88
| | 1015.385
|-
| | 89
| | 1026.92
|-
| | 90
| | 1038.46
|-
| | '''91'''
| | '''1050'''
|-
| | 92
| | 1061.54
|-
| | 93
| | 1073.08
|-
| | 95
| | 1096.15
|-
| | 96
| | 1107.69
|-
| | '''97'''
| | '''1119.23'''
|-
| | 99
| | 1142.31
|-
| | '''100'''
| | '''1153.85'''
|-
| | 101
| | 1165.385
|-
| | '''102'''
| | '''1176.92'''
|}
[[Category:apollo]]
[[Category:diaschismic]]
[[Category:edo]]
[[Category:magic]]
[[Category:necromancy]]</text>
      <sha1>80e40tn9q73nc93ptnn5bwp5ib0gu17</sha1>
    </revision>
    <revision>
      <id>4063</id>
      <timestamp>2016-11-14T17:06:24Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 599442450 - Original comment: **</comment>
      <origin>4063</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19482" sha1="akb3y0hygucjcnw513urtcw6dcdyfiy" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-11-14 17:06:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;599442450&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//104edo// divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, &amp;lt;104 165 241|, tempers out 3125/3072, and supports [[Magic family|magic temperament]]. The sharp major third val, &amp;lt;104 165 242|, tempers out 2048/2025 and supports [[Diaschismic family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for [[Magic family|magic temperament]], providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic family#Magic-13-limit-Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the [[optimal patent val]].

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

**17-limit Regular Temperaments**
||~ Degree ||~ Cents ||
|| **2** || **23.08** ||
|| 3 || 34.615 ||
|| 4 || 46.15 ||
|| **5** || **57.69** ||
|| **7** || **80.77** ||
|| 8 || 92.31 ||
|| 9 || 103.85 ||
|| 10 || 115.385 ||
|| 11 || 126.92 ||
|| 12 || 138.46 ||
|| **13** || **150** ||
|| 14 || 161.54 ||
|| 15 || 173.08 ||
|| 16 || 184.615 ||
|| 17 || 196.15 ||
|| **18** || **207.69** ||
|| **20** || **230.77** ||
|| 21 || 242.31 ||
|| 22 || 253.85 ||
|| **23** || **265.385** ||
|| **25** || **288.46** ||
|| 26 || 300 ||
|| 27 || 311.54 ||
|| 28 || 323.08 ||
|| 29 || 334.615 ||
|| **30** || **346.15** ||
|| 31 || 357.69 ||
|| 32 || 369.23 ||
|| 33 || 380.77 ||
|| 34 || 392.31 ||
|| 35 || 403.85 ||
|| 36 || 415.385 ||
|| 38 || 438.46 ||
|| 39 || 450 ||
|| 40 || 461.54 ||
|| **41** || **473.08** ||
|| **43** || **496.15** ||
|| **45** || **519.23** ||
|| 46 || 530.77 ||
|| 47 || 542.31 ||
|| **48** || **553.85** ||
|| 50 || 576.92 ||
|| 51 || 588.45 ||
|| 52 || 600 ||
|| 53 || 611.54 ||
|| **54** || **623.08** ||
|| 56 || 646.15 ||
|| 57 || 657.69 ||
|| 58 || 669.23 ||
|| 59 || 680.77 ||
|| **61** || **703.85** ||
|| 63 || 726.92 ||
|| 64 || 738.46 ||
|| 65 || 750 ||
|| **66** || **761.54** ||
|| 67 || 773.08 ||
|| **68** || **784.615** ||
|| 69 || 796.15 ||
|| 70 || 807.69 ||
|| 71 || 819.23 ||
|| 72 || 830.77 ||
|| 73 || 842.31 ||
|| **74** || **853.85** ||
|| 75 || 865.385 ||
|| 76 || 876.92 ||
|| 77 || 888.46 ||
|| 78 || 900 ||
|| 79 || 911.54 ||
|| **81** || **934.615** ||
|| 82 || 946.15 ||
|| 83 || 957.69 ||
|| **84** || **969.23** ||
|| 86 || 992.31 ||
|| 87 || 1003.85 ||
|| 88 || 1015.385 ||
|| 89 || 1026.92 ||
|| 90 || 1038.46 ||
|| **91** || **1050** ||
|| 92 || 1061.54 ||
|| 93 || 1073.08 ||
|| 95 || 1096.15 ||
|| 96 || 1107.69 ||
|| **97** || **1119.23** ||
|| 99 || 1142.31 ||
|| **100** || **1153.85** ||
|| 101 || 1165.385 ||
|| **102** || **1176.92** ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;104edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;em&amp;gt;104edo&amp;lt;/em&amp;gt; divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit &amp;lt;a class="wiki_link" href="/val"&amp;gt;val&amp;lt;/a&amp;gt;s, and both are useful. The flat major third val, &amp;amp;lt;104 165 241|, tempers out 3125/3072, and supports &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;. The sharp major third val, &amp;amp;lt;104 165 242|, tempers out 2048/2025 and supports &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;diaschismic temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104edo with the flat third is especially notable as an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;, providing the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit magic and the 13-limit magic extension &amp;lt;a class="wiki_link" href="/Magic%20family#Magic-13-limit-Necromancy"&amp;gt;necromancy&amp;lt;/a&amp;gt;. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17-limit Regular Temperaments&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;2&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;23.08&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34.615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46.15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;84&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;969.23&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;992.31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;87&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1003.85&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1015.385&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;89&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1026.92&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1038.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;91&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;1050&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;92&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1061.54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1073.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1096.15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1107.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;97&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;1119.23&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1142.31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;100&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;1153.85&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1165.385&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;102&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;1176.92&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>akb3y0hygucjcnw513urtcw6dcdyfiy</sha1>
    </revision>
    <revision>
      <id>4064</id>
      <timestamp>2016-10-28T00:13:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 597358160 - Original comment: **</comment>
      <origin>4064</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19441" sha1="me0n1ftvq4rio1kvt8f7z2dyymejex8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-28 00:13:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;597358160&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//104edo// divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit [[#|vals]], and both are useful. The flat major third val, &amp;lt;104 165 241|, tempers out 3125/3072, and supports [[Magic family|magic temperament]]. The sharp major third val, &amp;lt;104 165 242|, tempers out 2048/2025 and supports [[Diaschismic family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for [[Magic family|magic temperament]], providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic family#Magic-13-limit-Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the [[optimal patent val]].

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

**17-limit Regular Temperaments**
||~ Degree ||~ Cents ||
|| **2** || **23.08** ||
|| 3 || 34.615 ||
|| 4 || 46.15 ||
|| **5** || **57.69** ||
|| **7** || **80.77** ||
|| 8 || 92.31 ||
|| 9 || 103.85 ||
|| 10 || 115.385 ||
|| 11 || 126.92 ||
|| 12 || 138.46 ||
|| **13** || **150** ||
|| 14 || 161.54 ||
|| 15 || 173.08 ||
|| 16 || 184.615 ||
|| 17 || 196.15 ||
|| **18** || **207.69** ||
|| **20** || **230.77** ||
|| 21 || 242.31 ||
|| 22 || 253.85 ||
|| **23** || **265.385** ||
|| **25** || **288.46** ||
|| 26 || 300 ||
|| 27 || 311.54 ||
|| 28 || 323.08 ||
|| 29 || 334.615 ||
|| **30** || **346.15** ||
|| 31 || 357.69 ||
|| 32 || 369.23 ||
|| 33 || 380.77 ||
|| 34 || 392.31 ||
|| 35 || 403.85 ||
|| 36 || 415.385 ||
|| 38 || 438.46 ||
|| 39 || 450 ||
|| 40 || 461.54 ||
|| **41** || **473.08** ||
|| **43** || **496.15** ||
|| **45** || **519.23** ||
|| 46 || 530.77 ||
|| 47 || 542.31 ||
|| **48** || **553.85** ||
|| 50 || 576.92 ||
|| 51 || 588.45 ||
|| 52 || 600 ||
|| 53 || 611.54 ||
|| **54** || **623.08** ||
|| 56 || 646.15 ||
|| 57 || 657.69 ||
|| 58 || 669.23 ||
|| 59 || 680.77 ||
|| **61** || **703.85** ||
|| 63 || 726.92 ||
|| 64 || 738.46 ||
|| 65 || 750 ||
|| **66** || **761.54** ||
|| 67 || 773.08 ||
|| **68** || **784.615** ||
|| 69 || 796.15 ||
|| 70 || 807.69 ||
|| 71 || 819.23 ||
|| 72 || 830.77 ||
|| 73 || 842.31 ||
|| **74** || **853.85** ||
|| 75 || 865.385 ||
|| 76 || 876.92 ||
|| 77 || 888.46 ||
|| 78 || 900 ||
|| 79 || 911.54 ||
|| **81** || **934.615** ||
|| 82 || 946.15 ||
|| 83 || 957.69 ||
|| **84** || **969.23** ||
|| 86 || 992.31 ||
|| 87 || 1003.85 ||
|| 88 || 1015.385 ||
|| 89 || 1026.92 ||
|| 90 || 1038.46 ||
|| **91** || **1050** ||
|| 92 || 1061.54 ||
|| 93 || 1073.08 ||
|| 95 || 1096.15 ||
|| 96 || 1107.69 ||
|| **97** || **1119.23** ||
|| 99 || 1142.31 ||
|| **100** || **1153.85** ||
|| 101 || 1165.385 ||
|| **102** || **1176.92** ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;104edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;em&amp;gt;104edo&amp;lt;/em&amp;gt; divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit [[#|vals]], and both are useful. The flat major third val, &amp;amp;lt;104 165 241|, tempers out 3125/3072, and supports &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;. The sharp major third val, &amp;amp;lt;104 165 242|, tempers out 2048/2025 and supports &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;diaschismic temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104edo with the flat third is especially notable as an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;, providing the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit magic and the 13-limit magic extension &amp;lt;a class="wiki_link" href="/Magic%20family#Magic-13-limit-Necromancy"&amp;gt;necromancy&amp;lt;/a&amp;gt;. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17-limit Regular Temperaments&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;2&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;23.08&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34.615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46.15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;5&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;57.69&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;80.77&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92.31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.85&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.385&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;126.92&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;138.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;13&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;150&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;161.54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;173.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;184.615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;196.15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;18&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;207.69&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;20&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;230.77&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;242.31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;253.85&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;23&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;265.385&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;25&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;288.46&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;300&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;311.54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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      <sha1>me0n1ftvq4rio1kvt8f7z2dyymejex8</sha1>
    </revision>
    <revision>
      <id>4065</id>
      <timestamp>2016-08-08T11:22:03Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 588914016 - Original comment: **</comment>
      <origin>4065</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19453" sha1="fpwe2mpzw9s05r2nrgxb7lrg08jwyjn" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-08-08 11:22:03 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;588914016&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//104edo// divides the octave into 104 parts of size 11.54 [[#|cents]] each. It has two different equally viable 5-limit [[#|vals]], and both are useful. The flat major third val, &amp;lt;104 165 241|, tempers out 3125/3072, and supports [[Magic family|magic temperament]]. The sharp major third val, &amp;lt;104 165 242|, tempers out 2048/2025 and supports [[Diaschismic family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for [[Magic family|magic temperament]], providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic family#Magic-13-limit-Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the [[optimal patent val]].

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

**17-limit Regular Temperaments**
||~ Degree ||~ Cents ||
|| **2** || **23.08** ||
|| 3 || 34.615 ||
|| 4 || 46.15 ||
|| **5** || **57.69** ||
|| **7** || **80.77** ||
|| 8 || 92.31 ||
|| 9 || 103.85 ||
|| 10 || 115.385 ||
|| 11 || 126.92 ||
|| 12 || 138.46 ||
|| **13** || **150** ||
|| 14 || 161.54 ||
|| 15 || 173.08 ||
|| 16 || 184.615 ||
|| 17 || 196.15 ||
|| **18** || **207.69** ||
|| **20** || **230.77** ||
|| 21 || 242.31 ||
|| 22 || 253.85 ||
|| **23** || **265.385** ||
|| **25** || **288.46** ||
|| 26 || 300 ||
|| 27 || 311.54 ||
|| 28 || 323.08 ||
|| 29 || 334.615 ||
|| **30** || **346.15** ||
|| 31 || 357.69 ||
|| 32 || 369.23 ||
|| 33 || 380.77 ||
|| 34 || 392.31 ||
|| 35 || 403.85 ||
|| 36 || 415.385 ||
|| 38 || 438.46 ||
|| 39 || 450 ||
|| 40 || 461.54 ||
|| **41** || **473.08** ||
|| **43** || **496.15** ||
|| **45** || **519.23** ||
|| 46 || 530.77 ||
|| 47 || 542.31 ||
|| **48** || **553.85** ||
|| 50 || 576.92 ||
|| 51 || 588.45 ||
|| 52 || 600 ||
|| 53 || 611.54 ||
|| **54** || **623.08** ||
|| 56 || 646.15 ||
|| 57 || 657.69 ||
|| 58 || 669.23 ||
|| 59 || 680.77 ||
|| **61** || **703.85** ||
|| 63 || 726.92 ||
|| 64 || 738.46 ||
|| 65 || 750 ||
|| **66** || **761.54** ||
|| 67 || 773.08 ||
|| **68** || **784.615** ||
|| 69 || 796.15 ||
|| 70 || 807.69 ||
|| 71 || 819.23 ||
|| 72 || 830.77 ||
|| 73 || 842.31 ||
|| **74** || **853.85** ||
|| 75 || 865.385 ||
|| 76 || 876.92 ||
|| 77 || 888.46 ||
|| 78 || 900 ||
|| 79 || 911.54 ||
|| **81** || **934.615** ||
|| 82 || 946.15 ||
|| 83 || 957.69 ||
|| **84** || **969.23** ||
|| 86 || 992.31 ||
|| 87 || 1003.85 ||
|| 88 || 1015.385 ||
|| 89 || 1026.92 ||
|| 90 || 1038.46 ||
|| **91** || **1050** ||
|| 92 || 1061.54 ||
|| 93 || 1073.08 ||
|| 95 || 1096.15 ||
|| 96 || 1107.69 ||
|| **97** || **1119.23** ||
|| 99 || 1142.31 ||
|| **100** || **1153.85** ||
|| 101 || 1165.385 ||
|| **102** || **1176.92** ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;104edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;em&amp;gt;104edo&amp;lt;/em&amp;gt; divides the octave into 104 parts of size 11.54 [[#|cents]] each. It has two different equally viable 5-limit [[#|vals]], and both are useful. The flat major third val, &amp;amp;lt;104 165 241|, tempers out 3125/3072, and supports &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;. The sharp major third val, &amp;amp;lt;104 165 242|, tempers out 2048/2025 and supports &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;diaschismic temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104edo with the flat third is especially notable as an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;, providing the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit magic and the 13-limit magic extension &amp;lt;a class="wiki_link" href="/Magic%20family#Magic-13-limit-Necromancy"&amp;gt;necromancy&amp;lt;/a&amp;gt;. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17-limit Regular Temperaments&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;2&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;23.08&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34.615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46.15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;5&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;57.69&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;80.77&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92.31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.85&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.385&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;126.92&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;138.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;13&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;150&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;161.54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;173.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;184.615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;196.15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;18&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;207.69&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;20&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;230.77&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;242.31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;253.85&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;23&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;265.385&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;25&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;288.46&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;300&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;311.54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;28&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;323.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;334.615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;30&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;346.15&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;32&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;369.23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;380.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;392.31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;403.85&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;415.385&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;38&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;438.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;450&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;461.54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;41&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;473.08&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;43&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;496.15&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;45&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;519.23&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;530.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;542.31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;48&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;553.85&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;576.92&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;588.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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    <revision>
      <id>4066</id>
      <timestamp>2013-09-12T02:42:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 450498666 - Original comment: **</comment>
      <origin>4066</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5002" sha1="b7ls8i47vpgqpyzs87cfdbgngv2w61m" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2013-09-12 02:42:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;450498666&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//104edo// divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, &amp;lt;104 165 241|, tempers out 3125/3072, and supports [[Magic family|magic temperament]]. The sharp major third val, &amp;lt;104 165 242|, tempers out 2048/2025 and supports [[Diaschismic family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for [[Magic family|magic temperament]], providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic family#Magic-13-limit-Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the [[optimal patent val]].

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891,  2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;104edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;em&amp;gt;104edo&amp;lt;/em&amp;gt; divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, &amp;amp;lt;104 165 241|, tempers out 3125/3072, and supports &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;. The sharp major third val, &amp;amp;lt;104 165 242|, tempers out 2048/2025 and supports &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;diaschismic temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104edo with the flat third is especially notable as an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;, providing the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit magic and the 13-limit magic extension &amp;lt;a class="wiki_link" href="/Magic%20family#Magic-13-limit-Necromancy"&amp;gt;necromancy&amp;lt;/a&amp;gt;. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891,  2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>b7ls8i47vpgqpyzs87cfdbgngv2w61m</sha1>
    </revision>
    <revision>
      <id>4067</id>
      <timestamp>2013-09-12T02:30:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 450497198 - Original comment: **</comment>
      <origin>4067</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4640" sha1="pkz32wp50fqhkkbbvxmhvif8dc1yvc4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2013-09-12 02:30:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;450497198&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//104edo// divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, &amp;lt;104 165 241|, tempers out 3125/3072, and supports [[Magic family|magic temperament]]. The sharp major third val, &amp;lt;104 165 242|, tempers out 2048/2025 and supports [[Diaschismic family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for [[Magic family|magic temperament]], providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic family#Magic-13-limit-Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the [[optimal patent val]].

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891,  2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;104edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;em&amp;gt;104edo&amp;lt;/em&amp;gt; divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, &amp;amp;lt;104 165 241|, tempers out 3125/3072, and supports &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;. The sharp major third val, &amp;amp;lt;104 165 242|, tempers out 2048/2025 and supports &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;diaschismic temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104edo with the flat third is especially notable as an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;, providing the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit magic and the 13-limit magic extension &amp;lt;a class="wiki_link" href="/Magic%20family#Magic-13-limit-Necromancy"&amp;gt;necromancy&amp;lt;/a&amp;gt;. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891,  2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4068</id>
      <timestamp>2011-10-18T15:01:30Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 266087584 - Original comment: **</comment>
      <origin>4068</origin>
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      <text bytes="4254" sha1="i1cw1dh1z1lansg9oxq8k200466q3jk" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-10-18 15:01:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;266087584&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//104edo// divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, &amp;lt;104 165 241|, tempers out 3125/3072, and supports [[Magic family|magic temperament]]. The sharp major third val, &amp;lt;104 165 242|, tempers out 2048/2025 and supports [[Diaschismic family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for [[Magic family|magic temperament]], providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic family#Magic-13-limit-Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the [[optimal patent val]].

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;104edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;em&amp;gt;104edo&amp;lt;/em&amp;gt; divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, &amp;amp;lt;104 165 241|, tempers out 3125/3072, and supports &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;. The sharp major third val, &amp;amp;lt;104 165 242|, tempers out 2048/2025 and supports &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;diaschismic temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104edo with the flat third is especially notable as an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;, providing the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit magic and the 13-limit magic extension &amp;lt;a class="wiki_link" href="/Magic%20family#Magic-13-limit-Necromancy"&amp;gt;necromancy&amp;lt;/a&amp;gt;. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 (apollo temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4069</id>
      <timestamp>2011-10-18T14:58:40Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 266086410 - Original comment: **</comment>
      <origin>4069</origin>
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      <text bytes="4212" sha1="i1ymgj2nw4sjy1r13rkclxdeq7dwzhr" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-10-18 14:58:40 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;266086410&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//104edo// divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, &amp;lt;104 165 241|, tempers out 3125/3072, and supports [[Magic family|magic temperament]]. The sharp major third val, &amp;lt;104 165 242|, tempers out 2048/2025 and supports [[Diaschismic family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for [[Magic family|magic temperament]], providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic family#Magic-13-limit-Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224, 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the [[optimal patent val]].

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;104edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;em&amp;gt;104edo&amp;lt;/em&amp;gt; divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, &amp;amp;lt;104 165 241|, tempers out 3125/3072, and supports &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;. The sharp major third val, &amp;amp;lt;104 165 242|, tempers out 2048/2025 and supports &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;diaschismic temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104edo with the flat third is especially notable as an excellent tuning for &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;, providing the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit magic and the 13-limit magic extension &amp;lt;a class="wiki_link" href="/Magic%20family#Magic-13-limit-Necromancy"&amp;gt;necromancy&amp;lt;/a&amp;gt;. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224, 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>i1ymgj2nw4sjy1r13rkclxdeq7dwzhr</sha1>
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    <revision>
      <id>4070</id>
      <timestamp>2011-07-03T15:07:02Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 239839617 - Original comment: **</comment>
      <origin>4070</origin>
      <model>wikitext</model>
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      <text bytes="3859" sha1="0kv1zqau047cyvv56n8007msjpuhf81" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-07-03 15:07:02 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239839617&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//104edo// divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, &amp;lt;104 165 241|, tempers out 3125/3072, and supports [[Magic family|magic temperament]]. The sharp major third val, &amp;lt;104 165 242|, tempers out 2048/2025 and supports [[Diaschismic family|diaschismic temperament]].

104 with the flat third is especially notable as an excellent tuning for the 11-limit version of [[Magic family|magic temperament]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224, 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the [[optimal patent val]].

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;104edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;em&amp;gt;104edo&amp;lt;/em&amp;gt; divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, &amp;amp;lt;104 165 241|, tempers out 3125/3072, and supports &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;. The sharp major third val, &amp;amp;lt;104 165 242|, tempers out 2048/2025 and supports &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;diaschismic temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 with the flat third is especially notable as an excellent tuning for the 11-limit version of &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224, 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>0kv1zqau047cyvv56n8007msjpuhf81</sha1>
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    <revision>
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      <timestamp>2010-12-10T21:40:49Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 187273313 - Original comment: **</comment>
      <origin>4071</origin>
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      <text bytes="3701" sha1="dyb33ka1m6637kwvjs1nr5zp56hcjno" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2010-12-10 21:40:49 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;187273313&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//104edo// divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, &amp;lt;104 165 241|, tempers out 3125/3072, and supports [[Magic family|magic temperament]]. The sharp major third val, &amp;lt;104 165 242|, tempers out 2048/2025 and supports [[Diaschismic family|diaschismic temperament]].

104 with the flat third is especially notable as an excellent tuning for the 11-limit version of [[Magic family|magic temperament]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224, 245/243 or 875/864, or the rank four temperament tempering out 100/99.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;104edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;em&amp;gt;104edo&amp;lt;/em&amp;gt; divides the octave into 104 parts of size 11.54 cents each. It has two different equally viable 5-limit vals, and both are useful. The flat major third val, &amp;amp;lt;104 165 241|, tempers out 3125/3072, and supports &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;. The sharp major third val, &amp;amp;lt;104 165 242|, tempers out 2048/2025 and supports &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;diaschismic temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 with the flat third is especially notable as an excellent tuning for the 11-limit version of &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224, 245/243 or 875/864, or the rank four temperament tempering out 100/99.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, 126/125 and 5120/5103 in the 7-limit, 176/175 and 896/891 in the 11-limit, 196/195 and 364/363 in the 13-limit and 136/135 and 256/255 in the 17-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>dyb33ka1m6637kwvjs1nr5zp56hcjno</sha1>
    </revision>
    <revision>
      <id>4072</id>
      <timestamp>2010-08-10T16:23:57Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 155976857 - Original comment: **</comment>
      <origin>4072</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2349" sha1="pht1js6ojs4jzvh2aiamzkm46hbauf4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2010-08-10 16:23:57 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;155976857&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//104edo// divides the octave into 104 parts of size 11.54 cents each, and is especially notable as an excellent tuning for the 11-limit version of [[Magic family|magic temperament]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224, 245/243 or 875/864, or the rank four temperament tempering out 100/99.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;104edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;em&amp;gt;104edo&amp;lt;/em&amp;gt; divides the octave into 104 parts of size 11.54 cents each, and is especially notable as an excellent tuning for the 11-limit version of &amp;lt;a class="wiki_link" href="/Magic%20family"&amp;gt;magic temperament&amp;lt;/a&amp;gt;. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out 225/224, 245/243 and 875/864; and in the 11-limit, 100/99, 896/891, 385/384 and 540/539. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224, 245/243 or 875/864, or the rank four temperament tempering out 100/99.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>pht1js6ojs4jzvh2aiamzkm46hbauf4</sha1>
    </revision>
    <revision>
      <id>48328</id>
      <parentid>21</parentid>
      <timestamp>2020-08-27T06:02:55Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Cleanup</comment>
      <origin>48328</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4145" sha1="lpv6hjistuy5hyp3t8kobxyxdmlgsop" xml:space="preserve">'''104edo''' divides the [[octave]] into 104 parts of size 11.54 [[cent|cents]] each. 

== Theory ==
104edo has two different equally viable 5-limit [[val|vals]], and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[Magic_family|magic temperament]]. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[Diaschismic_family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic_family #Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[Marvel family #Apollo|apollo temperament]]), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

== 17-limit Regular Temperaments ==
''todo: clarify this table''

{| class="wikitable"
|-
! | Degree
! | Cents
|-
| | '''2'''
| | '''23.08'''
|-
| | 3
| | 34.615
|-
| | 4
| | 46.15
|-
| | '''5'''
| | '''57.69'''
|-
| | '''7'''
| | '''80.77'''
|-
| | 8
| | 92.31
|-
| | 9
| | 103.85
|-
| | 10
| | 115.385
|-
| | 11
| | 126.92
|-
| | 12
| | 138.46
|-
| | '''13'''
| | '''150'''
|-
| | 14
| | 161.54
|-
| | 15
| | 173.08
|-
| | 16
| | 184.615
|-
| | 17
| | 196.15
|-
| | '''18'''
| | '''207.69'''
|-
| | '''20'''
| | '''230.77'''
|-
| | 21
| | 242.31
|-
| | 22
| | 253.85
|-
| | '''23'''
| | '''265.385'''
|-
| | '''25'''
| | '''288.46'''
|-
| | 26
| | 300
|-
| | 27
| | 311.54
|-
| | 28
| | 323.08
|-
| | 29
| | 334.615
|-
| | '''30'''
| | '''346.15'''
|-
| | 31
| | 357.69
|-
| | 32
| | 369.23
|-
| | 33
| | 380.77
|-
| | 34
| | 392.31
|-
| | 35
| | 403.85
|-
| | 36
| | 415.385
|-
| | 38
| | 438.46
|-
| | 39
| | 450
|-
| | 40
| | 461.54
|-
| | '''41'''
| | '''473.08'''
|-
| | '''43'''
| | '''496.15'''
|-
| | '''45'''
| | '''519.23'''
|-
| | 46
| | 530.77
|-
| | 47
| | 542.31
|-
| | '''48'''
| | '''553.85'''
|-
| | 50
| | 576.92
|-
| | 51
| | 588.45
|-
| | 52
| | 600
|-
| | 53
| | 611.54
|-
| | '''54'''
| | '''623.08'''
|-
| | 56
| | 646.15
|-
| | 57
| | 657.69
|-
| | 58
| | 669.23
|-
| | 59
| | 680.77
|-
| | '''61'''
| | '''703.85'''
|-
| | 63
| | 726.92
|-
| | 64
| | 738.46
|-
| | 65
| | 750
|-
| | '''66'''
| | '''761.54'''
|-
| | 67
| | 773.08
|-
| | '''68'''
| | '''784.615'''
|-
| | 69
| | 796.15
|-
| | 70
| | 807.69
|-
| | 71
| | 819.23
|-
| | 72
| | 830.77
|-
| | 73
| | 842.31
|-
| | '''74'''
| | '''853.85'''
|-
| | 75
| | 865.385
|-
| | 76
| | 876.92
|-
| | 77
| | 888.46
|-
| | 78
| | 900
|-
| | 79
| | 911.54
|-
| | '''81'''
| | '''934.615'''
|-
| | 82
| | 946.15
|-
| | 83
| | 957.69
|-
| | '''84'''
| | '''969.23'''
|-
| | 86
| | 992.31
|-
| | 87
| | 1003.85
|-
| | 88
| | 1015.385
|-
| | 89
| | 1026.92
|-
| | 90
| | 1038.46
|-
| | '''91'''
| | '''1050'''
|-
| | 92
| | 1061.54
|-
| | 93
| | 1073.08
|-
| | 95
| | 1096.15
|-
| | 96
| | 1107.69
|-
| | '''97'''
| | '''1119.23'''
|-
| | 99
| | 1142.31
|-
| | '''100'''
| | '''1153.85'''
|-
| | 101
| | 1165.385
|-
| | '''102'''
| | '''1176.92'''
|}
[[Category:apollo]]
[[Category:diaschismic]]
[[Category:edo]]
[[Category:magic]]
[[Category:necromancy]]</text>
      <sha1>lpv6hjistuy5hyp3t8kobxyxdmlgsop</sha1>
    </revision>
    <revision>
      <id>48329</id>
      <parentid>48328</parentid>
      <timestamp>2020-08-27T07:16:31Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+rank-2 temperaments</comment>
      <origin>48329</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4915" sha1="7wqrjln9ipgfol5unvmhb2p7lxuzoyq" xml:space="preserve">'''104edo''' divides the [[octave]] into 104 parts of size 11.54 [[cent|cents]] each. 

== Theory ==
104edo has two different equally viable 5-limit [[val|vals]], and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[Magic_family|magic temperament]]. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[Diaschismic_family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic_family #Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[Marvel family #Apollo|apollo temperament]]), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

== Rank two temperaments ==
=== In patent val ===
{| class="wikitable center-all right-3 left-5"
! Periods&lt;br&gt;per octave
! Generator
! Cents
! Associated&lt;br&gt;ratio
! Temperament
|-
| 1
| 33\104
| 380.769
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.462
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.846
| 
| [[Undim]]
|}

=== In 104c val ===
{| class="wikitable center-all right-3 left-5"
! Periods&lt;br&gt;per octave
! Generator
! Cents
! Associated&lt;br&gt;ratio
! Temperament
|-
| 1
| 21\104
| 242.308
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.538
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.308
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.154
| 4/3
| [[Diaschismic]]
|-
| 8
| 50\104
| 576.923
| 121/84
| [[Octowerck]] (7- or 11-limit)
|}

== 17-limit Regular Temperaments ==
''todo: clarify this table''

{| class="wikitable"
|-
! | Degree
! | Cents
|-
| | '''2'''
| | '''23.08'''
|-
| | 3
| | 34.615
|-
| | 4
| | 46.15
|-
| | '''5'''
| | '''57.69'''
|-
| | '''7'''
| | '''80.77'''
|-
| | 8
| | 92.31
|-
| | 9
| | 103.85
|-
| | 10
| | 115.385
|-
| | 11
| | 126.92
|-
| | 12
| | 138.46
|-
| | '''13'''
| | '''150'''
|-
| | 14
| | 161.54
|-
| | 15
| | 173.08
|-
| | 16
| | 184.615
|-
| | 17
| | 196.15
|-
| | '''18'''
| | '''207.69'''
|-
| | '''20'''
| | '''230.77'''
|-
| | 21
| | 242.31
|-
| | 22
| | 253.85
|-
| | '''23'''
| | '''265.385'''
|-
| | '''25'''
| | '''288.46'''
|-
| | 26
| | 300
|-
| | 27
| | 311.54
|-
| | 28
| | 323.08
|-
| | 29
| | 334.615
|-
| | '''30'''
| | '''346.15'''
|-
| | 31
| | 357.69
|-
| | 32
| | 369.23
|-
| | 33
| | 380.77
|-
| | 34
| | 392.31
|-
| | 35
| | 403.85
|-
| | 36
| | 415.385
|-
| | 38
| | 438.46
|-
| | 39
| | 450
|-
| | 40
| | 461.54
|-
| | '''41'''
| | '''473.08'''
|-
| | '''43'''
| | '''496.15'''
|-
| | '''45'''
| | '''519.23'''
|-
| | 46
| | 530.77
|-
| | 47
| | 542.31
|-
| | '''48'''
| | '''553.85'''
|-
| | 50
| | 576.92
|-
| | 51
| | 588.45
|-
| | 52
| | 600
|-
| | 53
| | 611.54
|-
| | '''54'''
| | '''623.08'''
|-
| | 56
| | 646.15
|-
| | 57
| | 657.69
|-
| | 58
| | 669.23
|-
| | 59
| | 680.77
|-
| | '''61'''
| | '''703.85'''
|-
| | 63
| | 726.92
|-
| | 64
| | 738.46
|-
| | 65
| | 750
|-
| | '''66'''
| | '''761.54'''
|-
| | 67
| | 773.08
|-
| | '''68'''
| | '''784.615'''
|-
| | 69
| | 796.15
|-
| | 70
| | 807.69
|-
| | 71
| | 819.23
|-
| | 72
| | 830.77
|-
| | 73
| | 842.31
|-
| | '''74'''
| | '''853.85'''
|-
| | 75
| | 865.385
|-
| | 76
| | 876.92
|-
| | 77
| | 888.46
|-
| | 78
| | 900
|-
| | 79
| | 911.54
|-
| | '''81'''
| | '''934.615'''
|-
| | 82
| | 946.15
|-
| | 83
| | 957.69
|-
| | '''84'''
| | '''969.23'''
|-
| | 86
| | 992.31
|-
| | 87
| | 1003.85
|-
| | 88
| | 1015.385
|-
| | 89
| | 1026.92
|-
| | 90
| | 1038.46
|-
| | '''91'''
| | '''1050'''
|-
| | 92
| | 1061.54
|-
| | 93
| | 1073.08
|-
| | 95
| | 1096.15
|-
| | 96
| | 1107.69
|-
| | '''97'''
| | '''1119.23'''
|-
| | 99
| | 1142.31
|-
| | '''100'''
| | '''1153.85'''
|-
| | 101
| | 1165.385
|-
| | '''102'''
| | '''1176.92'''
|}
[[Category:apollo]]
[[Category:diaschismic]]
[[Category:edo]]
[[Category:magic]]
[[Category:necromancy]]</text>
      <sha1>7wqrjln9ipgfol5unvmhb2p7lxuzoyq</sha1>
    </revision>
    <revision>
      <id>48483</id>
      <parentid>48329</parentid>
      <timestamp>2020-09-04T06:48:35Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Nobody understands that table; replace with a normal (and shorter) one</comment>
      <origin>48483</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5281" sha1="3hj4xt9ciym10uqj9ljy1vqeydzilet" xml:space="preserve">'''104edo''' divides the [[octave]] into 104 parts of size 11.54 [[cent|cents]] each. 

== Theory ==
104edo has two different equally viable 5-limit [[val|vals]], and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[Magic_family|magic temperament]]. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[Diaschismic_family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic_family #Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[Marvel family #Apollo|apollo temperament]]), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

== Rank two temperaments ==
=== In patent val ===
{| class="wikitable center-all right-3 left-5"
! Periods&lt;br&gt;per octave
! Generator
! Cents
! Associated&lt;br&gt;ratio
! Temperament
|-
| 1
| 33\104
| 380.769
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.462
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.846
| 
| [[Undim]]
|}

=== In 104c val ===
{| class="wikitable center-all right-3 left-5"
! Periods&lt;br&gt;per octave
! Generator
! Cents
! Associated&lt;br&gt;ratio
! Temperament
|-
| 1
| 21\104
| 242.308
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.538
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.308
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.154
| 4/3
| [[Diaschismic]]
|-
| 8
| 50\104
| 576.923
| 121/84
| [[Octowerck]] (7- or 11-limit)
|}

== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4 left-5"
|-
! rowspan="2"| #
! rowspan="2"| Cents
! colspan="3"| Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25&lt;br&gt;Subgroup
! Additional Ratios of 5&lt;br&gt;Tending Sharp (104c Val)
! Additional Ratios of 5&lt;br&gt;Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
| 
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]]
|-
| 29
| 334.615
| [[17/14]]
|
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
|…
|…
|…
|…
|…
|}

[[Category:apollo]]
[[Category:diaschismic]]
[[Category:edo]]
[[Category:magic]]
[[Category:necromancy]]</text>
      <sha1>3hj4xt9ciym10uqj9ljy1vqeydzilet</sha1>
    </revision>
    <revision>
      <id>54951</id>
      <parentid>48483</parentid>
      <timestamp>2020-12-04T23:13:05Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54951</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5307" sha1="8iug30u95xmc0jjv5381ha70bxpm7dv" xml:space="preserve">'''104edo''' divides the [[octave]] into 104 parts of size 11.54 [[cent|cents]] each. 

== Theory ==
104edo has two different equally viable 5-limit [[val|vals]], and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[Magic_family|magic temperament]]. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[Diaschismic_family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic_family #Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[Marvel family #Apollo|apollo temperament]]), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

== Rank two temperaments ==
=== In patent val ===
{| class="wikitable center-all right-3 left-5"
! Periods&lt;br&gt;per octave
! Generator
! Cents
! Associated&lt;br&gt;ratio
! Temperament
|-
| 1
| 33\104
| 380.769
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.462
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.846
| 
| [[Undim]]
|}

=== In 104c val ===
{| class="wikitable center-all right-3 left-5"
! Periods&lt;br&gt;per octave
! Generator
! Cents
! Associated&lt;br&gt;ratio
! Temperament
|-
| 1
| 21\104
| 242.308
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.538
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.308
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.154
| 4/3
| [[Diaschismic]]
|-
| 8
| 50\104
| 576.923
| 121/84
| [[Octowerck]] (7- or 11-limit)
|}

== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4 left-5"
|-
! rowspan="2"| #
! rowspan="2"| Cents
! colspan="3"| Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25&lt;br&gt;Subgroup
! Additional Ratios of 5&lt;br&gt;Tending Sharp (104c Val)
! Additional Ratios of 5&lt;br&gt;Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
| 
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]]
|-
| 29
| 334.615
| [[17/14]]
|
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
|…
|…
|…
|…
|…
|}

[[Category:apollo]]
[[Category:diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:magic]]
[[Category:necromancy]]</text>
      <sha1>8iug30u95xmc0jjv5381ha70bxpm7dv</sha1>
    </revision>
    <revision>
      <id>67313</id>
      <parentid>54951</parentid>
      <timestamp>2021-04-14T20:18:32Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Rank two temperaments */</comment>
      <origin>67313</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5367" sha1="dk106jfyh16r31bi8di0iha36sk2grq" xml:space="preserve">'''104edo''' divides the [[octave]] into 104 parts of size 11.54 [[cent|cents]] each. 

== Theory ==
104edo has two different equally viable 5-limit [[val|vals]], and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[Magic_family|magic temperament]]. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[Diaschismic_family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic_family #Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[Marvel family #Apollo|apollo temperament]]), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

== Rank two temperaments ==
=== In patent val ===
{| class="wikitable center-all right-3 left-5"
! Periods&lt;br&gt;per octave
! Generator
! Cents
! Associated&lt;br&gt;ratio
! Temperament
|-
| 1
| 33\104
| 380.769
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.462
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.846
| 18/17
| [[Undim]]
|}

=== In 104c val ===
{| class="wikitable center-all right-3 left-5"
! Periods&lt;br&gt;per octave
! Generator (reduced)
! Cents (reduced)
! Associated&lt;br&gt;ratio (reduced)
! Temperament
|-
| 1
| 21\104
| 242.308
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.538
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.308
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.154
| 4/3
| [[Diaschismic]]
|-
| 8
| 50\104 (2\104)
| 576.923 (23.077)
| 121/84 (78/77)
| [[Octowerck]] (7- or 11-limit)
|}

== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4 left-5"
|-
! rowspan="2"| #
! rowspan="2"| Cents
! colspan="3"| Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25&lt;br&gt;Subgroup
! Additional Ratios of 5&lt;br&gt;Tending Sharp (104c Val)
! Additional Ratios of 5&lt;br&gt;Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
| 
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]]
|-
| 29
| 334.615
| [[17/14]]
|
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
|…
|…
|…
|…
|…
|}

[[Category:apollo]]
[[Category:diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:magic]]
[[Category:necromancy]]</text>
      <sha1>dk106jfyh16r31bi8di0iha36sk2grq</sha1>
    </revision>
    <revision>
      <id>67891</id>
      <parentid>67313</parentid>
      <timestamp>2021-04-19T14:58:41Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Rank two temperaments */ table cleanup</comment>
      <origin>67891</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5379" sha1="7kkv7631xjl196t3jmkacyfkixzse7k" xml:space="preserve">'''104edo''' divides the [[octave]] into 104 parts of size 11.54 [[cent|cents]] each. 

== Theory ==
104edo has two different equally viable 5-limit [[val|vals]], and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[Magic_family|magic temperament]]. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[Diaschismic_family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic_family #Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[Marvel family #Apollo|apollo temperament]]), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

== Rank two temperaments ==
=== In patent val ===
{| class="wikitable center-all right-3 left-5"
! Periods&lt;br&gt;per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| 1
| 33\104
| 380.769
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.462
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.846
| 18/17
| [[Undim]]
|}

=== In 104c val ===
{| class="wikitable center-all right-3 left-5"
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 21\104
| 242.308
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.538
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.308
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.154
| 4/3
| [[Diaschismic]]
|-
| 8
| 50\104&lt;br&gt;(2\104)
| 576.923&lt;br&gt;(23.077)
| 121/84&lt;br&gt;(78/77)
| [[Octowerck]] (7- or 11-limit)
|}

== Intervals ==
{| class="wikitable center-all right-2 left-3 left-4 left-5"
|-
! rowspan="2"| #
! rowspan="2"| Cents
! colspan="3"| Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25&lt;br&gt;Subgroup
! Additional Ratios of 5&lt;br&gt;Tending Sharp (104c Val)
! Additional Ratios of 5&lt;br&gt;Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
| 
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]]
|-
| 29
| 334.615
| [[17/14]]
|
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
|…
|…
|…
|…
|…
|}

[[Category:apollo]]
[[Category:diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:magic]]
[[Category:necromancy]]</text>
      <sha1>7kkv7631xjl196t3jmkacyfkixzse7k</sha1>
    </revision>
    <revision>
      <id>68031</id>
      <parentid>67891</parentid>
      <timestamp>2021-04-20T21:49:13Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Rank two temperaments */</comment>
      <origin>68031</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6907" sha1="f5f1ywej4smgkf37jc6crokd45vjjqy" xml:space="preserve">'''104edo''' divides the [[octave]] into 104 parts of size 11.5385 [[cent|cents]] each. 

==Theory ==
104edo has two different equally viable 5-limit [[val|vals]], and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[Magic_family|magic temperament]]. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[Diaschismic_family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic_family #Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[Marvel family #Apollo|apollo temperament]]), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

== Rank two temperaments==
===In patent val ===
{| class="wikitable center-all"
!Periods&lt;br&gt;per octave
!Generator
! Cents
!Associated ratio
!Temperament
|-
| rowspan="2" |1
|33\104
|380.769
| 5/4
|[[Magic]] / necromancy / divination
|-
|51\104
|588.462
|7/5
|[[Untriton]]
|-
|4
|9\104
|103.846
|18/17
|[[Undim]]
|}

===In 104c val===
{| class="wikitable center-all"
!Periods&lt;br&gt;per octave
!Generator&lt;br&gt;(reduced)
!Cents&lt;br&gt;(reduced)
!Associated ratio&lt;br&gt;(reduced)
!Temperament
|-
| rowspan="3" |1
|21\104
| 242.308
|147/128
|[[Septiquarter]]
|-
|27\104
|311.538
|6/5
|[[Oolong]]
|-
|47\104
| 542.308
| 15/11
|[[Casablanca]] / marrakesh
|-
|2
|43\104
|496.154
|4/3
|[[Diaschismic]]
|-
|8
|50\104&lt;br&gt;(2\104)
|576.923&lt;br&gt;(23.077)
|121/84&lt;br&gt;(78/77)
|[[Octowerck]] (7- or 11-limit)
|}

==Intervals==
{| class="wikitable center-all"
|-
! rowspan="2" |#
! rowspan="2" |Cents
! colspan="3" | Approximate Ratios
|-
!of 2.3.7.11.13.17.19.25&lt;br&gt;Subgroup
!Additional Ratios of 5&lt;br&gt;Tending Sharp (104c Val)
!Additional Ratios of 5&lt;br&gt;Tending Flat (Patent Val)
|-
| 0
|0.000
|[[1/1]]
|[[126/125]]
|[[225/224]], [[100/99]]
|-
|1
|11.538
| [[225/224]], [[100/99]]
|
|
|-
|2
|23.077
|[[64/63]]
|[[81/80]], [[225/224]]
|[[50/49]]
|-
|3
|34.615
|[[49/48]], [[50/49]]
|
|[[81/80]], [[126/125]]
|-
|4
|46.154
|
|[[36/35]], [[50/49]]
|
|-
|5
|57.692
|[[28/27]], [[33/32]]
|
|[[25/24]], [[36/35]]
|-
|6
|69.231
|[[25/24]]
|
|
|-
|7
|80.769
|[[22/21]]
|[[25/24]], [[21/20]]
|[[20/19]]
|-
|8
|92.308
|[[19/18]]
|
[[20/19]]
|
[[21/20]]
|-
|9
|103.846
|[[17/16]], [[18/17]]
|
[[16/15]]
|
|-
|10
|115.385
|
|
|[[16/15]], [[15/14]]
|-
|11
|126.923
|[[14/13]]
|[[15/14]]
|
|-
|12
|138.462
|[[13/12]]
|
|
|-
|13
|150.000
|[[12/11]]
|
|
|-
|14
|161.538
|
|[[11/10]]
|
|-
|15
|173.077
|[[21/19]]
|
|[[10/9]], [[11/10]]
|-
|16
|184.615
|
|[[10/9]]
|
|-
|17
|196.154
|[[28/25]], [[19/17]]
|
|
|-
|18
|207.692
|9/8
|[[17/15]]
|
|-
|19
|219.231
|[[25/22]]
|
|[[17/15]]
|-
|20
|230.769
|[[8/7]]
|
|
|-
| 21
|242.308
|
|
|[[15/13]]
|-
|22
|253.846
|[[22/19]]
|[[15/13]]
|
|-
|23
|265.385
|[[7/6]]
|
|
|-
|24
|276.923
|[[75/64]]
|
|[[20/17]]
|-
| 25
|288.462
|[[32/27]], [[13/11]]
|[[20/17]]
|
|-
| 26
|300.000
|[[25/21]], [[19/16]]
|
|
|-
|27
|311.538
|
|[[6/5]]
|
|-
|28
|323.077
|
|
|[[6/5]]
|-
|29
|334.615
|[[17/14]]
|
|
|-
|30
|346.154
|[[11/9]], [[39/32]]
|
|
|-
|31
|357.692
|[[27/22]], [[16/13]]
|
|
|-
|32
|369.231
|[[26/21]], [[21/17]]
|
|
|-
| 33
|380.769
|
|
|[[5/4]]
|-
|34
|392.308
|
|[[5/4]]
|
|-
|35
|403.846
|[[63/50]], [[24/19]]
|[[19/15]]
|
|-
| 36
|415.385
|[[81/64]], [[14/11]]
|
|
[[19/15]]
|-
|37
|426.923
|[[32/25]]
|
|
|-
|38
|438.462
|[[9/7]]
|
|
|-
|39
|450.000
|[[22/17]]
|[[13/10]]
|
|-
|40
|461.538
|[[17/13]]
|
|[[13/10]]
|-
|41
|473.077
|[[21/16]]
|
|
|-
|42
|484.615
|
|
|
|-
|43
|496.154
|[[4/3]]
|
|
|-
|44
|507.692
|
|
|
|-
|45
|519.231
|
|[[27/20]]
|
|-
|46
|530.769
|[[19/14]]
|
|[[27/20]], [[15/11]]
|-
|47
|542.308
|[[26/19]]
|[[15/11]]
|
|-
|48
|553.846
|[[11/8]]
|
|
|-
|49
|565.385
|[[18/13]]
|
|
|-
|50
|576.923
|
|[[7/5]]
|
|-
|51
|588.462
|
|
|[[45/32]], [[7/5]]
|-
|52
|600.000
|[[17/12]], [[24/17]]
|[[45/32]], [[64/45]]
|
|-
|…
|…
|…
|…
|…
|}
Since 104edo has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*[[13edo]], it is the first edo where two smaller edos it allows one to use as circulating temperaments are Fibonacci edos.
{| class="wikitable"
|+Circulating temperaments in 104edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|21:20
|-
|6
|[[2L 6s]]
|18:17
|-
|7
|[[6L 1s]]
|15:14
|-
|8
|[[8edo]]
|equal
|-
|9
|[[5L 4s]]
|12:11
|-
|10
|[[4L 6s]]
|11:10
|-
|11
|[[5L 6s]]
|10:9
|-
|12
|[[8L 4s]]
|9:8
|-
|13
|[[13edo]]
|equal
|-
|14
|[[4L 10s]]
|8:7
|-
|15
|[[14L 1s]]
| rowspan="3" |7:6
|-
|16
|8L 8s
|-
|17
|[[2L 15s]]
|-
|18
|12L 6s
| rowspan="3" |6:5
|-
|19
|[[9L 10s]]
|-
|20
|4L 16s
|-
|21
|20L 1s
| rowspan="5" |5:4
|-
|22
|16L 6s
|-
|23
|[[12L 11s]]
|-
|24
|8L 16s
|-
|25
|4L 21s
|-
|26
|[[26edo]]
|equal
|-
|27
|23L 4s
| rowspan="8" |4:3
|-
|28
|20L 8s
|-
|29
|[[17L 12s]]
|-
|30
|14L 16s
|-
|31
|11L 20s
|-
|32
|8L 24s
|-
|33
|5L 28s
|-
|34
|2L 32s
|-
|35
|34L 1s
| rowspan="17" |3:2
|-
|36
|32L 4s
|-
|37
|30L 7s
|-
|38
|28L 10s
|-
|39
|26L 13s
|-
|40
|24L 16s
|-
|41
|22L 19s
|-
|42
|20L 22s
|-
|43
|18L 25s
|-
|44
|16L 28s
|-
|45
|14L 31s
|-
|46
|12L 34s
|-
|47
|10L 37s
|-
|48
|8L 40s
|-
|49
|6L 43s
|-
|50
|4L 46s
|-
|51
|2L 46s
|-
|52
|[[52edo]]
|equal
|-
|53
|51L 2s
| rowspan="31" |2:1
|-
|54
|50L 4s
|-
|55
|49L 6s
|-
|56
|48L 8s
|-
|57
|47L 10s
|-
|58
|46L 12s
|-
|59
|45L 14s
|-
|60
|44L 16s
|-
|61
|43L 18s
|-
|62
|42L 20s
|-
|63
|41L 22s
|-
|64
|40L 24s
|-
|65
|39L 26s
|-
|66
|38L 28s
|-
|67
|37L 30s
|-
|68
|36L 32s
|-
|69
|35L 34s
|-
|70
|34L 36s
|-
|71
|33L 38s
|-
|72
|32L 40s
|-
|73
|31L 42s
|-
|74
|30L 44s
|-
|75
|29L 46s
|-
|76
|28L 48s
|-
|77
|27L 50s
|-
|78
|26L 52s
|-
|79
|25L 54s
|-
|80
|24L 56s
|-
|81
|23L 58s
|-
|82
|22L 60s
|-
|83
|21L 62s
|}
[[Category:apollo]]
[[Category:diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:magic]]
[[Category:necromancy]]</text>
      <sha1>f5f1ywej4smgkf37jc6crokd45vjjqy</sha1>
    </revision>
    <revision>
      <id>68032</id>
      <parentid>68031</parentid>
      <timestamp>2021-04-20T21:55:15Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68032</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6907" sha1="83i1egy29x9i6640envvn1o50dlnzfh" xml:space="preserve">'''104edo''' divides the [[octave]] into 104 parts of size 11.5385 [[cent|cents]] each. 

==Theory ==
104edo has two different equally viable 5-limit [[val|vals]], and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[Magic_family|magic temperament]]. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[Diaschismic_family|diaschismic temperament]].

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[Magic_family #Necromancy|necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[Marvel family #Apollo|apollo temperament]]), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

== Rank two temperaments==
===In patent val ===
{| class="wikitable center-all"
!Periods&lt;br&gt;per octave
!Generator
! Cents
!Associated ratio
!Temperament
|-
| rowspan="2" |1
|33\104
|380.769
| 5/4
|[[Magic]] / necromancy / divination
|-
|51\104
|588.462
|7/5
|[[Untriton]]
|-
|4
|9\104
|103.846
|18/17
|[[Undim]]
|}

===In 104c val===
{| class="wikitable center-all"
!Periods&lt;br&gt;per octave
!Generator&lt;br&gt;(reduced)
!Cents&lt;br&gt;(reduced)
!Associated ratio&lt;br&gt;(reduced)
!Temperament
|-
| rowspan="3" |1
|21\104
| 242.308
|147/128
|[[Septiquarter]]
|-
|27\104
|311.538
|6/5
|[[Oolong]]
|-
|47\104
| 542.308
| 15/11
|[[Casablanca]] / marrakesh
|-
|2
|43\104
|496.154
|4/3
|[[Diaschismic]]
|-
|8
|50\104&lt;br&gt;(2\104)
|576.923&lt;br&gt;(23.077)
|121/84&lt;br&gt;(78/77)
|[[Octowerck]] (7- or 11-limit)
|}

==Intervals==
{| class="wikitable center-all"
|-
! rowspan="2" |#
! rowspan="2" |Cents
! colspan="3" | Approximate Ratios
|-
!of 2.3.7.11.13.17.19.25&lt;br&gt;Subgroup
!Additional Ratios of 5&lt;br&gt;Tending Sharp (104c Val)
!Additional Ratios of 5&lt;br&gt;Tending Flat (Patent Val)
|-
| 0
|0.000
|[[1/1]]
|[[126/125]]
|[[225/224]], [[100/99]]
|-
|1
|11.538
| [[225/224]], [[100/99]]
|
|
|-
|2
|23.077
|[[64/63]]
|[[81/80]], [[225/224]]
|[[50/49]]
|-
|3
|34.615
|[[49/48]], [[50/49]]
|
|[[81/80]], [[126/125]]
|-
|4
|46.154
|
|[[36/35]], [[50/49]]
|
|-
|5
|57.692
|[[28/27]], [[33/32]]
|
|[[25/24]], [[36/35]]
|-
|6
|69.231
|[[25/24]]
|
|
|-
|7
|80.769
|[[22/21]]
|[[25/24]], [[21/20]]
|[[20/19]]
|-
|8
|92.308
|[[19/18]]
|
[[20/19]]
|
[[21/20]]
|-
|9
|103.846
|[[17/16]], [[18/17]]
|
[[16/15]]
|
|-
|10
|115.385
|
|
|[[16/15]], [[15/14]]
|-
|11
|126.923
|[[14/13]]
|[[15/14]]
|
|-
|12
|138.462
|[[13/12]]
|
|
|-
|13
|150.000
|[[12/11]]
|
|
|-
|14
|161.538
|
|[[11/10]]
|
|-
|15
|173.077
|[[21/19]]
|
|[[10/9]], [[11/10]]
|-
|16
|184.615
|
|[[10/9]]
|
|-
|17
|196.154
|[[28/25]], [[19/17]]
|
|
|-
|18
|207.692
|9/8
|[[17/15]]
|
|-
|19
|219.231
|[[25/22]]
|
|[[17/15]]
|-
|20
|230.769
|[[8/7]]
|
|
|-
| 21
|242.308
|
|
|[[15/13]]
|-
|22
|253.846
|[[22/19]]
|[[15/13]]
|
|-
|23
|265.385
|[[7/6]]
|
|
|-
|24
|276.923
|[[75/64]]
|
|[[20/17]]
|-
| 25
|288.462
|[[32/27]], [[13/11]]
|[[20/17]]
|
|-
| 26
|300.000
|[[25/21]], [[19/16]]
|
|
|-
|27
|311.538
|
|[[6/5]]
|
|-
|28
|323.077
|
|
|[[6/5]]
|-
|29
|334.615
|[[17/14]]
|
|
|-
|30
|346.154
|[[11/9]], [[39/32]]
|
|
|-
|31
|357.692
|[[27/22]], [[16/13]]
|
|
|-
|32
|369.231
|[[26/21]], [[21/17]]
|
|
|-
| 33
|380.769
|
|
|[[5/4]]
|-
|34
|392.308
|
|[[5/4]]
|
|-
|35
|403.846
|[[63/50]], [[24/19]]
|[[19/15]]
|
|-
| 36
|415.385
|[[81/64]], [[14/11]]
|
|
[[19/15]]
|-
|37
|426.923
|[[32/25]]
|
|
|-
|38
|438.462
|[[9/7]]
|
|
|-
|39
|450.000
|[[22/17]]
|[[13/10]]
|
|-
|40
|461.538
|[[17/13]]
|
|[[13/10]]
|-
|41
|473.077
|[[21/16]]
|
|
|-
|42
|484.615
|
|
|
|-
|43
|496.154
|[[4/3]]
|
|
|-
|44
|507.692
|
|
|
|-
|45
|519.231
|
|[[27/20]]
|
|-
|46
|530.769
|[[19/14]]
|
|[[27/20]], [[15/11]]
|-
|47
|542.308
|[[26/19]]
|[[15/11]]
|
|-
|48
|553.846
|[[11/8]]
|
|
|-
|49
|565.385
|[[18/13]]
|
|
|-
|50
|576.923
|
|[[7/5]]
|
|-
|51
|588.462
|
|
|[[45/32]], [[7/5]]
|-
|52
|600.000
|[[17/12]], [[24/17]]
|[[45/32]], [[64/45]]
|
|-
|…
|…
|…
|…
|…
|}
Since 104edo has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*[[13edo]], it is the first edo where two smaller edos it allows one to use as circulating temperaments are Fibonacci edos.
{| class="wikitable"
|+Circulating temperaments in 104edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|21:20
|-
|6
|[[2L 4s]]
|18:17
|-
|7
|[[6L 1s]]
|15:14
|-
|8
|[[8edo]]
|equal
|-
|9
|[[5L 4s]]
|12:11
|-
|10
|[[4L 6s]]
|11:10
|-
|11
|[[5L 6s]]
|10:9
|-
|12
|[[8L 4s]]
|9:8
|-
|13
|[[13edo]]
|equal
|-
|14
|[[4L 10s]]
|8:7
|-
|15
|[[14L 1s]]
| rowspan="3" |7:6
|-
|16
|8L 8s
|-
|17
|[[2L 15s]]
|-
|18
|12L 6s
| rowspan="3" |6:5
|-
|19
|[[9L 10s]]
|-
|20
|4L 16s
|-
|21
|20L 1s
| rowspan="5" |5:4
|-
|22
|16L 6s
|-
|23
|[[12L 11s]]
|-
|24
|8L 16s
|-
|25
|4L 21s
|-
|26
|[[26edo]]
|equal
|-
|27
|23L 4s
| rowspan="8" |4:3
|-
|28
|20L 8s
|-
|29
|[[17L 12s]]
|-
|30
|14L 16s
|-
|31
|11L 20s
|-
|32
|8L 24s
|-
|33
|5L 28s
|-
|34
|2L 32s
|-
|35
|34L 1s
| rowspan="17" |3:2
|-
|36
|32L 4s
|-
|37
|30L 7s
|-
|38
|28L 10s
|-
|39
|26L 13s
|-
|40
|24L 16s
|-
|41
|22L 19s
|-
|42
|20L 22s
|-
|43
|18L 25s
|-
|44
|16L 28s
|-
|45
|14L 31s
|-
|46
|12L 34s
|-
|47
|10L 37s
|-
|48
|8L 40s
|-
|49
|6L 43s
|-
|50
|4L 46s
|-
|51
|2L 46s
|-
|52
|[[52edo]]
|equal
|-
|53
|51L 2s
| rowspan="31" |2:1
|-
|54
|50L 4s
|-
|55
|49L 6s
|-
|56
|48L 8s
|-
|57
|47L 10s
|-
|58
|46L 12s
|-
|59
|45L 14s
|-
|60
|44L 16s
|-
|61
|43L 18s
|-
|62
|42L 20s
|-
|63
|41L 22s
|-
|64
|40L 24s
|-
|65
|39L 26s
|-
|66
|38L 28s
|-
|67
|37L 30s
|-
|68
|36L 32s
|-
|69
|35L 34s
|-
|70
|34L 36s
|-
|71
|33L 38s
|-
|72
|32L 40s
|-
|73
|31L 42s
|-
|74
|30L 44s
|-
|75
|29L 46s
|-
|76
|28L 48s
|-
|77
|27L 50s
|-
|78
|26L 52s
|-
|79
|25L 54s
|-
|80
|24L 56s
|-
|81
|23L 58s
|-
|82
|22L 60s
|-
|83
|21L 62s
|}
[[Category:apollo]]
[[Category:diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:magic]]
[[Category:necromancy]]</text>
      <sha1>83i1egy29x9i6640envvn1o50dlnzfh</sha1>
    </revision>
    <revision>
      <id>72334</id>
      <parentid>68032</parentid>
      <timestamp>2021-06-10T18:10:04Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cleanup 1/2</comment>
      <origin>72334</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7282" sha1="c40ge8b9anq1tsv17uuea4cqsp93unw" xml:space="preserve">'''104 EDO''' divides the [[octave]] into 104 parts of size 11.5385 [[cent]]s each. 

== Theory ==

{{Primes in edo|104|prec=2}}

104 EDO has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[magic]] temperament. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

== Rank two temperaments ==

=== In patent val ===

{| class="wikitable center-all"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| rowspan="2" | 1
| 33\104
| 380.769
| 5/4
| [[Magic]] / necromancy / divination
|-
| 51\104
| 588.462
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.846
| 18/17
| [[Undim]]
|}

=== In 104c val ===

{| class="wikitable center-all"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| rowspan="3" | 1
| 21\104
| 242.308
| 147/128
| [[Septiquarter]]
|-
| 27\104
| 311.538
| 6/5
| [[Oolong]]
|-
| 47\104
| 542.308
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.154
| 4/3
| [[Diaschismic]]
|-
| 8
| 50\104 &lt;br&gt; (2\104)
| 576.923 &lt;br&gt; (23.077)
| 121/84 &lt;br&gt; (78/77)
| [[Octowerck]] (7- or 11-limit)
|}

==Intervals==

{| class="wikitable center-all"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
|
[[20/19]]
|
[[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
|
[[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]]
|-
| 29
| 334.615
| [[17/14]]
|
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
|
[[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

Since 104edo has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*[[13edo]], it is the first edo where two smaller edos it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable"
|+Circulating temperaments in 104edo
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8edo]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13edo]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" |7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" |6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" |5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26edo]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" |4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" |3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52edo]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" |2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>c40ge8b9anq1tsv17uuea4cqsp93unw</sha1>
    </revision>
    <revision>
      <id>72337</id>
      <parentid>72334</parentid>
      <timestamp>2021-06-10T18:19:23Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cleanup 2/2</comment>
      <origin>72337</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7321" sha1="dsbc500604maynqba8p5v9crr849ewj" xml:space="preserve">'''104 EDO''' divides the [[octave]] into 104 parts of size 11.5385 [[cent]]s each. 

== Theory ==

{{Primes in edo|104|prec=2}}

104 EDO has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[magic]] temperament. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104 EDO with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

== Rank two temperaments ==

=== In patent val ===

{| class="wikitable center-all"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| rowspan="2" | 1
| 33\104
| 380.769
| 5/4
| [[Magic]] / necromancy / divination
|-
| 51\104
| 588.462
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.846
| 18/17
| [[Undim]]
|}

=== In 104c val ===

{| class="wikitable center-all"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| rowspan="3" | 1
| 21\104
| 242.308
| 147/128
| [[Septiquarter]]
|-
| 27\104
| 311.538
| 6/5
| [[Oolong]]
|-
| 47\104
| 542.308
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.154
| 4/3
| [[Diaschismic]]
|-
| 8
| 50\104 &lt;br&gt; (2\104)
| 576.923 &lt;br&gt; (23.077)
| 121/84 &lt;br&gt; (78/77)
| [[Octowerck]] (7- or 11-limit)
|}

== Intervals ==

{| class="wikitable center-all right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]]
|-
| 29
| 334.615
| [[17/14]]
|
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" | 3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52 EDO]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" | 2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>dsbc500604maynqba8p5v9crr849ewj</sha1>
    </revision>
    <revision>
      <id>74107</id>
      <parentid>72337</parentid>
      <timestamp>2021-07-11T10:37:07Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ prec is now by default 2 between 53 and 526 EDOs</comment>
      <origin>74107</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7314" sha1="gcoa6qk0wx4rr0bkaplh3tr8cwsy80c" xml:space="preserve">'''104 EDO''' divides the [[octave]] into 104 parts of size 11.5385 [[cent]]s each. 

== Theory ==

{{Primes in edo|104}}

104 EDO has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[magic]] temperament. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104 EDO with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11, 13, or 17 limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

== Rank two temperaments ==

=== In patent val ===

{| class="wikitable center-all"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| rowspan="2" | 1
| 33\104
| 380.769
| 5/4
| [[Magic]] / necromancy / divination
|-
| 51\104
| 588.462
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.846
| 18/17
| [[Undim]]
|}

=== In 104c val ===

{| class="wikitable center-all"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| rowspan="3" | 1
| 21\104
| 242.308
| 147/128
| [[Septiquarter]]
|-
| 27\104
| 311.538
| 6/5
| [[Oolong]]
|-
| 47\104
| 542.308
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.154
| 4/3
| [[Diaschismic]]
|-
| 8
| 50\104 &lt;br&gt; (2\104)
| 576.923 &lt;br&gt; (23.077)
| 121/84 &lt;br&gt; (78/77)
| [[Octowerck]] (7- or 11-limit)
|}

== Intervals ==

{| class="wikitable center-all right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]]
|-
| 29
| 334.615
| [[17/14]]
|
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" | 3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52 EDO]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" | 2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>gcoa6qk0wx4rr0bkaplh3tr8cwsy80c</sha1>
    </revision>
    <revision>
      <id>75224</id>
      <parentid>74107</parentid>
      <timestamp>2021-07-29T10:54:20Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Improve intro and sectioning</comment>
      <origin>75224</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7564" sha1="e7panecbi1p7uiez01c0gu4wybj2hlf" xml:space="preserve">The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size 11.5385 [[cent]]s each. 

== Theory ==

104 EDO has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[magic]] temperament. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104 EDO with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Primes in edo|104}}

== Regular temperament properties ==
=== Rank-2 temperaments ===

==== In patent val ====

{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| 1
| 33\104
| 380.769
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.462
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.846
| 18/17
| [[Undim]]
|}

==== In 104c val ====

{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| 1
| 21\104
| 242.308
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.538
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.308
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.154
| 4/3
| [[Diaschismic]]
|-
| 8
| 50\104 &lt;br&gt; (2\104)
| 576.923 &lt;br&gt; (23.077)
| 121/84 &lt;br&gt; (78/77)
| [[Octowerck]] (7- or 11-limit)
|}

== Intervals ==

{| class="wikitable center-all right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]]
|-
| 29
| 334.615
| [[17/14]]
|
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

== Scales ==
Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" | 3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52 EDO]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" | 2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>e7panecbi1p7uiez01c0gu4wybj2hlf</sha1>
    </revision>
    <revision>
      <id>76550</id>
      <parentid>75224</parentid>
      <timestamp>2021-08-18T13:07:22Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+RTT table and important corrections</comment>
      <origin>76550</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8390" sha1="hshvps7t690kry9kow7f6j2j07gofub" xml:space="preserve">The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size 11.5385 [[cent]]s each. 

== Theory ==

104 EDO has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[magic]] temperament. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104 EDO with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Primes in edo|104}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| 1
| 33\104
| 380.769
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.462
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.846
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| 1
| 21\104
| 242.308
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.538
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.308
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.154
| 4/3
| [[Diaschismic]]
|-
| 8
| 50\104 &lt;br&gt; (2\104)
| 565.385 &lt;br&gt; (34.615)
| 168/121 &lt;br&gt; (55/54)
| [[Octowerck]]
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

== Scales ==
Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" | 3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52 EDO]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" | 2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>hshvps7t690kry9kow7f6j2j07gofub</sha1>
    </revision>
    <revision>
      <id>76551</id>
      <parentid>76550</parentid>
      <timestamp>2021-08-18T13:08:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Correction +1</comment>
      <origin>76551</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8390" sha1="s073gfin1wr8uthsw5rboxe9nm8cfry" xml:space="preserve">The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size 11.5385 [[cent]]s each. 

== Theory ==

104 EDO has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[magic]] temperament. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104 EDO with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Primes in edo|104}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| 1
| 33\104
| 380.769
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.462
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.846
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| 1
| 21\104
| 242.308
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.538
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.308
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.154
| 4/3
| [[Diaschismic]]
|-
| 8
| 49\104 &lt;br&gt; (2\104)
| 565.385 &lt;br&gt; (34.615)
| 168/121 &lt;br&gt; (55/54)
| [[Octowerck]]
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

== Scales ==
Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" | 3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52 EDO]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" | 2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Equal divisions of the octave]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>s073gfin1wr8uthsw5rboxe9nm8cfry</sha1>
    </revision>
    <revision>
      <id>76553</id>
      <parentid>76551</parentid>
      <timestamp>2021-08-18T13:19:02Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+infobox</comment>
      <origin>76553</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8597" sha1="6qjh8oo0xhjqjhu5ds5yhg1sfy8ut6l" xml:space="preserve">{{Infobox ET
| Prime factorization = 2&lt;sup&gt;3&lt;/sup&gt; × 13
| Step size = 11.53846¢
| Fifth = 61\104 (703.85¢)
| Major 2nd = 18\104 (208¢)
| Minor 2nd = 7\104 (81¢)
| Augmented 1sn = 11\104  (127¢)
}}

The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size about 11.5 [[cent]]s each. 

== Theory ==

104 EDO has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[magic]] temperament. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104 EDO with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Primes in edo|104}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| 1
| 33\104
| 380.769
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.462
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.846
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| 1
| 21\104
| 242.308
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.538
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.308
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.154
| 4/3
| [[Diaschismic]]
|-
| 8
| 49\104 &lt;br&gt; (2\104)
| 565.385 &lt;br&gt; (34.615)
| 168/121 &lt;br&gt; (55/54)
| [[Octowerck]]
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

== Scales ==
Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" | 3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52 EDO]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" | 2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Equal divisions of the octave]]
[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>6qjh8oo0xhjqjhu5ds5yhg1sfy8ut6l</sha1>
    </revision>
    <revision>
      <id>76597</id>
      <parentid>76553</parentid>
      <timestamp>2021-08-20T07:47:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Rank-2 temperaments */ +mowgli</comment>
      <origin>76597</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8634" sha1="tfvrw9dbcazybriemgl9mli5ori2cj1" xml:space="preserve">{{Infobox ET
| Prime factorization = 2&lt;sup&gt;3&lt;/sup&gt; × 13
| Step size = 11.53846¢
| Fifth = 61\104 (703.85¢)
| Major 2nd = 18\104 (208¢)
| Minor 2nd = 7\104 (81¢)
| Augmented 1sn = 11\104  (127¢)
}}

The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size about 11.5 [[cent]]s each. 

== Theory ==

104 EDO has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val|104 165 241}} ([[patent val]]), tempers out [[3125/3072]], and supports [[magic]] temperament. The sharp major third val, {{val|104 165 242}} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104 EDO with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank three temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank four temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, 2197/2187, 16807/16731, 20449/20412, 21632/21609, 26411/26364 and 10648/10647. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Primes in edo|104}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 43\104
| 496.15
| 4/3
| [[Diaschismic]]
|-
| 8
| 49\104 &lt;br&gt; (2\104)
| 565.38 &lt;br&gt; (34.62)
| 168/121 &lt;br&gt; (55/54)
| [[Octowerck]]
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

== Scales ==
Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" | 3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52 EDO]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" | 2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Equal divisions of the octave]]
[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>tfvrw9dbcazybriemgl9mli5ori2cj1</sha1>
    </revision>
    <revision>
      <id>79773</id>
      <parentid>76597</parentid>
      <timestamp>2021-10-22T10:33:21Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Update infobox and some cleanup</comment>
      <origin>79773</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8740" sha1="73cshs1gv5de81cnb111yov56d5hfqh" xml:space="preserve">{{Infobox ET
| Prime factorization = 2&lt;sup&gt;3&lt;/sup&gt; × 13
| Step size = 11.53846¢
| Fifth = 61\104 (703.85¢)
| Major 2nd = 18\104 (208¢)
| Semitones = 11:7 (127¢ : 81¢)
}}
The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size about 11.5 [[cent]]s each. 

== Theory ==

104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and supports [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Primes in edo|104}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br&gt;(9\104)
| 496.15&lt;br&gt;(103.85)
| 4/3&lt;br&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104 &lt;br&gt; (2\104)
| 565.38 &lt;br&gt; (34.62)
| 168/121 &lt;br&gt; (55/54)
| [[Octowerck]] / octowerckis
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

== Scales ==
Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" | 3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52 EDO]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" | 2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Equal divisions of the octave]]
[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>73cshs1gv5de81cnb111yov56d5hfqh</sha1>
    </revision>
    <revision>
      <id>79774</id>
      <parentid>79773</parentid>
      <timestamp>2021-10-22T10:38:09Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Unify precision and consistency :)</comment>
      <origin>79774</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8757" sha1="4x0f42yxk7pep4o0x3j1v9tv6qzweu7" xml:space="preserve">{{Infobox ET
| Prime factorization = 2&lt;sup&gt;3&lt;/sup&gt; × 13
| Step size = 11.538¢
| Fifth = 61\104 (703.846¢)
| Major 2nd = 18\104 (208¢)
| Semitones = 11:7 (127¢ : 81¢)
| Consistency = 3
}}
The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size about 11.5 [[cent]]s each. 

== Theory ==

104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and supports [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Primes in edo|104}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br&gt;(9\104)
| 496.15&lt;br&gt;(103.85)
| 4/3&lt;br&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104 &lt;br&gt; (2\104)
| 565.38 &lt;br&gt; (34.62)
| 168/121 &lt;br&gt; (55/54)
| [[Octowerck]] / octowerckis
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

== Scales ==
Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" | 3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52 EDO]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" | 2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Equal divisions of the octave]]
[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>4x0f42yxk7pep4o0x3j1v9tv6qzweu7</sha1>
    </revision>
    <revision>
      <id>84829</id>
      <parentid>79774</parentid>
      <timestamp>2022-01-25T19:00:03Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>link to new page for Supports</comment>
      <origin>84829</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8761" sha1="k5l2iq5a9ktsb66y3p50n09v9j4a20t" xml:space="preserve">{{Infobox ET
| Prime factorization = 2&lt;sup&gt;3&lt;/sup&gt; × 13
| Step size = 11.538¢
| Fifth = 61\104 (703.846¢)
| Major 2nd = 18\104 (208¢)
| Semitones = 11:7 (127¢ : 81¢)
| Consistency = 3
}}
The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size about 11.5 [[cent]]s each. 

== Theory ==

104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Primes in edo|104}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br&gt;(9\104)
| 496.15&lt;br&gt;(103.85)
| 4/3&lt;br&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104 &lt;br&gt; (2\104)
| 565.38 &lt;br&gt; (34.62)
| 168/121 &lt;br&gt; (55/54)
| [[Octowerck]] / octowerckis
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

== Scales ==
Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" | 3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52 EDO]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" | 2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Equal divisions of the octave]]
[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>k5l2iq5a9ktsb66y3p50n09v9j4a20t</sha1>
    </revision>
    <revision>
      <id>92914</id>
      <parentid>84829</parentid>
      <timestamp>2022-07-02T16:16:42Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92914</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8789" sha1="9nhevnr2aorqamnqc0zb1an03tca71y" xml:space="preserve">{{Infobox ET
| Prime factorization = 2&lt;sup&gt;3&lt;/sup&gt; × 13
| Step size = 11.538¢
| Fifth = 61\104 (703.846¢)
| Major 2nd = 18\104 (208¢)
| Semitones = 11:7 (127¢ : 81¢)
| Consistency = 3
}}
The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size about 11.5 [[cent]]s each. 

== Theory ==

104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Primes in edo|104}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br&gt;(9\104)
| 496.15&lt;br&gt;(103.85)
| 4/3&lt;br&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104 &lt;br&gt; (2\104)
| 565.38 &lt;br&gt; (34.62)
| 168/121 &lt;br&gt; (55/54)
| [[Octowerck]] / octowerckis
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

== Scales ==
Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" | 3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52 EDO]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" | 2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>9nhevnr2aorqamnqc0zb1an03tca71y</sha1>
    </revision>
    <revision>
      <id>96080</id>
      <parentid>92914</parentid>
      <timestamp>2022-10-04T16:00:50Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET now computes most parameters automatically</comment>
      <origin>96080</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8611" sha1="8vcykto5y0vt12868v8qiw7x932je3z" xml:space="preserve">{{Infobox ET}}
The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size about 11.5 [[cent]]s each. 

== Theory ==

104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Primes in edo|104}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br&gt;(9\104)
| 496.15&lt;br&gt;(103.85)
| 4/3&lt;br&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104 &lt;br&gt; (2\104)
| 565.38 &lt;br&gt; (34.62)
| 168/121 &lt;br&gt; (55/54)
| [[Octowerck]] / octowerckis
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

== Scales ==
Since 104 EDO has a step of 11.5385 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. As 8*[[13 EDO]], it is the first EDO where two smaller EDOs it allows one to use as circulating temperaments are Fibonacci EDOs.

{| class="wikitable center-all"
|+ Circulating temperaments in 104 EDO
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[4L 1s]]
| 21:20
|-
| 6
| [[2L 4s]]
| 18:17
|-
| 7
| [[6L 1s]]
| 15:14
|-
| 8
| [[8 EDO]]
| equal
|-
| 9
| [[5L 4s]]
| 12:11
|-
| 10
| [[4L 6s]]
| 11:10
|-
| 11
| [[5L 6s]]
| 10:9
|-
| 12
| [[8L 4s]]
| 9:8
|-
| 13
| [[13 EDO]]
| equal
|-
| 14
| [[4L 10s]]
| 8:7
|-
| 15
| [[14L 1s]]
| rowspan="3" | 7:6
|-
| 16
| 8L 8s
|-
| 17
| [[2L 15s]]
|-
| 18
| 12L 6s
| rowspan="3" | 6:5
|-
| 19
| [[9L 10s]]
|-
| 20
| 4L 16s
|-
| 21
| 20L 1s
| rowspan="5" | 5:4
|-
| 22
| 16L 6s
|-
| 23
| [[12L 11s]]
|-
| 24
| 8L 16s
|-
| 25
| 4L 21s
|-
| 26
| [[26 EDO]]
| equal
|-
| 27
| 23L 4s
| rowspan="8" | 4:3
|-
| 28
| 20L 8s
|-
| 29
| [[17L 12s]]
|-
| 30
| 14L 16s
|-
| 31
| 11L 20s
|-
| 32
| 8L 24s
|-
| 33
| 5L 28s
|-
| 34
| 2L 32s
|-
| 35
| 34L 1s
| rowspan="17" | 3:2
|-
| 36
| 32L 4s
|-
| 37
| 30L 7s
|-
| 38
| 28L 10s
|-
| 39
| 26L 13s
|-
| 40
| 24L 16s
|-
| 41
| 22L 19s
|-
| 42
| 20L 22s
|-
| 43
| 18L 25s
|-
| 44
| 16L 28s
|-
| 45
| 14L 31s
|-
| 46
| 12L 34s
|-
| 47
| 10L 37s
|-
| 48
| 8L 40s
|-
| 49
| 6L 43s
|-
| 50
| 4L 46s
|-
| 51
| 2L 46s
|-
| 52
| [[52 EDO]]
| equal
|-
| 53
| 51L 2s
| rowspan="31" | 2:1
|-
| 54
| 50L 4s
|-
| 55
| 49L 6s
|-
| 56
| 48L 8s
|-
| 57
| 47L 10s
|-
| 58
| 46L 12s
|-
| 59
| 45L 14s
|-
| 60
| 44L 16s
|-
| 61
| 43L 18s
|-
| 62
| 42L 20s
|-
| 63
| 41L 22s
|-
| 64
| 40L 24s
|-
| 65
| 39L 26s
|-
| 66
| 38L 28s
|-
| 67
| 37L 30s
|-
| 68
| 36L 32s
|-
| 69
| 35L 34s
|-
| 70
| 34L 36s
|-
| 71
| 33L 38s
|-
| 72
| 32L 40s
|-
| 73
| 31L 42s
|-
| 74
| 30L 44s
|-
| 75
| 29L 46s
|-
| 76
| 28L 48s
|-
| 77
| 27L 50s
|-
| 78
| 26L 52s
|-
| 79
| 25L 54s
|-
| 80
| 24L 56s
|-
| 81
| 23L 58s
|-
| 82
| 22L 60s
|-
| 83
| 21L 62s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>8vcykto5y0vt12868v8qiw7x932je3z</sha1>
    </revision>
    <revision>
      <id>115189</id>
      <parentid>96080</parentid>
      <timestamp>2023-05-30T21:12:08Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Scales */</comment>
      <origin>115189</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6591" sha1="o68diyt92hdls7crblnzftpg6d8it42" xml:space="preserve">{{Infobox ET}}
The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size about 11.5 [[cent]]s each. 

== Theory ==

104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Primes in edo|104}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator
! Cents
! Associated ratio
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per octave
! Generator &lt;br&gt; (reduced)
! Cents &lt;br&gt; (reduced)
! Associated ratio &lt;br&gt; (reduced)
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br&gt;(9\104)
| 496.15&lt;br&gt;(103.85)
| 4/3&lt;br&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104 &lt;br&gt; (2\104)
| 565.38 &lt;br&gt; (34.62)
| 168/121 &lt;br&gt; (55/54)
| [[Octowerck]] / octowerckis
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>o68diyt92hdls7crblnzftpg6d8it42</sha1>
    </revision>
    <revision>
      <id>115231</id>
      <parentid>115189</parentid>
      <timestamp>2023-05-31T06:07:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Update the prime error table; +subsets and supersets; style</comment>
      <origin>115231</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6727" sha1="7lk8bis90khsa2jwoztlw6n2iv1gr0v" xml:space="preserve">{{Infobox ET}}
The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size about 11.5 [[cent]]s each. 

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per 8ve
! Generator
! Cents
! Associated Ratio
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated Ratio&lt;br&gt;(Reduced)
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br&gt;(9\104)
| 496.15&lt;br&gt;(103.85)
| 4/3&lt;br&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104 &lt;br&gt; (2\104)
| 565.38 &lt;br&gt; (34.62)
| 168/121 &lt;br&gt; (55/54)
| [[Octowerck]] / octowerckis
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>7lk8bis90khsa2jwoztlw6n2iv1gr0v</sha1>
    </revision>
    <revision>
      <id>115232</id>
      <parentid>115231</parentid>
      <timestamp>2023-05-31T06:12:06Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ +bosonic</comment>
      <origin>115232</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6802" sha1="qoslqahkf4zw9453botwn5z5642j13p" xml:space="preserve">{{Infobox ET}}
The '''104 equal divisions of the octave''' ('''104edo'''), or the '''104(-tone) equal temperament''' ('''104tet''', '''104et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 104 parts of size about 11.5 [[cent]]s each. 

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per 8ve
! Generator
! Cents
! Associated Ratio
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated Ratio&lt;br&gt;(Reduced)
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br&gt;(9\104)
| 496.15&lt;br&gt;(103.85)
| 4/3&lt;br&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br&gt;(2\104)
| 565.38&lt;br&gt;(34.62)
| 168/121&lt;br&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br&gt;(1\104)
| 496.15&lt;br&gt;(11.54)
| 4/3&lt;br&gt;(225/224)
| [[Bosonic]]
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>qoslqahkf4zw9453botwn5z5642j13p</sha1>
    </revision>
    <revision>
      <id>115233</id>
      <parentid>115232</parentid>
      <timestamp>2023-05-31T06:14:11Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Template for intro</comment>
      <origin>115233</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6560" sha1="0dnnzmedoytvxgagvzygtisp8s7vub4" xml:space="preserve">{{Infobox ET}}
{{EDO intro|104}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per 8ve
! Generator
! Cents
! Associated Ratio
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated Ratio&lt;br&gt;(Reduced)
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br&gt;(9\104)
| 496.15&lt;br&gt;(103.85)
| 4/3&lt;br&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br&gt;(2\104)
| 565.38&lt;br&gt;(34.62)
| 168/121&lt;br&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br&gt;(1\104)
| 496.15&lt;br&gt;(11.54)
| 4/3&lt;br&gt;(225/224)
| [[Bosonic]]
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| [[126/125]]
| [[225/224]], [[100/99]]
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], [[225/224]]
| [[50/49]]
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| [[81/80]], [[126/125]]
|-
| 4
| 46.154
|
| [[36/35]], [[50/49]]
|
|-
| 5
| 57.692
| [[28/27]], [[33/32]]
|
| [[25/24]], [[36/35]]
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| [[25/24]], [[21/20]]
| [[20/19]]
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| [[21/20]]
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| [[16/15]]
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| [[15/14]]
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| [[10/9]], [[11/10]]
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| 9/8
| [[17/15]]
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| [[15/13]]
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| [[20/17]]
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], [[40/33]]
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| [[5/4]]
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| [[19/15]]
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| [[13/10]]
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| [[27/20]], [[15/11]]
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], [[7/5]]
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| [[45/32]], [[64/45]]
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>0dnnzmedoytvxgagvzygtisp8s7vub4</sha1>
    </revision>
    <revision>
      <id>121943</id>
      <parentid>115233</parentid>
      <timestamp>2023-08-13T09:01:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ italicize inconsistent intervals; -redundant category</comment>
      <origin>121943</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6613" sha1="rg7c9ty359gz6ug15gaqj0ld83dg2x8" xml:space="preserve">{{Infobox ET}}
{{EDO intro|104}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| [{{val| 104 165 }}]
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| [{{val| 104 165 242 }}] (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| [{{val| 104 165 242 292 }}] (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| [{{val| 104 165 242 292 360 }}] (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| [{{val| 104 165 242 292 360 385 }}] (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods &lt;br&gt; per 8ve
! Generator
! Cents
! Associated Ratio
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated Ratio&lt;br&gt;(Reduced)
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br&gt;(9\104)
| 496.15&lt;br&gt;(103.85)
| 4/3&lt;br&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br&gt;(2\104)
| 565.38&lt;br&gt;(34.62)
| 168/121&lt;br&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br&gt;(1\104)
| 496.15&lt;br&gt;(11.54)
| 4/3&lt;br&gt;(225/224)
| [[Bosonic]]
|}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| ''[[126/125]]''
| ''[[225/224]]'', ''[[100/99]]''
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.154
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.692
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| ''[[20/19]]''
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| ''[[5/4]]''
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], ''[[7/5]]''
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>rg7c9ty359gz6ug15gaqj0ld83dg2x8</sha1>
    </revision>
    <revision>
      <id>145900</id>
      <parentid>121943</parentid>
      <timestamp>2024-06-08T08:58:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>145900</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6752" sha1="awgd3awt2h70wese4a30b57r5nmc2vd" xml:space="preserve">{{Infobox ET}}
{{EDO intro|104}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| -0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| -1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| -0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| -0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| -0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
==== In patent val ====
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

==== In 104c val ====
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br&gt;(9\104)
| 496.15&lt;br&gt;(103.85)
| 4/3&lt;br&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br&gt;(2\104)
| 565.38&lt;br&gt;(34.62)
| 168/121&lt;br&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br&gt;(1\104)
| 496.15&lt;br&gt;(11.54)
| 4/3&lt;br&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25 &lt;br&gt; Subgroup
! Additional Ratios of 5 &lt;br&gt; Tending Sharp (104c Val)
! Additional Ratios of 5 &lt;br&gt; Tending Flat (Patent Val)
|-
| 0
| 0.000
| [[1/1]]
| ''[[126/125]]''
| ''[[225/224]]'', ''[[100/99]]''
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.154
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.692
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| ''[[20/19]]''
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| ''[[5/4]]''
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], ''[[7/5]]''
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Diaschismic]]
[[Category:Apollo]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>awgd3awt2h70wese4a30b57r5nmc2vd</sha1>
    </revision>
    <revision>
      <id>165987</id>
      <parentid>145900</parentid>
      <timestamp>2024-11-16T06:13:09Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>165987</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6452" sha1="1og0uaqmoasgs3q5gorewhm36l8pclr" xml:space="preserve">{{Infobox ET}}
{{EDO intro|104}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{{comma basis begin}}
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| &amp;minus;0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| &amp;minus;1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| &amp;minus;0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| &amp;minus;0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| &amp;minus;0.855
| 0.864
| 7.49
{{comma basis end}}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
{{orf}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | &amp;#35;
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! of 2.3.7.11.13.17.19.25&lt;br /&gt;subgroup
! Additional ratios of 5&lt;br /&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br /&gt;tending flat (patent val)
|-
| 0
| 0.000
| [[1/1]]
| ''[[126/125]]''
| ''[[225/224]]'', ''[[100/99]]''
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.154
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.692
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| ''[[20/19]]''
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| ''[[5/4]]''
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], ''[[7/5]]''
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>1og0uaqmoasgs3q5gorewhm36l8pclr</sha1>
    </revision>
    <revision>
      <id>165988</id>
      <parentid>165987</parentid>
      <timestamp>2024-11-16T06:13:26Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */</comment>
      <origin>165988</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6452" sha1="7ii71uu9kkoxcd4xten9oq7njr6jyj3" xml:space="preserve">{{Infobox ET}}
{{EDO intro|104}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{{comma basis begin}}
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| &amp;minus;0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| &amp;minus;1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| &amp;minus;0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| &amp;minus;0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| &amp;minus;0.855
| 0.864
| 7.49
{{comma basis end}}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
{{orf}}

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | &amp;#35;
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br /&gt;subgroup
! Additional ratios of 5&lt;br /&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br /&gt;tending flat (patent val)
|-
| 0
| 0.000
| [[1/1]]
| ''[[126/125]]''
| ''[[225/224]]'', ''[[100/99]]''
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.154
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.692
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| ''[[20/19]]''
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| ''[[5/4]]''
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], ''[[7/5]]''
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>7ii71uu9kkoxcd4xten9oq7njr6jyj3</sha1>
    </revision>
    <revision>
      <id>166093</id>
      <parentid>165988</parentid>
      <timestamp>2024-11-16T11:41:07Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Partial undo</comment>
      <origin>166093</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6839" sha1="hlnczf0vktd3qz4vjpgjios6tdccign" xml:space="preserve">{{Infobox ET}}
{{EDO intro|104}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| &amp;minus;0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| &amp;minus;1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| &amp;minus;0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| &amp;minus;0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| &amp;minus;0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | &amp;#35;
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br /&gt;subgroup
! Additional ratios of 5&lt;br /&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br /&gt;tending flat (patent val)
|-
| 0
| 0.000
| [[1/1]]
| ''[[126/125]]''
| ''[[225/224]]'', ''[[100/99]]''
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.154
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.692
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| ''[[20/19]]''
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| ''[[5/4]]''
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], ''[[7/5]]''
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>hlnczf0vktd3qz4vjpgjios6tdccign</sha1>
    </revision>
    <revision>
      <id>167921</id>
      <parentid>166093</parentid>
      <timestamp>2024-11-30T01:39:36Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Add scales section</comment>
      <origin>167921</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6866" sha1="brrtpgororptfqczrrebrc2sdhdkhij" xml:space="preserve">{{Infobox ET}}
{{EDO intro|104}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| &amp;minus;0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| &amp;minus;1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| &amp;minus;0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| &amp;minus;0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| &amp;minus;0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | &amp;#35;
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br /&gt;subgroup
! Additional ratios of 5&lt;br /&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br /&gt;tending flat (patent val)
|-
| 0
| 0.000
| [[1/1]]
| ''[[126/125]]''
| ''[[225/224]]'', ''[[100/99]]''
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.154
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.692
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| ''[[20/19]]''
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| ''[[5/4]]''
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], ''[[7/5]]''
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

== Scales ==
* [[MET-24]]

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>brrtpgororptfqczrrebrc2sdhdkhij</sha1>
    </revision>
    <revision>
      <id>167957</id>
      <parentid>167921</parentid>
      <timestamp>2024-11-30T15:00:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Undo revision 167921 by [[Special:Contributions/BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]). MET-24 is a scale in 2048edo</comment>
      <origin>167957</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6839" sha1="hlnczf0vktd3qz4vjpgjios6tdccign" xml:space="preserve">{{Infobox ET}}
{{EDO intro|104}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the 17&amp;amp;87 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| &amp;minus;0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| &amp;minus;1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| &amp;minus;0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| &amp;minus;0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| &amp;minus;0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | &amp;#35;
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br /&gt;subgroup
! Additional ratios of 5&lt;br /&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br /&gt;tending flat (patent val)
|-
| 0
| 0.000
| [[1/1]]
| ''[[126/125]]''
| ''[[225/224]]'', ''[[100/99]]''
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.154
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.692
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| ''[[20/19]]''
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| ''[[5/4]]''
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], ''[[7/5]]''
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>hlnczf0vktd3qz4vjpgjios6tdccign</sha1>
    </revision>
    <revision>
      <id>176515</id>
      <parentid>167957</parentid>
      <timestamp>2025-01-16T15:33:43Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176515</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6852" sha1="q6ukltz81rq8m380f6xs0z3wzsnsoqu" xml:space="preserve">{{Infobox ET}}
{{EDO intro|104}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap|17 &amp;amp; 87}} 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| &amp;minus;0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| &amp;minus;1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| &amp;minus;0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| &amp;minus;0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| &amp;minus;0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | &amp;#35;
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br /&gt;subgroup
! Additional ratios of 5&lt;br /&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br /&gt;tending flat (patent val)
|-
| 0
| 0.000
| [[1/1]]
| ''[[126/125]]''
| ''[[225/224]]'', ''[[100/99]]''
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.154
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.692
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| ''[[20/19]]''
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| ''[[5/4]]''
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], ''[[7/5]]''
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>q6ukltz81rq8m380f6xs0z3wzsnsoqu</sha1>
    </revision>
    <revision>
      <id>176519</id>
      <parentid>176515</parentid>
      <timestamp>2025-01-16T15:35:10Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176519</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6846" sha1="by167issrutvmq2ctiiyxpkpffjsaxg" xml:space="preserve">{{Infobox ET}}
{{EDO intro|104}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap|17 &amp;amp; 87}} 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| &amp;minus;0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| &amp;minus;1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| &amp;minus;0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| &amp;minus;0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| &amp;minus;0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | &amp;#35;
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br /&gt;subgroup
! Additional ratios of 5&lt;br /&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br /&gt;tending flat (patent val)
|-
| 0
| 0.000
| [[1/1]]
| ''[[126/125]]''
| ''[[225/224]]'', ''[[100/99]]''
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.154
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.692
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| ''[[20/19]]''
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| ''[[5/4]]''
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], ''[[7/5]]''
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>by167issrutvmq2ctiiyxpkpffjsaxg</sha1>
    </revision>
    <revision>
      <id>181307</id>
      <parentid>176519</parentid>
      <timestamp>2025-02-18T16:46:51Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181307</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6841" sha1="dqfqijk652z4q99o8rclqqqti54gqwi" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap|17 &amp;amp; 87}} 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| &amp;minus;0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| &amp;minus;1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| &amp;minus;0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| &amp;minus;0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| &amp;minus;0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | &amp;#35;
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br /&gt;subgroup
! Additional ratios of 5&lt;br /&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br /&gt;tending flat (patent val)
|-
| 0
| 0.000
| [[1/1]]
| ''[[126/125]]''
| ''[[225/224]]'', ''[[100/99]]''
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.154
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.692
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| ''[[20/19]]''
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| ''[[5/4]]''
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], ''[[7/5]]''
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>dqfqijk652z4q99o8rclqqqti54gqwi</sha1>
    </revision>
    <revision>
      <id>183493</id>
      <parentid>181307</parentid>
      <timestamp>2025-02-27T16:31:24Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>183493</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6821" sha1="cac7bj44n41dvzusmevpbnvvn9wtf6i" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap|17 &amp;amp; 87}} 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Subsets and supersets ===
Since 104 factors into 2&lt;sup&gt;3&lt;/sup&gt; × 13, it has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| −0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| −1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| −0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| −0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| −0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | &amp;#35;
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br /&gt;subgroup
! Additional ratios of 5&lt;br /&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br /&gt;tending flat (patent val)
|-
| 0
| 0.000
| [[1/1]]
| ''[[126/125]]''
| ''[[225/224]]'', ''[[100/99]]''
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.154
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.692
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| ''[[20/19]]''
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| ''[[5/4]]''
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], ''[[7/5]]''
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>cac7bj44n41dvzusmevpbnvvn9wtf6i</sha1>
    </revision>
    <revision>
      <id>190341</id>
      <parentid>183493</parentid>
      <timestamp>2025-04-09T12:18:45Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ +octave stretch</comment>
      <origin>190341</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7198" sha1="l8121j72gfgxpy187ojb5g4qd9r626y" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap| 17 &amp; 87 }} 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Octave stretch ===
104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as [[269ed6]], which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as [[165edt]]. 

=== Subsets and supersets ===
Since 104 factors into primes as {{nowrap| 2&lt;sup&gt;3&lt;/sup&gt; × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| −0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| −1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| −0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| −0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| −0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Intervals ==

{| class="wikitable center-1 right-2"
|-
! rowspan="2" | &amp;#35;
! rowspan="2" | Cents
! colspan="3" | Approximate Ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br /&gt;subgroup
! Additional ratios of 5&lt;br /&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br /&gt;tending flat (patent val)
|-
| 0
| 0.000
| [[1/1]]
| ''[[126/125]]''
| ''[[225/224]]'', ''[[100/99]]''
|-
| 1
| 11.538
| [[225/224]], [[100/99]]
|
|
|-
| 2
| 23.077
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.615
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.154
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.692
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.231
| [[25/24]]
|
|
|-
| 7
| 80.769
| [[22/21]]
| ''[[25/24]]'', [[21/20]]
| ''[[20/19]]''
|-
| 8
| 92.308
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.846
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.385
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.923
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.462
| [[13/12]]
|
|
|-
| 13
| 150.000
| [[12/11]]
|
|
|-
| 14
| 161.538
|
| [[11/10]]
|
|-
| 15
| 173.077
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.615
|
| [[10/9]]
|
|-
| 17
| 196.154
| [[28/25]], [[19/17]]
|
|
|-
| 18
| 207.692
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.231
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.769
| [[8/7]]
|
|
|-
| 21
| 242.308
|
|
| [[15/13]]
|-
| 22
| 253.846
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.385
| [[7/6]]
|
|
|-
| 24
| 276.923
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.462
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.000
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.538
|
| [[6/5]]
|
|-
| 28
| 323.077
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.615
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.154
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.692
| [[27/22]], [[16/13]]
|
|
|-
| 32
| 369.231
| [[26/21]], [[21/17]]
|
|
|-
| 33
| 380.769
|
|
| [[5/4]]
|-
| 34
| 392.308
|
| ''[[5/4]]''
|
|-
| 35
| 403.846
| [[63/50]], [[24/19]]
| [[19/15]]
|
|-
| 36
| 415.385
| [[81/64]], [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.923
| [[32/25]]
|
|
|-
| 38
| 438.462
| [[9/7]]
|
|
|-
| 39
| 450.000
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.538
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.077
| [[21/16]]
|
|
|-
| 42
| 484.615
|
|
|
|-
| 43
| 496.154
| [[4/3]]
|
|
|-
| 44
| 507.692
|
|
|
|-
| 45
| 519.231
|
| [[27/20]]
|
|-
| 46
| 530.769
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.308
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.846
| [[11/8]]
|
|
|-
| 49
| 565.385
| [[18/13]]
|
|
|-
| 50
| 576.923
|
| [[7/5]]
|
|-
| 51
| 588.462
|
|
| [[45/32]], ''[[7/5]]''
|-
| 52
| 600.000
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>l8121j72gfgxpy187ojb5g4qd9r626y</sha1>
    </revision>
    <revision>
      <id>190352</id>
      <parentid>190341</parentid>
      <timestamp>2025-04-09T13:36:29Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ reduce cent values to one decimal place, per discussion on Discord; sort ratios; -some excessively complex ratios</comment>
      <origin>190352</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7024" sha1="lls84zl9dexw794lq827phnk491qekx" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap| 17 &amp; 87 }} 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Octave stretch ===
104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as [[269ed6]], which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as [[165edt]]. 

=== Subsets and supersets ===
Since 104 factors into primes as {{nowrap| 2&lt;sup&gt;3&lt;/sup&gt; × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| −0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| −1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| −0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| −0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| −0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br&gt;subgroup
! Additional ratios of 5&lt;br&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br&gt;tending flat (patent val)
|-
| 0
| 0.0
| [[1/1]]
| 
| 
|-
| 1
| 11.5
| [[100/99]], [[225/224]]
|
|
|-
| 2
| 23.1
| [[64/63]]
| [[81/80]], ''[[225/224]]''
| ''[[50/49]]''
|-
| 3
| 34.6
| [[49/48]], [[50/49]]
|
| ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.2
|
| [[36/35]], ''[[50/49]]''
|
|-
| 5
| 57.7
| ''[[28/27]]'', [[33/32]]
|
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.2
| [[25/24]]
|
|
|-
| 7
| 80.8
| [[22/21]]
| [[21/20]], ''[[25/24]]''
| ''[[20/19]]''
|-
| 8
| 92.3
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.8
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.4
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.9
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.5
| [[13/12]]
|
|
|-
| 13
| 150.0
| [[12/11]]
|
|
|-
| 14
| 161.5
|
| [[11/10]]
|
|-
| 15
| 173.1
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.6
|
| [[10/9]]
|
|-
| 17
| 196.2
| [[19/17]], [[28/25]]
|
|
|-
| 18
| 207.7
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.2
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.8
| [[8/7]]
|
|
|-
| 21
| 242.3
|
|
| [[15/13]]
|-
| 22
| 253.8
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.4
| [[7/6]]
|
|
|-
| 24
| 276.9
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.5
| [[32/27]], [[13/11]]
| ''[[20/17]]''
|
|-
| 26
| 300.0
| [[25/21]], [[19/16]]
|
|
|-
| 27
| 311.5
|
| [[6/5]]
|
|-
| 28
| 323.1
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.6
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.2
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.7
| [[16/13]], [[27/22]]
|
|
|-
| 32
| 369.2
| [[21/17]], [[26/21]]
|
|
|-
| 33
| 380.8
|
|
| [[5/4]]
|-
| 34
| 392.3
|
| ''[[5/4]]''
|
|-
| 35
| 403.8
| [[24/19]], [[63/50]]
| [[19/15]]
|
|-
| 36
| 415.4
| [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.9
| [[32/25]]
|
|
|-
| 38
| 438.5
| [[9/7]]
|
|
|-
| 39
| 450.0
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.5
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.1
| [[21/16]]
|
|
|-
| 42
| 484.6
|
|
|
|-
| 43
| 496.2
| [[4/3]]
|
|
|-
| 44
| 507.7
|
|
|
|-
| 45
| 519.2
|
| [[27/20]]
|
|-
| 46
| 530.8
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.3
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.8
| [[11/8]]
|
|
|-
| 49
| 565.4
| [[18/13]]
|
|
|-
| 50
| 576.9
|
| [[7/5]]
|
|-
| 51
| 588.5
|
|
| ''[[7/5]]'', [[45/32]]
|-
| 52
| 600.0
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>lls84zl9dexw794lq827phnk491qekx</sha1>
    </revision>
    <revision>
      <id>190360</id>
      <parentid>190352</parentid>
      <timestamp>2025-04-09T14:05:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ +some ratios</comment>
      <origin>190360</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7267" sha1="jkc88ywmy6uj0nftleo3svsh0f7vdrc" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap| 17 &amp; 87 }} 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Octave stretch ===
104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as [[269ed6]], which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as [[165edt]]. 

=== Subsets and supersets ===
Since 104 factors into primes as {{nowrap| 2&lt;sup&gt;3&lt;/sup&gt; × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| −0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| −1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| −0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| −0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| −0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br&gt;subgroup
! Additional ratios of 5&lt;br&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br&gt;tending flat (patent val)
|-
| 0
| 0.0
| [[1/1]]
| 
| 
|-
| 1
| 11.5
| [[144/143]], [[169/168]]
| ''[[91/90]]'', [[121/120]]
| [[105/104]], [[196/195]]
|-
| 2
| 23.1
| [[64/63]], [[99/98]]
| [[81/80]], [[100/99]], ''[[105/104]]''
| ''[[50/49]]'', ''[[55/54]]'', [[91/90]], ''[[121/120]]''
|-
| 3
| 34.6
| [[49/48]], [[50/49]]
| [[55/54]]
| ''[[40/39]]'', [[45/44]], ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.2
|
| [[36/35]], [[40/39]], ''[[45/44]]'', ''[[50/49]]''
|
|-
| 5
| 57.7
| ''[[28/27]]'', [[33/32]]
| ''[[26/25]]''
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.2
| [[25/24]], [[26/25]], [[27/26]]
|
|
|-
| 7
| 80.8
| [[22/21]]
| [[21/20]], ''[[25/24]]''
| ''[[20/19]]'', ''[[26/25]]''
|-
| 8
| 92.3
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.8
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.4
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.9
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.5
| [[13/12]]
|
|
|-
| 13
| 150.0
| [[12/11]]
|
|
|-
| 14
| 161.5
|
| [[11/10]]
|
|-
| 15
| 173.1
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.6
|
| [[10/9]]
|
|-
| 17
| 196.2
| [[19/17]], [[28/25]]
|
|
|-
| 18
| 207.7
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.2
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.8
| [[8/7]]
|
|
|-
| 21
| 242.3
| [[38/33]]
| 
| [[15/13]]
|-
| 22
| 253.8
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.4
| [[7/6]]
|
|
|-
| 24
| 276.9
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.5
| [[13/11]], [[32/27]]
| ''[[20/17]]''
|
|-
| 26
| 300.0
| [[19/16]], [[25/21]]
|
|
|-
| 27
| 311.5
|
| [[6/5]]
|
|-
| 28
| 323.1
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.6
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.2
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.7
| [[16/13]], [[27/22]]
|
|
|-
| 32
| 369.2
| [[21/17]], [[26/21]]
|
|
|-
| 33
| 380.8
|
|
| [[5/4]]
|-
| 34
| 392.3
|
| ''[[5/4]]''
|
|-
| 35
| 403.8
| [[24/19]], [[63/50]]
| [[19/15]]
|
|-
| 36
| 415.4
| [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.9
| [[32/25]]
|
|
|-
| 38
| 438.5
| [[9/7]]
|
|
|-
| 39
| 450.0
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.5
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.1
| [[21/16]]
|
|
|-
| 42
| 484.6
|
|
|
|-
| 43
| 496.2
| [[4/3]]
|
|
|-
| 44
| 507.7
|
|
|
|-
| 45
| 519.2
|
| [[27/20]]
|
|-
| 46
| 530.8
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.3
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.8
| [[11/8]]
|
|
|-
| 49
| 565.4
| [[18/13]]
|
|
|-
| 50
| 576.9
|
| [[7/5]]
|
|-
| 51
| 588.5
|
|
| ''[[7/5]]'', [[45/32]]
|-
| 52
| 600.0
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>jkc88ywmy6uj0nftleo3svsh0f7vdrc</sha1>
    </revision>
    <revision>
      <id>218727</id>
      <parentid>190360</parentid>
      <timestamp>2025-12-07T19:59:20Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ 28/27 isn't inconsistent</comment>
      <origin>218727</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7263" sha1="5t1fte5sj07es6pko3k1o29n8bbcczb" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap| 17 &amp; 87 }} 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Octave stretch ===
104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as [[269ed6]], which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as [[165edt]]. 

=== Subsets and supersets ===
Since 104 factors into primes as {{nowrap| 2&lt;sup&gt;3&lt;/sup&gt; × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| −0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| −1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| −0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| −0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| −0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br&gt;subgroup
! Additional ratios of 5&lt;br&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br&gt;tending flat (patent val)
|-
| 0
| 0.0
| [[1/1]]
| 
| 
|-
| 1
| 11.5
| [[144/143]], [[169/168]]
| ''[[91/90]]'', [[121/120]]
| [[105/104]], [[196/195]]
|-
| 2
| 23.1
| [[64/63]], [[99/98]]
| [[81/80]], [[100/99]], ''[[105/104]]''
| ''[[50/49]]'', ''[[55/54]]'', [[91/90]], ''[[121/120]]''
|-
| 3
| 34.6
| [[49/48]], [[50/49]]
| [[55/54]]
| ''[[40/39]]'', [[45/44]], ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.2
|
| [[36/35]], [[40/39]], ''[[45/44]]'', ''[[50/49]]''
|
|-
| 5
| 57.7
| [[28/27]], [[33/32]]
| ''[[26/25]]''
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.2
| [[25/24]], [[26/25]], [[27/26]]
|
|
|-
| 7
| 80.8
| [[22/21]]
| [[21/20]], ''[[25/24]]''
| ''[[20/19]]'', ''[[26/25]]''
|-
| 8
| 92.3
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.8
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.4
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.9
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.5
| [[13/12]]
|
|
|-
| 13
| 150.0
| [[12/11]]
|
|
|-
| 14
| 161.5
|
| [[11/10]]
|
|-
| 15
| 173.1
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.6
|
| [[10/9]]
|
|-
| 17
| 196.2
| [[19/17]], [[28/25]]
|
|
|-
| 18
| 207.7
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.2
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.8
| [[8/7]]
|
|
|-
| 21
| 242.3
| [[38/33]]
| 
| [[15/13]]
|-
| 22
| 253.8
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.4
| [[7/6]]
|
|
|-
| 24
| 276.9
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.5
| [[13/11]], [[32/27]]
| ''[[20/17]]''
|
|-
| 26
| 300.0
| [[19/16]], [[25/21]]
|
|
|-
| 27
| 311.5
|
| [[6/5]]
|
|-
| 28
| 323.1
|
|
| [[6/5]], ''[[40/33]]''
|-
| 29
| 334.6
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.2
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.7
| [[16/13]], [[27/22]]
|
|
|-
| 32
| 369.2
| [[21/17]], [[26/21]]
|
|
|-
| 33
| 380.8
|
|
| [[5/4]]
|-
| 34
| 392.3
|
| ''[[5/4]]''
|
|-
| 35
| 403.8
| [[24/19]], [[63/50]]
| [[19/15]]
|
|-
| 36
| 415.4
| [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.9
| [[32/25]]
|
|
|-
| 38
| 438.5
| [[9/7]]
|
|
|-
| 39
| 450.0
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.5
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.1
| [[21/16]]
|
|
|-
| 42
| 484.6
|
|
|
|-
| 43
| 496.2
| [[4/3]]
|
|
|-
| 44
| 507.7
|
|
|
|-
| 45
| 519.2
|
| [[27/20]]
|
|-
| 46
| 530.8
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.3
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.8
| [[11/8]]
|
|
|-
| 49
| 565.4
| [[18/13]]
|
|
|-
| 50
| 576.9
|
| [[7/5]]
|
|-
| 51
| 588.5
|
|
| ''[[7/5]]'', [[45/32]]
|-
| 52
| 600.0
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>5t1fte5sj07es6pko3k1o29n8bbcczb</sha1>
    </revision>
    <revision>
      <id>219209</id>
      <parentid>218727</parentid>
      <timestamp>2025-12-18T00:56:23Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Intervals */ mark inconsistent 6/5</comment>
      <origin>219209</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7267" sha1="abnqea0ekejbqhxhkxsveddpn51c68v" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap| 17 &amp; 87 }} 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Octave stretch ===
104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as [[269ed6]], which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as [[165edt]]. 

=== Subsets and supersets ===
Since 104 factors into primes as {{nowrap| 2&lt;sup&gt;3&lt;/sup&gt; × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| −0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| −1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| −0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| −0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| −0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br&gt;subgroup
! Additional ratios of 5&lt;br&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br&gt;tending flat (patent val)
|-
| 0
| 0.0
| [[1/1]]
| 
| 
|-
| 1
| 11.5
| [[144/143]], [[169/168]]
| ''[[91/90]]'', [[121/120]]
| [[105/104]], [[196/195]]
|-
| 2
| 23.1
| [[64/63]], [[99/98]]
| [[81/80]], [[100/99]], ''[[105/104]]''
| ''[[50/49]]'', ''[[55/54]]'', [[91/90]], ''[[121/120]]''
|-
| 3
| 34.6
| [[49/48]], [[50/49]]
| [[55/54]]
| ''[[40/39]]'', [[45/44]], ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.2
|
| [[36/35]], [[40/39]], ''[[45/44]]'', ''[[50/49]]''
|
|-
| 5
| 57.7
| [[28/27]], [[33/32]]
| ''[[26/25]]''
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.2
| [[25/24]], [[26/25]], [[27/26]]
|
|
|-
| 7
| 80.8
| [[22/21]]
| [[21/20]], ''[[25/24]]''
| ''[[20/19]]'', ''[[26/25]]''
|-
| 8
| 92.3
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.8
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.4
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.9
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.5
| [[13/12]]
|
|
|-
| 13
| 150.0
| [[12/11]]
|
|
|-
| 14
| 161.5
|
| [[11/10]]
|
|-
| 15
| 173.1
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.6
|
| [[10/9]]
|
|-
| 17
| 196.2
| [[19/17]], [[28/25]]
|
|
|-
| 18
| 207.7
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.2
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.8
| [[8/7]]
|
|
|-
| 21
| 242.3
| [[38/33]]
| 
| [[15/13]]
|-
| 22
| 253.8
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.4
| [[7/6]]
|
|
|-
| 24
| 276.9
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.5
| [[13/11]], [[32/27]]
| ''[[20/17]]''
|
|-
| 26
| 300.0
| [[19/16]], [[25/21]]
|
|
|-
| 27
| 311.5
|
| [[6/5]]
|
|-
| 28
| 323.1
|
|
| ''[[6/5]]'', ''[[40/33]]''
|-
| 29
| 334.6
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.2
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.7
| [[16/13]], [[27/22]]
|
|
|-
| 32
| 369.2
| [[21/17]], [[26/21]]
|
|
|-
| 33
| 380.8
|
|
| [[5/4]]
|-
| 34
| 392.3
|
| ''[[5/4]]''
|
|-
| 35
| 403.8
| [[24/19]], [[63/50]]
| [[19/15]]
|
|-
| 36
| 415.4
| [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.9
| [[32/25]]
|
|
|-
| 38
| 438.5
| [[9/7]]
|
|
|-
| 39
| 450.0
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.5
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.1
| [[21/16]]
|
|
|-
| 42
| 484.6
|
|
|
|-
| 43
| 496.2
| [[4/3]]
|
|
|-
| 44
| 507.7
|
|
|
|-
| 45
| 519.2
|
| [[27/20]]
|
|-
| 46
| 530.8
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.3
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.8
| [[11/8]]
|
|
|-
| 49
| 565.4
| [[18/13]]
|
|
|-
| 50
| 576.9
|
| [[7/5]]
|
|-
| 51
| 588.5
|
|
| ''[[7/5]]'', [[45/32]]
|-
| 52
| 600.0
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>abnqea0ekejbqhxhkxsveddpn51c68v</sha1>
    </revision>
    <revision>
      <id>225942</id>
      <parentid>219209</parentid>
      <timestamp>2026-03-13T13:33:14Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct" to "[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct"</comment>
      <origin>225942</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7322" sha1="m35c7s6kr0fq052r1f8cay79vkushjl" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], [[896/891]], [[385/384]] and [[540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap| 17 &amp; 87 }} 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Octave stretch ===
104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as [[269ed6]], which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as [[165edt]]. 

=== Subsets and supersets ===
Since 104 factors into primes as {{nowrap| 2&lt;sup&gt;3&lt;/sup&gt; × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| −0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| −1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| −0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| −0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| −0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br&gt;subgroup
! Additional ratios of 5&lt;br&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br&gt;tending flat (patent val)
|-
| 0
| 0.0
| [[1/1]]
| 
| 
|-
| 1
| 11.5
| [[144/143]], [[169/168]]
| ''[[91/90]]'', [[121/120]]
| [[105/104]], [[196/195]]
|-
| 2
| 23.1
| [[64/63]], [[99/98]]
| [[81/80]], [[100/99]], ''[[105/104]]''
| ''[[50/49]]'', ''[[55/54]]'', [[91/90]], ''[[121/120]]''
|-
| 3
| 34.6
| [[49/48]], [[50/49]]
| [[55/54]]
| ''[[40/39]]'', [[45/44]], ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.2
|
| [[36/35]], [[40/39]], ''[[45/44]]'', ''[[50/49]]''
|
|-
| 5
| 57.7
| [[28/27]], [[33/32]]
| ''[[26/25]]''
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.2
| [[25/24]], [[26/25]], [[27/26]]
|
|
|-
| 7
| 80.8
| [[22/21]]
| [[21/20]], ''[[25/24]]''
| ''[[20/19]]'', ''[[26/25]]''
|-
| 8
| 92.3
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.8
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.4
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.9
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.5
| [[13/12]]
|
|
|-
| 13
| 150.0
| [[12/11]]
|
|
|-
| 14
| 161.5
|
| [[11/10]]
|
|-
| 15
| 173.1
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.6
|
| [[10/9]]
|
|-
| 17
| 196.2
| [[19/17]], [[28/25]]
|
|
|-
| 18
| 207.7
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.2
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.8
| [[8/7]]
|
|
|-
| 21
| 242.3
| [[38/33]]
| 
| [[15/13]]
|-
| 22
| 253.8
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.4
| [[7/6]]
|
|
|-
| 24
| 276.9
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.5
| [[13/11]], [[32/27]]
| ''[[20/17]]''
|
|-
| 26
| 300.0
| [[19/16]], [[25/21]]
|
|
|-
| 27
| 311.5
|
| [[6/5]]
|
|-
| 28
| 323.1
|
|
| ''[[6/5]]'', ''[[40/33]]''
|-
| 29
| 334.6
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.2
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.7
| [[16/13]], [[27/22]]
|
|
|-
| 32
| 369.2
| [[21/17]], [[26/21]]
|
|
|-
| 33
| 380.8
|
|
| [[5/4]]
|-
| 34
| 392.3
|
| ''[[5/4]]''
|
|-
| 35
| 403.8
| [[24/19]], [[63/50]]
| [[19/15]]
|
|-
| 36
| 415.4
| [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.9
| [[32/25]]
|
|
|-
| 38
| 438.5
| [[9/7]]
|
|
|-
| 39
| 450.0
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.5
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.1
| [[21/16]]
|
|
|-
| 42
| 484.6
|
|
|
|-
| 43
| 496.2
| [[4/3]]
|
|
|-
| 44
| 507.7
|
|
|
|-
| 45
| 519.2
|
| [[27/20]]
|
|-
| 46
| 530.8
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.3
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.8
| [[11/8]]
|
|
|-
| 49
| 565.4
| [[18/13]]
|
|
|-
| 50
| 576.9
|
| [[7/5]]
|
|-
| 51
| 588.5
|
|
| ''[[7/5]]'', [[45/32]]
|-
| 52
| 600.0
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>m35c7s6kr0fq052r1f8cay79vkushjl</sha1>
    </revision>
    <revision>
      <id>230030</id>
      <parentid>225942</parentid>
      <timestamp>2026-05-11T22:52:51Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Theory */ no-fives first; note 26edo's 2.7.11 and orgone</comment>
      <origin>230030</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8133" sha1="q63glc2zbhtv19t0vebd1zz9bg4524k" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
104edo is a strong no-fives system, with good approximations up to the no-5 19-limit. In the [[2.3.7.11.13 subgroup|2.3.7.11.13-subgroup]], it tempers out [[352/351]], [[364/363]], [[896/891]], [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364.&lt;!-- Add commas in 2.3.7.11.13.17.19 as well --&gt; It is an excellent tuning for the 2.3.7.11.13-subgroup [[rank]]-3 [[parapyth]] temperament tempering out 352/351, 364/363, and 896/891, which maps [[14/11]] to the diatonic major third and [[13/11]] to the diatonic minor third, in fact providing the [[optimal patent val]]. Additionally, it supports the extension to prime 17 known as [[etypyth]], which maps 17/14 to the augmented second, though [[121edo]] is a more optimal tuning of it. It also provides the optimal patent val for the 2.3.7.11.13-subgroup {{nowrap| 17 &amp; 87 }} temperament tempering out 352/351, 364/363 and 2197/2187, which splits 3/1 into three ~13/9's, and can be considered a rank-2 reduction of parapyth.

Notably, 104edo inherits [[26edo]]'s accurate representation of the [[2.7.11 subgroup|2.7.11-subgroup]], and thus supports [[orgone]] temperament in that subgroup.

If prime 5 is desired, 104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], 896/891, [[385/384]] and [[540/539]]. It also provides an excellent tuning for the rank-3 temperament pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, and the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104edo with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], 352/351 and 364/363 in the 13-limit, and [[136/135]] and [[256/255]] in the 17-limit.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Octave stretch ===
104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as [[269ed6]], which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as [[165edt]]. 

=== Subsets and supersets ===
Since 104 factors into primes as {{nowrap| 2&lt;sup&gt;3&lt;/sup&gt; × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| −0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| −1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| −0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| −0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| −0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br&gt;subgroup
! Additional ratios of 5&lt;br&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br&gt;tending flat (patent val)
|-
| 0
| 0.0
| [[1/1]]
| 
| 
|-
| 1
| 11.5
| [[144/143]], [[169/168]]
| ''[[91/90]]'', [[121/120]]
| [[105/104]], [[196/195]]
|-
| 2
| 23.1
| [[64/63]], [[99/98]]
| [[81/80]], [[100/99]], ''[[105/104]]''
| ''[[50/49]]'', ''[[55/54]]'', [[91/90]], ''[[121/120]]''
|-
| 3
| 34.6
| [[49/48]], [[50/49]]
| [[55/54]]
| ''[[40/39]]'', [[45/44]], ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.2
|
| [[36/35]], [[40/39]], ''[[45/44]]'', ''[[50/49]]''
|
|-
| 5
| 57.7
| [[28/27]], [[33/32]]
| ''[[26/25]]''
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.2
| [[25/24]], [[26/25]], [[27/26]]
|
|
|-
| 7
| 80.8
| [[22/21]]
| [[21/20]], ''[[25/24]]''
| ''[[20/19]]'', ''[[26/25]]''
|-
| 8
| 92.3
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.8
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.4
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.9
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.5
| [[13/12]]
|
|
|-
| 13
| 150.0
| [[12/11]]
|
|
|-
| 14
| 161.5
|
| [[11/10]]
|
|-
| 15
| 173.1
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.6
|
| [[10/9]]
|
|-
| 17
| 196.2
| [[19/17]], [[28/25]]
|
|
|-
| 18
| 207.7
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.2
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.8
| [[8/7]]
|
|
|-
| 21
| 242.3
| [[38/33]]
| 
| [[15/13]]
|-
| 22
| 253.8
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.4
| [[7/6]]
|
|
|-
| 24
| 276.9
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.5
| [[13/11]], [[32/27]]
| ''[[20/17]]''
|
|-
| 26
| 300.0
| [[19/16]], [[25/21]]
|
|
|-
| 27
| 311.5
|
| [[6/5]]
|
|-
| 28
| 323.1
|
|
| ''[[6/5]]'', ''[[40/33]]''
|-
| 29
| 334.6
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.2
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.7
| [[16/13]], [[27/22]]
|
|
|-
| 32
| 369.2
| [[21/17]], [[26/21]]
|
|
|-
| 33
| 380.8
|
|
| [[5/4]]
|-
| 34
| 392.3
|
| ''[[5/4]]''
|
|-
| 35
| 403.8
| [[24/19]], [[63/50]]
| [[19/15]]
|
|-
| 36
| 415.4
| [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.9
| [[32/25]]
|
|
|-
| 38
| 438.5
| [[9/7]]
|
|
|-
| 39
| 450.0
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.5
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.1
| [[21/16]]
|
|
|-
| 42
| 484.6
|
|
|
|-
| 43
| 496.2
| [[4/3]]
|
|
|-
| 44
| 507.7
|
|
|
|-
| 45
| 519.2
|
| [[27/20]]
|
|-
| 46
| 530.8
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.3
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.8
| [[11/8]]
|
|
|-
| 49
| 565.4
| [[18/13]]
|
|
|-
| 50
| 576.9
|
| [[7/5]]
|
|-
| 51
| 588.5
|
|
| ''[[7/5]]'', [[45/32]]
|-
| 52
| 600.0
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>q63glc2zbhtv19t0vebd1zz9bg4524k</sha1>
    </revision>
    <revision>
      <id>230033</id>
      <parentid>230030</parentid>
      <timestamp>2026-05-11T23:01:12Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Theory */ dual-5</comment>
      <origin>230033</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8238" sha1="6nphba5mjebrymd59pidkdk104fduyi" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
104edo is a strong no-fives system, with good approximations up to the no-5 19-limit. In the [[2.3.7.11.13 subgroup|2.3.7.11.13-subgroup]], it tempers out [[352/351]], [[364/363]], [[896/891]], [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364.&lt;!-- Add commas in 2.3.7.11.13.17.19 as well --&gt; It is an excellent tuning for the 2.3.7.11.13-subgroup [[rank]]-3 [[parapyth]] temperament tempering out 352/351, 364/363, and 896/891, which maps [[14/11]] to the diatonic major third and [[13/11]] to the diatonic minor third, in fact providing the [[optimal patent val]]. Additionally, it supports the extension to prime 17 known as [[etypyth]], which maps 17/14 to the augmented second, though [[121edo]] is a more optimal tuning of it. It also provides the optimal patent val for the 2.3.7.11.13-subgroup {{nowrap| 17 &amp; 87 }} temperament tempering out 352/351, 364/363 and 2197/2187, which splits 3/1 into three ~13/9's, and can be considered a rank-2 reduction of parapyth.

Notably, 104edo inherits [[26edo]]'s accurate representation of the [[2.7.11 subgroup|2.7.11-subgroup]], and thus supports [[orgone]] temperament in that subgroup.

If prime 5 is desired, 104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament. Additionally, it is viable to treat 104edo as dual-5, or as a 2.3.25.7.11.13.17.19 subgroup temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], 896/891, [[385/384]] and [[540/539]]. It also provides an excellent tuning for the rank-3 temperament pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, and the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104edo with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], 352/351 and 364/363 in the 13-limit, and [[136/135]] and [[256/255]] in the 17-limit.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Octave stretch ===
104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as [[269ed6]], which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as [[165edt]]. 

=== Subsets and supersets ===
Since 104 factors into primes as {{nowrap| 2&lt;sup&gt;3&lt;/sup&gt; × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| −0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| −1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| −0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| −0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| −0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate ratios
|-
! Of 2.3.7.11.13.17.19.25&lt;br&gt;subgroup
! Additional ratios of 5&lt;br&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br&gt;tending flat (patent val)
|-
| 0
| 0.0
| [[1/1]]
| 
| 
|-
| 1
| 11.5
| [[144/143]], [[169/168]]
| ''[[91/90]]'', [[121/120]]
| [[105/104]], [[196/195]]
|-
| 2
| 23.1
| [[64/63]], [[99/98]]
| [[81/80]], [[100/99]], ''[[105/104]]''
| ''[[50/49]]'', ''[[55/54]]'', [[91/90]], ''[[121/120]]''
|-
| 3
| 34.6
| [[49/48]], [[50/49]]
| [[55/54]]
| ''[[40/39]]'', [[45/44]], ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.2
|
| [[36/35]], [[40/39]], ''[[45/44]]'', ''[[50/49]]''
|
|-
| 5
| 57.7
| [[28/27]], [[33/32]]
| ''[[26/25]]''
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.2
| [[25/24]], [[26/25]], [[27/26]]
|
|
|-
| 7
| 80.8
| [[22/21]]
| [[21/20]], ''[[25/24]]''
| ''[[20/19]]'', ''[[26/25]]''
|-
| 8
| 92.3
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.8
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.4
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.9
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.5
| [[13/12]]
|
|
|-
| 13
| 150.0
| [[12/11]]
|
|
|-
| 14
| 161.5
|
| [[11/10]]
|
|-
| 15
| 173.1
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.6
|
| [[10/9]]
|
|-
| 17
| 196.2
| [[19/17]], [[28/25]]
|
|
|-
| 18
| 207.7
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.2
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.8
| [[8/7]]
|
|
|-
| 21
| 242.3
| [[38/33]]
| 
| [[15/13]]
|-
| 22
| 253.8
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.4
| [[7/6]]
|
|
|-
| 24
| 276.9
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.5
| [[13/11]], [[32/27]]
| ''[[20/17]]''
|
|-
| 26
| 300.0
| [[19/16]], [[25/21]]
|
|
|-
| 27
| 311.5
|
| [[6/5]]
|
|-
| 28
| 323.1
|
|
| ''[[6/5]]'', ''[[40/33]]''
|-
| 29
| 334.6
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.2
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.7
| [[16/13]], [[27/22]]
|
|
|-
| 32
| 369.2
| [[21/17]], [[26/21]]
|
|
|-
| 33
| 380.8
|
|
| [[5/4]]
|-
| 34
| 392.3
|
| ''[[5/4]]''
|
|-
| 35
| 403.8
| [[24/19]], [[63/50]]
| [[19/15]]
|
|-
| 36
| 415.4
| [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.9
| [[32/25]]
|
|
|-
| 38
| 438.5
| [[9/7]]
|
|
|-
| 39
| 450.0
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.5
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.1
| [[21/16]]
|
|
|-
| 42
| 484.6
|
|
|
|-
| 43
| 496.2
| [[4/3]]
|
|
|-
| 44
| 507.7
|
|
|
|-
| 45
| 519.2
|
| [[27/20]]
|
|-
| 46
| 530.8
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.3
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.8
| [[11/8]]
|
|
|-
| 49
| 565.4
| [[18/13]]
|
|
|-
| 50
| 576.9
|
| [[7/5]]
|
|-
| 51
| 588.5
|
|
| ''[[7/5]]'', [[45/32]]
|-
| 52
| 600.0
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>6nphba5mjebrymd59pidkdk104fduyi</sha1>
    </revision>
    <revision>
      <id>230034</id>
      <parentid>230033</parentid>
      <timestamp>2026-05-11T23:01:30Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ proper subgroup ordering</comment>
      <origin>230034</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8238" sha1="191dl7oc60m8dup05ddgbo7714xm7un" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
104edo is a strong no-fives system, with good approximations up to the no-5 19-limit. In the [[2.3.7.11.13 subgroup|2.3.7.11.13-subgroup]], it tempers out [[352/351]], [[364/363]], [[896/891]], [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364.&lt;!-- Add commas in 2.3.7.11.13.17.19 as well --&gt; It is an excellent tuning for the 2.3.7.11.13-subgroup [[rank]]-3 [[parapyth]] temperament tempering out 352/351, 364/363, and 896/891, which maps [[14/11]] to the diatonic major third and [[13/11]] to the diatonic minor third, in fact providing the [[optimal patent val]]. Additionally, it supports the extension to prime 17 known as [[etypyth]], which maps 17/14 to the augmented second, though [[121edo]] is a more optimal tuning of it. It also provides the optimal patent val for the 2.3.7.11.13-subgroup {{nowrap| 17 &amp; 87 }} temperament tempering out 352/351, 364/363 and 2197/2187, which splits 3/1 into three ~13/9's, and can be considered a rank-2 reduction of parapyth.

Notably, 104edo inherits [[26edo]]'s accurate representation of the [[2.7.11 subgroup|2.7.11-subgroup]], and thus supports [[orgone]] temperament in that subgroup.

If prime 5 is desired, 104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament. Additionally, it is viable to treat 104edo as dual-5, or as a 2.3.25.7.11.13.17.19 subgroup temperament.

104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], 896/891, [[385/384]] and [[540/539]]. It also provides an excellent tuning for the rank-3 temperament pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, and the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

104edo with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], 352/351 and 364/363 in the 13-limit, and [[136/135]] and [[256/255]] in the 17-limit.

=== Prime harmonics ===
{{Harmonics in equal|104}}

=== Octave stretch ===
104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as [[269ed6]], which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as [[165edt]]. 

=== Subsets and supersets ===
Since 104 factors into primes as {{nowrap| 2&lt;sup&gt;3&lt;/sup&gt; × 13 }}, 104edo has subset edos {{EDOs| 2, 4, 8, 13, 26, and 52 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 165 -104 }}
| {{mapping| 104 165 }}
| −0.597
| 0.596
| 5.17
|-
| 2.3.5
| 2048/2025, {{monzo| 0 22 -15 }}
| {{mapping| 104 165 242 }} (104c)
| −1.258
| 1.054
| 9.14
|-
| 2.3.5.7
| 126/125, 2048/2025, 117649/116640
| {{mapping| 104 165 242 292 }} (104c)
| −0.980
| 1.032
| 8.95
|-
| 2.3.5.7.11
| 126/125, 176/175, 896/891, 14641/14580
| {{mapping| 104 165 242 292 360 }} (104c)
| −0.930
| 0.929
| 8.05
|-
| 2.3.5.7.11.13
| 126/125, 176/175, 196/195, 364/363, 2197/2187
| {{mapping| 104 165 242 292 360 385 }} (104c)
| −0.855
| 0.864
| 7.49
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Patent val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 33\104
| 380.77
| 5/4
| [[Magic]] / necromancy / divination
|-
| 1
| 51\104
| 588.46
| 7/5
| [[Untriton]]
|-
| 4
| 9\104
| 103.85
| 18/17
| [[Undim]]
|}

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | 104c val
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\104
| 126.92
| 27/25
| [[Mowgli]]
|-
| 1
| 21\104
| 242.31
| 147/128
| [[Septiquarter]]
|-
| 1
| 27\104
| 311.54
| 6/5
| [[Oolong]]
|-
| 1
| 47\104
| 542.31
| 15/11
| [[Casablanca]] / marrakesh
|-
| 2
| 21\104
| 242.31
| 121/105
| [[Semiseptiquarter]]
|-
| 2
| 43\104&lt;br /&gt;(9\104)
| 496.15&lt;br /&gt;(103.85)
| 4/3&lt;br /&gt;(17/16)
| [[Diaschismic]]
|-
| 8
| 49\104&lt;br /&gt;(2\104)
| 565.38&lt;br /&gt;(34.62)
| 168/121&lt;br /&gt;(55/54)
| [[Octowerck]] / octowerckis
|-
| 26
| 43\104&lt;br /&gt;(1\104)
| 496.15&lt;br /&gt;(11.54)
| 4/3&lt;br /&gt;(225/224)
| [[Bosonic]]
|}
&lt;nowiki /&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! colspan="3" | Approximate ratios
|-
! Of 2.3.25.7.11.13.17.19&lt;br&gt;subgroup
! Additional ratios of 5&lt;br&gt;tending sharp (104c val)
! Additional ratios of 5&lt;br&gt;tending flat (patent val)
|-
| 0
| 0.0
| [[1/1]]
| 
| 
|-
| 1
| 11.5
| [[144/143]], [[169/168]]
| ''[[91/90]]'', [[121/120]]
| [[105/104]], [[196/195]]
|-
| 2
| 23.1
| [[64/63]], [[99/98]]
| [[81/80]], [[100/99]], ''[[105/104]]''
| ''[[50/49]]'', ''[[55/54]]'', [[91/90]], ''[[121/120]]''
|-
| 3
| 34.6
| [[49/48]], [[50/49]]
| [[55/54]]
| ''[[40/39]]'', [[45/44]], ''[[81/80]]'', ''[[126/125]]''
|-
| 4
| 46.2
|
| [[36/35]], [[40/39]], ''[[45/44]]'', ''[[50/49]]''
|
|-
| 5
| 57.7
| [[28/27]], [[33/32]]
| ''[[26/25]]''
| ''[[25/24]]'', ''[[36/35]]''
|-
| 6
| 69.2
| [[25/24]], [[26/25]], [[27/26]]
|
|
|-
| 7
| 80.8
| [[22/21]]
| [[21/20]], ''[[25/24]]''
| ''[[20/19]]'', ''[[26/25]]''
|-
| 8
| 92.3
| [[19/18]]
| [[20/19]]
| ''[[21/20]]''
|-
| 9
| 103.8
| [[17/16]], [[18/17]]
| ''[[16/15]]''
|
|-
| 10
| 115.4
|
|
| [[16/15]], [[15/14]]
|-
| 11
| 126.9
| [[14/13]]
| ''[[15/14]]''
|
|-
| 12
| 138.5
| [[13/12]]
|
|
|-
| 13
| 150.0
| [[12/11]]
|
|
|-
| 14
| 161.5
|
| [[11/10]]
|
|-
| 15
| 173.1
| [[21/19]]
|
| ''[[10/9]]'', ''[[11/10]]''
|-
| 16
| 184.6
|
| [[10/9]]
|
|-
| 17
| 196.2
| [[19/17]], [[28/25]]
|
|
|-
| 18
| 207.7
| [[9/8]]
| ''[[17/15]]''
|
|-
| 19
| 219.2
| [[25/22]]
|
| [[17/15]]
|-
| 20
| 230.8
| [[8/7]]
|
|
|-
| 21
| 242.3
| [[38/33]]
| 
| [[15/13]]
|-
| 22
| 253.8
| [[22/19]]
| ''[[15/13]]''
|
|-
| 23
| 265.4
| [[7/6]]
|
|
|-
| 24
| 276.9
| [[75/64]]
|
| [[20/17]]
|-
| 25
| 288.5
| [[13/11]], [[32/27]]
| ''[[20/17]]''
|
|-
| 26
| 300.0
| [[19/16]], [[25/21]]
|
|
|-
| 27
| 311.5
|
| [[6/5]]
|
|-
| 28
| 323.1
|
|
| ''[[6/5]]'', ''[[40/33]]''
|-
| 29
| 334.6
| [[17/14]]
| [[40/33]]
|
|-
| 30
| 346.2
| [[11/9]], [[39/32]]
|
|
|-
| 31
| 357.7
| [[16/13]], [[27/22]]
|
|
|-
| 32
| 369.2
| [[21/17]], [[26/21]]
|
|
|-
| 33
| 380.8
|
|
| [[5/4]]
|-
| 34
| 392.3
|
| ''[[5/4]]''
|
|-
| 35
| 403.8
| [[24/19]], [[63/50]]
| [[19/15]]
|
|-
| 36
| 415.4
| [[14/11]]
|
| ''[[19/15]]''
|-
| 37
| 426.9
| [[32/25]]
|
|
|-
| 38
| 438.5
| [[9/7]]
|
|
|-
| 39
| 450.0
| [[22/17]]
| [[13/10]]
|
|-
| 40
| 461.5
| [[17/13]]
|
| ''[[13/10]]''
|-
| 41
| 473.1
| [[21/16]]
|
|
|-
| 42
| 484.6
|
|
|
|-
| 43
| 496.2
| [[4/3]]
|
|
|-
| 44
| 507.7
|
|
|
|-
| 45
| 519.2
|
| [[27/20]]
|
|-
| 46
| 530.8
| [[19/14]]
|
| ''[[27/20]]'', ''[[15/11]]''
|-
| 47
| 542.3
| [[26/19]]
| [[15/11]]
|
|-
| 48
| 553.8
| [[11/8]]
|
|
|-
| 49
| 565.4
| [[18/13]]
|
|
|-
| 50
| 576.9
|
| [[7/5]]
|
|-
| 51
| 588.5
|
|
| ''[[7/5]]'', [[45/32]]
|-
| 52
| 600.0
| [[17/12]], [[24/17]]
| ''[[45/32]]'', ''[[64/45]]''
|
|-
| …
| …
| …
| …
| …
|}

[[Category:Apollo]]
[[Category:Diaschismic]]
[[Category:Magic]]
[[Category:Necromancy]]</text>
      <sha1>191dl7oc60m8dup05ddgbo7714xm7un</sha1>
    </revision>
  </page>
  <page>
    <title>105edo</title>
    <ns>0</ns>
    <id>22</id>
    <revision>
      <id>22</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>22</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1157" sha1="fm6kgx1vrsfnusr8xg5yha5hhwkvi33" xml:space="preserve">'''105edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 105 equal parts of 11.429 [[cent|cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, [[tempering_out|tempering out]] [[81/80|81/80]] in the [[5-limit|5-limit]]; 81/80, [[126/125|126/125]] and hence 225/224 in the [[7-limit|7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit|11-limit]]; and if we want to push that far, 144/143 in the [[13-limit|13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[Optimal_patent_val|optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo|74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes closest to 100, 105 is a perfect substitute for it when a "[[cent|cent]]" is desired to include them all.
[[Category:edo]]
[[Category:huygens]]
[[Category:meantone]]
[[Category:theory]]</text>
      <sha1>fm6kgx1vrsfnusr8xg5yha5hhwkvi33</sha1>
    </revision>
    <revision>
      <id>4073</id>
      <timestamp>2016-12-07T17:34:16Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 601660756 - Original comment: **</comment>
      <origin>4073</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3852" sha1="1t2j6z1sz99wkhg3zfvzaee3dk3vygz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-12-07 17:34:16 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;601660756&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**105edo** is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, [[tempering out]] [[81_80|81/80]] in the [[5-limit]]; 81/80, [[126_125|126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes closest to 100, 105 is a perfect substitute for it when a "[[cent]]" is desired to include them all.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;105edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;105edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 105 equal parts of 11.429 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempering out&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/81_80"&amp;gt;81/80&amp;lt;/a&amp;gt; in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 81/80, &amp;lt;a class="wiki_link" href="/126_125"&amp;gt;126/125&amp;lt;/a&amp;gt; and hence 225/224 in the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;; 99/98, 176/175 and 441/440 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; and if we want to push that far, 144/143 in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt;. This is the sharper fifth mapping (aka &amp;amp;quot;huygens&amp;amp;quot;) of 11-limit meantone.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
105edo gives the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though &amp;lt;a class="wiki_link" href="/74edo"&amp;gt;74edo&amp;lt;/a&amp;gt; is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes closest to 100, 105 is a perfect substitute for it when a &amp;amp;quot;&amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;&amp;amp;quot; is desired to include them all.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>1t2j6z1sz99wkhg3zfvzaee3dk3vygz</sha1>
    </revision>
    <revision>
      <id>4074</id>
      <timestamp>2011-06-29T08:19:53Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 239302685 - Original comment: **</comment>
      <origin>4074</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3490" sha1="leu2iqnkrx7wzoozzzbck86onf6ojku" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 08:19:53 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239302685&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**105edo** is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, [[tempering out]] [[81_80|81/80]] in the [[5-limit]]; 81/80, [[126_125|126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 of the three smallest odd primes, with other divisors being 15, 21 and 35.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;105edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;105edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 105 equal parts of 11.429 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempering out&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/81_80"&amp;gt;81/80&amp;lt;/a&amp;gt; in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 81/80, &amp;lt;a class="wiki_link" href="/126_125"&amp;gt;126/125&amp;lt;/a&amp;gt; and hence 225/224 in the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;; 99/98, 176/175 and 441/440 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; and if we want to push that far, 144/143 in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt;. This is the sharper fifth mapping (aka &amp;amp;quot;huygens&amp;amp;quot;) of 11-limit meantone.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
105edo gives the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though &amp;lt;a class="wiki_link" href="/74edo"&amp;gt;74edo&amp;lt;/a&amp;gt; is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 of the three smallest odd primes, with other divisors being 15, 21 and 35.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4075</id>
      <timestamp>2011-06-29T08:11:18Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 239301961 - Original comment: **</comment>
      <origin>4075</origin>
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      <text bytes="3474" sha1="juv5gvzr3gcksy2v2twxrb4nib2buoj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 08:11:18 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239301961&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**105edo**, the 105 equal division divides the [[octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, [[tempering out]] [[81_80|81/80]] in the [[5-limit]]; 81/80, [[126_125|126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 of the three smallest odd primes, with other divisors being 15, 21 and 35.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;105edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;105edo&amp;lt;/strong&amp;gt;, the 105 equal division divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 105 equal parts of 11.429 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempering out&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/81_80"&amp;gt;81/80&amp;lt;/a&amp;gt; in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 81/80, &amp;lt;a class="wiki_link" href="/126_125"&amp;gt;126/125&amp;lt;/a&amp;gt; and hence 225/224 in the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;; 99/98, 176/175 and 441/440 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; and if we want to push that far, 144/143 in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt;. This is the sharper fifth mapping (aka &amp;amp;quot;huygens&amp;amp;quot;) of 11-limit meantone.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
105edo gives the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though &amp;lt;a class="wiki_link" href="/74edo"&amp;gt;74edo&amp;lt;/a&amp;gt; is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 of the three smallest odd primes, with other divisors being 15, 21 and 35.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <timestamp>2011-03-22T17:59:48Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 212979694 - Original comment: **</comment>
      <origin>4076</origin>
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      <text bytes="2906" sha1="ehqg85qc40a0ihs679f238kw19nsi8j" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-22 17:59:48 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;212979694&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 105 equal division divides the octave into 105 equal parts of 11.429 cents each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, tempering out 81/80 in the 5-limit; 81/80, 126/125 and hence 225/224 in the 7-limit; 99/98, 176/175 and 441/440 in the 11-limit; and if we want to push that far, 144/143 in the 13-limit. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 of the three smallest odd primes, with other divisors being 15, 21 and 35.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;105edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 105 equal division divides the octave into 105 equal parts of 11.429 cents each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, tempering out 81/80 in the 5-limit; 81/80, 126/125 and hence 225/224 in the 7-limit; 99/98, 176/175 and 441/440 in the 11-limit; and if we want to push that far, 144/143 in the 13-limit. This is the sharper fifth mapping (aka &amp;amp;quot;huygens&amp;amp;quot;) of 11-limit meantone.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
105edo gives the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though &amp;lt;a class="wiki_link" href="/74edo"&amp;gt;74edo&amp;lt;/a&amp;gt; is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 of the three smallest odd primes, with other divisors being 15, 21 and 35.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>33754</id>
      <parentid>22</parentid>
      <timestamp>2018-09-26T18:09:31Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
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      <text bytes="1231" sha1="g0tz2dek76klfxv93x9ka83s8t9c79f" xml:space="preserve">'''105edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 105 equal parts of 11.429 [[cent|cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, [[tempering_out|tempering out]] [[81/80|81/80]] in the [[5-limit|5-limit]]; 81/80, [[126/125|126/125]] and hence 225/224 in the [[7-limit|7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit|11-limit]]; and if we want to push that far, 144/143 in the [[13-limit|13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[Optimal_patent_val|optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo|74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes closest to 100, 105 is a perfect substitute for it when a "[[cent|cent]]" is desired to include them all.

== 105edo close-up ==

&lt;pre&gt;C . . . . . . C# . . Db . . . . . . D&lt;/pre&gt;

[[Category:edo]]
[[Category:huygens]]
[[Category:meantone]]
[[Category:theory]]</text>
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    <revision>
      <id>33755</id>
      <parentid>33754</parentid>
      <timestamp>2018-09-26T18:11:41Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <comment>/* 105edo close-up */</comment>
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      <text bytes="1237" sha1="6surhlkx48ih2rkadofozrvbpv7rgsa" xml:space="preserve">'''105edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 105 equal parts of 11.429 [[cent|cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, [[tempering_out|tempering out]] [[81/80|81/80]] in the [[5-limit|5-limit]]; 81/80, [[126/125|126/125]] and hence 225/224 in the [[7-limit|7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit|11-limit]]; and if we want to push that far, 144/143 in the [[13-limit|13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[Optimal_patent_val|optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo|74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes closest to 100, 105 is a perfect substitute for it when a "[[cent|cent]]" is desired to include them all.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;

[[Category:edo]]
[[Category:huygens]]
[[Category:meantone]]
[[Category:theory]]</text>
      <sha1>6surhlkx48ih2rkadofozrvbpv7rgsa</sha1>
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    <revision>
      <id>40211</id>
      <parentid>33755</parentid>
      <timestamp>2019-02-13T01:55:41Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
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      <text bytes="1309" sha1="hpsewn7tlkjmxu395h9p7d2i53e2ydz" xml:space="preserve">'''105edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 105 equal parts of 11.429 [[cent|cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, [[tempering_out|tempering out]] [[81/80|81/80]] in the [[5-limit|5-limit]]; 81/80, [[126/125|126/125]] and hence 225/224 in the [[7-limit|7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit|11-limit]]; and if we want to push that far, 144/143 in the [[13-limit|13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[Optimal_patent_val|optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo|74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "[[cent|cent]]" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;

[[Category:edo]]
[[Category:huygens]]
[[Category:meantone]]
[[Category:theory]]</text>
      <sha1>hpsewn7tlkjmxu395h9p7d2i53e2ydz</sha1>
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    <revision>
      <id>45640</id>
      <parentid>40211</parentid>
      <timestamp>2020-06-01T14:11:22Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>recat</comment>
      <origin>45640</origin>
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      <text bytes="1330" sha1="b8bl5k2qtgeti2wr1msyy7n895sspzm" xml:space="preserve">'''105edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 105 equal parts of 11.429 [[cent|cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, [[tempering_out|tempering out]] [[81/80|81/80]] in the [[5-limit|5-limit]]; 81/80, [[126/125|126/125]] and hence 225/224 in the [[7-limit|7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit|11-limit]]; and if we want to push that far, 144/143 in the [[13-limit|13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[Optimal_patent_val|optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo|74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "[[cent|cent]]" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;


[[Category:Theory]]
[[Category:Meantone]]
[[Category:Edo]]
[[Category:105edo]]
[[Category:Huygens]]</text>
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    <revision>
      <id>47447</id>
      <parentid>45640</parentid>
      <timestamp>2020-07-02T19:48:15Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
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      <comment>Explain in natural language as well as math why it is the highest Edo Meantone temperament.</comment>
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      <text bytes="1524" sha1="7xsosbwo7wskn0hsy7fd4rwbz23nsko" xml:space="preserve">'''105edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 105 equal parts of 11.429 [[cent|cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfils both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempering_out|tempers out]] [[81/80|81/80]] in the [[5-limit|5-limit]]; 81/80, [[126/125|126/125]] and hence 225/224 in the [[7-limit|7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit|11-limit]]; and if we want to push that far, 144/143 in the [[13-limit|13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[Optimal_patent_val|optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo|74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "[[cent|cent]]" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;


[[Category:Theory]]
[[Category:Meantone]]
[[Category:Edo]]
[[Category:105edo]]
[[Category:Huygens]]</text>
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    <revision>
      <id>54949</id>
      <parentid>47447</parentid>
      <timestamp>2020-12-04T23:13:05Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54949</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1550" sha1="dpzkxscn2v7ps50qimzlneo4x877in1" xml:space="preserve">'''105edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 105 equal parts of 11.429 [[cent|cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfils both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempering_out|tempers out]] [[81/80|81/80]] in the [[5-limit|5-limit]]; 81/80, [[126/125|126/125]] and hence 225/224 in the [[7-limit|7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit|11-limit]]; and if we want to push that far, 144/143 in the [[13-limit|13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[Optimal_patent_val|optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo|74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "[[cent|cent]]" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;


[[Category:Theory]]
[[Category:Meantone]]
[[Category:Equal divisions of the octave]]
[[Category:105edo]]
[[Category:Huygens]]</text>
      <sha1>dpzkxscn2v7ps50qimzlneo4x877in1</sha1>
    </revision>
    <revision>
      <id>55800</id>
      <parentid>54949</parentid>
      <timestamp>2020-12-06T10:10:46Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cat sorting, links simplified</comment>
      <origin>55800</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1445" sha1="k4w31twqiz53zmpdyiq50n2k8mglo8z" xml:space="preserve">'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;


[[Category:Theory]]
[[Category:Meantone]]
[[Category:Equal divisions of the octave]]
[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Huygens]]</text>
      <sha1>k4w31twqiz53zmpdyiq50n2k8mglo8z</sha1>
    </revision>
    <revision>
      <id>58694</id>
      <parentid>55800</parentid>
      <timestamp>2021-01-11T21:16:10Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>+ prime intervals</comment>
      <origin>58694</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1467" sha1="o0vsiq5yi43ap6sk68dzob84aaphmkk" xml:space="preserve">'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
{{Primes in edo|105}}

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;


[[Category:Theory]]
[[Category:Meantone]]
[[Category:Equal divisions of the octave]]
[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Huygens]]</text>
      <sha1>o0vsiq5yi43ap6sk68dzob84aaphmkk</sha1>
    </revision>
    <revision>
      <id>68033</id>
      <parentid>58694</parentid>
      <timestamp>2021-04-20T22:18:55Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68033</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3157" sha1="ijwvj8e2nod8gclplbdl2veca46epoj" xml:space="preserve">'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
{{Primes in edo|105}}

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;


[[Category:Theory]]
[[Category:Meantone]]
[[Category:Equal divisions of the octave]]
[[Category:105edo| ]]
Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
|9:8
|-
|13
|[[1L 12s]]
| rowspan="2" |8:7
|-
|14
|[[7L 7s]]
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|13L 5s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="32" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|-
|84
|21L 63s
|}&lt;!-- main article --&gt;
[[Category:Huygens]]</text>
      <sha1>ijwvj8e2nod8gclplbdl2veca46epoj</sha1>
    </revision>
    <revision>
      <id>68034</id>
      <parentid>68033</parentid>
      <timestamp>2021-04-20T22:29:06Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* 105edo close-up */</comment>
      <origin>68034</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3157" sha1="35p8uft8560phdkbw74v2h8w8fl8n0n" xml:space="preserve">'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
{{Primes in edo|105}}

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;


[[Category:Theory]]
[[Category:Meantone]]
[[Category:Equal divisions of the octave]]
[[Category:105edo| ]]
Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
|9:8
|-
|13
|[[1L 12s]]
| rowspan="2" |8:7
|-
|14
|[[7L 7s]]
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="32" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|-
|84
|21L 63s
|}&lt;!-- main article --&gt;
[[Category:Huygens]]</text>
      <sha1>35p8uft8560phdkbw74v2h8w8fl8n0n</sha1>
    </revision>
    <revision>
      <id>68046</id>
      <parentid>68034</parentid>
      <timestamp>2021-04-21T00:41:30Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68046</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3157" sha1="09efpx5zeoq9164b3xs4kxa6p0g8cwr" xml:space="preserve">'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
{{Primes in edo|105}}

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;


[[Category:Theory]]
[[Category:Meantone]]
[[Category:Equal divisions of the octave]]
[[Category:105edo| ]]
Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
| rowspan="2" |9:8
|-
|13
|[[1L 12s]]
|-
|14
|[[7L 7s]]
|8:7
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="32" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|-
|84
|21L 63s
|}&lt;!-- main article --&gt;
[[Category:Huygens]]</text>
      <sha1>09efpx5zeoq9164b3xs4kxa6p0g8cwr</sha1>
    </revision>
    <revision>
      <id>68075</id>
      <parentid>68046</parentid>
      <timestamp>2021-04-21T02:52:54Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* 105edo close-up */</comment>
      <origin>68075</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3141" sha1="3an0cn5yilkfj0yrp7l7iacc2ltimpm" xml:space="preserve">'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
{{Primes in edo|105}}

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;


[[Category:Theory]]
[[Category:Meantone]]
[[Category:Equal divisions of the octave]]
[[Category:105edo| ]]
Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
| rowspan="2" |9:8
|-
|13
|[[1L 12s]]
|-
|14
|[[7L 7s]]
|8:7
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="31" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|}&lt;!-- main article --&gt;
[[Category:Huygens]]</text>
      <sha1>3an0cn5yilkfj0yrp7l7iacc2ltimpm</sha1>
    </revision>
    <revision>
      <id>68292</id>
      <parentid>68075</parentid>
      <timestamp>2021-04-22T16:30:11Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68292</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3185" sha1="p7ibh14egxu99g3vmufar4q25xl5px6" xml:space="preserve">'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
{{Primes in edo|105}}

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the triangular number closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;


[[Category:Theory]]
[[Category:Meantone]]
[[Category:Equal divisions of the octave]]
[[Category:105edo| ]]
Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments, which it is the first triangular edo to do. 
{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
| rowspan="2" |9:8
|-
|13
|[[1L 12s]]
|-
|14
|[[7L 7s]]
|8:7
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="31" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|}&lt;!-- main article --&gt;
[[Category:Huygens]]</text>
      <sha1>p7ibh14egxu99g3vmufar4q25xl5px6</sha1>
    </revision>
    <revision>
      <id>84269</id>
      <parentid>68292</parentid>
      <timestamp>2022-01-19T22:10:42Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>direct triangular number link to definition within the xen wiki, to provide justifiably relevant xen-related information</comment>
      <origin>84269</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3189" sha1="18nimqm5jfrhl5jv9l3leyjl34vkjt4" xml:space="preserve">'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
{{Primes in edo|105}}

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the [[triangular number]] closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;


[[Category:Theory]]
[[Category:Meantone]]
[[Category:Equal divisions of the octave]]
[[Category:105edo| ]]
Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments, which it is the first triangular edo to do. 
{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
| rowspan="2" |9:8
|-
|13
|[[1L 12s]]
|-
|14
|[[7L 7s]]
|8:7
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="31" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|}&lt;!-- main article --&gt;
[[Category:Huygens]]</text>
      <sha1>18nimqm5jfrhl5jv9l3leyjl34vkjt4</sha1>
    </revision>
    <revision>
      <id>87431</id>
      <parentid>84269</parentid>
      <timestamp>2022-02-26T16:48:48Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87431</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3169" sha1="ihqxizc7xymysrk6vgx6qbfmxx1kwg3" xml:space="preserve">'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
{{Primes in edo|105}}

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the [[triangular number]] closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== 105edo close-up ==

&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;


[[Category:Meantone]]
[[Category:Equal divisions of the octave]]
[[Category:105edo| ]]
Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments, which it is the first triangular edo to do. 
{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
| rowspan="2" |9:8
|-
|13
|[[1L 12s]]
|-
|14
|[[7L 7s]]
|8:7
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="31" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|}&lt;!-- main article --&gt;
[[Category:Huygens]]</text>
      <sha1>ihqxizc7xymysrk6vgx6qbfmxx1kwg3</sha1>
    </revision>
    <revision>
      <id>92915</id>
      <parentid>87431</parentid>
      <timestamp>2022-07-02T16:17:43Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key, move categories at the bottom of the page source</comment>
      <origin>92915</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3197" sha1="0bgs4jl2damct564hbwpehhfuryszrh" xml:space="preserve">'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
{{Primes in edo|105}}

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the [[triangular number]] closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== 105edo close-up ==
&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;

Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments, which it is the first triangular edo to do. 
{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
| rowspan="2" |9:8
|-
|13
|[[1L 12s]]
|-
|14
|[[7L 7s]]
|8:7
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="31" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|}

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>0bgs4jl2damct564hbwpehhfuryszrh</sha1>
    </revision>
    <revision>
      <id>96187</id>
      <parentid>92915</parentid>
      <timestamp>2022-10-04T18:48:33Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96187</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3212" sha1="e4smps8pkwrx1zggjr8g40qpzc3x41t" xml:space="preserve">{{Infobox ET}}
'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. It is most notable as a tuning of meantone and in particular higher limit extensions of meantone, as it is the highest edo that strictly fulfills both criteria of meantone - ie, all intervals can be reached by stacking it's best fifth, and stacking four of them equals it's best major third. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping (aka "huygens") of 11-limit meantone.
{{Primes in edo|105}}

105edo gives the [[optimal patent val]] for 11-limit meantone (ie huygens rather than meanpop) and provides a good tuning in the 13-limit, though [[74edo]] is in that case the optimal patent val. 105 is highly composite, being the product 3*5*7 (i. e. (14+1)*14/2) of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the [[triangular number]] closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== 105edo close-up ==
&lt;pre&gt;C . . Dbb B## . . C# . . Db . . . C## . . D&lt;/pre&gt;

Since 105edo has a step of 11.429 cents, it also allows one to use its MOS scales as circulating temperaments, which it is the first triangular edo to do. 
{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
| rowspan="2" |9:8
|-
|13
|[[1L 12s]]
|-
|14
|[[7L 7s]]
|8:7
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="31" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|}

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>e4smps8pkwrx1zggjr8g40qpzc3x41t</sha1>
    </revision>
    <revision>
      <id>99320</id>
      <parentid>96187</parentid>
      <timestamp>2022-11-27T12:53:45Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup and -factual errors (127edo seems to be the last meantone pval edo)</comment>
      <origin>99320</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3011" sha1="37e4nd394icbz4y07ttr3d72rxkivon" xml:space="preserve">{{Infobox ET}}
'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 

=== Odd harmonics ===
{{Harmonics in equal|105}}

=== Miscellany ===
105 is fairly composite, being the product 3 × 5 × 7 of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the [[triangular number]] closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== Scales ==
Since 105edo has a step of 11.429 cents, it also allows one to use its [[mos scale]]s as [[circulating temperament]]s, which it is the first triangular edo to do{{clarify}}. 

{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
| rowspan="2" |9:8
|-
|13
|[[1L 12s]]
|-
|14
|[[7L 7s]]
|8:7
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="31" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|}

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>37e4nd394icbz4y07ttr3d72rxkivon</sha1>
    </revision>
    <revision>
      <id>106515</id>
      <parentid>99320</parentid>
      <timestamp>2023-03-31T17:50:45Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <minor/>
      <comment>De-orphanising</comment>
      <origin>106515</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3060" sha1="bahikajv8esebq92s9pk8z737mhh1z1" xml:space="preserve">{{Infobox ET}}
'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 

=== Odd harmonics ===
{{Harmonics in equal|105}}

=== Intervals ===
[[Table of 105edo intervals]]

=== Miscellany ===
105 is fairly composite, being the product 3 × 5 × 7 of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the [[triangular number]] closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== Scales ==
Since 105edo has a step of 11.429 cents, it also allows one to use its [[mos scale]]s as [[circulating temperament]]s, which it is the first triangular edo to do{{clarify}}. 

{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
| rowspan="2" |9:8
|-
|13
|[[1L 12s]]
|-
|14
|[[7L 7s]]
|8:7
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="31" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|}

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>bahikajv8esebq92s9pk8z737mhh1z1</sha1>
    </revision>
    <revision>
      <id>113769</id>
      <parentid>106515</parentid>
      <timestamp>2023-05-20T03:29:51Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>113769</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3153" sha1="0d4oz00gsv2k4ny5h9dk8hivmy65esv" xml:space="preserve">{{Infobox ET}}
'''105edo''' is the [[equal division of the octave]] into 105 equal parts of 11.429 [[cent]]s each. 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.

=== Odd harmonics ===
{{Harmonics in equal|105}}

=== Intervals ===
[[Table of 105edo intervals]]

=== Miscellany ===
105 is fairly composite, being the product 3 × 5 × 7 of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the [[triangular number]] closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== Scales ==
Since 105edo has a step of 11.429 cents, it also allows one to use its [[mos scale]]s as [[circulating temperament]]s, which it is the first triangular edo to do{{clarify}}. 

{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
| rowspan="2" |9:8
|-
|13
|[[1L 12s]]
|-
|14
|[[7L 7s]]
|8:7
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="31" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|}

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>0d4oz00gsv2k4ny5h9dk8hivmy65esv</sha1>
    </revision>
    <revision>
      <id>114539</id>
      <parentid>113769</parentid>
      <timestamp>2023-05-24T21:09:50Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Add 15-odd-limit table, mention two examples of meantone extensions relevant to this tuning, misc. edits</comment>
      <origin>114539</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3174" sha1="kf1fkume1gc1ymdcrce7iu4gncvbyjp" xml:space="preserve">{{Infobox ET}}
{{EDO intro|105}} 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[Huygens]]. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.

=== Odd harmonics ===
{{Harmonics in equal|105}}

== Intervals ==
{{Main|Table of 105edo intervals}}

=== 15-odd-limit interval mappings ===
{{15-odd-limit|105}}

=== Miscellany ===
105 is fairly composite, being the product 3 × 5 × 7 of the three smallest odd primes, with other divisors being 15, 21 and 35. As the common multiple of these three primes and the [[triangular number]] closest to 100, 105 is a perfect substitute for it when a "cent" is desired to include them all or be a triangular number.

== Scales ==
Since 105edo has a step of 11.429 cents, it also allows one to use its [[mos scale]]s as [[circulating temperament]]s, which it is the first triangular edo to do{{clarify}}. 

{| class="wikitable"
|+Circulating temperaments in 105edo
!Tones
!Pattern
!L:s
|-
|5
|[[5edo]]
|equal
|-
|6
|[[3L 3s]]
|18:17
|-
|7
|[[7edo]]
|equal
|-
|8
|[[1L 7s]]
|14:13
|-
|9
|[[6L 3s]]
|12:11
|-
|10
|[[5L 5s]]
|11:10
|-
|11
|[[6L 5s]]
|10:9
|-
|12
|[[9L 3s]]
| rowspan="2" |9:8
|-
|13
|[[1L 12s]]
|-
|14
|[[7L 7s]]
|8:7
|-
|15
|[[15edo]]
|equal
|-
|16
|[[9L 7s]]
| rowspan="2" |7:6
|-
|17
|[[3L 14s]]
|-
|18
|15L 3s
| rowspan="3" |6:5
|-
|19
|[[10L 9s]]
|-
|20
|5L 15s
|-
|21
|[[21edo]]
|equal
|-
|22
|[[17L 5s]]
| rowspan="5" |5:4
|-
|23
|13L 10s
|-
|24
|9L 15s
|-
|25
|5L 20s
|-
|26
|1L 25s
|-
|27
|24L 3s
| rowspan="8" |4:3
|-
|28
|21L 7s
|-
|29
|18L 11s
|-
|30
|15L 15s
|-
|31
|12L 19s
|-
|32
|9L 23s
|-
|33
|6L 27s
|-
|34
|3L 31s
|-
|35
|[[35edo]]
|equal
|-
|36
|33L 3s
| rowspan="17" |3:2
|-
|37
|31L 6s
|-
|38
|29L 9s
|-
|39
|27L 12s
|-
|40
|25L 15s
|-
|41
|23L 18s
|-
|42
|21L 21s
|-
|43
|19L 24s
|-
|44
|17L 27s
|-
|45
|15L 30s
|-
|46
|13L 33s
|-
|47
|11L 36s
|-
|48
|9L 39s
|-
|49
|7L 42s
|-
|50
|5L 45s
|-
|51
|3L 48s
|-
|52
|1L 51s
|-
|53
|52L 1s
| rowspan="31" |2:1
|-
|54
|51L 3s
|-
|55
|50L 5s
|-
|56
|49L 7s
|-
|57
|48L 9s
|-
|58
|47L 11s
|-
|59
|46L 13s
|-
|60
|45L 15s
|-
|61
|44L 17s
|-
|62
|43L 19s
|-
|63
|42L 21s
|-
|64
|41L 23s
|-
|65
|40L 25s
|-
|66
|39L 27s
|-
|67
|38L 29s
|-
|68
|37L 31s
|-
|69
|36L 33s
|-
|70
|35L 35s
|-
|71
|34L 37s
|-
|72
|33L 39s
|-
|73
|32L 41s
|-
|74
|31L 43s
|-
|75
|30L 45s
|-
|76
|29L 47s
|-
|77
|28L 49s
|-
|78
|27L 51s
|-
|79
|26L 53s
|-
|80
|25L 55s
|-
|81
|24L 57s
|-
|82
|23L 59s
|-
|83
|22L 61s
|}

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>kf1fkume1gc1ymdcrce7iu4gncvbyjp</sha1>
    </revision>
    <revision>
      <id>115190</id>
      <parentid>114539</parentid>
      <timestamp>2023-05-30T21:13:13Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>115190</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1212" sha1="jf2vma1g3w830lngdnoozh3oda3zxhq" xml:space="preserve">{{Infobox ET}}
{{EDO intro|105}} 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[Huygens]]. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.

=== Odd harmonics ===
{{Harmonics in equal|105}}

=== Subsets and supersets ===
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35. 

== Intervals ==
{{Main|Table of 105edo intervals}}

=== 15-odd-limit interval mappings ===
{{15-odd-limit|105}}

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>jf2vma1g3w830lngdnoozh3oda3zxhq</sha1>
    </revision>
    <revision>
      <id>137413</id>
      <parentid>115190</parentid>
      <timestamp>2024-02-28T02:10:28Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>137413</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1230" sha1="fzubz48jljdxfbafb8hglmrcuxcdaw1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|105}} 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[Huygens]]. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.

=== Odd harmonics ===
{{Harmonics in equal|105}}

=== Subsets and supersets ===
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35. 

== Intervals ==
{{Main|Table of 105edo intervals}}
{{Interval table}}
=== 15-odd-limit interval mappings ===
{{15-odd-limit|105}}

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>fzubz48jljdxfbafb8hglmrcuxcdaw1</sha1>
    </revision>
    <revision>
      <id>144953</id>
      <parentid>137413</parentid>
      <timestamp>2024-05-24T13:21:03Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>144953</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1250" sha1="9wzb5sim0bo4lum877j5xm97n36vxjm" xml:space="preserve">{{Infobox ET}}
{{EDO intro|105}} 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[Huygens]]. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.

=== Odd harmonics ===
{{Harmonics in equal|105}}

=== Subsets and supersets ===
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35. 

== Intervals ==
{{Main|Table of 105edo intervals}}
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|105}}
{{Q-odd-limit intervals|105}}

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>9wzb5sim0bo4lum877j5xm97n36vxjm</sha1>
    </revision>
    <revision>
      <id>144954</id>
      <parentid>144953</parentid>
      <timestamp>2024-05-24T13:21:17Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>144954</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1220" sha1="211em1i7tqbysjnulcut77t39yhorbz" xml:space="preserve">{{Infobox ET}}
{{EDO intro|105}} 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[Huygens]]. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.

=== Odd harmonics ===
{{Harmonics in equal|105}}

=== Subsets and supersets ===
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35. 

== Intervals ==
{{Main|Table of 105edo intervals}}
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|105}}

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>211em1i7tqbysjnulcut77t39yhorbz</sha1>
    </revision>
    <revision>
      <id>175419</id>
      <parentid>144954</parentid>
      <timestamp>2025-01-09T14:04:54Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Link to lumatone mapping page</comment>
      <origin>175419</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1357" sha1="k33tz135yx9swax22shzztlxvcv9wli" xml:space="preserve">{{Infobox ET}}
{{EDO intro|105}} 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[Huygens]]. It [[tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (aka huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.

=== Odd harmonics ===
{{Harmonics in equal|105}}

=== Subsets and supersets ===
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35. 

== Intervals ==
{{Main|Table of 105edo intervals}}
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|105}}

== Instruments ==
=== Lumatone ===
The [[lumatone]] can be used to play 105edo. For key mappings, see: [[Lumatone mapping for 105edo]].

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>k33tz135yx9swax22shzztlxvcv9wli</sha1>
    </revision>
    <revision>
      <id>176750</id>
      <parentid>175419</parentid>
      <timestamp>2025-01-17T15:16:42Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ cleanup</comment>
      <origin>176750</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1373" sha1="101pzov7yhair3qm1vxiz1d8qkanstf" xml:space="preserve">{{Infobox ET}}
{{EDO intro|105}} 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[huygens]]. It [[tempering out|tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (a.k.a. huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.

=== Odd harmonics ===
{{Harmonics in equal|105}}

=== Subsets and supersets ===
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35.

== Intervals ==
{{Main|Table of 105edo intervals}}
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|105}}

== Instruments ==
=== Lumatone ===
The [[lumatone]] can be used to play 105edo. For key mappings, see: [[Lumatone mapping for 105edo]].

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>101pzov7yhair3qm1vxiz1d8qkanstf</sha1>
    </revision>
    <revision>
      <id>181308</id>
      <parentid>176750</parentid>
      <timestamp>2025-02-18T16:47:03Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181308</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1368" sha1="bwods8glepb3g7logqyu485ss9uukw1" xml:space="preserve">{{Infobox ET}}
{{ED intro}} 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[huygens]]. It [[tempering out|tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (a.k.a. huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.

=== Odd harmonics ===
{{Harmonics in equal|105}}

=== Subsets and supersets ===
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35.

== Intervals ==
{{Main|Table of 105edo intervals}}
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|105}}

== Instruments ==
=== Lumatone ===
The [[lumatone]] can be used to play 105edo. For key mappings, see: [[Lumatone mapping for 105edo]].

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>bwods8glepb3g7logqyu485ss9uukw1</sha1>
    </revision>
    <revision>
      <id>194826</id>
      <parentid>181308</parentid>
      <timestamp>2025-04-30T19:26:53Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>194826</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1594" sha1="64okut1keidfxgycnmqd0eircy1viu8" xml:space="preserve">{{Infobox ET}}
{{ED intro}} 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[huygens]]. It [[tempering out|tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (a.k.a. huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.

=== Odd harmonics ===
{{Harmonics in equal|105}}

=== Subsets and supersets ===
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35.

As such, the val [105 165 245 294], which is contorted in 2.n for each prime n in the subgroup, may be used to extend the concept of 15edo's 5-limit harmony to the 7-limit, producing an independent dimension for each prime. 

== Intervals ==
{{Main|Table of 105edo intervals}}
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|105}}

== Instruments ==
=== Lumatone ===
The [[lumatone]] can be used to play 105edo. For key mappings, see: [[Lumatone mapping for 105edo]].

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
      <sha1>64okut1keidfxgycnmqd0eircy1viu8</sha1>
    </revision>
    <revision>
      <id>194827</id>
      <parentid>194826</parentid>
      <timestamp>2025-04-30T19:44:12Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <origin>194827</origin>
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{{ED intro}} 

== Theory ==
105edo is most notable as a tuning of [[meantone]] and in particular higher-limit extensions of meantone, such as [[grosstone]] and [[huygens]]. It [[tempering out|tempers out]] [[81/80]] in the [[5-limit]]; 81/80, [[126/125]] and hence 225/224 in the [[7-limit]]; 99/98, 176/175 and 441/440 in the [[11-limit]]; and if we want to push that far, 144/143 in the [[13-limit]]. This is the sharper fifth mapping of 11-limit meantone (a.k.a. huygens rather than meanpop), for which it gives the [[optimal patent val]], and provides a good tuning for the 13-limit extension, though [[74edo]] is in that case the optimal patent val. 105edo's meantone fifth is nearly identical to the [[CTE tuning|CTE generator]] for meantone.

=== Odd harmonics ===
{{Harmonics in equal|105}}

=== Subsets and supersets ===
105 is the product of 3 × 5 × 7, the three smallest odd primes, with other divisors being 15, 21 and 35.

As such, the val [105 165 245 294], which is contorted in 2.n for each prime n in the subgroup, may be used to extend the concept of 21edo's 5-limit harmony to the 7-limit, producing an independent dimension for each prime.

== Intervals ==
{{Main|Table of 105edo intervals}}
=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|105}}

== Instruments ==
=== Lumatone ===
The [[lumatone]] can be used to play 105edo. For key mappings, see: [[Lumatone mapping for 105edo]].

[[Category:105edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Huygens]]
[[Category:Meantone]]</text>
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[[Category:interval_measure]]</text>
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Dividing the octave into 10600 equal parts gives a [[turkish cent]].

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Dividing the octave into 10600 equal parts gives a [[turkish cent]].

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{{EDO intro}}

Dividing the octave into 10600 equal parts gives a [[turkish cent]].

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Dividing the octave into 10600 equal parts gives a [[turkish cent]].

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{{ED intro}}

Dividing the octave into 10600 equal parts gives a [[turkish cent]].

{{Harmonics in equal|10600}}

{{Stub}}

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  <page>
    <title>106edo</title>
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      <text bytes="1271" sha1="5t7ehy42rjnbdmkm8fq8jw413wesvm6" xml:space="preserve">The 106 equal division divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo|53edo]], and is [[Saturation|contorted]] through the 7-limit, tempering out the same commas (32805/32768, 15625/15552, 1600000/1594323, 2109375/2097152 in the 5-limit, 3125/3097, 225/224, 4000/3969, 1728/1715, 2430/2401, 4375/4374 in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out 243/242, 3025/3024 and 9801/9800, so that it supports [[Marvel_family#Spectacle|spectacle temperament]] and [[Semicomma_family#Borwell|borwell temperament]].

The division is notable for the fact that it is related to the [[turkish_cent|turkish cent]], ot türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[Relative_cent|relative cent]] division for 106edo.

Artists using 106 et:

&lt;ul&gt;&lt;li&gt;[[Dolores_Catherino|Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]&lt;/li&gt;&lt;li&gt;[http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]&lt;/li&gt;&lt;/ul&gt;      [[Category:53edo]]
[[Category:polychromatic]]</text>
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      <comment>**Imported revision 571231689 - Original comment: **</comment>
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      <text bytes="4168" sha1="2dg8n22m4mzkms137f7i3c6it25pbgl" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2016-01-06 05:07:10 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;571231689&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 106 equal division divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[Saturation|contorted]] through the 7-limit, tempering out the same commas (32805/32768, 15625/15552, 1600000/1594323, 2109375/2097152 in the 5-limit, 3125/3097, 225/224, 4000/3969, 1728/1715, 2430/2401, 4375/4374 in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out 243/242, 3025/3024 and 9801/9800, so that it supports [[Marvel family#Spectacle|spectacle temperament]] and [[Semicomma family#Borwell|borwell temperament]].

The division is notable for the fact that it is related to the [[turkish cent]], ot türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo.

Artists using 106 et:
* [[Dolores Catherino]] -- [[http://dolorescatherino.com|her website]], [[https://www.youtube.com/user/dolomuse|YouTube profile: dolomuse]]
* [[http://chrisvaisvil.com/still-life-in-106-notes-per-octave/|Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;106edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 106 equal division divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53edo&amp;lt;/a&amp;gt;, and is &amp;lt;a class="wiki_link" href="/Saturation"&amp;gt;contorted&amp;lt;/a&amp;gt; through the 7-limit, tempering out the same commas (32805/32768, 15625/15552, 1600000/1594323, 2109375/2097152 in the 5-limit, 3125/3097, 225/224, 4000/3969, 1728/1715, 2430/2401, 4375/4374 in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out 243/242, 3025/3024 and 9801/9800, so that it supports &amp;lt;a class="wiki_link" href="/Marvel%20family#Spectacle"&amp;gt;spectacle temperament&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Semicomma%20family#Borwell"&amp;gt;borwell temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The division is notable for the fact that it is related to the &amp;lt;a class="wiki_link" href="/turkish%20cent"&amp;gt;turkish cent&amp;lt;/a&amp;gt;, ot türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the &amp;lt;a class="wiki_link" href="/relative%20cent"&amp;gt;relative cent&amp;lt;/a&amp;gt; division for 106edo.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Artists using 106 et:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a class="wiki_link" href="/Dolores%20Catherino"&amp;gt;Dolores Catherino&amp;lt;/a&amp;gt; -- &amp;lt;a class="wiki_link_ext" href="http://dolorescatherino.com" rel="nofollow"&amp;gt;her website&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="https://www.youtube.com/user/dolomuse" rel="nofollow"&amp;gt;YouTube profile: dolomuse&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/still-life-in-106-notes-per-octave/" rel="nofollow"&amp;gt;Still Life in 106 Notes Per Octave « Music &amp;amp;amp; Techniques by Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 571231271 - Original comment: **</comment>
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      <text bytes="3799" sha1="14kx5mcxmpujguw43143gwnqzomdzds" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2016-01-06 04:50:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;571231271&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 106 equal division divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[Saturation|contorted]] through the 7-limit, tempering out the same commas (32805/32768, 15625/15552, 1600000/1594323, 2109375/2097152 in the 5-limit, 3125/3097, 225/224, 4000/3969, 1728/1715, 2430/2401, 4375/4374 in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out 243/242, 3025/3024 and 9801/9800, so that it supports [[Marvel family#Spectacle|spectacle temperament]] and [[Semicomma family#Borwell|borwell temperament]].

The division is notable for the fact that it is related to the [[turkish cent]], ot türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo.

Artists using 106 et:
* [[Dolores Catherino]] -- [[http://dolorescatherino.com|her website]], [[https://www.youtube.com/user/dolomuse|YouTube profile: dolomuse]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;106edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 106 equal division divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53edo&amp;lt;/a&amp;gt;, and is &amp;lt;a class="wiki_link" href="/Saturation"&amp;gt;contorted&amp;lt;/a&amp;gt; through the 7-limit, tempering out the same commas (32805/32768, 15625/15552, 1600000/1594323, 2109375/2097152 in the 5-limit, 3125/3097, 225/224, 4000/3969, 1728/1715, 2430/2401, 4375/4374 in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out 243/242, 3025/3024 and 9801/9800, so that it supports &amp;lt;a class="wiki_link" href="/Marvel%20family#Spectacle"&amp;gt;spectacle temperament&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Semicomma%20family#Borwell"&amp;gt;borwell temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The division is notable for the fact that it is related to the &amp;lt;a class="wiki_link" href="/turkish%20cent"&amp;gt;turkish cent&amp;lt;/a&amp;gt;, ot türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the &amp;lt;a class="wiki_link" href="/relative%20cent"&amp;gt;relative cent&amp;lt;/a&amp;gt; division for 106edo.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Artists using 106 et:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a class="wiki_link" href="/Dolores%20Catherino"&amp;gt;Dolores Catherino&amp;lt;/a&amp;gt; -- &amp;lt;a class="wiki_link_ext" href="http://dolorescatherino.com" rel="nofollow"&amp;gt;her website&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="https://www.youtube.com/user/dolomuse" rel="nofollow"&amp;gt;YouTube profile: dolomuse&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>14kx5mcxmpujguw43143gwnqzomdzds</sha1>
    </revision>
    <revision>
      <id>4081</id>
      <timestamp>2011-09-19T12:15:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 255686812 - Original comment: **</comment>
      <origin>4081</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3230" sha1="nlfx1kgd209a9xchsikesrs6xg4x0g0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-09-19 12:15:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;255686812&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 106 equal division divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[Saturation|contorted]] through the 7-limit, tempering out the same commas (32805/32768, 15625/15552, 1600000/1594323, 2109375/2097152 in the 5-limit, 3125/3097, 225/224, 4000/3969, 1728/1715, 2430/2401, 4375/4374 in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out 243/242, 3025/3024 and 9801/9800, so that it supports [[Marvel family#Spectacle|spectacle temperament]] and [[Semicomma family#Borwell|borwell temperament]].

The division is notable for the fact that it is related to the [[turkish cent]], ot türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;106edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 106 equal division divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53edo&amp;lt;/a&amp;gt;, and is &amp;lt;a class="wiki_link" href="/Saturation"&amp;gt;contorted&amp;lt;/a&amp;gt; through the 7-limit, tempering out the same commas (32805/32768, 15625/15552, 1600000/1594323, 2109375/2097152 in the 5-limit, 3125/3097, 225/224, 4000/3969, 1728/1715, 2430/2401, 4375/4374 in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out 243/242, 3025/3024 and 9801/9800, so that it supports &amp;lt;a class="wiki_link" href="/Marvel%20family#Spectacle"&amp;gt;spectacle temperament&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Semicomma%20family#Borwell"&amp;gt;borwell temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The division is notable for the fact that it is related to the &amp;lt;a class="wiki_link" href="/turkish%20cent"&amp;gt;turkish cent&amp;lt;/a&amp;gt;, ot türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the &amp;lt;a class="wiki_link" href="/relative%20cent"&amp;gt;relative cent&amp;lt;/a&amp;gt; division for 106edo.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>nlfx1kgd209a9xchsikesrs6xg4x0g0</sha1>
    </revision>
    <revision>
      <id>40649</id>
      <parentid>24</parentid>
      <timestamp>2019-03-05T19:37:42Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>40649</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1416" sha1="p2sj97agbk7hodcosbcufoqvqv20vmp" xml:space="preserve">The 106 equal division divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo|53edo]], and is [[Saturation|contorted]] through the 7-limit, tempering out the same commas (32805/32768, 15625/15552, 1600000/1594323, 2109375/2097152 in the 5-limit, 3125/3097, 225/224, 4000/3969, 1728/1715, 2430/2401, 4375/4374 in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out 243/242, 3025/3024 and 9801/9800, so that it supports [[Marvel_family#Spectacle|spectacle temperament]] and [[Semicomma_family#Borwell|borwell temperament]].

The division is notable for the fact that it is related to the [[turkish_cent|turkish cent]], ot türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[Relative_cent|relative cent]] division for 106edo. Conversely, it makes the [[Relative_cent|relative cent]] which most closely approximates dividing an exact 3/2, if you care about such a thing.

Artists using 106 et:

&lt;ul&gt;&lt;li&gt;[[Dolores_Catherino|Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]&lt;/li&gt;&lt;li&gt;[http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]&lt;/li&gt;&lt;/ul&gt;      
[[Category:53edo]]
[[Category:polychromatic]]</text>
      <sha1>p2sj97agbk7hodcosbcufoqvqv20vmp</sha1>
    </revision>
    <revision>
      <id>41179</id>
      <parentid>40649</parentid>
      <timestamp>2019-03-26T03:21:25Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>41179</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1489" sha1="r317ejc6b3xrotpmeavbhsdqvrpmtmp" xml:space="preserve">The 106 equal division divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo|53edo]], and is [[Saturation|contorted]] through the 7-limit, tempering out the same commas (32805/32768, 15625/15552, 1600000/1594323, 2109375/2097152 in the 5-limit, 3125/3097, 225/224, 4000/3969, 1728/1715, 2430/2401, 4375/4374 in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out 243/242, 3025/3024 and 9801/9800, so that it supports [[Marvel_family#Spectacle|spectacle temperament]] and [[Semicomma_family#Borwell|borwell temperament]].

The division is notable for the fact that it is related to the [[turkish_cent|turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[Relative_cent|relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[Relative_cent|relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact 3/2, if you care about such a thing.

Artists using 106 et:

&lt;ul&gt;&lt;li&gt;[[Dolores_Catherino|Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]&lt;/li&gt;&lt;li&gt;[http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]&lt;/li&gt;&lt;/ul&gt;      
[[Category:53edo]]
[[Category:polychromatic]]</text>
      <sha1>r317ejc6b3xrotpmeavbhsdqvrpmtmp</sha1>
    </revision>
    <revision>
      <id>50801</id>
      <parentid>41179</parentid>
      <timestamp>2020-10-05T19:57:34Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>simplified wiki markup, linked to temperaments via dedicated redirect pages</comment>
      <origin>50801</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1356" sha1="1bahj9bypxtociqpuib31m04xc2rtp8" xml:space="preserve">The 106 equal division divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[Saturation|contorted]] through the [[7-limit]], tempering out the same commas (32805/32768, 15625/15552, 1600000/1594323, 2109375/2097152 in the [[5-limit]], 3125/3097, 225/224, 4000/3969, 1728/1715, 2430/2401, 4375/4374 in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out 243/242, 3025/3024 and 9801/9800, so that it supports [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact 3/2, if you care about such a thing.

Artists using 106 et:

* [[Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Polychromatic]]</text>
      <sha1>1bahj9bypxtociqpuib31m04xc2rtp8</sha1>
    </revision>
    <revision>
      <id>52724</id>
      <parentid>50801</parentid>
      <timestamp>2020-11-18T16:43:53Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>added some links and cat</comment>
      <origin>52724</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1407" sha1="enijqc37f4wyd3zb07lgy4hchae93gu" xml:space="preserve">The '''106 equal division''' divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[Contorsion|contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], 1600000/1594323, 2109375/2097152 in the [[5-limit]], 3125/3097, [[225/224]], 4000/3969, 1728/1715, 2430/2401, [[4375/4374]] in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out [[243/242]], 3025/3024 and [[9801/9800]], so that it supports [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.

Artists using 106 et:

* [[Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:Edo]]
[[Category:53edo]]
[[Category:Polychromatic]]</text>
      <sha1>enijqc37f4wyd3zb07lgy4hchae93gu</sha1>
    </revision>
    <revision>
      <id>54952</id>
      <parentid>52724</parentid>
      <timestamp>2020-12-04T23:13:05Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54952</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1433" sha1="981ap6n40qq07hzeyplb9tzueduq6kl" xml:space="preserve">The '''106 equal division''' divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[Contorsion|contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], 1600000/1594323, 2109375/2097152 in the [[5-limit]], 3125/3097, [[225/224]], 4000/3969, 1728/1715, 2430/2401, [[4375/4374]] in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out [[243/242]], 3025/3024 and [[9801/9800]], so that it supports [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.

Artists using 106 et:

* [[Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:Equal divisions of the octave]]
[[Category:53edo]]
[[Category:Polychromatic]]</text>
      <sha1>981ap6n40qq07hzeyplb9tzueduq6kl</sha1>
    </revision>
    <revision>
      <id>68040</id>
      <parentid>54952</parentid>
      <timestamp>2021-04-20T23:41:18Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68040</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3126" sha1="oxvg6kidpkjiwrna5qkfb2p9rkglitj" xml:space="preserve">The '''106 equal division''' divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[Contorsion|contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], 1600000/1594323, 2109375/2097152 in the [[5-limit]], 3125/3097, [[225/224]], 4000/3969, 1728/1715, 2430/2401, [[4375/4374]] in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out [[243/242]], 3025/3024 and [[9801/9800]], so that it supports [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.


Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
|9:8
|-
|13
|[[2L 11s]]
| rowspan="3" |8:7
|-
|14
|[[8L 6s]]
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="32" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|-
|85
|21L 64s
|}Artists using 106 et:

*[[Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
*[http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:Equal divisions of the octave]]
[[Category:53edo]]
[[Category:Polychromatic]]</text>
      <sha1>oxvg6kidpkjiwrna5qkfb2p9rkglitj</sha1>
    </revision>
    <revision>
      <id>68047</id>
      <parentid>68040</parentid>
      <timestamp>2021-04-21T00:42:56Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68047</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3140" sha1="gwpxr9r8hccyl7u4tfhdvnhw4yatr19" xml:space="preserve">The '''106 equal division''' divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[Contorsion|contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], 1600000/1594323, 2109375/2097152 in the [[5-limit]], 3125/3097, [[225/224]], 4000/3969, 1728/1715, 2430/2401, [[4375/4374]] in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out [[243/242]], 3025/3024 and [[9801/9800]], so that it supports [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.


Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="32" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|-
|85
|21L 64s
|}Artists using 106 et:

*[[Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
*[http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:Equal divisions of the octave]]
[[Category:53edo]]
[[Category:Polychromatic]]</text>
      <sha1>gwpxr9r8hccyl7u4tfhdvnhw4yatr19</sha1>
    </revision>
    <revision>
      <id>68076</id>
      <parentid>68047</parentid>
      <timestamp>2021-04-21T02:53:26Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68076</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3124" sha1="nsf4bv2kvopyac19pmuz888jg4wo9t0" xml:space="preserve">The '''106 equal division''' divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[Contorsion|contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], 1600000/1594323, 2109375/2097152 in the [[5-limit]], 3125/3097, [[225/224]], 4000/3969, 1728/1715, 2430/2401, [[4375/4374]] in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out [[243/242]], 3025/3024 and [[9801/9800]], so that it supports [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.


Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}Artists using 106 et:

*[[Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
*[http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:Equal divisions of the octave]]
[[Category:53edo]]
[[Category:Polychromatic]]</text>
      <sha1>nsf4bv2kvopyac19pmuz888jg4wo9t0</sha1>
    </revision>
    <revision>
      <id>78774</id>
      <parentid>68076</parentid>
      <timestamp>2021-09-29T19:46:51Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>add link for defactoring</comment>
      <origin>78774</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3133" sha1="362oppkeddturo2dyldwoqd3tilhmkl" xml:space="preserve">The '''106 equal division''' divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[contorted]] (or [[enfactored]]) through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], 1600000/1594323, 2109375/2097152 in the [[5-limit]], 3125/3097, [[225/224]], 4000/3969, 1728/1715, 2430/2401, [[4375/4374]] in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out [[243/242]], 3025/3024 and [[9801/9800]], so that it supports [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.


Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}Artists using 106 et:

*[[Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
*[http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:Equal divisions of the octave]]
[[Category:53edo]]
[[Category:Polychromatic]]</text>
      <sha1>362oppkeddturo2dyldwoqd3tilhmkl</sha1>
    </revision>
    <revision>
      <id>80696</id>
      <parentid>78774</parentid>
      <timestamp>2021-11-12T23:06:05Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>remove temporary link to defactoring</comment>
      <origin>80696</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3113" sha1="9ia8f55yc96svjfrpulma5vsgbnezfr" xml:space="preserve">The '''106 equal division''' divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], 1600000/1594323, 2109375/2097152 in the [[5-limit]], 3125/3097, [[225/224]], 4000/3969, 1728/1715, 2430/2401, [[4375/4374]] in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out [[243/242]], 3025/3024 and [[9801/9800]], so that it supports [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.


Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}Artists using 106 et:

*[[Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
*[http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:Equal divisions of the octave]]
[[Category:53edo]]
[[Category:Polychromatic]]</text>
      <sha1>9ia8f55yc96svjfrpulma5vsgbnezfr</sha1>
    </revision>
    <revision>
      <id>84144</id>
      <parentid>80696</parentid>
      <timestamp>2022-01-16T01:26:38Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <origin>84144</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3147" sha1="ef2206ps3ghtoptsw46jxs5sa6ndah5" xml:space="preserve">The '''106 equal division''' divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], 1600000/1594323, 2109375/2097152 in the [[5-limit]], 3125/3097, [[225/224]], 4000/3969, 1728/1715, 2430/2401, [[4375/4374]] in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out [[243/242]], 3025/3024 and [[9801/9800]], so that it supports [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.


{{Primes in edo|106|columns=24}}

Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}Artists using 106 et:

*[[Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
*[http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:Equal divisions of the octave]]
[[Category:53edo]]
[[Category:Polychromatic]]</text>
      <sha1>ef2206ps3ghtoptsw46jxs5sa6ndah5</sha1>
    </revision>
    <revision>
      <id>84828</id>
      <parentid>84144</parentid>
      <timestamp>2022-01-25T18:59:55Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>link to new page for Supports</comment>
      <origin>84828</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3151" sha1="cl3sfnbnkk187cxlfitnhr41mv1fb35" xml:space="preserve">The '''106 equal division''' divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], 1600000/1594323, 2109375/2097152 in the [[5-limit]], 3125/3097, [[225/224]], 4000/3969, 1728/1715, 2430/2401, [[4375/4374]] in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out [[243/242]], 3025/3024 and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.


{{Primes in edo|106|columns=24}}

Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}Artists using 106 et:

*[[Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
*[http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:Equal divisions of the octave]]
[[Category:53edo]]
[[Category:Polychromatic]]</text>
      <sha1>cl3sfnbnkk187cxlfitnhr41mv1fb35</sha1>
    </revision>
    <revision>
      <id>85649</id>
      <parentid>84828</parentid>
      <timestamp>2022-02-11T10:02:51Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Improve readability</comment>
      <origin>85649</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3221" sha1="e62cv6wt2ac1ivwg7eat1d0tplceatu" xml:space="preserve">The '''106 equal division''' divides the octave into 106 equal parts of 11.321 cents each. Since 106 = 2*53 it is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], 1600000/1594323, 2109375/2097152 in the [[5-limit]], 3125/3097, [[225/224]], 4000/3969, 1728/1715, 2430/2401, [[4375/4374]] in the 7-limit) as the patent val for 53edo. In the 11-limit it also tempers out [[243/242]], 3025/3024 and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

== Scales ==
Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] -- [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:Equal divisions of the octave]]
[[Category:53edo]]
[[Category:Polychromatic]]</text>
      <sha1>e62cv6wt2ac1ivwg7eat1d0tplceatu</sha1>
    </revision>
    <revision>
      <id>85650</id>
      <parentid>85649</parentid>
      <timestamp>2022-02-11T10:06:52Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Links and corrections</comment>
      <origin>85650</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3293" sha1="h4svoa81d46lw2simb01n16fh1eijvc" xml:space="preserve">The '''106 equal division of the octave''' ('''106edo''') divides the [[octave]] into 106 [[equal]] parts of 11.321 [[cent]]s each. 

Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], 2109375/2097152 in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

== Scales ==
Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:Equal divisions of the octave]]
[[Category:53edo]]
[[Category:Polychromatic]]</text>
      <sha1>h4svoa81d46lw2simb01n16fh1eijvc</sha1>
    </revision>
    <revision>
      <id>92916</id>
      <parentid>85650</parentid>
      <timestamp>2022-07-02T16:18:34Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92916</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3321" sha1="2e7lhbtpbx1t06dgefgou15fjrnq4jg" xml:space="preserve">The '''106 equal division of the octave''' ('''106edo''') divides the [[octave]] into 106 [[equal]] parts of 11.321 [[cent]]s each. 

Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], 2109375/2097152 in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

== Scales ==
Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>2e7lhbtpbx1t06dgefgou15fjrnq4jg</sha1>
    </revision>
    <revision>
      <id>95135</id>
      <parentid>92916</parentid>
      <timestamp>2022-08-29T20:19:15Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+infobox, sectioning</comment>
      <origin>95135</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3403" sha1="2tmzhpj1gbkp1nhg10k6n9zupsmx5iw" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 53
| Step size = 12.12121¢
| Fifth = 62\106 (701.89¢) (→[[53edo|31\53]])
| Semitones = 10:8 (113.21¢ : 90.57¢)
| Consistency = 5
}}
{{EDO intro|106}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

== Scales ==
Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>2tmzhpj1gbkp1nhg10k6n9zupsmx5iw</sha1>
    </revision>
    <revision>
      <id>95136</id>
      <parentid>95135</parentid>
      <timestamp>2022-08-29T20:19:53Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Correction</comment>
      <origin>95136</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3403" sha1="jw83vxe0tft6og03b7fschblxlp8x9w" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 53
| Step size = 11.32075¢
| Fifth = 62\106 (701.89¢) (→[[53edo|31\53]])
| Semitones = 10:8 (113.21¢ : 90.57¢)
| Consistency = 5
}}
{{EDO intro|106}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]], if you care about such a thing.

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

== Scales ==
Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>jw83vxe0tft6og03b7fschblxlp8x9w</sha1>
    </revision>
    <revision>
      <id>95335</id>
      <parentid>95136</parentid>
      <timestamp>2022-09-11T06:46:58Z</timestamp>
      <contributor>
        <username>Stavats</username>
        <id>339</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ Remove irrelevant text</comment>
      <origin>95335</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3371" sha1="orvu3gjcc8swdj3wh2kvmlqjycft6gg" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 53
| Step size = 11.32075¢
| Fifth = 62\106 (701.89¢) (→[[53edo|31\53]])
| Semitones = 10:8 (113.21¢ : 90.57¢)
| Consistency = 5
}}
{{EDO intro|106}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]].

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

== Scales ==
Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>orvu3gjcc8swdj3wh2kvmlqjycft6gg</sha1>
    </revision>
    <revision>
      <id>96081</id>
      <parentid>95335</parentid>
      <timestamp>2022-10-04T16:01:14Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET now computes most parameters automatically</comment>
      <origin>96081</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3206" sha1="sqv3yu7i4uqg4uow5sqywhb7smjylxh" xml:space="preserve">{{Infobox ET}}
{{EDO intro|106}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]].

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

== Scales ==
Since 106edo has a step of 11.321 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. 
{| class="wikitable"
|+Circulating temperaments in 106edo
!Tones
!Pattern
!L:s
|-
|5
|[[1L 4s]]
|22:21
|-
|6
|[[4L 2s]]
|18:17
|-
|7
|[[1L 6s]]
|16:15
|-
|8
|[[2L 6s]]
| 14:13
|-
|9
|[[7L 2s]]
|12:11
|-
|10
|[[6L 4s]]
| 11:10
|-
|11
|[[7L 4s]]
|10:9
|-
|12
|[[10L 2s]]
| rowspan="2" |9:8
|-
|13
|[[2L 11s]]
|-
|14
|[[8L 6s]]
| rowspan="2" |8:7
|-
|15
|[[1L 14s]]
|-
|16
|[[10L 6s]]
| rowspan="2" |7:6
|-
|17
|4L 13s
|-
|18
|16L 2s
| rowspan="4" |6:5
|-
|19
|[[11L 8s]]
|-
|20
|6L 14s
|-
|21
|1L 20s
|-
|22
|18L 4s
| rowspan="5" |5:4
|-
|23
|14L 9s
|-
|24
|10L 14s
|-
|25
|[[6L 19s]]
|-
|26
|2L 24s
|-
|27
|25L 2s
| rowspan="9" |4:3
|-
|28
|22L 6s
|-
|29
|19L 10s
|-
|30
|16L 14s
|-
|31
|13L 18s
|-
|32
|10L 22s
|-
|33
|7L 26s
|-
|34
|4L 30s
|-
|35
|1L 34s
|-
|36
|34L 2s
| rowspan="17" |3:2
|-
|37
|32L 5s
|-
|38
|30L 8s
|-
|39
|28L 11s
|-
|40
|26L 14s
|-
|41
|24L 17s
|-
|42
|22L 20s
|-
|43
|20L 23s
|-
|44
|18L 26s
|-
|45
|16L 29s
|-
|46
|14L 32s
|-
|47
|12L 35s
|-
|48
|10L 38s
|-
|49
|8L 41s
|-
|50
|6L 44s
|-
|51
|4L 47s
|-
|52
|2L 50s
|-
|53
|[[53edo]]
|equal
|-
|54
|52L 2s
| rowspan="31" |2:1
|-
|55
|51L 4s
|-
|56
|50L 6s
|-
|57
|49L 8s
|-
|58
|48L 10s
|-
|59
|47L 12s
|-
|60
|46L 14s
|-
|61
|45L 16s
|-
|62
|44L 18s
|-
|63
|43L 20s
|-
|64
|42L 22s
|-
|65
|41L 24s
|-
|66
|40L 26s
|-
|67
|39L 28s
|-
|68
|38L 30s
|-
|69
|37L 32s
|-
|70
|36L 34s
|-
|71
|35L 36s
|-
|72
|34L 38s
|-
|73
|33L 40s
|-
|74
|32L 42s
|-
|75
|31L 44s
|-
|76
|30L 46s
|-
|77
|29L 48s
|-
|78
|28L 50s
|-
|79
|27L 52s
|-
|80
|26L 54s
|-
|81
|25L 56s
|-
|82
|24L 58s
|-
|83
|23L 60s
|-
|84
|22L 62s
|}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>sqv3yu7i4uqg4uow5sqywhb7smjylxh</sha1>
    </revision>
    <revision>
      <id>115191</id>
      <parentid>96081</parentid>
      <timestamp>2023-05-30T21:13:31Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Scales */</comment>
      <origin>115191</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1488" sha1="6ez8rgr8mq3damnlfht11ej94xr8rs4" xml:space="preserve">{{Infobox ET}}
{{EDO intro|106}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]].

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>6ez8rgr8mq3damnlfht11ej94xr8rs4</sha1>
    </revision>
    <revision>
      <id>132172</id>
      <parentid>115191</parentid>
      <timestamp>2024-01-20T00:55:27Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added interval table</comment>
      <origin>132172</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1524" sha1="bfmkxq33nmg7vwrdz45mkgrg71x7zip" xml:space="preserve">{{Infobox ET}}
{{EDO intro|106}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]].

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

== Intervals ==
{{Interval table}}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>bfmkxq33nmg7vwrdz45mkgrg71x7zip</sha1>
    </revision>
    <revision>
      <id>132896</id>
      <parentid>132172</parentid>
      <timestamp>2024-01-25T09:41:57Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>/* Prime harmonics */ Added collapsed 53edo table for comparison</comment>
      <origin>132896</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1596" sha1="33lo4x0ds3x2tdffid4wdh0fich1cjy" xml:space="preserve">{{Infobox ET}}
{{EDO intro|106}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]].

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

53edo for comparison:
{{Harmonics in equal|53|collapsed=1|columns=16}}

== Intervals ==
{{Interval table}}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>33lo4x0ds3x2tdffid4wdh0fich1cjy</sha1>
    </revision>
    <revision>
      <id>181310</id>
      <parentid>132896</parentid>
      <timestamp>2025-02-18T16:47:39Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181310</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1591" sha1="nxpz643qpad44ngg75ayd1usgq975lf" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]].

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

53edo for comparison:
{{Harmonics in equal|53|collapsed=1|columns=16}}

== Intervals ==
{{Interval table}}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>nxpz643qpad44ngg75ayd1usgq975lf</sha1>
    </revision>
    <revision>
      <id>191035</id>
      <parentid>181310</parentid>
      <timestamp>2025-04-11T07:41:06Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ (''See [[regular temperament]] for more about what all this means and how to use it.'')</comment>
      <origin>191035</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1679" sha1="7wa0wng9w69d9taz11g184t73fv2tp9" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament. (''See [[regular temperament]] for more about what all this means and how to use it.'')

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]].

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

53edo for comparison:
{{Harmonics in equal|53|collapsed=1|columns=16}}

== Intervals ==
{{Interval table}}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>7wa0wng9w69d9taz11g184t73fv2tp9</sha1>
    </revision>
    <revision>
      <id>191150</id>
      <parentid>191035</parentid>
      <timestamp>2025-04-11T11:16:50Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>No need to remind readers of what a regular temperament is everywhere</comment>
      <origin>191150</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1591" sha1="nxpz643qpad44ngg75ayd1usgq975lf" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]].

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

53edo for comparison:
{{Harmonics in equal|53|collapsed=1|columns=16}}

== Intervals ==
{{Interval table}}

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>nxpz643qpad44ngg75ayd1usgq975lf</sha1>
    </revision>
    <revision>
      <id>205055</id>
      <parentid>191150</parentid>
      <timestamp>2025-07-24T05:26:59Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <minor/>
      <comment>Add lumatone mapping link.</comment>
      <origin>205055</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1644" sha1="icj4uo92ta2i73auck1n9rrog5jyd3m" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament.

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]].

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

53edo for comparison:
{{Harmonics in equal|53|collapsed=1|columns=16}}

== Intervals ==
{{Interval table}}

== Instruments ==
* [[Lumatone mapping for 106edo]]

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>icj4uo92ta2i73auck1n9rrog5jyd3m</sha1>
    </revision>
    <revision>
      <id>217262</id>
      <parentid>205055</parentid>
      <timestamp>2025-11-18T00:53:08Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>Note about inconsistency</comment>
      <origin>217262</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1864" sha1="pkurbnus5w6q4zkzh22xkrbi7e8fhc8" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament. Unfortunately, it is now only consistent to the [[5-odd-limit]] due to 7/5 being closer to 51 steps rather than its patent val mapping of 52 steps. A superset of 53edo with much a higher consistency limit is [[159edo]].

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo. Conversely, it makes the Pythagorean [[relative cent]] (or pion, symbol π&lt;sup&gt;¢&lt;/sup&gt;, π&lt;sup&gt;r¢&lt;/sup&gt;), which most closely approximates equally dividing an exact [[3/2]].

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

53edo for comparison:
{{Harmonics in equal|53|collapsed=1|columns=16}}

== Intervals ==
{{Interval table}}

== Instruments ==
* [[Lumatone mapping for 106edo]]

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>pkurbnus5w6q4zkzh22xkrbi7e8fhc8</sha1>
    </revision>
    <revision>
      <id>217293</id>
      <parentid>217262</parentid>
      <timestamp>2025-11-18T22:41:49Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Theory */ Removed confusing MMTM sentence from years ago</comment>
      <origin>217293</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1689" sha1="5z3yq5rmjt3xmmwehpfsk1fy8y0c84g" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 106 = 2 × 53, 106edo is closely related to [[53edo]], and is [[contorted]] through the [[7-limit]], tempering out the same commas ([[32805/32768]], [[15625/15552]], [[1600000/1594323]], [[2109375/2097152]] in the [[5-limit]], 3125/3087, [[225/224]], 4000/3969, [[1728/1715]], [[2430/2401]], [[4375/4374]] in the 7-limit) as the [[patent val]] for 53edo. In the 11-limit it also tempers out [[243/242]], [[3025/3024]] and [[9801/9800]], so that it [[support]]s [[spectacle]] temperament and [[borwell]] temperament. Unfortunately, it is now only consistent to the [[5-odd-limit]] due to 7/5 being closer to 51 steps rather than its patent val mapping of 52 steps. A superset of 53edo with much a higher consistency limit is [[159edo]].

The division is notable for the fact that it is related to the [[turkish cent]], or türk sent, which divides 106edo into 100 parts just as ordinary cents divides 12edo into 100 parts, thereby making it the [[relative cent]] division for 106edo.

=== Prime harmonics ===
{{Harmonics in equal|106|columns=16}}

53edo for comparison:
{{Harmonics in equal|53|collapsed=1|columns=16}}

== Intervals ==
{{Interval table}}

== Instruments ==
* [[Lumatone mapping for 106edo]]

== See also ==
Artists using 106 et:
* [[Dolores Catherino]] – [http://dolorescatherino.com her website], [https://www.youtube.com/user/dolomuse YouTube profile: dolomuse]
* [http://chrisvaisvil.com/still-life-in-106-notes-per-octave/ Still Life in 106 Notes Per Octave « Music &amp;amp; Techniques by Chris Vaisvil]

[[Category:53edo]]
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Polychromatic]]</text>
      <sha1>5z3yq5rmjt3xmmwehpfsk1fy8y0c84g</sha1>
    </revision>
  </page>
  <page>
    <title>108/77</title>
    <ns>0</ns>
    <id>25</id>
    <revision>
      <id>25</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>25</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="149" sha1="aes56di86s0k4flyzl4g3rjiy9nki0x" xml:space="preserve">'''108/77'''
|2 3 0 -1 -1&amp;gt;

585.721154 cents

'''108/77''' is 540/539 (3.2 cents) sharp of [[7/5|7/5]], and is the interval between 11/9 and 12/7.</text>
      <sha1>aes56di86s0k4flyzl4g3rjiy9nki0x</sha1>
    </revision>
    <revision>
      <id>4082</id>
      <timestamp>2014-07-01T14:14:40Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 515366180 - Original comment: **</comment>
      <origin>4082</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1660" sha1="d88no7mz0v5ijlepz7ppmlgjwabsggr" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2014-07-01 14:14:40 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;515366180&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**108/77**
|2 3 0 -1 -1&amp;gt;
585.721154 cents

**108/77** is 540/539 (3.2 cents) sharp of [[7_5|7/5]], and is the interval between 11/9 and 12/7.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;108_77&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;108/77&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|2 3 0 -1 -1&amp;amp;gt;&amp;lt;br /&amp;gt;
585.721154 cents&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;108/77&amp;lt;/strong&amp;gt; is 540/539 (3.2 cents) sharp of &amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5&amp;lt;/a&amp;gt;, and is the interval between 11/9 and 12/7.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>d88no7mz0v5ijlepz7ppmlgjwabsggr</sha1>
    </revision>
    <revision>
      <id>38179</id>
      <parentid>25</parentid>
      <timestamp>2018-10-29T17:36:56Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>added interval infobox and categories</comment>
      <origin>38179</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="503" sha1="3114tsv33wzyy5or9vp3yl913km381g" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 108/77
| Monzo = 2 3 0 -1 -1
| Cents = 585.72115
| Name = swetismic augmented fourth
| Color name = 1ur4, luru 4th
| Sound = 
}}

'''108/77''' is 540/539 (3.2 cents) sharp of [[7/5]], and is the interval between [[11/9]] and [[12/7]]. Its [[inverse interval]] is [[77/56]].

:''See also [[Gallery of just intervals]]''

[[Category:11-limit]]
[[Category:Interval]]
[[Category:Ratio]]
[[Category:Tritone]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>3114tsv33wzyy5or9vp3yl913km381g</sha1>
    </revision>
    <revision>
      <id>46316</id>
      <parentid>38179</parentid>
      <timestamp>2020-06-10T07:11:57Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cats</comment>
      <origin>46316</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="490" sha1="mic1r6poaz9z0lpbqk16y4fhv4a25r3" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 108/77
| Monzo = 2 3 0 -1 -1
| Cents = 585.72115
| Name = swetismic augmented fourth
| Color name = 1ur4, luru 4th
| Sound = 
}}

'''108/77''' is 540/539 (3.2 cents) sharp of [[7/5]], and is the interval between [[11/9]] and [[12/7]]. Its [[inverse interval]] is [[77/56]].

:''See also [[Gallery of just intervals]]''

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Tritone]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>mic1r6poaz9z0lpbqk16y4fhv4a25r3</sha1>
    </revision>
    <revision>
      <id>46878</id>
      <parentid>46316</parentid>
      <timestamp>2020-06-15T19:01:34Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>added sound</comment>
      <origin>46878</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="543" sha1="20slt958ubpa69yj6yn2yvm6qcwcthu" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 108/77
| Monzo = 2 3 0 -1 -1
| Cents = 585.72115
| Name = swetismic augmented fourth
| Color name = 1ur4, luru 4th
| Sound = Ji-108-77-csound-foscil-220hz.mp3
}}

'''108/77''' is 540/539 (3.2 cents) sharp of [[7/5]], and is the interval between [[11/9]] and [[12/7]]. Its [[inverse interval]] is [[77/56]].

:''See also [[Gallery of just intervals]]''

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Tritone]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>20slt958ubpa69yj6yn2yvm6qcwcthu</sha1>
    </revision>
    <revision>
      <id>49695</id>
      <parentid>46878</parentid>
      <timestamp>2020-09-16T21:13:15Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>+cat</comment>
      <origin>49695</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="567" sha1="1akrrz5sb8mt28cglsr56motvgtmbyy" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 108/77
| Monzo = 2 3 0 -1 -1
| Cents = 585.72115
| Name = swetismic augmented fourth
| Color name = 1ur4, luru 4th
| Sound = Ji-108-77-csound-foscil-220hz.mp3
}}

'''108/77''' is 540/539 (3.2 cents) sharp of [[7/5]], and is the interval between [[11/9]] and [[12/7]]. Its [[inverse interval]] is [[77/56]].

:''See also [[Gallery of just intervals]]''

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Tritone]]
[[Category:Listen]]
[[Category:Sweetismic]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>1akrrz5sb8mt28cglsr56motvgtmbyy</sha1>
    </revision>
    <revision>
      <id>57613</id>
      <parentid>49695</parentid>
      <timestamp>2020-12-26T14:23:56Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+FJS name; styling; correction</comment>
      <origin>57613</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="643" sha1="pythn2o18r19p7u7na88o5wlyqaprah" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 108/77
| Monzo = 2 3 0 -1 -1
| Cents = 585.72115
| Name = swetismic augmented fourth
| Color name = 1ur4, luru 4th
| FJS name = A4&lt;sub&gt;77&lt;/sub&gt;
| Sound = Ji-108-77-csound-foscil-220hz.mp3
}}

'''108/77''' is 540/539 (3.2 cents) sharp of [[7/5]], and is the interval between [[11/9]] and [[12/7]]. 

== See also ==
* [[77/54]] – its [[octave complement]]
* [[77/72]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Tritone]]
[[Category:Listen]]
[[Category:Swetismic]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>pythn2o18r19p7u7na88o5wlyqaprah</sha1>
    </revision>
    <revision>
      <id>64434</id>
      <parentid>57613</parentid>
      <timestamp>2021-03-27T03:59:03Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64434</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="691" sha1="751m49ponyl864kzt7d8x2ox6i9mpjx" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 108/77
| Monzo = 2 3 0 -1 -1
| Cents = 585.72115
| Name = swetismic augmented fourth
| Color name = 1ur4, luru 4th
| FJS name = A4&lt;sub&gt;77&lt;/sub&gt;
| Sound = Ji-108-77-csound-foscil-220hz.mp3
}}

'''108/77''' is 540/539 (3.2 cents) sharp of [[7/5]], and is the interval between [[11/9]] and [[12/7]]. 

== See also ==
* [[77/54]] – its [[octave complement]]
* [[77/72]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Tritone]]
[[Category:Listen]]
[[Category:Swetismic]]
[[Category:Pages with internal sound examples]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>751m49ponyl864kzt7d8x2ox6i9mpjx</sha1>
    </revision>
    <revision>
      <id>86101</id>
      <parentid>64434</parentid>
      <timestamp>2022-02-17T20:53:09Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Misc. edits, categories</comment>
      <origin>86101</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="610" sha1="0u9ejetqv28wexocb0uzjffokncrukg" xml:space="preserve">{{Infobox Interval
| Ratio = 108/77
| Monzo = 2 3 0 -1 -1
| Cents = 585.72115
| Name = swetismic augmented fourth
| Color name = 1ur4, luru 4th
| FJS name = A4&lt;sub&gt;77&lt;/sub&gt;
| Sound = Ji-108-77-csound-foscil-220hz.mp3
}}

'''108/77''' is [[540/539]] (3.2 cents) sharp of [[7/5]], and is the interval between [[11/9]] and [[12/7]]. 

== See also ==
* [[77/54]] – its [[octave complement]]
* [[77/72]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Tritone]]
[[Category:Swetismic]]
[[Category:Pages with internal sound examples]]

{{todo|expand|improve synopsis}}</text>
      <sha1>0u9ejetqv28wexocb0uzjffokncrukg</sha1>
    </revision>
    <revision>
      <id>86116</id>
      <parentid>86101</parentid>
      <timestamp>2022-02-18T12:19:07Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Interval name</comment>
      <origin>86116</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="839" sha1="iwdicfn3mkd64x136uc5fvh7c9chogr" xml:space="preserve">{{Infobox Interval
| Ratio = 108/77
| Monzo = 2 3 0 -1 -1
| Cents = 585.72115
| Name = symbiotic diminished fifth
| Color name = 1ur4, luru 4th
| FJS name = A4&lt;sub&gt;77&lt;/sub&gt;
| Sound = Ji-108-77-csound-foscil-220hz.mp3
}}

'''108/77''' is the 11-limit interval between [[11/9]] and [[12/7]]. It is [[19712/19683]] (symbiotic comma, 2.5 [[¢]]) sharp of the Pythagorean diminished fifth [[1024/729]]. Meanwhile, it is [[540/539]] (swetisma, 3.2 ¢) sharp of [[7/5]]. 

== Terminology ==
This interval could also be called the ''swetismic augmented fourth'' if viewed as fudging from 7/5 instead. 

== See also ==
* [[77/54]] – its [[octave complement]]
* [[77/72]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Tritone]]
[[Category:Symbiotic]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>iwdicfn3mkd64x136uc5fvh7c9chogr</sha1>
    </revision>
    <revision>
      <id>86281</id>
      <parentid>86116</parentid>
      <timestamp>2022-02-22T23:46:38Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86281</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="791" sha1="pshvjf9ms4iq7dxtgsygsp39flgl2bp" xml:space="preserve">{{Infobox Interval
| Ratio = 108/77
| Monzo = 2 3 0 -1 -1
| Cents = 585.72115
| Name = symbiotic diminished fifth
| Color name = 1ur4, luru 4th
| FJS name = A4&lt;sub&gt;77&lt;/sub&gt;
| Sound = Ji-108-77-csound-foscil-220hz.mp3
}}

'''108/77''' is the 11-limit interval between [[11/9]] and [[12/7]]. It is [[19712/19683]] (symbiotic comma, 2.5 [[¢]]) sharp of the Pythagorean diminished fifth [[1024/729]]. Meanwhile, it is [[540/539]] (swetisma, 3.2 ¢) sharp of [[7/5]]. 

== Terminology ==
This interval could also be called the ''swetismic augmented fourth'' if viewed as fudging from 7/5 instead. 

== See also ==
* [[77/54]] – its [[octave complement]]
* [[77/72]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Tritone]]
[[Category:Symbiotic]]</text>
      <sha1>pshvjf9ms4iq7dxtgsygsp39flgl2bp</sha1>
    </revision>
    <revision>
      <id>97947</id>
      <parentid>86281</parentid>
      <timestamp>2022-10-25T11:32:53Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97947</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="681" sha1="h5344d3xw8zc5jv8f4he0aa5wrldp9q" xml:space="preserve">{{Infobox Interval
| Name = symbiotic diminished fifth
| Color name = 1ur4, luru 4th
| Sound = Ji-108-77-csound-foscil-220hz.mp3
}}

'''108/77''' is the 11-limit interval between [[11/9]] and [[12/7]]. It is [[19712/19683]] (symbiotic comma, 2.5 [[¢]]) sharp of the Pythagorean diminished fifth [[1024/729]]. Meanwhile, it is [[540/539]] (swetisma, 3.2 ¢) sharp of [[7/5]]. 

== Terminology ==
This interval could also be called the ''swetismic augmented fourth'' if viewed as fudging from 7/5 instead. 

== See also ==
* [[77/54]] – its [[octave complement]]
* [[77/72]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Tritone]]
[[Category:Symbiotic]]</text>
      <sha1>h5344d3xw8zc5jv8f4he0aa5wrldp9q</sha1>
    </revision>
  </page>
  <page>
    <title>108edo</title>
    <ns>0</ns>
    <id>27</id>
    <revision>
      <id>27</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>27</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="657" sha1="bhcesspbmrj3xtw2qetdsup74jzl70g" xml:space="preserve">The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.
[[Category:valentine]]</text>
      <sha1>bhcesspbmrj3xtw2qetdsup74jzl70g</sha1>
    </revision>
    <revision>
      <id>4083</id>
      <timestamp>2011-07-06T15:45:47Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 240246981 - Original comment: **</comment>
      <origin>4083</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2557" sha1="daaoshnpp502pecdc0akt0z0f99iqby" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-07-06 15:45:47 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;240246981&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //108 equal division// divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;108edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;108 equal division&amp;lt;/em&amp;gt; divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting &amp;lt;a class="wiki_link" href="/Starling%20temperaments#Valentine temperament"&amp;gt;valentine temperament&amp;lt;/a&amp;gt;, and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>daaoshnpp502pecdc0akt0z0f99iqby</sha1>
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    <revision>
      <id>40644</id>
      <parentid>27</parentid>
      <timestamp>2019-03-05T18:51:45Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>40644</origin>
      <model>wikitext</model>
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      <text bytes="961" sha1="24uwndvoq3objui1vil86fujdo4y1k6" xml:space="preserve">The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p \sup p \cdot q \sup q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.
[[Category:valentine]]</text>
      <sha1>24uwndvoq3objui1vil86fujdo4y1k6</sha1>
    </revision>
    <revision>
      <id>40645</id>
      <parentid>40644</parentid>
      <timestamp>2019-03-05T18:59:04Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>40645</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="961" sha1="a9nyfkjbchxfpf3kbx9pej3g768l0v0" xml:space="preserve">The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p \pow p \cdot q \pow q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.
[[Category:valentine]]</text>
      <sha1>a9nyfkjbchxfpf3kbx9pej3g768l0v0</sha1>
    </revision>
    <revision>
      <id>40646</id>
      <parentid>40645</parentid>
      <timestamp>2019-03-05T18:59:29Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>40646</origin>
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      <text bytes="961" sha1="m2hfkm2bn8u0wcd6dn9dhgfdk6vhrkd" xml:space="preserve">The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p \exp p \cdot q \exp q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.
[[Category:valentine]]</text>
      <sha1>m2hfkm2bn8u0wcd6dn9dhgfdk6vhrkd</sha1>
    </revision>
    <revision>
      <id>40647</id>
      <parentid>40646</parentid>
      <timestamp>2019-03-05T19:03:12Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>40647</origin>
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      <text bytes="951" sha1="ho72fzcxwunus7632tj9z2weobu0sss" xml:space="preserve">The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p^p \cdot q^q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.
[[Category:valentine]]</text>
      <sha1>ho72fzcxwunus7632tj9z2weobu0sss</sha1>
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    <revision>
      <id>68067</id>
      <parentid>40647</parentid>
      <timestamp>2021-04-21T01:27:14Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68067</origin>
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      <text bytes="2707" sha1="krtxmbby6j8xbhfpybxg50vu2d1ex26" xml:space="preserve">The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p^p \cdot q^q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.


Since 108edo has a step of 11.111 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 108edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|22:21
|-
|6
|[[6edo]]
|equal
|-
|7
|[[3L 4s]]
|16:15
|-
|8
|[[4L 4s]]
|14:13
|-
|9
|[[9edo]]
|equal
|-
|10
|[[8L 2s]]
|11:10
|-
|11
|[[9L 2s]]
|10:9
|-
|12
|[[12edo]]
|equal
|-
|13
|[[3L 10s]]
|9:8
|-
|14
|[[10L 4s]]
| rowspan="2" |8:7
|-
|15
|[[3L 12s]]
|-
|16
|12L 4s
| rowspan="2" |7:6
|-
|17
|[[6L 11s]]
|-
|18
|[[18edo]]
|equal
|-
|19
|[[13L 6s]]
| rowspan="3" |6:5
|-
|20
|8L 12s
|-
|21
|3L 18s
|-
|22
|20L 2s
| rowspan="5" |5:4
|-
|23
|16L 7s
|-
|24
|12L 12s
|-
|25
|8L 17s
|-
|26
|4L 22s
|-
|27
|[[27edo]]
|equal
|-
|28
|24L 4s
| rowspan="8" |4:3
|-
|29
|21L 8s
|-
|30
|18L 12s
|-
|31
|15L 16s
|-
|32
|12L 20s
|-
|33
|9L 24s
|-
|34
|6L 28s
|-
|35
|3L 32s
|-
|36
|[[36edo]]
|equal
|-
|37
|34L 3s
| rowspan="17" |3:2
|-
|38
|32L 6s
|-
|39
|30L 9s
|-
|40
|28L 12s
|-
|41
|26L 15s
|-
|42
|24L 18s
|-
|43
|22L 21s
|-
|44
|20L 24s
|-
|45
|18L 27s
|-
|46
|16L 30s
|-
|47
|14L 33s
|-
|48
|12L 36s
|-
|49
|10L 39s
|-
|50
|8L 42s
|-
|51
|6L 45s
|-
|52
|4L 48s
|-
|53
|2L 51s
|-
|54
|[[54edo]]
|equal
|-
|55
|53L 2s
| rowspan="33" |2:1
|-
|56
|52L 4s
|-
|57
|51L 6s
|-
|58
|50L 8s
|-
|59
|49L 10s
|-
|60
|48L 12s
|-
|61
|47L 14s
|-
|62
|46L 16s
|-
|63
|45L 18s
|-
|64
|44L 20s
|-
|65
|43L 22s
|-
|66
|42L 24s
|-
|67
|41L 26s
|-
|68
|40L 28s
|-
|69
|39L 30s
|-
|70
|38L 32s
|-
|71
|37L 34s
|-
|72
|36L 36s
|-
|73
|35L 38s
|-
|74
|34L 40s
|-
|75
|33L 42s
|-
|76
|32L 44s
|-
|77
|31L 46s
|-
|78
|30L 48s
|-
|79
|29L 50s
|-
|80
|28L 52s
|-
|81
|27L 54s
|-
|82
|26L 56s
|-
|83
|25L 58s
|-
|84
|24L 60s
|-
|85
|23L 62s
|-
|86
|22L 64s
|-
|87
|21L 66s
|}
[[Category:valentine]]</text>
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    </revision>
    <revision>
      <id>68078</id>
      <parentid>68067</parentid>
      <timestamp>2021-04-21T02:55:02Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68078</origin>
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      <text bytes="2691" sha1="ahi9i1nrwtuslk5b9j5kj0g4hki7vgs" xml:space="preserve">The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p^p \cdot q^q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.


Since 108edo has a step of 11.111 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 108edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|22:21
|-
|6
|[[6edo]]
|equal
|-
|7
|[[3L 4s]]
|16:15
|-
|8
|[[4L 4s]]
|14:13
|-
|9
|[[9edo]]
|equal
|-
|10
|[[8L 2s]]
|11:10
|-
|11
|[[9L 2s]]
|10:9
|-
|12
|[[12edo]]
|equal
|-
|13
|[[3L 10s]]
|9:8
|-
|14
|[[10L 4s]]
| rowspan="2" |8:7
|-
|15
|[[3L 12s]]
|-
|16
|12L 4s
| rowspan="2" |7:6
|-
|17
|[[6L 11s]]
|-
|18
|[[18edo]]
|equal
|-
|19
|[[13L 6s]]
| rowspan="3" |6:5
|-
|20
|8L 12s
|-
|21
|3L 18s
|-
|22
|20L 2s
| rowspan="5" |5:4
|-
|23
|16L 7s
|-
|24
|12L 12s
|-
|25
|8L 17s
|-
|26
|4L 22s
|-
|27
|[[27edo]]
|equal
|-
|28
|24L 4s
| rowspan="8" |4:3
|-
|29
|21L 8s
|-
|30
|18L 12s
|-
|31
|15L 16s
|-
|32
|12L 20s
|-
|33
|9L 24s
|-
|34
|6L 28s
|-
|35
|3L 32s
|-
|36
|[[36edo]]
|equal
|-
|37
|34L 3s
| rowspan="17" |3:2
|-
|38
|32L 6s
|-
|39
|30L 9s
|-
|40
|28L 12s
|-
|41
|26L 15s
|-
|42
|24L 18s
|-
|43
|22L 21s
|-
|44
|20L 24s
|-
|45
|18L 27s
|-
|46
|16L 30s
|-
|47
|14L 33s
|-
|48
|12L 36s
|-
|49
|10L 39s
|-
|50
|8L 42s
|-
|51
|6L 45s
|-
|52
|4L 48s
|-
|53
|2L 51s
|-
|54
|[[54edo]]
|equal
|-
|55
|53L 2s
| rowspan="32" |2:1
|-
|56
|52L 4s
|-
|57
|51L 6s
|-
|58
|50L 8s
|-
|59
|49L 10s
|-
|60
|48L 12s
|-
|61
|47L 14s
|-
|62
|46L 16s
|-
|63
|45L 18s
|-
|64
|44L 20s
|-
|65
|43L 22s
|-
|66
|42L 24s
|-
|67
|41L 26s
|-
|68
|40L 28s
|-
|69
|39L 30s
|-
|70
|38L 32s
|-
|71
|37L 34s
|-
|72
|36L 36s
|-
|73
|35L 38s
|-
|74
|34L 40s
|-
|75
|33L 42s
|-
|76
|32L 44s
|-
|77
|31L 46s
|-
|78
|30L 48s
|-
|79
|29L 50s
|-
|80
|28L 52s
|-
|81
|27L 54s
|-
|82
|26L 56s
|-
|83
|25L 58s
|-
|84
|24L 60s
|-
|85
|23L 62s
|-
|86
|22L 64s
|}
[[Category:valentine]]</text>
      <sha1>ahi9i1nrwtuslk5b9j5kj0g4hki7vgs</sha1>
    </revision>
    <revision>
      <id>92918</id>
      <parentid>68078</parentid>
      <timestamp>2022-07-02T16:19:57Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Categories</comment>
      <origin>92918</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2762" sha1="jcgdte3us49sjypdv8yny5pryil9rlg" xml:space="preserve">The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p^p \cdot q^q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.

Since 108edo has a step of 11.111 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 108edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|22:21
|-
|6
|[[6edo]]
|equal
|-
|7
|[[3L 4s]]
|16:15
|-
|8
|[[4L 4s]]
|14:13
|-
|9
|[[9edo]]
|equal
|-
|10
|[[8L 2s]]
|11:10
|-
|11
|[[9L 2s]]
|10:9
|-
|12
|[[12edo]]
|equal
|-
|13
|[[3L 10s]]
|9:8
|-
|14
|[[10L 4s]]
| rowspan="2" |8:7
|-
|15
|[[3L 12s]]
|-
|16
|12L 4s
| rowspan="2" |7:6
|-
|17
|[[6L 11s]]
|-
|18
|[[18edo]]
|equal
|-
|19
|[[13L 6s]]
| rowspan="3" |6:5
|-
|20
|8L 12s
|-
|21
|3L 18s
|-
|22
|20L 2s
| rowspan="5" |5:4
|-
|23
|16L 7s
|-
|24
|12L 12s
|-
|25
|8L 17s
|-
|26
|4L 22s
|-
|27
|[[27edo]]
|equal
|-
|28
|24L 4s
| rowspan="8" |4:3
|-
|29
|21L 8s
|-
|30
|18L 12s
|-
|31
|15L 16s
|-
|32
|12L 20s
|-
|33
|9L 24s
|-
|34
|6L 28s
|-
|35
|3L 32s
|-
|36
|[[36edo]]
|equal
|-
|37
|34L 3s
| rowspan="17" |3:2
|-
|38
|32L 6s
|-
|39
|30L 9s
|-
|40
|28L 12s
|-
|41
|26L 15s
|-
|42
|24L 18s
|-
|43
|22L 21s
|-
|44
|20L 24s
|-
|45
|18L 27s
|-
|46
|16L 30s
|-
|47
|14L 33s
|-
|48
|12L 36s
|-
|49
|10L 39s
|-
|50
|8L 42s
|-
|51
|6L 45s
|-
|52
|4L 48s
|-
|53
|2L 51s
|-
|54
|[[54edo]]
|equal
|-
|55
|53L 2s
| rowspan="32" |2:1
|-
|56
|52L 4s
|-
|57
|51L 6s
|-
|58
|50L 8s
|-
|59
|49L 10s
|-
|60
|48L 12s
|-
|61
|47L 14s
|-
|62
|46L 16s
|-
|63
|45L 18s
|-
|64
|44L 20s
|-
|65
|43L 22s
|-
|66
|42L 24s
|-
|67
|41L 26s
|-
|68
|40L 28s
|-
|69
|39L 30s
|-
|70
|38L 32s
|-
|71
|37L 34s
|-
|72
|36L 36s
|-
|73
|35L 38s
|-
|74
|34L 40s
|-
|75
|33L 42s
|-
|76
|32L 44s
|-
|77
|31L 46s
|-
|78
|30L 48s
|-
|79
|29L 50s
|-
|80
|28L 52s
|-
|81
|27L 54s
|-
|82
|26L 56s
|-
|83
|25L 58s
|-
|84
|24L 60s
|-
|85
|23L 62s
|-
|86
|22L 64s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Valentine]]</text>
      <sha1>jcgdte3us49sjypdv8yny5pryil9rlg</sha1>
    </revision>
    <revision>
      <id>96189</id>
      <parentid>92918</parentid>
      <timestamp>2022-10-04T18:50:35Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96189</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2777" sha1="0vqi044l3181k0hxnyqulf9eu6vnypq" xml:space="preserve">{{Infobox ET}}
The ''108 equal division'' divides the octave into 108 equal parts of 11.111 cents each. It tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p^p \cdot q^q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.

Since 108edo has a step of 11.111 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 108edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|22:21
|-
|6
|[[6edo]]
|equal
|-
|7
|[[3L 4s]]
|16:15
|-
|8
|[[4L 4s]]
|14:13
|-
|9
|[[9edo]]
|equal
|-
|10
|[[8L 2s]]
|11:10
|-
|11
|[[9L 2s]]
|10:9
|-
|12
|[[12edo]]
|equal
|-
|13
|[[3L 10s]]
|9:8
|-
|14
|[[10L 4s]]
| rowspan="2" |8:7
|-
|15
|[[3L 12s]]
|-
|16
|12L 4s
| rowspan="2" |7:6
|-
|17
|[[6L 11s]]
|-
|18
|[[18edo]]
|equal
|-
|19
|[[13L 6s]]
| rowspan="3" |6:5
|-
|20
|8L 12s
|-
|21
|3L 18s
|-
|22
|20L 2s
| rowspan="5" |5:4
|-
|23
|16L 7s
|-
|24
|12L 12s
|-
|25
|8L 17s
|-
|26
|4L 22s
|-
|27
|[[27edo]]
|equal
|-
|28
|24L 4s
| rowspan="8" |4:3
|-
|29
|21L 8s
|-
|30
|18L 12s
|-
|31
|15L 16s
|-
|32
|12L 20s
|-
|33
|9L 24s
|-
|34
|6L 28s
|-
|35
|3L 32s
|-
|36
|[[36edo]]
|equal
|-
|37
|34L 3s
| rowspan="17" |3:2
|-
|38
|32L 6s
|-
|39
|30L 9s
|-
|40
|28L 12s
|-
|41
|26L 15s
|-
|42
|24L 18s
|-
|43
|22L 21s
|-
|44
|20L 24s
|-
|45
|18L 27s
|-
|46
|16L 30s
|-
|47
|14L 33s
|-
|48
|12L 36s
|-
|49
|10L 39s
|-
|50
|8L 42s
|-
|51
|6L 45s
|-
|52
|4L 48s
|-
|53
|2L 51s
|-
|54
|[[54edo]]
|equal
|-
|55
|53L 2s
| rowspan="32" |2:1
|-
|56
|52L 4s
|-
|57
|51L 6s
|-
|58
|50L 8s
|-
|59
|49L 10s
|-
|60
|48L 12s
|-
|61
|47L 14s
|-
|62
|46L 16s
|-
|63
|45L 18s
|-
|64
|44L 20s
|-
|65
|43L 22s
|-
|66
|42L 24s
|-
|67
|41L 26s
|-
|68
|40L 28s
|-
|69
|39L 30s
|-
|70
|38L 32s
|-
|71
|37L 34s
|-
|72
|36L 36s
|-
|73
|35L 38s
|-
|74
|34L 40s
|-
|75
|33L 42s
|-
|76
|32L 44s
|-
|77
|31L 46s
|-
|78
|30L 48s
|-
|79
|29L 50s
|-
|80
|28L 52s
|-
|81
|27L 54s
|-
|82
|26L 56s
|-
|83
|25L 58s
|-
|84
|24L 60s
|-
|85
|23L 62s
|-
|86
|22L 64s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Valentine]]</text>
      <sha1>0vqi044l3181k0hxnyqulf9eu6vnypq</sha1>
    </revision>
    <revision>
      <id>101100</id>
      <parentid>96189</parentid>
      <timestamp>2023-01-08T12:05:16Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>sectioning</comment>
      <origin>101100</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2739" sha1="kgxvy2xjh508yhr9f3c8it0mr9rvmjz" xml:space="preserve">{{Infobox ET}}
{{EDO intro|180}}
==Theory==
108edo tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.
=== Miscellany ===
Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p^p \cdot q^q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.

Since 108edo has a step of 11.111 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 108edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|22:21
|-
|6
|[[6edo]]
|equal
|-
|7
|[[3L 4s]]
|16:15
|-
|8
|[[4L 4s]]
|14:13
|-
|9
|[[9edo]]
|equal
|-
|10
|[[8L 2s]]
|11:10
|-
|11
|[[9L 2s]]
|10:9
|-
|12
|[[12edo]]
|equal
|-
|13
|[[3L 10s]]
|9:8
|-
|14
|[[10L 4s]]
| rowspan="2" |8:7
|-
|15
|[[3L 12s]]
|-
|16
|12L 4s
| rowspan="2" |7:6
|-
|17
|[[6L 11s]]
|-
|18
|[[18edo]]
|equal
|-
|19
|[[13L 6s]]
| rowspan="3" |6:5
|-
|20
|8L 12s
|-
|21
|3L 18s
|-
|22
|20L 2s
| rowspan="5" |5:4
|-
|23
|16L 7s
|-
|24
|12L 12s
|-
|25
|8L 17s
|-
|26
|4L 22s
|-
|27
|[[27edo]]
|equal
|-
|28
|24L 4s
| rowspan="8" |4:3
|-
|29
|21L 8s
|-
|30
|18L 12s
|-
|31
|15L 16s
|-
|32
|12L 20s
|-
|33
|9L 24s
|-
|34
|6L 28s
|-
|35
|3L 32s
|-
|36
|[[36edo]]
|equal
|-
|37
|34L 3s
| rowspan="17" |3:2
|-
|38
|32L 6s
|-
|39
|30L 9s
|-
|40
|28L 12s
|-
|41
|26L 15s
|-
|42
|24L 18s
|-
|43
|22L 21s
|-
|44
|20L 24s
|-
|45
|18L 27s
|-
|46
|16L 30s
|-
|47
|14L 33s
|-
|48
|12L 36s
|-
|49
|10L 39s
|-
|50
|8L 42s
|-
|51
|6L 45s
|-
|52
|4L 48s
|-
|53
|2L 51s
|-
|54
|[[54edo]]
|equal
|-
|55
|53L 2s
| rowspan="32" |2:1
|-
|56
|52L 4s
|-
|57
|51L 6s
|-
|58
|50L 8s
|-
|59
|49L 10s
|-
|60
|48L 12s
|-
|61
|47L 14s
|-
|62
|46L 16s
|-
|63
|45L 18s
|-
|64
|44L 20s
|-
|65
|43L 22s
|-
|66
|42L 24s
|-
|67
|41L 26s
|-
|68
|40L 28s
|-
|69
|39L 30s
|-
|70
|38L 32s
|-
|71
|37L 34s
|-
|72
|36L 36s
|-
|73
|35L 38s
|-
|74
|34L 40s
|-
|75
|33L 42s
|-
|76
|32L 44s
|-
|77
|31L 46s
|-
|78
|30L 48s
|-
|79
|29L 50s
|-
|80
|28L 52s
|-
|81
|27L 54s
|-
|82
|26L 56s
|-
|83
|25L 58s
|-
|84
|24L 60s
|-
|85
|23L 62s
|-
|86
|22L 64s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Valentine]]</text>
      <sha1>kgxvy2xjh508yhr9f3c8it0mr9rvmjz</sha1>
    </revision>
    <revision>
      <id>111194</id>
      <parentid>101100</parentid>
      <timestamp>2023-05-02T10:59:03Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>The EDO intro mistakenly had 180edo instead of 108, so I corrected this</comment>
      <origin>111194</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2741" sha1="9m88ustrfwzoc95cb1alvcjc1fjomti" xml:space="preserve">{{Infobox ET}}
{{EDO intro|108}}

==Theory==
108edo tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

=== Miscellany ===
Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p^p \cdot q^q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.

Since 108edo has a step of 11.111 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 108edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|22:21
|-
|6
|[[6edo]]
|equal
|-
|7
|[[3L 4s]]
|16:15
|-
|8
|[[4L 4s]]
|14:13
|-
|9
|[[9edo]]
|equal
|-
|10
|[[8L 2s]]
|11:10
|-
|11
|[[9L 2s]]
|10:9
|-
|12
|[[12edo]]
|equal
|-
|13
|[[3L 10s]]
|9:8
|-
|14
|[[10L 4s]]
| rowspan="2" |8:7
|-
|15
|[[3L 12s]]
|-
|16
|12L 4s
| rowspan="2" |7:6
|-
|17
|[[6L 11s]]
|-
|18
|[[18edo]]
|equal
|-
|19
|[[13L 6s]]
| rowspan="3" |6:5
|-
|20
|8L 12s
|-
|21
|3L 18s
|-
|22
|20L 2s
| rowspan="5" |5:4
|-
|23
|16L 7s
|-
|24
|12L 12s
|-
|25
|8L 17s
|-
|26
|4L 22s
|-
|27
|[[27edo]]
|equal
|-
|28
|24L 4s
| rowspan="8" |4:3
|-
|29
|21L 8s
|-
|30
|18L 12s
|-
|31
|15L 16s
|-
|32
|12L 20s
|-
|33
|9L 24s
|-
|34
|6L 28s
|-
|35
|3L 32s
|-
|36
|[[36edo]]
|equal
|-
|37
|34L 3s
| rowspan="17" |3:2
|-
|38
|32L 6s
|-
|39
|30L 9s
|-
|40
|28L 12s
|-
|41
|26L 15s
|-
|42
|24L 18s
|-
|43
|22L 21s
|-
|44
|20L 24s
|-
|45
|18L 27s
|-
|46
|16L 30s
|-
|47
|14L 33s
|-
|48
|12L 36s
|-
|49
|10L 39s
|-
|50
|8L 42s
|-
|51
|6L 45s
|-
|52
|4L 48s
|-
|53
|2L 51s
|-
|54
|[[54edo]]
|equal
|-
|55
|53L 2s
| rowspan="32" |2:1
|-
|56
|52L 4s
|-
|57
|51L 6s
|-
|58
|50L 8s
|-
|59
|49L 10s
|-
|60
|48L 12s
|-
|61
|47L 14s
|-
|62
|46L 16s
|-
|63
|45L 18s
|-
|64
|44L 20s
|-
|65
|43L 22s
|-
|66
|42L 24s
|-
|67
|41L 26s
|-
|68
|40L 28s
|-
|69
|39L 30s
|-
|70
|38L 32s
|-
|71
|37L 34s
|-
|72
|36L 36s
|-
|73
|35L 38s
|-
|74
|34L 40s
|-
|75
|33L 42s
|-
|76
|32L 44s
|-
|77
|31L 46s
|-
|78
|30L 48s
|-
|79
|29L 50s
|-
|80
|28L 52s
|-
|81
|27L 54s
|-
|82
|26L 56s
|-
|83
|25L 58s
|-
|84
|24L 60s
|-
|85
|23L 62s
|-
|86
|22L 64s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Valentine]]</text>
      <sha1>9m88ustrfwzoc95cb1alvcjc1fjomti</sha1>
    </revision>
    <revision>
      <id>111195</id>
      <parentid>111194</parentid>
      <timestamp>2023-05-02T11:00:29Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added harmonics table</comment>
      <origin>111195</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2793" sha1="pg31cqzve62cpynfrgpzt4sjhqsdmd3" xml:space="preserve">{{Infobox ET}}
{{EDO intro|108}}

==Theory==
108edo tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

=== Prime harmonics ===
{{Harmonics in equal|108}}

=== Miscellany ===
Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p^p \cdot q^q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.

Since 108edo has a step of 11.111 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 108edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|22:21
|-
|6
|[[6edo]]
|equal
|-
|7
|[[3L 4s]]
|16:15
|-
|8
|[[4L 4s]]
|14:13
|-
|9
|[[9edo]]
|equal
|-
|10
|[[8L 2s]]
|11:10
|-
|11
|[[9L 2s]]
|10:9
|-
|12
|[[12edo]]
|equal
|-
|13
|[[3L 10s]]
|9:8
|-
|14
|[[10L 4s]]
| rowspan="2" |8:7
|-
|15
|[[3L 12s]]
|-
|16
|12L 4s
| rowspan="2" |7:6
|-
|17
|[[6L 11s]]
|-
|18
|[[18edo]]
|equal
|-
|19
|[[13L 6s]]
| rowspan="3" |6:5
|-
|20
|8L 12s
|-
|21
|3L 18s
|-
|22
|20L 2s
| rowspan="5" |5:4
|-
|23
|16L 7s
|-
|24
|12L 12s
|-
|25
|8L 17s
|-
|26
|4L 22s
|-
|27
|[[27edo]]
|equal
|-
|28
|24L 4s
| rowspan="8" |4:3
|-
|29
|21L 8s
|-
|30
|18L 12s
|-
|31
|15L 16s
|-
|32
|12L 20s
|-
|33
|9L 24s
|-
|34
|6L 28s
|-
|35
|3L 32s
|-
|36
|[[36edo]]
|equal
|-
|37
|34L 3s
| rowspan="17" |3:2
|-
|38
|32L 6s
|-
|39
|30L 9s
|-
|40
|28L 12s
|-
|41
|26L 15s
|-
|42
|24L 18s
|-
|43
|22L 21s
|-
|44
|20L 24s
|-
|45
|18L 27s
|-
|46
|16L 30s
|-
|47
|14L 33s
|-
|48
|12L 36s
|-
|49
|10L 39s
|-
|50
|8L 42s
|-
|51
|6L 45s
|-
|52
|4L 48s
|-
|53
|2L 51s
|-
|54
|[[54edo]]
|equal
|-
|55
|53L 2s
| rowspan="32" |2:1
|-
|56
|52L 4s
|-
|57
|51L 6s
|-
|58
|50L 8s
|-
|59
|49L 10s
|-
|60
|48L 12s
|-
|61
|47L 14s
|-
|62
|46L 16s
|-
|63
|45L 18s
|-
|64
|44L 20s
|-
|65
|43L 22s
|-
|66
|42L 24s
|-
|67
|41L 26s
|-
|68
|40L 28s
|-
|69
|39L 30s
|-
|70
|38L 32s
|-
|71
|37L 34s
|-
|72
|36L 36s
|-
|73
|35L 38s
|-
|74
|34L 40s
|-
|75
|33L 42s
|-
|76
|32L 44s
|-
|77
|31L 46s
|-
|78
|30L 48s
|-
|79
|29L 50s
|-
|80
|28L 52s
|-
|81
|27L 54s
|-
|82
|26L 56s
|-
|83
|25L 58s
|-
|84
|24L 60s
|-
|85
|23L 62s
|-
|86
|22L 64s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Valentine]]</text>
      <sha1>pg31cqzve62cpynfrgpzt4sjhqsdmd3</sha1>
    </revision>
    <revision>
      <id>111331</id>
      <parentid>111195</parentid>
      <timestamp>2023-05-02T21:51:02Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>111331</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2920" sha1="dge8wr5nh7ebhw417gciyc7n6huensx" xml:space="preserve">{{Infobox ET}}
{{EDO intro|108}}

==Theory==
108edo tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12n-edo which offers a decent alternative fifth to 12edo's fifth, which is [[27edo]]'s superpyth fifth.
=== Prime harmonics ===
{{Harmonics in equal|108}}

=== Miscellany ===
Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p^p \cdot q^q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.

Since 108edo has a step of 11.111 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 108edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|22:21
|-
|6
|[[6edo]]
|equal
|-
|7
|[[3L 4s]]
|16:15
|-
|8
|[[4L 4s]]
|14:13
|-
|9
|[[9edo]]
|equal
|-
|10
|[[8L 2s]]
|11:10
|-
|11
|[[9L 2s]]
|10:9
|-
|12
|[[12edo]]
|equal
|-
|13
|[[3L 10s]]
|9:8
|-
|14
|[[10L 4s]]
| rowspan="2" |8:7
|-
|15
|[[3L 12s]]
|-
|16
|12L 4s
| rowspan="2" |7:6
|-
|17
|[[6L 11s]]
|-
|18
|[[18edo]]
|equal
|-
|19
|[[13L 6s]]
| rowspan="3" |6:5
|-
|20
|8L 12s
|-
|21
|3L 18s
|-
|22
|20L 2s
| rowspan="5" |5:4
|-
|23
|16L 7s
|-
|24
|12L 12s
|-
|25
|8L 17s
|-
|26
|4L 22s
|-
|27
|[[27edo]]
|equal
|-
|28
|24L 4s
| rowspan="8" |4:3
|-
|29
|21L 8s
|-
|30
|18L 12s
|-
|31
|15L 16s
|-
|32
|12L 20s
|-
|33
|9L 24s
|-
|34
|6L 28s
|-
|35
|3L 32s
|-
|36
|[[36edo]]
|equal
|-
|37
|34L 3s
| rowspan="17" |3:2
|-
|38
|32L 6s
|-
|39
|30L 9s
|-
|40
|28L 12s
|-
|41
|26L 15s
|-
|42
|24L 18s
|-
|43
|22L 21s
|-
|44
|20L 24s
|-
|45
|18L 27s
|-
|46
|16L 30s
|-
|47
|14L 33s
|-
|48
|12L 36s
|-
|49
|10L 39s
|-
|50
|8L 42s
|-
|51
|6L 45s
|-
|52
|4L 48s
|-
|53
|2L 51s
|-
|54
|[[54edo]]
|equal
|-
|55
|53L 2s
| rowspan="32" |2:1
|-
|56
|52L 4s
|-
|57
|51L 6s
|-
|58
|50L 8s
|-
|59
|49L 10s
|-
|60
|48L 12s
|-
|61
|47L 14s
|-
|62
|46L 16s
|-
|63
|45L 18s
|-
|64
|44L 20s
|-
|65
|43L 22s
|-
|66
|42L 24s
|-
|67
|41L 26s
|-
|68
|40L 28s
|-
|69
|39L 30s
|-
|70
|38L 32s
|-
|71
|37L 34s
|-
|72
|36L 36s
|-
|73
|35L 38s
|-
|74
|34L 40s
|-
|75
|33L 42s
|-
|76
|32L 44s
|-
|77
|31L 46s
|-
|78
|30L 48s
|-
|79
|29L 50s
|-
|80
|28L 52s
|-
|81
|27L 54s
|-
|82
|26L 56s
|-
|83
|25L 58s
|-
|84
|24L 60s
|-
|85
|23L 62s
|-
|86
|22L 64s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Valentine]]</text>
      <sha1>dge8wr5nh7ebhw417gciyc7n6huensx</sha1>
    </revision>
    <revision>
      <id>111332</id>
      <parentid>111331</parentid>
      <timestamp>2023-05-02T21:51:08Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>111332</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2919" sha1="9xsqzl5945ooawxq7t5wx4zg61b2pzo" xml:space="preserve">{{Infobox ET}}
{{EDO intro|108}}

==Theory==
108edo tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12n-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.
=== Prime harmonics ===
{{Harmonics in equal|108}}

=== Miscellany ===
Aside from tuning, 108 is the smallest number with a prime factorization of the form &lt;math&gt;p^p \cdot q^q&lt;/math&gt;. Being close to 100, it is a good substitute for a relative cent if you desire the measure to have such a property. Also, multiplying it by 12 gives the fourth power of an integer.

Since 108edo has a step of 11.111 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 108edo
!Tones
!Pattern
!L:s
|-
|5
|[[3L 2s]]
|22:21
|-
|6
|[[6edo]]
|equal
|-
|7
|[[3L 4s]]
|16:15
|-
|8
|[[4L 4s]]
|14:13
|-
|9
|[[9edo]]
|equal
|-
|10
|[[8L 2s]]
|11:10
|-
|11
|[[9L 2s]]
|10:9
|-
|12
|[[12edo]]
|equal
|-
|13
|[[3L 10s]]
|9:8
|-
|14
|[[10L 4s]]
| rowspan="2" |8:7
|-
|15
|[[3L 12s]]
|-
|16
|12L 4s
| rowspan="2" |7:6
|-
|17
|[[6L 11s]]
|-
|18
|[[18edo]]
|equal
|-
|19
|[[13L 6s]]
| rowspan="3" |6:5
|-
|20
|8L 12s
|-
|21
|3L 18s
|-
|22
|20L 2s
| rowspan="5" |5:4
|-
|23
|16L 7s
|-
|24
|12L 12s
|-
|25
|8L 17s
|-
|26
|4L 22s
|-
|27
|[[27edo]]
|equal
|-
|28
|24L 4s
| rowspan="8" |4:3
|-
|29
|21L 8s
|-
|30
|18L 12s
|-
|31
|15L 16s
|-
|32
|12L 20s
|-
|33
|9L 24s
|-
|34
|6L 28s
|-
|35
|3L 32s
|-
|36
|[[36edo]]
|equal
|-
|37
|34L 3s
| rowspan="17" |3:2
|-
|38
|32L 6s
|-
|39
|30L 9s
|-
|40
|28L 12s
|-
|41
|26L 15s
|-
|42
|24L 18s
|-
|43
|22L 21s
|-
|44
|20L 24s
|-
|45
|18L 27s
|-
|46
|16L 30s
|-
|47
|14L 33s
|-
|48
|12L 36s
|-
|49
|10L 39s
|-
|50
|8L 42s
|-
|51
|6L 45s
|-
|52
|4L 48s
|-
|53
|2L 51s
|-
|54
|[[54edo]]
|equal
|-
|55
|53L 2s
| rowspan="32" |2:1
|-
|56
|52L 4s
|-
|57
|51L 6s
|-
|58
|50L 8s
|-
|59
|49L 10s
|-
|60
|48L 12s
|-
|61
|47L 14s
|-
|62
|46L 16s
|-
|63
|45L 18s
|-
|64
|44L 20s
|-
|65
|43L 22s
|-
|66
|42L 24s
|-
|67
|41L 26s
|-
|68
|40L 28s
|-
|69
|39L 30s
|-
|70
|38L 32s
|-
|71
|37L 34s
|-
|72
|36L 36s
|-
|73
|35L 38s
|-
|74
|34L 40s
|-
|75
|33L 42s
|-
|76
|32L 44s
|-
|77
|31L 46s
|-
|78
|30L 48s
|-
|79
|29L 50s
|-
|80
|28L 52s
|-
|81
|27L 54s
|-
|82
|26L 56s
|-
|83
|25L 58s
|-
|84
|24L 60s
|-
|85
|23L 62s
|-
|86
|22L 64s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Valentine]]</text>
      <sha1>9xsqzl5945ooawxq7t5wx4zg61b2pzo</sha1>
    </revision>
    <revision>
      <id>115193</id>
      <parentid>111332</parentid>
      <timestamp>2023-05-30T21:14:16Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Miscellany */</comment>
      <origin>115193</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="868" sha1="refvmfumjdbxs06dfl05t4dycj13elx" xml:space="preserve">{{Infobox ET}}
{{EDO intro|108}}

==Theory==
108edo tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12n-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.
=== Prime harmonics ===
{{Harmonics in equal|108}}

 [[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Valentine]]</text>
      <sha1>refvmfumjdbxs06dfl05t4dycj13elx</sha1>
    </revision>
    <revision>
      <id>132296</id>
      <parentid>115193</parentid>
      <timestamp>2024-01-21T07:03:24Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added interval table</comment>
      <origin>132296</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="904" sha1="3bece3it7wo44ffa1y3kze5d05ak7mz" xml:space="preserve">{{Infobox ET}}
{{EDO intro|108}}

==Theory==
108edo tempers out the Pythagorean comma, 531441/524288, in the 3-limit and 1990656/1953125, the valensixthtone comma, in the 5-limit. In the 7-limit it tempers out 126/125 and 1029/1024, supporting [[Starling_temperaments#Valentine temperament|valentine temperament]], and making for a good tuning for it and for starling temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out 540/539 and the 108e val tempers out 121/120 and 176/175, supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12n-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.
=== Prime harmonics ===
{{Harmonics in equal|108}}

== Intervals ==
{{Interval table}}

 [[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Valentine]]</text>
      <sha1>3bece3it7wo44ffa1y3kze5d05ak7mz</sha1>
    </revision>
    <revision>
      <id>145894</id>
      <parentid>132296</parentid>
      <timestamp>2024-06-08T08:31:28Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>145894</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="811" sha1="l6ghm8eezvpxhd7wdvhhrc4lep1uywx" xml:space="preserve">{{Infobox ET}}
{{EDO intro|108}}

== Theory ==
108et [[tempering out|tempers out]] the [[Pythagorean comma]] in the 3-limit and 1990656/1953125 ([[valentine comma]]) in the 5-limit. In the 7-limit it tempers out [[126/125]] and [[1029/1024]], [[support]]ing [[valentine]], and making for a good tuning for it and for [[starling]] temperament, the planar temperament tempering out 126/125. In the 11-limit the patent val tempers out [[540/539]] and the 108e val tempers out [[121/120]] and [[176/175]], supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12''n''-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.

=== Prime harmonics ===
{{Harmonics in equal|108}}

== Intervals ==
{{Interval table}}

[[Category:Valentine]]</text>
      <sha1>l6ghm8eezvpxhd7wdvhhrc4lep1uywx</sha1>
    </revision>
    <revision>
      <id>145895</id>
      <parentid>145894</parentid>
      <timestamp>2024-06-08T08:33:27Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+subsets and supersets</comment>
      <origin>145895</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="966" sha1="2ormvn2h893s01ty2gr06oqxjp8y5y1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|108}}

== Theory ==
108et [[tempering out|tempers out]] the [[Pythagorean comma]] in the 3-limit and 1990656/1953125 ([[valentine comma]]) in the 5-limit. In the 7-limit it tempers out [[126/125]] and [[1029/1024]], [[support]]ing [[valentine]], and making for a good tuning for it and for [[starling]] temperament, the planar temperament tempering out 126/125. In the 11-limit the [[patent val]] tempers out [[540/539]] and the 108e val tempers out [[121/120]] and [[176/175]], supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12''n''-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.

=== Prime harmonics ===
{{Harmonics in equal|108}}

=== Subsets and supersets ===
Since 108 factors into {{factorization|108}}, 108edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 27, 36, and 54 }}. 

== Intervals ==
{{Interval table}}

[[Category:Valentine]]</text>
      <sha1>2ormvn2h893s01ty2gr06oqxjp8y5y1</sha1>
    </revision>
    <revision>
      <id>145896</id>
      <parentid>145895</parentid>
      <timestamp>2024-06-08T08:39:06Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+more commas</comment>
      <origin>145896</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1008" sha1="qlh5nosc1yxc7m1h153yxn0nv7joywh" xml:space="preserve">{{Infobox ET}}
{{EDO intro|108}}

== Theory ==
108et [[tempering out|tempers out]] the [[Pythagorean comma]] in the 3-limit and 1990656/1953125 ([[valentine comma]]) in the 5-limit. In the 7-limit it tempers out [[126/125]] and [[1029/1024]], [[support]]ing [[valentine]], and making for a good tuning for it and for [[starling]] temperament, the planar temperament tempering out 126/125. In the 11-limit the [[patent val]] tempers out [[540/539]], and 1344/1331 and the 108e val tempers out [[121/120]], [[176/175]], [[385/384]], and [[441/440]], supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12''n''-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.

=== Prime harmonics ===
{{Harmonics in equal|108}}

=== Subsets and supersets ===
Since 108 factors into {{factorization|108}}, 108edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 27, 36, and 54 }}. 

== Intervals ==
{{Interval table}}

[[Category:Valentine]]</text>
      <sha1>qlh5nosc1yxc7m1h153yxn0nv7joywh</sha1>
    </revision>
    <revision>
      <id>145897</id>
      <parentid>145896</parentid>
      <timestamp>2024-06-08T08:40:19Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <origin>145897</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1008" sha1="9p3ecoidnp06ejf7wkewbssvbuqospu" xml:space="preserve">{{Infobox ET}}
{{EDO intro|108}}

== Theory ==
108et [[tempering out|tempers out]] the [[Pythagorean comma]] in the 3-limit and 1990656/1953125 ([[valentine comma]]) in the 5-limit. In the 7-limit it tempers out [[126/125]] and [[1029/1024]], [[support]]ing [[valentine]], and making for a good tuning for it and for [[starling]] temperament, the planar temperament tempering out 126/125. In the 11-limit the [[patent val]] tempers out [[540/539]] and 1344/1331, and the 108e val tempers out [[121/120]], [[176/175]], [[385/384]], and [[441/440]], supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12''n''-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.

=== Prime harmonics ===
{{Harmonics in equal|108}}

=== Subsets and supersets ===
Since 108 factors into {{factorization|108}}, 108edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 27, 36, and 54 }}. 

== Intervals ==
{{Interval table}}

[[Category:Valentine]]</text>
      <sha1>9p3ecoidnp06ejf7wkewbssvbuqospu</sha1>
    </revision>
    <revision>
      <id>181311</id>
      <parentid>145897</parentid>
      <timestamp>2025-02-18T16:47:56Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181311</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1003" sha1="sdp4ba5j2xqqukgksjpx6splvgdil2r" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
108et [[tempering out|tempers out]] the [[Pythagorean comma]] in the 3-limit and 1990656/1953125 ([[valentine comma]]) in the 5-limit. In the 7-limit it tempers out [[126/125]] and [[1029/1024]], [[support]]ing [[valentine]], and making for a good tuning for it and for [[starling]] temperament, the planar temperament tempering out 126/125. In the 11-limit the [[patent val]] tempers out [[540/539]] and 1344/1331, and the 108e val tempers out [[121/120]], [[176/175]], [[385/384]], and [[441/440]], supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12''n''-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.

=== Prime harmonics ===
{{Harmonics in equal|108}}

=== Subsets and supersets ===
Since 108 factors into {{factorization|108}}, 108edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 27, 36, and 54 }}. 

== Intervals ==
{{Interval table}}

[[Category:Valentine]]</text>
      <sha1>sdp4ba5j2xqqukgksjpx6splvgdil2r</sha1>
    </revision>
    <revision>
      <id>207322</id>
      <parentid>181311</parentid>
      <timestamp>2025-08-17T11:27:07Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <comment>Add music.</comment>
      <origin>207322</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1111" sha1="pqq7iyj889wldzioabkv2w7txkhoojw" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
108et [[tempering out|tempers out]] the [[Pythagorean comma]] in the 3-limit and 1990656/1953125 ([[valentine comma]]) in the 5-limit. In the 7-limit it tempers out [[126/125]] and [[1029/1024]], [[support]]ing [[valentine]], and making for a good tuning for it and for [[starling]] temperament, the planar temperament tempering out 126/125. In the 11-limit the [[patent val]] tempers out [[540/539]] and 1344/1331, and the 108e val tempers out [[121/120]], [[176/175]], [[385/384]], and [[441/440]], supporting 11-limit valentine for which it is again a good tuning.

108edo is the smallest 12''n''-edo which offers a decent alternative fifth to 12edo's fifth, that is [[27edo]]'s superpyth fifth.

=== Prime harmonics ===
{{Harmonics in equal|108}}

=== Subsets and supersets ===
Since 108 factors into {{factorization|108}}, 108edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 27, 36, and 54 }}. 

== Intervals ==
{{Interval table}}

== Music ==
; [[Billy Stiltner]]
* [https://billystiltner.bandcamp.com/track/108-tone-equal 108 tone equal]
[[Category:Valentine]]</text>
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    <title>109edo</title>
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      <text bytes="676" sha1="amhw8bx1gp3d2uts7pdr4nnp5ztc3ta" xml:space="preserve">'''109edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 109 parts of 11.009 [[cent|cent]]s each. It [[tempering_out|tempers out]] 20000/19683 in the [[5-limit|5-limit]]; 245/243, 2401/2400 and 65625/65536 in the [[7-limit|7-limit]]; 385/384, 1375/1372, and 4000/3993 in the [[11-limit|11-limit]]. It provides the [[Optimal_patent_val|optimal patent val]] for 7-limit [[Tetracot_family|octacot temperament]], and 11 and 13 limit [[Sensamagic_clan#Leapweek|leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is the 29th [[prime_numbers|prime]] edo.
[[Category:edo]]
[[Category:prime_edo]]
[[Category:theory]]</text>
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      <comment>**Imported revision 535942364 - Original comment: **</comment>
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109edo is the 29th [[prime numbers|prime]] edo.&lt;/pre&gt;&lt;/div&gt;
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&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 109 equal division divides the octave into 109 equal parts of 11.009 cents each. It tempers out 20000/19683 in the 5-limit;  245/243, 2401/2400 and 65625/65536 in the 7-limit; 385/384, 1375/1372, and 4000/3993 in the 11-limit. It provides the [[optimal patent val]] for 11-limit [[Tetracot family|octacot temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;109edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 109 equal division divides the octave into 109 equal parts of 11.009 cents each. It tempers out 20000/19683 in the 5-limit;  245/243, 2401/2400 and 65625/65536 in the 7-limit; 385/384, 1375/1372, and 4000/3993 in the 11-limit. It provides the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit &amp;lt;a class="wiki_link" href="/Tetracot%20family"&amp;gt;octacot temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>3jeye8iffbya9n4e11liz2kd4qigd9d</sha1>
    </revision>
    <revision>
      <id>38331</id>
      <parentid>28</parentid>
      <timestamp>2018-11-02T10:22:53Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>cat rename, better link</comment>
      <origin>38331</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="663" sha1="fauiftsdgp5a7qvi2ejnntv6iphly31" xml:space="preserve">'''109edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 109 parts of 11.009 [[cent|cent]]s each. It [[tempering_out|tempers out]] 20000/19683 in the [[5-limit|5-limit]]; 245/243, 2401/2400 and 65625/65536 in the [[7-limit|7-limit]]; 385/384, 1375/1372, and 4000/3993 in the [[11-limit|11-limit]]. It provides the [[Optimal_patent_val|optimal patent val]] for 7-limit [[Tetracot_family|octacot temperament]], and 11 and 13 limit [[Sensamagic_clan#Leapweek|leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is the 29th [[prime EDO]].

[[Category:Edo]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>fauiftsdgp5a7qvi2ejnntv6iphly31</sha1>
    </revision>
    <revision>
      <id>46235</id>
      <parentid>38331</parentid>
      <timestamp>2020-06-09T19:50:06Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>simplified wiki markup</comment>
      <origin>46235</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="552" sha1="1ge4b27itqgkzdg1npg94rt69ru3kwl" xml:space="preserve">'''109edo''' is the [[equal division of the octave]] into 109 parts of 11.009 [[cent]]s each. It [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is the 29th [[prime EDO]].

[[Category:Edo]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>1ge4b27itqgkzdg1npg94rt69ru3kwl</sha1>
    </revision>
    <revision>
      <id>54956</id>
      <parentid>46235</parentid>
      <timestamp>2020-12-04T23:13:05Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54956</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="578" sha1="c72jhfg0a5cakkujbs05ygvchxlnvy8" xml:space="preserve">'''109edo''' is the [[equal division of the octave]] into 109 parts of 11.009 [[cent]]s each. It [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is the 29th [[prime EDO]].

[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>c72jhfg0a5cakkujbs05ygvchxlnvy8</sha1>
    </revision>
    <revision>
      <id>68072</id>
      <parentid>54956</parentid>
      <timestamp>2021-04-21T02:13:34Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68072</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2315" sha1="tm37b0jwjs8w2mh77v40yo7yo7hphtd" xml:space="preserve">'''109edo''' is the [[equal division of the octave]] into 109 parts of 11.009 [[cent]]s each. It [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is the 29th [[prime EDO]].


Since 109edo has a step of 11.009 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 109edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|22:21
|-
|6
|[[1L 5s]]
|19:18
|-
|7
|[[4L 3s]]
|16:15
|-
|8
|[[5L 3s]]
|14:13
|-
|9
|[[1L 8s]]
|13:12
|-
|10
|[[9L 1s]]
|11:10
|-
|11
|[[10L 1s]]
| rowspan="2" |10:9
|-
|12
|[[1L 11s]]
|-
|13
|[[4L 9s]]
|9:8
|-
|14
|[[11L 3s]]
| rowspan="2" |8:7
|-
|15
|[[4L 11s]]
|-
|16
|13L 3s
| rowspan="3" |7:6
|-
|17
|[[7L 10s]]
|-
|18
|1L 17s
|-
|19
|14L 5s
| rowspan="3" |6:5
|-
|20
|9L 11s
|-
|21
|4L 17s
|-
|22
|21L 1s
| rowspan="6" |5:4
|-
|23
|17L 6s
|-
|24
|13L 11s
|-
|25
|9L 16s
|-
|26
|5L 21s
|-
|27
|1L 26s
|-
|28
|25L 3s
| rowspan="9" |4:3
|-
|29
|22L 7s
|-
|30
|19L 11s
|-
|31
|16L 15s
|-
|32
|13L 19s
|-
|33
|10L 23s
|-
|34
|7L 27s
|-
|35
|4L 31s
|-
|36
|1L 35s
|-
|37
|35L 2s
| rowspan="18" |3:2
|-
|38
|33L 5s
|-
|39
|31L 8s
|-
|40
|29L 11s
|-
|41
|27L 14s
|-
|42
|25L 17s
|-
|43
|23L 20s
|-
|44
|21L 23s
|-
|45
|19L 26s
|-
|46
|17L 29s
|-
|47
|15L 32s
|-
|48
|13L 35s
|-
|49
|11L 38L
|-
|50
|9L 41s
|-
|51
|7L 44s
|-
|52
|5L 47s
|-
|53
|3L 50s
|-
|54
|1L 53s
|-
|55
|54L 1s
| rowspan="34" |2:1
|-
|56
|53L 3s
|-
|57
|52L 5s
|-
|58
|51L 7s
|-
|59
|50L 9s
|-
|60
|49L 11s
|-
|61
|48L 13s
|-
|62
|47L 15s
|-
|63
|46L 17s
|-
|64
|45L 19s
|-
|65
|44L 21s
|-
|66
|43L 23s
|-
|67
|42L 25s
|-
|68
|41L 27s
|-
|69
|40L 29s
|-
|70
|39L 31s
|-
|71
|38L 33s
|-
|72
|37L 35s
|-
|73
|36L 37s
|-
|74
|35L 39s
|-
|75
|34L 41s
|-
|76
|33L 43s
|-
|77
|32L 45s
|-
|78
|31L 47s
|-
|79
|30L 49s
|-
|80
|29L 51s
|-
|81
|28L 53s
|-
|82
|27L 55s
|-
|83
|26L 57s
|-
|84
|25L 59s
|-
|85
|24L 61s
|-
|86
|23L 63s
|-
|87
|22L 65s
|-
|88
|21L 67s
|}
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>tm37b0jwjs8w2mh77v40yo7yo7hphtd</sha1>
    </revision>
    <revision>
      <id>68080</id>
      <parentid>68072</parentid>
      <timestamp>2021-04-21T02:55:36Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68080</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2299" sha1="pub5os8ktvnzxbfaoarxn715tbefzda" xml:space="preserve">'''109edo''' is the [[equal division of the octave]] into 109 parts of 11.009 [[cent]]s each. It [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is the 29th [[prime EDO]].


Since 109edo has a step of 11.009 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 109edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|22:21
|-
|6
|[[1L 5s]]
|19:18
|-
|7
|[[4L 3s]]
|16:15
|-
|8
|[[5L 3s]]
|14:13
|-
|9
|[[1L 8s]]
|13:12
|-
|10
|[[9L 1s]]
|11:10
|-
|11
|[[10L 1s]]
| rowspan="2" |10:9
|-
|12
|[[1L 11s]]
|-
|13
|[[4L 9s]]
|9:8
|-
|14
|[[11L 3s]]
| rowspan="2" |8:7
|-
|15
|[[4L 11s]]
|-
|16
|13L 3s
| rowspan="3" |7:6
|-
|17
|[[7L 10s]]
|-
|18
|1L 17s
|-
|19
|14L 5s
| rowspan="3" |6:5
|-
|20
|9L 11s
|-
|21
|4L 17s
|-
|22
|21L 1s
| rowspan="6" |5:4
|-
|23
|17L 6s
|-
|24
|13L 11s
|-
|25
|9L 16s
|-
|26
|5L 21s
|-
|27
|1L 26s
|-
|28
|25L 3s
| rowspan="9" |4:3
|-
|29
|22L 7s
|-
|30
|19L 11s
|-
|31
|16L 15s
|-
|32
|13L 19s
|-
|33
|10L 23s
|-
|34
|7L 27s
|-
|35
|4L 31s
|-
|36
|1L 35s
|-
|37
|35L 2s
| rowspan="18" |3:2
|-
|38
|33L 5s
|-
|39
|31L 8s
|-
|40
|29L 11s
|-
|41
|27L 14s
|-
|42
|25L 17s
|-
|43
|23L 20s
|-
|44
|21L 23s
|-
|45
|19L 26s
|-
|46
|17L 29s
|-
|47
|15L 32s
|-
|48
|13L 35s
|-
|49
|11L 38L
|-
|50
|9L 41s
|-
|51
|7L 44s
|-
|52
|5L 47s
|-
|53
|3L 50s
|-
|54
|1L 53s
|-
|55
|54L 1s
| rowspan="33" |2:1
|-
|56
|53L 3s
|-
|57
|52L 5s
|-
|58
|51L 7s
|-
|59
|50L 9s
|-
|60
|49L 11s
|-
|61
|48L 13s
|-
|62
|47L 15s
|-
|63
|46L 17s
|-
|64
|45L 19s
|-
|65
|44L 21s
|-
|66
|43L 23s
|-
|67
|42L 25s
|-
|68
|41L 27s
|-
|69
|40L 29s
|-
|70
|39L 31s
|-
|71
|38L 33s
|-
|72
|37L 35s
|-
|73
|36L 37s
|-
|74
|35L 39s
|-
|75
|34L 41s
|-
|76
|33L 43s
|-
|77
|32L 45s
|-
|78
|31L 47s
|-
|79
|30L 49s
|-
|80
|29L 51s
|-
|81
|28L 53s
|-
|82
|27L 55s
|-
|83
|26L 57s
|-
|84
|25L 59s
|-
|85
|24L 61s
|-
|86
|23L 63s
|-
|87
|22L 65s
|}
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>pub5os8ktvnzxbfaoarxn715tbefzda</sha1>
    </revision>
    <revision>
      <id>68090</id>
      <parentid>68080</parentid>
      <timestamp>2021-04-21T04:54:36Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68090</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2448" sha1="9xds3x3zj2ijrs40kb9uyxd197omio6" xml:space="preserve">'''109edo''' is the [[equal division of the octave]] into 109 parts of 11.009 [[cent]]s each. It [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is the 29th [[prime EDO]].


Since 109edo has a step of 11.009 cents, it also allows one to use its MOS scales as circulating temperaments. It is also the first edo which also allows one to use an MOS scale one octave of which fills a standard piano keyboard as a circulating temperament. 
{| class="wikitable"
|+Circulating temperaments in 109edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|22:21
|-
|6
|[[1L 5s]]
|19:18
|-
|7
|[[4L 3s]]
|16:15
|-
|8
|[[5L 3s]]
|14:13
|-
|9
|[[1L 8s]]
|13:12
|-
|10
|[[9L 1s]]
|11:10
|-
|11
|[[10L 1s]]
| rowspan="2" |10:9
|-
|12
|[[1L 11s]]
|-
|13
|[[4L 9s]]
|9:8
|-
|14
|[[11L 3s]]
| rowspan="2" |8:7
|-
|15
|[[4L 11s]]
|-
|16
|13L 3s
| rowspan="3" |7:6
|-
|17
|[[7L 10s]]
|-
|18
|1L 17s
|-
|19
|14L 5s
| rowspan="3" |6:5
|-
|20
|9L 11s
|-
|21
|4L 17s
|-
|22
|21L 1s
| rowspan="6" |5:4
|-
|23
|17L 6s
|-
|24
|13L 11s
|-
|25
|9L 16s
|-
|26
|5L 21s
|-
|27
|1L 26s
|-
|28
|25L 3s
| rowspan="9" |4:3
|-
|29
|22L 7s
|-
|30
|19L 11s
|-
|31
|16L 15s
|-
|32
|13L 19s
|-
|33
|10L 23s
|-
|34
|7L 27s
|-
|35
|4L 31s
|-
|36
|1L 35s
|-
|37
|35L 2s
| rowspan="18" |3:2
|-
|38
|33L 5s
|-
|39
|31L 8s
|-
|40
|29L 11s
|-
|41
|27L 14s
|-
|42
|25L 17s
|-
|43
|23L 20s
|-
|44
|21L 23s
|-
|45
|19L 26s
|-
|46
|17L 29s
|-
|47
|15L 32s
|-
|48
|13L 35s
|-
|49
|11L 38L
|-
|50
|9L 41s
|-
|51
|7L 44s
|-
|52
|5L 47s
|-
|53
|3L 50s
|-
|54
|1L 53s
|-
|55
|54L 1s
| rowspan="33" |2:1
|-
|56
|53L 3s
|-
|57
|52L 5s
|-
|58
|51L 7s
|-
|59
|50L 9s
|-
|60
|49L 11s
|-
|61
|48L 13s
|-
|62
|47L 15s
|-
|63
|46L 17s
|-
|64
|45L 19s
|-
|65
|44L 21s
|-
|66
|43L 23s
|-
|67
|42L 25s
|-
|68
|41L 27s
|-
|69
|40L 29s
|-
|70
|39L 31s
|-
|71
|38L 33s
|-
|72
|37L 35s
|-
|73
|36L 37s
|-
|74
|35L 39s
|-
|75
|34L 41s
|-
|76
|33L 43s
|-
|77
|32L 45s
|-
|78
|31L 47s
|-
|79
|30L 49s
|-
|80
|29L 51s
|-
|81
|28L 53s
|-
|82
|27L 55s
|-
|83
|26L 57s
|-
|84
|25L 59s
|-
|85
|24L 61s
|-
|86
|23L 63s
|-
|87
|22L 65s
|}
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]
[[Category:Theory]]</text>
      <sha1>9xds3x3zj2ijrs40kb9uyxd197omio6</sha1>
    </revision>
    <revision>
      <id>87435</id>
      <parentid>68090</parentid>
      <timestamp>2022-02-26T16:48:49Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87435</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2428" sha1="gfxzizhi28m1gzxkt0st39muf12sijc" xml:space="preserve">'''109edo''' is the [[equal division of the octave]] into 109 parts of 11.009 [[cent]]s each. It [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is the 29th [[prime EDO]].


Since 109edo has a step of 11.009 cents, it also allows one to use its MOS scales as circulating temperaments. It is also the first edo which also allows one to use an MOS scale one octave of which fills a standard piano keyboard as a circulating temperament. 
{| class="wikitable"
|+Circulating temperaments in 109edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|22:21
|-
|6
|[[1L 5s]]
|19:18
|-
|7
|[[4L 3s]]
|16:15
|-
|8
|[[5L 3s]]
|14:13
|-
|9
|[[1L 8s]]
|13:12
|-
|10
|[[9L 1s]]
|11:10
|-
|11
|[[10L 1s]]
| rowspan="2" |10:9
|-
|12
|[[1L 11s]]
|-
|13
|[[4L 9s]]
|9:8
|-
|14
|[[11L 3s]]
| rowspan="2" |8:7
|-
|15
|[[4L 11s]]
|-
|16
|13L 3s
| rowspan="3" |7:6
|-
|17
|[[7L 10s]]
|-
|18
|1L 17s
|-
|19
|14L 5s
| rowspan="3" |6:5
|-
|20
|9L 11s
|-
|21
|4L 17s
|-
|22
|21L 1s
| rowspan="6" |5:4
|-
|23
|17L 6s
|-
|24
|13L 11s
|-
|25
|9L 16s
|-
|26
|5L 21s
|-
|27
|1L 26s
|-
|28
|25L 3s
| rowspan="9" |4:3
|-
|29
|22L 7s
|-
|30
|19L 11s
|-
|31
|16L 15s
|-
|32
|13L 19s
|-
|33
|10L 23s
|-
|34
|7L 27s
|-
|35
|4L 31s
|-
|36
|1L 35s
|-
|37
|35L 2s
| rowspan="18" |3:2
|-
|38
|33L 5s
|-
|39
|31L 8s
|-
|40
|29L 11s
|-
|41
|27L 14s
|-
|42
|25L 17s
|-
|43
|23L 20s
|-
|44
|21L 23s
|-
|45
|19L 26s
|-
|46
|17L 29s
|-
|47
|15L 32s
|-
|48
|13L 35s
|-
|49
|11L 38L
|-
|50
|9L 41s
|-
|51
|7L 44s
|-
|52
|5L 47s
|-
|53
|3L 50s
|-
|54
|1L 53s
|-
|55
|54L 1s
| rowspan="33" |2:1
|-
|56
|53L 3s
|-
|57
|52L 5s
|-
|58
|51L 7s
|-
|59
|50L 9s
|-
|60
|49L 11s
|-
|61
|48L 13s
|-
|62
|47L 15s
|-
|63
|46L 17s
|-
|64
|45L 19s
|-
|65
|44L 21s
|-
|66
|43L 23s
|-
|67
|42L 25s
|-
|68
|41L 27s
|-
|69
|40L 29s
|-
|70
|39L 31s
|-
|71
|38L 33s
|-
|72
|37L 35s
|-
|73
|36L 37s
|-
|74
|35L 39s
|-
|75
|34L 41s
|-
|76
|33L 43s
|-
|77
|32L 45s
|-
|78
|31L 47s
|-
|79
|30L 49s
|-
|80
|29L 51s
|-
|81
|28L 53s
|-
|82
|27L 55s
|-
|83
|26L 57s
|-
|84
|25L 59s
|-
|85
|24L 61s
|-
|86
|23L 63s
|-
|87
|22L 65s
|}
[[Category:Equal divisions of the octave]]
[[Category:Prime EDO]]</text>
      <sha1>gfxzizhi28m1gzxkt0st39muf12sijc</sha1>
    </revision>
    <revision>
      <id>92919</id>
      <parentid>87435</parentid>
      <timestamp>2022-07-02T16:20:07Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92919</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2456" sha1="0ld4m07qxw2ugupb7at22nbwzzoohkn" xml:space="preserve">'''109edo''' is the [[equal division of the octave]] into 109 parts of 11.009 [[cent]]s each. It [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is the 29th [[prime EDO]].

Since 109edo has a step of 11.009 cents, it also allows one to use its MOS scales as circulating temperaments. It is also the first edo which also allows one to use an MOS scale one octave of which fills a standard piano keyboard as a circulating temperament. 
{| class="wikitable"
|+Circulating temperaments in 109edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|22:21
|-
|6
|[[1L 5s]]
|19:18
|-
|7
|[[4L 3s]]
|16:15
|-
|8
|[[5L 3s]]
|14:13
|-
|9
|[[1L 8s]]
|13:12
|-
|10
|[[9L 1s]]
|11:10
|-
|11
|[[10L 1s]]
| rowspan="2" |10:9
|-
|12
|[[1L 11s]]
|-
|13
|[[4L 9s]]
|9:8
|-
|14
|[[11L 3s]]
| rowspan="2" |8:7
|-
|15
|[[4L 11s]]
|-
|16
|13L 3s
| rowspan="3" |7:6
|-
|17
|[[7L 10s]]
|-
|18
|1L 17s
|-
|19
|14L 5s
| rowspan="3" |6:5
|-
|20
|9L 11s
|-
|21
|4L 17s
|-
|22
|21L 1s
| rowspan="6" |5:4
|-
|23
|17L 6s
|-
|24
|13L 11s
|-
|25
|9L 16s
|-
|26
|5L 21s
|-
|27
|1L 26s
|-
|28
|25L 3s
| rowspan="9" |4:3
|-
|29
|22L 7s
|-
|30
|19L 11s
|-
|31
|16L 15s
|-
|32
|13L 19s
|-
|33
|10L 23s
|-
|34
|7L 27s
|-
|35
|4L 31s
|-
|36
|1L 35s
|-
|37
|35L 2s
| rowspan="18" |3:2
|-
|38
|33L 5s
|-
|39
|31L 8s
|-
|40
|29L 11s
|-
|41
|27L 14s
|-
|42
|25L 17s
|-
|43
|23L 20s
|-
|44
|21L 23s
|-
|45
|19L 26s
|-
|46
|17L 29s
|-
|47
|15L 32s
|-
|48
|13L 35s
|-
|49
|11L 38L
|-
|50
|9L 41s
|-
|51
|7L 44s
|-
|52
|5L 47s
|-
|53
|3L 50s
|-
|54
|1L 53s
|-
|55
|54L 1s
| rowspan="33" |2:1
|-
|56
|53L 3s
|-
|57
|52L 5s
|-
|58
|51L 7s
|-
|59
|50L 9s
|-
|60
|49L 11s
|-
|61
|48L 13s
|-
|62
|47L 15s
|-
|63
|46L 17s
|-
|64
|45L 19s
|-
|65
|44L 21s
|-
|66
|43L 23s
|-
|67
|42L 25s
|-
|68
|41L 27s
|-
|69
|40L 29s
|-
|70
|39L 31s
|-
|71
|38L 33s
|-
|72
|37L 35s
|-
|73
|36L 37s
|-
|74
|35L 39s
|-
|75
|34L 41s
|-
|76
|33L 43s
|-
|77
|32L 45s
|-
|78
|31L 47s
|-
|79
|30L 49s
|-
|80
|29L 51s
|-
|81
|28L 53s
|-
|82
|27L 55s
|-
|83
|26L 57s
|-
|84
|25L 59s
|-
|85
|24L 61s
|-
|86
|23L 63s
|-
|87
|22L 65s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Prime EDO]]</text>
      <sha1>0ld4m07qxw2ugupb7at22nbwzzoohkn</sha1>
    </revision>
    <revision>
      <id>96190</id>
      <parentid>92919</parentid>
      <timestamp>2022-10-04T18:50:52Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96190</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2471" sha1="o7gdtnwaogxsx6cd80yneb85fkpagoj" xml:space="preserve">{{Infobox ET}}
'''109edo''' is the [[equal division of the octave]] into 109 parts of 11.009 [[cent]]s each. It [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is the 29th [[prime EDO]].

Since 109edo has a step of 11.009 cents, it also allows one to use its MOS scales as circulating temperaments. It is also the first edo which also allows one to use an MOS scale one octave of which fills a standard piano keyboard as a circulating temperament. 
{| class="wikitable"
|+Circulating temperaments in 109edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|22:21
|-
|6
|[[1L 5s]]
|19:18
|-
|7
|[[4L 3s]]
|16:15
|-
|8
|[[5L 3s]]
|14:13
|-
|9
|[[1L 8s]]
|13:12
|-
|10
|[[9L 1s]]
|11:10
|-
|11
|[[10L 1s]]
| rowspan="2" |10:9
|-
|12
|[[1L 11s]]
|-
|13
|[[4L 9s]]
|9:8
|-
|14
|[[11L 3s]]
| rowspan="2" |8:7
|-
|15
|[[4L 11s]]
|-
|16
|13L 3s
| rowspan="3" |7:6
|-
|17
|[[7L 10s]]
|-
|18
|1L 17s
|-
|19
|14L 5s
| rowspan="3" |6:5
|-
|20
|9L 11s
|-
|21
|4L 17s
|-
|22
|21L 1s
| rowspan="6" |5:4
|-
|23
|17L 6s
|-
|24
|13L 11s
|-
|25
|9L 16s
|-
|26
|5L 21s
|-
|27
|1L 26s
|-
|28
|25L 3s
| rowspan="9" |4:3
|-
|29
|22L 7s
|-
|30
|19L 11s
|-
|31
|16L 15s
|-
|32
|13L 19s
|-
|33
|10L 23s
|-
|34
|7L 27s
|-
|35
|4L 31s
|-
|36
|1L 35s
|-
|37
|35L 2s
| rowspan="18" |3:2
|-
|38
|33L 5s
|-
|39
|31L 8s
|-
|40
|29L 11s
|-
|41
|27L 14s
|-
|42
|25L 17s
|-
|43
|23L 20s
|-
|44
|21L 23s
|-
|45
|19L 26s
|-
|46
|17L 29s
|-
|47
|15L 32s
|-
|48
|13L 35s
|-
|49
|11L 38L
|-
|50
|9L 41s
|-
|51
|7L 44s
|-
|52
|5L 47s
|-
|53
|3L 50s
|-
|54
|1L 53s
|-
|55
|54L 1s
| rowspan="33" |2:1
|-
|56
|53L 3s
|-
|57
|52L 5s
|-
|58
|51L 7s
|-
|59
|50L 9s
|-
|60
|49L 11s
|-
|61
|48L 13s
|-
|62
|47L 15s
|-
|63
|46L 17s
|-
|64
|45L 19s
|-
|65
|44L 21s
|-
|66
|43L 23s
|-
|67
|42L 25s
|-
|68
|41L 27s
|-
|69
|40L 29s
|-
|70
|39L 31s
|-
|71
|38L 33s
|-
|72
|37L 35s
|-
|73
|36L 37s
|-
|74
|35L 39s
|-
|75
|34L 41s
|-
|76
|33L 43s
|-
|77
|32L 45s
|-
|78
|31L 47s
|-
|79
|30L 49s
|-
|80
|29L 51s
|-
|81
|28L 53s
|-
|82
|27L 55s
|-
|83
|26L 57s
|-
|84
|25L 59s
|-
|85
|24L 61s
|-
|86
|23L 63s
|-
|87
|22L 65s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Prime EDO]]</text>
      <sha1>o7gdtnwaogxsx6cd80yneb85fkpagoj</sha1>
    </revision>
    <revision>
      <id>103702</id>
      <parentid>96190</parentid>
      <timestamp>2023-02-25T17:27:57Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>103702</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2428" sha1="hg4yjd3e85a67afse6krb6h6d6zr3n1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}

109edo [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is the 29th [[prime EDO]].

Since 109edo has a step of 11.009 cents, it also allows one to use its MOS scales as circulating temperaments. It is also the first edo which also allows one to use an MOS scale one octave of which fills a standard piano keyboard as a circulating temperament. 
{| class="wikitable mw-collapsible mw-collapsed"
|+Circulating temperaments in 109edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|22:21
|-
|6
|[[1L 5s]]
|19:18
|-
|7
|[[4L 3s]]
|16:15
|-
|8
|[[5L 3s]]
|14:13
|-
|9
|[[1L 8s]]
|13:12
|-
|10
|[[9L 1s]]
|11:10
|-
|11
|[[10L 1s]]
| rowspan="2" |10:9
|-
|12
|[[1L 11s]]
|-
|13
|[[4L 9s]]
|9:8
|-
|14
|[[11L 3s]]
| rowspan="2" |8:7
|-
|15
|[[4L 11s]]
|-
|16
|13L 3s
| rowspan="3" |7:6
|-
|17
|[[7L 10s]]
|-
|18
|1L 17s
|-
|19
|14L 5s
| rowspan="3" |6:5
|-
|20
|9L 11s
|-
|21
|4L 17s
|-
|22
|21L 1s
| rowspan="6" |5:4
|-
|23
|17L 6s
|-
|24
|13L 11s
|-
|25
|9L 16s
|-
|26
|5L 21s
|-
|27
|1L 26s
|-
|28
|25L 3s
| rowspan="9" |4:3
|-
|29
|22L 7s
|-
|30
|19L 11s
|-
|31
|16L 15s
|-
|32
|13L 19s
|-
|33
|10L 23s
|-
|34
|7L 27s
|-
|35
|4L 31s
|-
|36
|1L 35s
|-
|37
|35L 2s
| rowspan="18" |3:2
|-
|38
|33L 5s
|-
|39
|31L 8s
|-
|40
|29L 11s
|-
|41
|27L 14s
|-
|42
|25L 17s
|-
|43
|23L 20s
|-
|44
|21L 23s
|-
|45
|19L 26s
|-
|46
|17L 29s
|-
|47
|15L 32s
|-
|48
|13L 35s
|-
|49
|11L 38L
|-
|50
|9L 41s
|-
|51
|7L 44s
|-
|52
|5L 47s
|-
|53
|3L 50s
|-
|54
|1L 53s
|-
|55
|54L 1s
| rowspan="33" |2:1
|-
|56
|53L 3s
|-
|57
|52L 5s
|-
|58
|51L 7s
|-
|59
|50L 9s
|-
|60
|49L 11s
|-
|61
|48L 13s
|-
|62
|47L 15s
|-
|63
|46L 17s
|-
|64
|45L 19s
|-
|65
|44L 21s
|-
|66
|43L 23s
|-
|67
|42L 25s
|-
|68
|41L 27s
|-
|69
|40L 29s
|-
|70
|39L 31s
|-
|71
|38L 33s
|-
|72
|37L 35s
|-
|73
|36L 37s
|-
|74
|35L 39s
|-
|75
|34L 41s
|-
|76
|33L 43s
|-
|77
|32L 45s
|-
|78
|31L 47s
|-
|79
|30L 49s
|-
|80
|29L 51s
|-
|81
|28L 53s
|-
|82
|27L 55s
|-
|83
|26L 57s
|-
|84
|25L 59s
|-
|85
|24L 61s
|-
|86
|23L 63s
|-
|87
|22L 65s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Prime EDO]]</text>
      <sha1>hg4yjd3e85a67afse6krb6h6d6zr3n1</sha1>
    </revision>
    <revision>
      <id>106021</id>
      <parentid>103702</parentid>
      <timestamp>2023-03-22T14:10:10Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>106021</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2704" sha1="i8zmuy8fvuatytaun2bfvxheciidwc8" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}
==Theory==
109edo [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is a strong 2.5.7.11.19.23.31 subgroup tuning, with errors of less than 10% on all harmonics and an exceptionally small error of 0.16% on the 7th harmonic.

=== Prime harmonics ===
{{harmonics in equal|109}}
=== Subsets and supersets ===
109edo is the 29th [[prime EDO]].
==Scales==
Since 109edo has a step of 11.009 cents, it also allows one to use its MOS scales as circulating temperaments. It is also the first edo which also allows one to use an MOS scale one octave of which fills a standard piano keyboard as a circulating temperament{{clarify}}. 
{| class="wikitable mw-collapsible mw-collapsed"
|+Circulating temperaments in 109edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|22:21
|-
|6
|[[1L 5s]]
|19:18
|-
|7
|[[4L 3s]]
|16:15
|-
|8
|[[5L 3s]]
|14:13
|-
|9
|[[1L 8s]]
|13:12
|-
|10
|[[9L 1s]]
|11:10
|-
|11
|[[10L 1s]]
| rowspan="2" |10:9
|-
|12
|[[1L 11s]]
|-
|13
|[[4L 9s]]
|9:8
|-
|14
|[[11L 3s]]
| rowspan="2" |8:7
|-
|15
|[[4L 11s]]
|-
|16
|13L 3s
| rowspan="3" |7:6
|-
|17
|[[7L 10s]]
|-
|18
|1L 17s
|-
|19
|14L 5s
| rowspan="3" |6:5
|-
|20
|9L 11s
|-
|21
|4L 17s
|-
|22
|21L 1s
| rowspan="6" |5:4
|-
|23
|17L 6s
|-
|24
|13L 11s
|-
|25
|9L 16s
|-
|26
|5L 21s
|-
|27
|1L 26s
|-
|28
|25L 3s
| rowspan="9" |4:3
|-
|29
|22L 7s
|-
|30
|19L 11s
|-
|31
|16L 15s
|-
|32
|13L 19s
|-
|33
|10L 23s
|-
|34
|7L 27s
|-
|35
|4L 31s
|-
|36
|1L 35s
|-
|37
|35L 2s
| rowspan="18" |3:2
|-
|38
|33L 5s
|-
|39
|31L 8s
|-
|40
|29L 11s
|-
|41
|27L 14s
|-
|42
|25L 17s
|-
|43
|23L 20s
|-
|44
|21L 23s
|-
|45
|19L 26s
|-
|46
|17L 29s
|-
|47
|15L 32s
|-
|48
|13L 35s
|-
|49
|11L 38L
|-
|50
|9L 41s
|-
|51
|7L 44s
|-
|52
|5L 47s
|-
|53
|3L 50s
|-
|54
|1L 53s
|-
|55
|54L 1s
| rowspan="33" |2:1
|-
|56
|53L 3s
|-
|57
|52L 5s
|-
|58
|51L 7s
|-
|59
|50L 9s
|-
|60
|49L 11s
|-
|61
|48L 13s
|-
|62
|47L 15s
|-
|63
|46L 17s
|-
|64
|45L 19s
|-
|65
|44L 21s
|-
|66
|43L 23s
|-
|67
|42L 25s
|-
|68
|41L 27s
|-
|69
|40L 29s
|-
|70
|39L 31s
|-
|71
|38L 33s
|-
|72
|37L 35s
|-
|73
|36L 37s
|-
|74
|35L 39s
|-
|75
|34L 41s
|-
|76
|33L 43s
|-
|77
|32L 45s
|-
|78
|31L 47s
|-
|79
|30L 49s
|-
|80
|29L 51s
|-
|81
|28L 53s
|-
|82
|27L 55s
|-
|83
|26L 57s
|-
|84
|25L 59s
|-
|85
|24L 61s
|-
|86
|23L 63s
|-
|87
|22L 65s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Prime EDO]]</text>
      <sha1>i8zmuy8fvuatytaun2bfvxheciidwc8</sha1>
    </revision>
    <revision>
      <id>115194</id>
      <parentid>106021</parentid>
      <timestamp>2023-05-30T21:14:33Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Scales */</comment>
      <origin>115194</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="784" sha1="p4txb78adnsjs15v9qjmo8icnwma2hl" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}
==Theory==
109edo [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is a strong 2.5.7.11.19.23.31 subgroup tuning, with errors of less than 10% on all harmonics and an exceptionally small error of 0.16% on the 7th harmonic.

=== Prime harmonics ===
{{harmonics in equal|109}}
=== Subsets and supersets ===
109edo is the 29th [[prime EDO]].
[[Category:Equal divisions of the octave|###]]&lt;!-- 3-digit number --&gt; 
[[Category:Prime EDO]]</text>
      <sha1>p4txb78adnsjs15v9qjmo8icnwma2hl</sha1>
    </revision>
    <revision>
      <id>132009</id>
      <parentid>115194</parentid>
      <timestamp>2024-01-16T21:47:37Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <origin>132009</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="895" sha1="9vd640y6xi6fp1hqv0u5c8g703h2uxo" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}
==Theory==
109edo [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is a strong 2.5.7.11.19.23.31 subgroup tuning, with errors of less than 10% on all harmonics and an exceptionally small error of 0.16% on the 7th harmonic.

=== Prime harmonics ===
{{harmonics in equal|109}}
=== Subsets and supersets ===
109edo is the 29th [[prime EDO]].

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].
[[Category:Equal divisions of the octave|###]]&lt;!-- 3-digit number --&gt; 
[[Category:Prime EDO]]</text>
      <sha1>9vd640y6xi6fp1hqv0u5c8g703h2uxo</sha1>
    </revision>
    <revision>
      <id>132297</id>
      <parentid>132009</parentid>
      <timestamp>2024-01-21T07:04:16Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added interval table</comment>
      <origin>132297</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="932" sha1="bfri7wfy99bthoofpmlwz5oa6c4kwc8" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}
==Theory==
109edo [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is a strong 2.5.7.11.19.23.31 subgroup tuning, with errors of less than 10% on all harmonics and an exceptionally small error of 0.16% on the 7th harmonic.

=== Prime harmonics ===
{{harmonics in equal|109}}
=== Subsets and supersets ===
109edo is the 29th [[prime EDO]].

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}

[[Category:Equal divisions of the octave|###]]&lt;!-- 3-digit number --&gt; 
[[Category:Prime EDO]]</text>
      <sha1>bfri7wfy99bthoofpmlwz5oa6c4kwc8</sha1>
    </revision>
    <revision>
      <id>134024</id>
      <parentid>132297</parentid>
      <timestamp>2024-02-02T15:04:07Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>134024</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="997" sha1="15sngs91j9vgfve8yyz4kl1k15c1bt0" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}
==Theory==
109edo [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is a strong 2.5.7.11.19.23.31 subgroup tuning, with errors of less than 10% on all harmonics. The error on the 7th harmonic is exceptionally small due to 109 being a denominator of [[semiconvergent]] to log&lt;sub&gt;2&lt;/sub&gt;7.

=== Prime harmonics ===
{{harmonics in equal|109}}
=== Subsets and supersets ===
109edo is the 29th [[prime EDO]].

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}

[[Category:Equal divisions of the octave|###]]&lt;!-- 3-digit number --&gt; 
[[Category:Prime EDO]]</text>
      <sha1>15sngs91j9vgfve8yyz4kl1k15c1bt0</sha1>
    </revision>
    <revision>
      <id>134025</id>
      <parentid>134024</parentid>
      <timestamp>2024-02-02T15:04:51Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Prime harmonics */</comment>
      <origin>134025</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="998" sha1="0k0fvcvyggaf544as4xqirau0m99gt5" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}
==Theory==
109edo [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo is a strong 2.5.7.11.19.23.31 subgroup tuning, with errors of less than 10% on all harmonics. The error on the 7th harmonic is exceptionally small due to 109 being a denominator of [[semiconvergent]] to log&lt;sub&gt;2&lt;/sub&gt;7.

=== Prime harmonics ===
{{harmonics in equal|109}}

=== Subsets and supersets ===
109edo is the 29th [[prime EDO]].

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}

[[Category:Equal divisions of the octave|###]]&lt;!-- 3-digit number --&gt; 
[[Category:Prime EDO]]</text>
      <sha1>0k0fvcvyggaf544as4xqirau0m99gt5</sha1>
    </revision>
    <revision>
      <id>134026</id>
      <parentid>134025</parentid>
      <timestamp>2024-02-02T15:08:28Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */ rework</comment>
      <origin>134026</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1109" sha1="qla557d81qha7cth87tn9jbalzrguca" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}
==Theory==
109edo [[tempering out|tempers out]] 20000/19683 in the [[5-limit]]; [[245/243]], 2401/2400 and 65625/65536 in the [[7-limit]]; [[385/384]], 1375/1372, and 4000/3993 in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11 and 13 limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent 7th harmonic, being a denominator of [[semiconvergent]] to log&lt;sub&gt;2&lt;/sub&gt;7, and it is overall a strong 2.5.7.11.19.23.31.41 subgroup tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1572, 2255/2244, 2300/2299, 6860/6859, 10241/10240. 

=== Prime harmonics ===
{{harmonics in equal|109}}

=== Subsets and supersets ===
109edo is the 29th [[prime EDO]].

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}

[[Category:Equal divisions of the octave|###]]&lt;!-- 3-digit number --&gt; 
[[Category:Prime EDO]]</text>
      <sha1>qla557d81qha7cth87tn9jbalzrguca</sha1>
    </revision>
    <revision>
      <id>145698</id>
      <parentid>134026</parentid>
      <timestamp>2024-06-04T13:12:19Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Cleanup and correct typos</comment>
      <origin>145698</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1067" sha1="d26btoehwdhslzm1z8c0eqw3ue2drmi" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}

== Theory ==
109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log&lt;sub&gt;2&lt;/sub&gt;7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2244, 2300/2299, 6860/6859, 10241/10240. 

=== Prime harmonics ===
{{Harmonics in equal|109}}

=== Subsets and supersets ===
109edo is the 29th [[prime edo]].

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}</text>
      <sha1>d26btoehwdhslzm1z8c0eqw3ue2drmi</sha1>
    </revision>
    <revision>
      <id>145699</id>
      <parentid>145698</parentid>
      <timestamp>2024-06-04T13:12:45Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Subsets and supersets */</comment>
      <origin>145699</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1111" sha1="n30ujhr1dbp5e92d3mi0l822tstgsdf" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}

== Theory ==
109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log&lt;sub&gt;2&lt;/sub&gt;7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2244, 2300/2299, 6860/6859, 10241/10240. 

=== Prime harmonics ===
{{Harmonics in equal|109}}

=== Subsets and supersets ===
109edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]].

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}</text>
      <sha1>n30ujhr1dbp5e92d3mi0l822tstgsdf</sha1>
    </revision>
    <revision>
      <id>145893</id>
      <parentid>145699</parentid>
      <timestamp>2024-06-08T08:28:33Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Note a notable superset</comment>
      <origin>145893</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1223" sha1="ibnyxiksnua6sorcxcn09spnlyg17b3" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}

== Theory ==
109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log&lt;sub&gt;2&lt;/sub&gt;7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2244, 2300/2299, 6860/6859, 10241/10240. 

=== Prime harmonics ===
{{Harmonics in equal|109}}

=== Subsets and supersets ===
109edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3. 

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}</text>
      <sha1>ibnyxiksnua6sorcxcn09spnlyg17b3</sha1>
    </revision>
    <revision>
      <id>155208</id>
      <parentid>145893</parentid>
      <timestamp>2024-09-17T12:13:00Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>+music</comment>
      <origin>155208</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1356" sha1="37g0pex06nkf9730hvgu85g9082yvwf" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}

== Theory ==
109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log&lt;sub&gt;2&lt;/sub&gt;7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2244, 2300/2299, 6860/6859, 10241/10240. 

=== Prime harmonics ===
{{Harmonics in equal|109}}

=== Subsets and supersets ===
109edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3. 

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}

== Music ==
; [[Francium]]
* [https://www.youtube.com/watch?v=v_J71U392_k ''Teenagerges''] (2024) – tetracot[13] in 109edo tuning</text>
      <sha1>37g0pex06nkf9730hvgu85g9082yvwf</sha1>
    </revision>
    <revision>
      <id>155321</id>
      <parentid>155208</parentid>
      <timestamp>2024-09-18T23:01:20Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added link to orphaned page</comment>
      <origin>155321</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1390" sha1="011jfkh4z23fcxjg26bhg9c9aeeddhf" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}

== Theory ==
109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log&lt;sub&gt;2&lt;/sub&gt;7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2244, 2300/2299, 6860/6859, 10241/10240. 

=== Prime harmonics ===
{{Harmonics in equal|109}}

=== Subsets and supersets ===
109edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3. 

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}

== Music ==
; [[Francium]]
* [https://www.youtube.com/watch?v=v_J71U392_k ''Teenagerges''] (2024) – tetracot[13] in 109edo tuning

== See also ==
* [[109-7-comma]]</text>
      <sha1>011jfkh4z23fcxjg26bhg9c9aeeddhf</sha1>
    </revision>
    <revision>
      <id>155512</id>
      <parentid>155321</parentid>
      <timestamp>2024-09-19T17:18:01Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <origin>155512</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1401" sha1="fgvh2om75np9wlsyfhpnvkah7xn0kxl" xml:space="preserve">{{Infobox ET}}
{{EDO intro|109}}

== Theory ==
109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log&lt;sub&gt;2&lt;/sub&gt;7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2244, 2300/2299, 6860/6859, 10241/10240. 

=== Prime harmonics ===
{{Harmonics in equal|109|columns=16}}

=== Subsets and supersets ===
109edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3. 

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}

== Music ==
; [[Francium]]
* [https://www.youtube.com/watch?v=v_J71U392_k ''Teenagerges''] (2024) – tetracot[13] in 109edo tuning

== See also ==
* [[109-7-comma]]</text>
      <sha1>fgvh2om75np9wlsyfhpnvkah7xn0kxl</sha1>
    </revision>
    <revision>
      <id>181312</id>
      <parentid>155512</parentid>
      <timestamp>2025-02-18T16:48:42Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181312</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1396" sha1="4jiucfuv329qzod6xadpd9dh8v2wfy0" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log&lt;sub&gt;2&lt;/sub&gt;7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2244, 2300/2299, 6860/6859, 10241/10240. 

=== Prime harmonics ===
{{Harmonics in equal|109|columns=16}}

=== Subsets and supersets ===
109edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3. 

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}

== Music ==
; [[Francium]]
* [https://www.youtube.com/watch?v=v_J71U392_k ''Teenagerges''] (2024) – tetracot[13] in 109edo tuning

== See also ==
* [[109-7-comma]]</text>
      <sha1>4jiucfuv329qzod6xadpd9dh8v2wfy0</sha1>
    </revision>
    <revision>
      <id>198927</id>
      <parentid>181312</parentid>
      <timestamp>2025-06-07T11:35:09Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Music */ +music; +links</comment>
      <origin>198927</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1825" sha1="7yfpfq8q9kje0b8l3i9ka8ujwi8vi72" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log&lt;sub&gt;2&lt;/sub&gt;7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2244, 2300/2299, 6860/6859, 10241/10240. 

=== Prime harmonics ===
{{Harmonics in equal|109|columns=16}}

=== Subsets and supersets ===
109edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3. 

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}

== Music ==
; [[Francium]]
* "Teenagerges" from ''TOTMC September to December 2024'' (2024) – [https://open.spotify.com/track/4oQglJSEyp6CsL5RNWuiBy Spotify] | [https://francium223.bandcamp.com/track/teenagerges Bandcamp] | [https://www.youtube.com/watch?v=v_J71U392_k YouTube] – in Tetracot[13], 109edo tuning
* "Catbabel" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/0T7nW3ziEFvjV8c7v1EaMB Spotify] | [https://francium223.bandcamp.com/track/catbabel Bandcamp] | [https://www.youtube.com/watch?v=gtnTdPqiTDQ YouTube]

== See also ==
* [[109-7-comma]]</text>
      <sha1>7yfpfq8q9kje0b8l3i9ka8ujwi8vi72</sha1>
    </revision>
    <revision>
      <id>213520</id>
      <parentid>198927</parentid>
      <timestamp>2025-10-19T03:56:56Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
      <origin>213520</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1825" sha1="oso9qb4ej182vfh1jt3k8ib057bned7" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
109edo [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) in the [[5-limit]]; [[245/243]], [[2401/2400]] and [[65625/65536]] in the [[7-limit]]; [[385/384]], [[1375/1372]], and [[4000/3993]] in the [[11-limit]]. It provides the [[optimal patent val]] for 7-limit [[octacot]] temperament, and 11- and 13-limit [[leapweek]]; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log&lt;sub&gt;2&lt;/sub&gt;7, and it is overall a strong 2.5.7.11.19.23.31.41 [[subgroup]] tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2254, 2300/2299, 6860/6859, 10241/10240. 

=== Prime harmonics ===
{{Harmonics in equal|109|columns=16}}

=== Subsets and supersets ===
109edo is the 29th [[prime edo]], following [[107edo]] and before [[113edo]]. [[436edo]], which slices each step of 109edo in four, provides correction for the approximation to harmonic 3. 

=== Nonoctave temperaments ===
Taking every 8 degree of 109edo produces a scale extremely close to [[88cET]].

== Intervals ==
{{Interval table}}

== Music ==
; [[Francium]]
* "Teenagerges" from ''TOTMC September to December 2024'' (2024) – [https://open.spotify.com/track/4oQglJSEyp6CsL5RNWuiBy Spotify] | [https://francium223.bandcamp.com/track/teenagerges Bandcamp] | [https://www.youtube.com/watch?v=v_J71U392_k YouTube] – in Tetracot[13], 109edo tuning
* "Catbabel" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/0T7nW3ziEFvjV8c7v1EaMB Spotify] | [https://francium223.bandcamp.com/track/catbabel Bandcamp] | [https://www.youtube.com/watch?v=gtnTdPqiTDQ YouTube]

== See also ==
* [[109-7-comma]]</text>
      <sha1>oso9qb4ej182vfh1jt3k8ib057bned7</sha1>
    </revision>
  </page>
  <page>
    <title>10/1</title>
    <ns>0</ns>
    <id>29</id>
    <revision>
      <id>29</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>29</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="180" sha1="1ib5d1d3hqdyyg6a6jrehrtyrb4nme9" xml:space="preserve">'''10/1'''
|1 0 1&amp;gt;

3986.3137 cents

[[File:jid_10_1_pluck_adu_dr55.mp3]] [[:File:jid_10_1_pluck_adu_dr55.mp3|sound sample]]

The 10th harmonic; three octaves above [[5/4|5/4]].</text>
      <sha1>1ib5d1d3hqdyyg6a6jrehrtyrb4nme9</sha1>
    </revision>
    <revision>
      <id>4090</id>
      <timestamp>2014-06-08T14:03:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513249346 - Original comment: **</comment>
      <origin>4090</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2907" sha1="k1vrpgddx2n11zrackhbrk8yyu7sb7o" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-08 14:03:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513249346&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**10/1**
|1 0 1&amp;gt;
3986.3137 cents
[[media type="file" key="jid_10_1_pluck_adu_dr55.mp3"]] [[file:xenharmonic/jid_10_1_pluck_adu_dr55.mp3|sound sample]]

The 10th harmonic; three octaves above [[5_4|5/4]].
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10_1&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;10/1&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|1 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
3986.3137 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_10_1_pluck_adu_dr55.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_10_1_pluck_adu_dr55.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_10_1_pluck_adu_dr55.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_10_1_pluck_adu_dr55.mp3/513246268/jid_10_1_pluck_adu_dr55.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_10_1_pluck_adu_dr55.mp3/513246268/jid_10_1_pluck_adu_dr55.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 10th harmonic; three octaves above &amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>k1vrpgddx2n11zrackhbrk8yyu7sb7o</sha1>
    </revision>
    <revision>
      <id>64418</id>
      <parentid>29</parentid>
      <timestamp>2021-03-27T03:48:39Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <comment>Making new wiki category</comment>
      <origin>64418</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="229" sha1="bpx2tzyk3nx6dp3y2e2tvcsk94sbjwq" xml:space="preserve">'''10/1'''
|1 0 1&amp;gt;

3986.3137 cents

[[File:jid_10_1_pluck_adu_dr55.mp3]] [[:File:jid_10_1_pluck_adu_dr55.mp3|sound sample]]

The 10th harmonic; three octaves above [[5/4|5/4]].

[[Category:Pages with internal sound examples]]</text>
      <sha1>bpx2tzyk3nx6dp3y2e2tvcsk94sbjwq</sha1>
    </revision>
    <revision>
      <id>77678</id>
      <parentid>64418</parentid>
      <timestamp>2021-09-09T11:36:25Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>77678</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="307" sha1="kp3en50gxxcj9wht7qkg5wkbu616r8f" xml:space="preserve">{{Infobox Interval
| JI glyph = 
| Ratio = 10/1
| Monzo = 1 0 1
| Cents = 3986.31371
| Name = 10th harmonic &lt;br&gt;Decatave
| Color name = 
| FJS name = 
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}

10/1 is the '''10th harmonic'''; three octaves above [[5/4|5/4]].

[[Category:Pages with internal sound examples]]</text>
      <sha1>kp3en50gxxcj9wht7qkg5wkbu616r8f</sha1>
    </revision>
    <revision>
      <id>77680</id>
      <parentid>77678</parentid>
      <timestamp>2021-09-09T11:39:29Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>77680</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="356" sha1="53lizt9aennyf9vcpa9avv8pjra4e63" xml:space="preserve">{{Infobox Interval
| JI glyph = 
| Ratio = 10/1
| Monzo = 1 0 1
| Cents = 3986.31371
| Name = 10th harmonic &lt;br&gt;decatave
| Color name = 
| FJS name = 
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}

'''10/1''' is the 10th harmonic; three octaves above [[5/4|5/4]].

[[Category:Interval]]
[[Category:Just interval]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>53lizt9aennyf9vcpa9avv8pjra4e63</sha1>
    </revision>
    <revision>
      <id>81730</id>
      <parentid>77680</parentid>
      <timestamp>2021-12-11T04:29:04Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Categories, misc. edits</comment>
      <origin>81730</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="330" sha1="a7bya09vbfda7pf0l2rnxg5c1ahs8mp" xml:space="preserve">{{Infobox Interval
| JI glyph = 
| Ratio = 10/1
| Monzo = 1 0 1
| Cents = 3986.31371
| Name = 10th harmonic &lt;br&gt;decatave
| Color name = 
| FJS name = 
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}

'''10/1''' is the 10th [[harmonic]]; three octaves above [[5/4]].

[[Category:Harmonics]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>a7bya09vbfda7pf0l2rnxg5c1ahs8mp</sha1>
    </revision>
    <revision>
      <id>81732</id>
      <parentid>81730</parentid>
      <timestamp>2021-12-11T04:31:43Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Typography edit</comment>
      <origin>81732</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="331" sha1="na789bq7quo6d12e7ns4u3lt6d5uufh" xml:space="preserve">{{Infobox Interval
| JI glyph = 
| Ratio = 10/1
| Monzo = 1 0 1
| Cents = 3986.31371
| Name = 10th harmonic, &lt;br&gt;decatave
| Color name = 
| FJS name = 
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}

'''10/1''' is the 10th [[harmonic]]; three octaves above [[5/4]].

[[Category:Harmonics]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>na789bq7quo6d12e7ns4u3lt6d5uufh</sha1>
    </revision>
    <revision>
      <id>82425</id>
      <parentid>81732</parentid>
      <timestamp>2021-12-22T01:34:49Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Added color and FJS names, misc. edits, categories</comment>
      <origin>82425</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="381" sha1="iuc33y5frbp4vz07v5gvxfnt6j6w6y3" xml:space="preserve">{{Infobox Interval
| Ratio = 10/1
| Monzo = 1 0 1
| Cents = 3986.31371
| Name = 10th harmonic, &lt;br&gt;decatave
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| FJS name = M24&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}

'''10/1''' is the 10th [[harmonic]]; three octaves above [[5/4]].

[[Category:5-limit]]
[[Category:Harmonics]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>iuc33y5frbp4vz07v5gvxfnt6j6w6y3</sha1>
    </revision>
    <revision>
      <id>82511</id>
      <parentid>82425</parentid>
      <timestamp>2021-12-23T12:43:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+see also</comment>
      <origin>82511</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="429" sha1="prgzzg91jrfiy17bosgxyqyfq3qe41r" xml:space="preserve">{{Infobox Interval
| Ratio = 10/1
| Monzo = 1 0 1
| Cents = 3986.31371
| Name = 10th harmonic, &lt;br&gt;decatave
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| FJS name = M24&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}

'''10/1''' is the 10th [[harmonic]]; three octaves above [[5/4]].

== See also ==
* [[Gallery of just intervals]]

[[Category:5-limit]]
[[Category:Harmonics]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>prgzzg91jrfiy17bosgxyqyfq3qe41r</sha1>
    </revision>
    <revision>
      <id>86277</id>
      <parentid>82511</parentid>
      <timestamp>2022-02-22T23:45:23Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86277</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="381" sha1="cyfqhvx76oxvl5e1x6fose2nrclwlyr" xml:space="preserve">{{Infobox Interval
| Ratio = 10/1
| Monzo = 1 0 1
| Cents = 3986.31371
| Name = 10th harmonic, &lt;br&gt;decatave
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| FJS name = M24&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}

'''10/1''' is the 10th [[harmonic]]; three octaves above [[5/4]].

== See also ==
* [[Gallery of just intervals]]

[[Category:5-limit]]
[[Category:Harmonics]]</text>
      <sha1>cyfqhvx76oxvl5e1x6fose2nrclwlyr</sha1>
    </revision>
    <revision>
      <id>97982</id>
      <parentid>86277</parentid>
      <timestamp>2022-10-25T11:54:11Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97982</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="251" sha1="cu5fbwsq3s6yid5ala1p2ofykm7670s" xml:space="preserve">{{Infobox Interval
| Name = 10th harmonic, decatave
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}

'''10/1''' is the 10th [[harmonic]]; three octaves above [[5/4]].

== See also ==
* [[Gallery of just intervals]]</text>
      <sha1>cu5fbwsq3s6yid5ala1p2ofykm7670s</sha1>
    </revision>
    <revision>
      <id>105495</id>
      <parentid>97982</parentid>
      <timestamp>2023-03-13T08:12:05Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>"decade" is an important term in audio engineering</comment>
      <origin>105495</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="315" sha1="447uqlp3ezli3r187ci09f0zjppvwuv" xml:space="preserve">{{Infobox Interval
| Name = 10th harmonic, decade, decatave
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}
{{Wikipedia| Decade (log scale) }}

'''10/1''' is the 10th [[harmonic]] or the '''decade'''; three octaves above [[5/4]]. 

== See also ==
* [[Gallery of just intervals]]</text>
      <sha1>447uqlp3ezli3r187ci09f0zjppvwuv</sha1>
    </revision>
    <revision>
      <id>118782</id>
      <parentid>105495</parentid>
      <timestamp>2023-07-06T02:24:23Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added to category “equivalence interval”</comment>
      <origin>118782</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="349" sha1="5ewqly2zbv4tph7fwtiz81fhm0q2771" xml:space="preserve">{{Infobox Interval
| Name = 10th harmonic, decade, decatave
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}
{{Wikipedia| Decade (log scale) }}

'''10/1''' is the 10th [[harmonic]] or the '''decade'''; three octaves above [[5/4]]. 

== See also ==
* [[Gallery of just intervals]]
[[Category:Equivalence interval]]</text>
      <sha1>5ewqly2zbv4tph7fwtiz81fhm0q2771</sha1>
    </revision>
    <revision>
      <id>118825</id>
      <parentid>118782</parentid>
      <timestamp>2023-07-06T02:56:13Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Equivalence interval]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>118825</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="315" sha1="447uqlp3ezli3r187ci09f0zjppvwuv" xml:space="preserve">{{Infobox Interval
| Name = 10th harmonic, decade, decatave
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}
{{Wikipedia| Decade (log scale) }}

'''10/1''' is the 10th [[harmonic]] or the '''decade'''; three octaves above [[5/4]]. 

== See also ==
* [[Gallery of just intervals]]</text>
      <sha1>447uqlp3ezli3r187ci09f0zjppvwuv</sha1>
    </revision>
    <revision>
      <id>126125</id>
      <parentid>118825</parentid>
      <timestamp>2023-10-18T08:54:44Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+symbol, applications</comment>
      <origin>126125</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="417" sha1="clrpqtzdl205nwr0nk0u76tuo8s4yf0" xml:space="preserve">{{Infobox Interval
| Name = 10th harmonic, decade, decatave
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}
{{Wikipedia| Decade (log scale) }}

'''10/1''' is the '''10th harmonic''' or the '''decade''' (symbol: '''dec'''); three octaves above [[5/4]]. As an [[interval size measure]] it sees many applications in audio engineering. 

== See also ==
* [[Gallery of just intervals]]</text>
      <sha1>clrpqtzdl205nwr0nk0u76tuo8s4yf0</sha1>
    </revision>
    <revision>
      <id>129169</id>
      <parentid>126125</parentid>
      <timestamp>2023-12-16T06:40:23Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Expansion</comment>
      <origin>129169</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="683" sha1="185js26o46ntor4l7ipuqm08qdw7dwz" xml:space="preserve">{{Infobox Interval
| Name = 10th harmonic, decade
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}
{{Wikipedia| Decade (log scale) }}

'''10/1''', the '''10th harmonic''' or the '''decade''' (symbol: '''dec'''), is the [[harmonic]] past [[9/1]] and before [[11/1]]. It is three [[octave]]s above [[5/4]]. As a coarse [[interval size measure]] it sees many applications in audio engineering. In harmony, it can be approached as a stack of [[2/1]] and [[5/1]], making chords like 1-2-10, 1-5-10, or 1-2-5-10. 

It is sometimes known as a ''decave'' or ''decatave''. 

== See also ==
* [[Gallery of just intervals]]
* [[Decave]] (disambiguation page)</text>
      <sha1>185js26o46ntor4l7ipuqm08qdw7dwz</sha1>
    </revision>
    <revision>
      <id>162433</id>
      <parentid>129169</parentid>
      <timestamp>2024-10-27T00:07:20Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* See also */ link</comment>
      <origin>162433</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="694" sha1="jgjrjle3gyxe46ya9e686des72f0egd" xml:space="preserve">{{Infobox Interval
| Name = 10th harmonic, decade
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}
{{Wikipedia| Decade (log scale) }}

'''10/1''', the '''10th harmonic''' or the '''decade''' (symbol: '''dec'''), is the [[harmonic]] past [[9/1]] and before [[11/1]]. It is three [[octave]]s above [[5/4]]. As a coarse [[interval size measure]] it sees many applications in audio engineering. In harmony, it can be approached as a stack of [[2/1]] and [[5/1]], making chords like 1-2-10, 1-5-10, or 1-2-5-10. 

It is sometimes known as a ''decave'' or ''decatave''. 

== See also ==
* [[Gallery of just intervals]]
* [[Decave]] (disambiguation page)
* [[Ed10]]</text>
      <sha1>jgjrjle3gyxe46ya9e686des72f0egd</sha1>
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    <revision>
      <id>168274</id>
      <parentid>162433</parentid>
      <timestamp>2024-12-02T14:08:06Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Taxicab</comment>
      <origin>168274</origin>
      <model>wikitext</model>
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      <text bytes="728" sha1="o14j8jsjr6cypdp2ala647lb4zzs16v" xml:space="preserve">{{Infobox Interval
| Name = 10th harmonic, decade
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}
{{Wikipedia| Decade (log scale) }}

'''10/1''', the '''10th harmonic''' or the '''decade''' (symbol: '''dec'''), is the [[harmonic]] past [[9/1]] and before [[11/1]]. It is three [[octave]]s above [[5/4]]. As a coarse [[interval size measure]] it sees many applications in audio engineering. In harmony, it can be approached as a stack of [[2/1]] and [[5/1]], making chords like 1-2-10, 1-5-10, or 1-2-5-10. 

It is sometimes known as a ''decave'' or ''decatave''. 

== See also ==
* [[Gallery of just intervals]]
* [[Decave]] (disambiguation page)
* [[Ed10]]

[[Category:Taxicab-2 intervals]]</text>
      <sha1>o14j8jsjr6cypdp2ala647lb4zzs16v</sha1>
    </revision>
    <revision>
      <id>189100</id>
      <parentid>168274</parentid>
      <timestamp>2025-03-31T10:32:19Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+"decuple"</comment>
      <origin>189100</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="748" sha1="dhgzw9dddjur40pi0s4wfwqaoakp100" xml:space="preserve">{{Infobox Interval
| Name = 10th harmonic, decade, decuple
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}
{{Wikipedia| Decade (log scale) }}

'''10/1''', the '''10th harmonic''', '''decade''' or '''decuple''' (symbol: '''dec'''), is the [[harmonic]] past [[9/1]] and before [[11/1]]. It is three [[octave]]s above [[5/4]]. As a coarse [[interval size measure]] it sees many applications in audio engineering. In harmony, it can be approached as a stack of [[2/1]] and [[5/1]], making chords like 1-2-10, 1-5-10, or 1-2-5-10. 

It is sometimes known as a ''decave'' or ''decatave''. 

== See also ==
* [[Gallery of just intervals]]
* [[Decave]] (disambiguation page)
* [[Ed10]]

[[Category:Taxicab-2 intervals]]</text>
      <sha1>dhgzw9dddjur40pi0s4wfwqaoakp100</sha1>
    </revision>
    <revision>
      <id>189594</id>
      <parentid>189100</parentid>
      <timestamp>2025-04-02T10:59:11Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Style</comment>
      <origin>189594</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="762" sha1="2i0k4lbttecrntay980vhtz3s8e5r65" xml:space="preserve">{{Infobox Interval
| Name = 10th harmonic, decade, decuple
| Color name = c&lt;sup&gt;3&lt;/sup&gt;y3, tricoyo 3rd
| Sound = jid_10_1_pluck_adu_dr55.mp3
}}
{{Wikipedia| Decade (log scale) }}

'''10/1''', the '''10th harmonic''', '''decade''' or '''decuple''' (symbol: '''dec'''), is the [[harmonic]] past [[9/1]] and before [[11/1]]. It is three [[octave]]s above [[5/4]]. As a coarse [[interval size measure]] it sees many applications in audio engineering. In harmony, it can be approached as a stack of [[2/1]] and [[5/1]], making chords like 1–2–10, 1–5–10, or 1–2–5–10. 

It is sometimes known as a ''decave'' or ''decatave''. 

== See also ==
* [[Gallery of just intervals]]
* [[Decave]] (disambiguation page)
* [[Ed10]]

[[Category:Taxicab-2 intervals]]</text>
      <sha1>2i0k4lbttecrntay980vhtz3s8e5r65</sha1>
    </revision>
  </page>
  <page>
    <title>10 1</title>
    <ns>0</ns>
    <id>30</id>
    <redirect title="10/1" />
    <revision>
      <id>30</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>30</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18" sha1="q3ul17kavmjnl0m000cj64qplj56hm4" xml:space="preserve">#REDIRECT [[10/1]]</text>
      <sha1>q3ul17kavmjnl0m000cj64qplj56hm4</sha1>
    </revision>
  </page>
  <page>
    <title>10/3</title>
    <ns>0</ns>
    <id>31</id>
    <revision>
      <id>31</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>31</origin>
      <model>wikitext</model>
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      <text bytes="207" sha1="pnpv87pe8qso01ojqpx7tfj3pc8cj5e" xml:space="preserve">'''10/3'''
|1 -1 1&amp;gt;

2084.3587 cents

[[File:jid_10_3_pluck_adu_dr110.mp3]] [[:File:jid_10_3_pluck_adu_dr110.mp3|sound sample]]

The ratio between the 3rd and 10th harmonics; one octave above [[5/3|5/3]].</text>
      <sha1>pnpv87pe8qso01ojqpx7tfj3pc8cj5e</sha1>
    </revision>
    <revision>
      <id>4091</id>
      <timestamp>2014-06-08T13:56:08Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513248768 - Original comment: **</comment>
      <origin>4091</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2966" sha1="7nfgir527x3vfh0zihxu7zcz7pp4qht" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-08 13:56:08 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513248768&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**10/3**
|1 -1 1&amp;gt;
2084.3587 cents
[[media type="file" key="jid_10_3_pluck_adu_dr110.mp3"]] [[file:xenharmonic/jid_10_3_pluck_adu_dr110.mp3|sound sample]]

The ratio between the 3rd and 10th harmonics; one octave above [[5_3|5/3]].
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10_3&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;10/3&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|1 -1 1&amp;amp;gt;&amp;lt;br /&amp;gt;
2084.3587 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_10_3_pluck_adu_dr110.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_10_3_pluck_adu_dr110.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_10_3_pluck_adu_dr110.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_10_3_pluck_adu_dr110.mp3/513246280/jid_10_3_pluck_adu_dr110.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_10_3_pluck_adu_dr110.mp3/513246280/jid_10_3_pluck_adu_dr110.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The ratio between the 3rd and 10th harmonics; one octave above &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>7nfgir527x3vfh0zihxu7zcz7pp4qht</sha1>
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    <revision>
      <id>64422</id>
      <parentid>31</parentid>
      <timestamp>2021-03-27T03:51:51Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
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      <text bytes="256" sha1="h7bzxdrpfr596i0un7j228j4pz6nmpj" xml:space="preserve">'''10/3'''
|1 -1 1&amp;gt;

2084.3587 cents

[[File:jid_10_3_pluck_adu_dr110.mp3]] [[:File:jid_10_3_pluck_adu_dr110.mp3|sound sample]]

The ratio between the 3rd and 10th harmonics; one octave above [[5/3|5/3]].

[[Category:Pages with internal sound examples]]</text>
      <sha1>h7bzxdrpfr596i0un7j228j4pz6nmpj</sha1>
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    <revision>
      <id>86278</id>
      <parentid>64422</parentid>
      <timestamp>2022-02-22T23:46:38Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86278</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="207" sha1="pnpv87pe8qso01ojqpx7tfj3pc8cj5e" xml:space="preserve">'''10/3'''
|1 -1 1&amp;gt;

2084.3587 cents

[[File:jid_10_3_pluck_adu_dr110.mp3]] [[:File:jid_10_3_pluck_adu_dr110.mp3|sound sample]]

The ratio between the 3rd and 10th harmonics; one octave above [[5/3|5/3]].</text>
      <sha1>pnpv87pe8qso01ojqpx7tfj3pc8cj5e</sha1>
    </revision>
    <revision>
      <id>93915</id>
      <parentid>86278</parentid>
      <timestamp>2022-08-04T18:57:08Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox</comment>
      <origin>93915</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="323" sha1="nkt4s6wkqi2f1reo0fhbhtpgctaoy58" xml:space="preserve">{{Infobox Interval
| Ratio = 10/3
| Monzo = 1 -1 1
| Cents = 2084.35871
| Name = classic/just major thirteenth
| Color name = y13, yo 13th
| FJS name = M13&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_3_pluck_adu_dr110.mp3
}}

'''10/3''' is the [[ratio]] between the 3rd and 10th [[harmonic]]s; one octave above [[5/3]].

[[Category:Stub]]</text>
      <sha1>nkt4s6wkqi2f1reo0fhbhtpgctaoy58</sha1>
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    <revision>
      <id>93919</id>
      <parentid>93915</parentid>
      <timestamp>2022-08-04T19:13:11Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Categories</comment>
      <origin>93919</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="344" sha1="236qbza4kygzwyc9a3hjcfvg6ha73e6" xml:space="preserve">{{Infobox Interval
| Ratio = 10/3
| Monzo = 1 -1 1
| Cents = 2084.35871
| Name = classic/just major thirteenth
| Color name = y13, yo 13th
| FJS name = M13&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_3_pluck_adu_dr110.mp3
}}

'''10/3''' is the [[ratio]] between the 3rd and 10th [[harmonic]]s; one octave above [[5/3]].

[[Category:5-limit]]
[[Category:Stub]]</text>
      <sha1>236qbza4kygzwyc9a3hjcfvg6ha73e6</sha1>
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    <revision>
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      <parentid>93919</parentid>
      <timestamp>2022-08-04T20:59:55Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>3 digits (→ [[Conventions]])</comment>
      <origin>93923</origin>
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      <text bytes="342" sha1="pkasy9jsuz1oma2xthg6s1p0u7akd6g" xml:space="preserve">{{Infobox Interval
| Ratio = 10/3
| Monzo = 1 -1 1
| Cents = 2084.359
| Name = classic/just major thirteenth
| Color name = y13, yo 13th
| FJS name = M13&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_3_pluck_adu_dr110.mp3
}}

'''10/3''' is the [[ratio]] between the 3rd and 10th [[harmonic]]s; one octave above [[5/3]].

[[Category:5-limit]]
[[Category:Stub]]</text>
      <sha1>pkasy9jsuz1oma2xthg6s1p0u7akd6g</sha1>
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      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97943</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="241" sha1="gcxpjctbtbrnyjr8s5td1w94e0pyjqr" xml:space="preserve">{{Infobox Interval
| Name = classic/just major thirteenth
| Color name = y13, yo 13th
| Sound = jid_10_3_pluck_adu_dr110.mp3
}}

'''10/3''' is the [[ratio]] between the 3rd and 10th [[harmonic]]s; one octave above [[5/3]].

[[Category:Stub]]</text>
      <sha1>gcxpjctbtbrnyjr8s5td1w94e0pyjqr</sha1>
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    <revision>
      <id>120007</id>
      <parentid>97943</parentid>
      <timestamp>2023-07-11T21:49:17Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Stub]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>120007</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="222" sha1="2hgy00j24d311wxgdp333abrh7kz1z7" xml:space="preserve">{{Infobox Interval
| Name = classic/just major thirteenth
| Color name = y13, yo 13th
| Sound = jid_10_3_pluck_adu_dr110.mp3
}}

'''10/3''' is the [[ratio]] between the 3rd and 10th [[harmonic]]s; one octave above [[5/3]].</text>
      <sha1>2hgy00j24d311wxgdp333abrh7kz1z7</sha1>
    </revision>
    <revision>
      <id>158729</id>
      <parentid>120007</parentid>
      <timestamp>2024-10-06T00:36:54Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>See also</comment>
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      <text bytes="243" sha1="6vrfx3k5ltqmdgcriceg5gvlw91u6r0" xml:space="preserve">{{Infobox Interval
| Name = classic/just major thirteenth
| Color name = y13, yo 13th
| Sound = jid_10_3_pluck_adu_dr110.mp3
}}

'''10/3''' is the [[ratio]] between the 3rd and 10th [[harmonic]]s; one octave above [[5/3]]. See also [[ed10/3]].</text>
      <sha1>6vrfx3k5ltqmdgcriceg5gvlw91u6r0</sha1>
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  </page>
  <page>
    <title>10 3</title>
    <ns>0</ns>
    <id>32</id>
    <redirect title="10/3" />
    <revision>
      <id>32</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>32</origin>
      <model>wikitext</model>
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      <text bytes="18" sha1="2oi68m68kyof9to8mjmcffsftnll1bz" xml:space="preserve">#REDIRECT [[10/3]]</text>
      <sha1>2oi68m68kyof9to8mjmcffsftnll1bz</sha1>
    </revision>
  </page>
  <page>
    <title>10/7</title>
    <ns>0</ns>
    <id>33</id>
    <revision>
      <id>33</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>33</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="961" sha1="qnpuv7axf19d2limobj5fexmn8oneyv" xml:space="preserve">{| class="wikitable"
|-
| | [[File:ji_glyph_10_7.png|alt=ji glyph 10 7.png|148x118px|ji glyph 10 7.png]]
|-
| | JI Glyph for 10/7
|}

'''10/7'''
|1 0 1 -1&amp;gt;

617.48781 cents

Type: Utonal 

[[File:jid_10_7_pluck_adu_dr220.mp3]] [[:File:jid_10_7_pluck_adu_dr220.mp3|sound sample]]

In [[7-limit|7-limit]] [[Just_intonation|Just Intonation]], 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, [[7/5|7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4|5/4]]) appears above the harmonic seventh ([[7/4|7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET|88cET]], which has a good approximation of 10/7 -- but no 7/5.)

See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]      [[Category:7-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:ratio]]
[[Category:tritone]]</text>
      <sha1>qnpuv7axf19d2limobj5fexmn8oneyv</sha1>
    </revision>
    <revision>
      <id>4092</id>
      <timestamp>2015-03-24T20:46:11Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;k9assassin</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 545215048 - Original comment: **</comment>
      <origin>4092</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5154" sha1="jmey6va5hqq1wxdgzz7d97d5ubh2n48" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:k9assassin|k9assassin]] and made on &lt;tt&gt;2015-03-24 20:46:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;545215048&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[image:ji glyph 10 7.png width="148" height="118" caption="JI Glyph for 10/7
"]]

**10/7**
|1 0 1 -1&amp;gt;
617.48781 cents
Type: Utonal 
[[media type="file" key="jid_10_7_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_10_7_pluck_adu_dr220.mp3|sound sample]]

In [[7-limit]] [[Just Intonation]], 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, [[7_5|7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5_4|5/4]]) appears above the harmonic seventh ([[7_4|7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10_7&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:1:&amp;amp;lt;img src=&amp;amp;quot;/file/view/ji%20glyph%2010%207.png/545214478/148x118/ji%20glyph%2010%207.png&amp;amp;quot; alt=&amp;amp;quot;JI Glyph for 10/7&amp;amp;quot; title=&amp;amp;quot;JI Glyph for 10/7&amp;amp;quot; style=&amp;amp;quot;height: 118px; width: 148px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/ji%20glyph%2010%207.png/545214478/148x118/ji%20glyph%2010%207.png" alt="ji glyph 10 7.png" title="ji glyph 10 7.png" style="height: 118px; width: 148px;" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;JI Glyph for 10/7&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;10/7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|1 0 1 -1&amp;amp;gt;&amp;lt;br /&amp;gt;
617.48781 cents&amp;lt;br /&amp;gt;
Type: Utonal &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_10_7_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_10_7_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_10_7_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_10_7_pluck_adu_dr220.mp3/513184772/jid_10_7_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_10_7_pluck_adu_dr220.mp3/513184772/jid_10_7_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, &amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5&amp;lt;/a&amp;gt;, but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third (&amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt;) appears above the harmonic seventh (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;), such as 4:6:7:10. (This particular chord is well-approximated in &amp;lt;a class="wiki_link" href="/88cET"&amp;gt;88cET&amp;lt;/a&amp;gt;, which has a good approximation of 10/7 -- but no 7/5.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>jmey6va5hqq1wxdgzz7d97d5ubh2n48</sha1>
    </revision>
    <revision>
      <id>4093</id>
      <timestamp>2014-06-07T16:45:41Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513197632 - Original comment: **</comment>
      <origin>4093</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4267" sha1="8g3mi7vu3fs0m3j0y9swgusifglnprj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-07 16:45:41 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513197632&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**10/7**
|1 0 1 -1&amp;gt;
617.48781 cents
[[media type="file" key="jid_10_7_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_10_7_pluck_adu_dr220.mp3|sound sample]]

In [[7-limit]] [[Just Intonation]], 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, [[7_5|7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5_4|5/4]]) appears above the harmonic seventh ([[7_4|7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10_7&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;10/7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|1 0 1 -1&amp;amp;gt;&amp;lt;br /&amp;gt;
617.48781 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_10_7_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_10_7_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_10_7_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_10_7_pluck_adu_dr220.mp3/513184772/jid_10_7_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_10_7_pluck_adu_dr220.mp3/513184772/jid_10_7_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, &amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5&amp;lt;/a&amp;gt;, but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third (&amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt;) appears above the harmonic seventh (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;), such as 4:6:7:10. (This particular chord is well-approximated in &amp;lt;a class="wiki_link" href="/88cET"&amp;gt;88cET&amp;lt;/a&amp;gt;, which has a good approximation of 10/7 -- but no 7/5.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>8g3mi7vu3fs0m3j0y9swgusifglnprj</sha1>
    </revision>
    <revision>
      <id>4094</id>
      <timestamp>2014-06-07T12:31:50Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513184806 - Original comment: **</comment>
      <origin>4094</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3803" sha1="ml1wyj11sgv85yen7wwh81z8tca304f" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-07 12:31:50 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513184806&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**10/7**
|1 0 1 -1&amp;gt;
617.48781 cents
[[media type="file" key="jid_10_7_pluck_adu_dr220.mp3"]]

In [[7-limit]] [[Just Intonation]], 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, [[7_5|7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5_4|5/4]]) appears above the harmonic seventh ([[7_4|7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10_7&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;10/7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|1 0 1 -1&amp;amp;gt;&amp;lt;br /&amp;gt;
617.48781 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_10_7_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_10_7_pluck_adu_dr220.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_10_7_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, &amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5&amp;lt;/a&amp;gt;, but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third (&amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt;) appears above the harmonic seventh (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;), such as 4:6:7:10. (This particular chord is well-approximated in &amp;lt;a class="wiki_link" href="/88cET"&amp;gt;88cET&amp;lt;/a&amp;gt;, which has a good approximation of 10/7 -- but no 7/5.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ml1wyj11sgv85yen7wwh81z8tca304f</sha1>
    </revision>
    <revision>
      <id>4095</id>
      <timestamp>2011-09-28T20:27:27Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 259413234 - Original comment: **</comment>
      <origin>4095</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2624" sha1="swg7hwx5rpp4hrgi5l13oq73qvydohw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-09-28 20:27:27 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;259413234&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[7-limit]] [[Just Intonation]], 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, [[7_5|7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5_4|5/4]]) appears above the harmonic seventh ([[7_4|7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10_7&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, &amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5&amp;lt;/a&amp;gt;, but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third (&amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt;) appears above the harmonic seventh (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;), such as 4:6:7:10. (This particular chord is well-approximated in &amp;lt;a class="wiki_link" href="/88cET"&amp;gt;88cET&amp;lt;/a&amp;gt;, which has a good approximation of 10/7 -- but no 7/5.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>swg7hwx5rpp4hrgi5l13oq73qvydohw</sha1>
    </revision>
    <revision>
      <id>36557</id>
      <parentid>33</parentid>
      <timestamp>2018-10-11T14:22:51Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>switched to infobox interval</comment>
      <origin>36557</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="948" sha1="d5mlv6yi8veuhervpf4k55i4g18k5hu" xml:space="preserve">{{Infobox Interval
| Icon = [[File:ji_glyph_10_7.png|alt=ji glyph 10 7.png|148x118px|ji glyph 10 7.png]] &lt;small&gt;&lt;br/&gt;JI Glyph&lt;/small&gt;
| Ratio = 10/7
| Cents = 617.48781 
| Monzo = 1 0 1 -1 
| Name = high tritone
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

Type: Utonal 

In [[7-limit|7-limit]] [[Just_intonation|Just Intonation]], 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, [[7/5|7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4|5/4]]) appears above the harmonic seventh ([[7/4|7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET|88cET]], which has a good approximation of 10/7 -- but no 7/5.)

See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]      [[Category:7-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:ratio]]
[[Category:tritone]]</text>
      <sha1>d5mlv6yi8veuhervpf4k55i4g18k5hu</sha1>
    </revision>
    <revision>
      <id>36558</id>
      <parentid>36557</parentid>
      <timestamp>2018-10-11T14:24:31Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>simplified links</comment>
      <origin>36558</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="885" sha1="djr0slo948vk4uurc1n3flkq50u7lsr" xml:space="preserve">{{Infobox Interval
| Icon = [[File:ji_glyph_10_7.png|alt=ji glyph 10 7.png|148x118px|ji glyph 10 7.png]] &lt;small&gt;&lt;br/&gt;JI Glyph&lt;/small&gt;
| Ratio = 10/7
| Cents = 617.48781 
| Monzo = 1 0 1 -1 
| Name = high tritone
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

Type: Utonal 

In [[7-limit]] [[Just Intonation]], 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

:''See also [[Gallery of Just Intervals]]''

[[Category:7-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:ratio]]
[[Category:tritone]]</text>
      <sha1>djr0slo948vk4uurc1n3flkq50u7lsr</sha1>
    </revision>
    <revision>
      <id>36560</id>
      <parentid>36558</parentid>
      <timestamp>2018-10-11T14:37:16Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>consistent parameter order in [[Template:Infobox Interval]]</comment>
      <origin>36560</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="885" sha1="m9mscjkxinmjmnxupn3hlergeyse92x" xml:space="preserve">{{Infobox Interval
| Icon = [[File:ji_glyph_10_7.png|alt=ji glyph 10 7.png|148x118px|ji glyph 10 7.png]] &lt;small&gt;&lt;br/&gt;JI Glyph&lt;/small&gt;
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

Type: Utonal 

In [[7-limit]] [[Just Intonation]], 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

:''See also [[Gallery of Just Intervals]]''

[[Category:7-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:ratio]]
[[Category:tritone]]</text>
      <sha1>m9mscjkxinmjmnxupn3hlergeyse92x</sha1>
    </revision>
    <revision>
      <id>37349</id>
      <parentid>36560</parentid>
      <timestamp>2018-10-17T22:21:34Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>unified glyph</comment>
      <origin>37349</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="855" sha1="m6ldademm47iqubzd4cvdjvycqrlr0w" xml:space="preserve">{{Infobox Interval
| Icon = [[File:ji_glyph_10_7.png|124px]] &lt;small&gt;&lt;br/&gt;[[JI Glyphs|JI Glyph]]&lt;/small&gt;
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

Type: Utonal 

In [[7-limit]] [[Just Intonation]], 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

:''See also [[Gallery of Just Intervals]]''

[[Category:7-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:ratio]]
[[Category:tritone]]</text>
      <sha1>m6ldademm47iqubzd4cvdjvycqrlr0w</sha1>
    </revision>
    <revision>
      <id>37350</id>
      <parentid>37349</parentid>
      <timestamp>2018-10-17T22:21:52Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>linkfix</comment>
      <origin>37350</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="855" sha1="ku3dosp6ld7e9un5wvwc8hoc9pr37dn" xml:space="preserve">{{Infobox Interval
| Icon = [[File:ji_glyph_10_7.png|124px]] &lt;small&gt;&lt;br/&gt;[[JI glyphs|JI glyph]]&lt;/small&gt;
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

Type: Utonal 

In [[7-limit]] [[Just Intonation]], 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

:''See also [[Gallery of Just Intervals]]''

[[Category:7-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:ratio]]
[[Category:tritone]]</text>
      <sha1>ku3dosp6ld7e9un5wvwc8hoc9pr37dn</sha1>
    </revision>
    <revision>
      <id>37518</id>
      <parentid>37350</parentid>
      <timestamp>2018-10-22T14:13:12Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>updated interval infobox</comment>
      <origin>37518</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="816" sha1="kvge0ms1y3r8xnukbbukjeizma75gzh" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

Type: Utonal 

In [[7-limit]] [[Just Intonation]], 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

:''See also [[Gallery of Just Intervals]]''

[[Category:7-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:ratio]]
[[Category:tritone]]</text>
      <sha1>kvge0ms1y3r8xnukbbukjeizma75gzh</sha1>
    </revision>
    <revision>
      <id>37577</id>
      <parentid>37518</parentid>
      <timestamp>2018-10-22T22:26:41Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>added the color name</comment>
      <origin>37577</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="845" sha1="cbkprs6q3k6w1rjtjkxqgytjc3k6c3m" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone
| Color name = ry4, ruyo 4th
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

Type: Utonal 

In [[7-limit]] [[Just Intonation]], 10/7 is a high tritone measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

:''See also [[Gallery of Just Intervals]]''

[[Category:7-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:ratio]]
[[Category:tritone]]</text>
      <sha1>cbkprs6q3k6w1rjtjkxqgytjc3k6c3m</sha1>
    </revision>
    <revision>
      <id>37644</id>
      <parentid>37577</parentid>
      <timestamp>2018-10-23T17:03:25Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>lemmata bold, expanded see also sections</comment>
      <origin>37644</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="962" sha1="mjuz5yziq78j2vm29kb5n934jlp5pfv" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone
| Color name = ry4, ruyo 4th
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

Type: Utonal 

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high tritone''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

== See also ==
* [[Gallery of Just Intervals]]
* [[Tritone]]
* [http://en.wikipedia.org/wiki/Tritone Tritone - Wikipedia]

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Just interval]]
[[Category:Ratio]]
[[Category:Tritone]]</text>
      <sha1>mjuz5yziq78j2vm29kb5n934jlp5pfv</sha1>
    </revision>
    <revision>
      <id>46313</id>
      <parentid>37644</parentid>
      <timestamp>2020-06-10T07:10:27Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cats</comment>
      <origin>46313</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="949" sha1="jramynemqsdxil3aj9mzjoruassurwy" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone
| Color name = ry4, ruyo 4th
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

Type: Utonal 

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high tritone''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

== See also ==
* [[Gallery of Just Intervals]]
* [[Tritone]]
* [http://en.wikipedia.org/wiki/Tritone Tritone - Wikipedia]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Tritone]]</text>
      <sha1>jramynemqsdxil3aj9mzjoruassurwy</sha1>
    </revision>
    <revision>
      <id>48559</id>
      <parentid>46313</parentid>
      <timestamp>2020-09-05T03:47:13Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>48559</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="969" sha1="cxlkwlbbm9o7hj1azi3lix9c7qcgy8c" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone
| Color name = ry4, ruyo 4th
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

Type: Utonal 

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high tritone''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

== See also ==
* [[Gallery of Just Intervals]]
* [[Tritone]]
* [http://en.wikipedia.org/wiki/Tritone Tritone - Wikipedia]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Tritone]]
[[Category:Over-7]]</text>
      <sha1>cxlkwlbbm9o7hj1azi3lix9c7qcgy8c</sha1>
    </revision>
    <revision>
      <id>49857</id>
      <parentid>48559</parentid>
      <timestamp>2020-09-19T14:35:59Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+FJS name; cleanup</comment>
      <origin>49857</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1097" sha1="8ic5c24yllzwmsuz2j7mvgk8afsgu8g" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone, &lt;br&gt;Euler's tritone
| Color name = ry4, ruyo 4th
| FJS name = A4&lt;sup&gt;5&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high tritone''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

== See also ==
* [[Tritone]]
* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal_tritone|Septimal tritone - Wikipedia]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Tritone]]
[[Category:Over-7]]</text>
      <sha1>8ic5c24yllzwmsuz2j7mvgk8afsgu8g</sha1>
    </revision>
    <revision>
      <id>50385</id>
      <parentid>49857</parentid>
      <timestamp>2020-09-27T15:25:41Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>tritone link in text, removed from see-also section</comment>
      <origin>50385</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1088" sha1="31uqk8y4a6ox8rc3w1h5nu9dupsaxfn" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone, &lt;br&gt;Euler's tritone
| Color name = ry4, ruyo 4th
| FJS name = A4&lt;sup&gt;5&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal_tritone|Septimal tritone - Wikipedia]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Tritone]]
[[Category:Over-7]]</text>
      <sha1>31uqk8y4a6ox8rc3w1h5nu9dupsaxfn</sha1>
    </revision>
    <revision>
      <id>57423</id>
      <parentid>50385</parentid>
      <timestamp>2020-12-23T23:03:46Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>57423</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1513" sha1="qgjkukaxau23a6yqi6g6lhjake93p78" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone, &lt;br&gt;Euler's tritone
| Color name = ry4, ruyo 4th
| FJS name = A4&lt;sup&gt;5&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10. (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7 -- but no 7/5.)  While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a scale that demonstrates [[Wikipedia:Rothenberg propriety|Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal_tritone|Septimal tritone - Wikipedia]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Tritone]]
[[Category:Over-7]]</text>
      <sha1>qgjkukaxau23a6yqi6g6lhjake93p78</sha1>
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| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone, &lt;br&gt;Euler's tritone
| Color name = ry4, ruyo 4th
| FJS name = A4&lt;sup&gt;5&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10 (This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5). While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a scale that demonstrates [[Wikipedia:Rothenberg propriety|Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal_tritone|Septimal tritone - Wikipedia]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Tritone]]
[[Category:Over-7]]</text>
      <sha1>9uoftzis24ybww2szpa03lqukoe6n39</sha1>
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      <timestamp>2020-12-23T23:04:33Z</timestamp>
      <contributor>
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| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone, &lt;br&gt;Euler's tritone
| Color name = ry4, ruyo 4th
| FJS name = A4&lt;sup&gt;5&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10- This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a scale that demonstrates [[Wikipedia:Rothenberg propriety|Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal_tritone|Septimal tritone - Wikipedia]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Tritone]]
[[Category:Over-7]]</text>
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      <contributor>
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| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone, &lt;br&gt;Euler's tritone
| Color name = ry4, ruyo 4th
| FJS name = A4&lt;sup&gt;5&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10- This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a heptatonic scale that demonstrates [[Wikipedia:Rothenberg propriety|Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal_tritone|Septimal tritone - Wikipedia]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Tritone]]
[[Category:Over-7]]</text>
      <sha1>02qcdxlhxlhgn6f5z8t17eum12g54kd</sha1>
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      <parentid>57427</parentid>
      <timestamp>2021-03-27T03:52:22Z</timestamp>
      <contributor>
        <username>Userminusone</username>
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      <origin>64423</origin>
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| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone, &lt;br&gt;Euler's tritone
| Color name = ry4, ruyo 4th
| FJS name = A4&lt;sup&gt;5&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10- This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a heptatonic scale that demonstrates [[Wikipedia:Rothenberg propriety|Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal_tritone|Septimal tritone - Wikipedia]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Tritone]]
[[Category:Over-7]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>654t5rujgwpw3gf6v2w3wu0y85vofhc</sha1>
    </revision>
    <revision>
      <id>66272</id>
      <parentid>64423</parentid>
      <timestamp>2021-04-12T02:39:46Z</timestamp>
      <contributor>
        <username>Inthar</username>
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| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone, &lt;br&gt;Euler's tritone
| Color name = ry4, ruyo 4th
| FJS name = A4&lt;sup&gt;5&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10- This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a heptatonic scale that demonstrates [[Wikipedia:Rothenberg propriety|Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal_tritone|Septimal tritone - Wikipedia]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Tritone]]
[[Category:Over-7]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>i35b71wtb1tsjw9vzv363unwegeokmc</sha1>
    </revision>
    <revision>
      <id>66329</id>
      <parentid>66272</parentid>
      <timestamp>2021-04-12T06:52:54Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>Undo revision 66272 by [[Special:Contributions/IlL|IlL]] ([[User talk:IlL|talk]])</comment>
      <origin>66329</origin>
      <model>wikitext</model>
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| JI glyph = [[File:ji_glyph_10_7.png|x48px]]
| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone, &lt;br&gt;Euler's tritone
| Color name = ry4, ruyo 4th
| FJS name = A4&lt;sup&gt;5&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10- This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a heptatonic scale that demonstrates [[Wikipedia:Rothenberg propriety|Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal_tritone|Septimal tritone - Wikipedia]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Tritone]]
[[Category:Over-7]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>654t5rujgwpw3gf6v2w3wu0y85vofhc</sha1>
    </revision>
    <revision>
      <id>66359</id>
      <parentid>66329</parentid>
      <timestamp>2021-04-12T09:10:52Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Undo revision 66329 by [[Special:Contributions/Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]])</comment>
      <origin>66359</origin>
      <model>wikitext</model>
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| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone, &lt;br&gt;Euler's tritone
| Color name = ry4, ruyo 4th
| FJS name = A4&lt;sup&gt;5&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10- This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a heptatonic scale that demonstrates [[Wikipedia:Rothenberg propriety|Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal_tritone|Septimal tritone - Wikipedia]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Tritone]]
[[Category:Over-7]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>i35b71wtb1tsjw9vzv363unwegeokmc</sha1>
    </revision>
    <revision>
      <id>83058</id>
      <parentid>66359</parentid>
      <timestamp>2022-01-01T21:31:12Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Added Wikipedia box, misc. edits, categories</comment>
      <origin>83058</origin>
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| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone, &lt;br&gt;greater septimal tritone, &lt;br&gt;Euler's tritone
| Color name = ry4, ruyo 4th
| FJS name = A4&lt;sup&gt;5&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}
{{Wikipedia|Septimal tritone}}

In [[7-limit]] [[just intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10- This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a heptatonic scale that demonstrates [[Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:7-limit]]
[[Category:Tritone]]
[[Category:Over-7]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>smtddy0qtz1zlgz0snrumlo68vfj4n1</sha1>
    </revision>
    <revision>
      <id>86290</id>
      <parentid>83058</parentid>
      <timestamp>2022-02-22T23:46:38Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86290</origin>
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| Ratio = 10/7
| Monzo = 1 0 1 -1 
| Cents = 617.48781 
| Name = high tritone, &lt;br&gt;greater septimal tritone, &lt;br&gt;Euler's tritone
| Color name = ry4, ruyo 4th
| FJS name = A4&lt;sup&gt;5&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}
{{Wikipedia|Septimal tritone}}

In [[7-limit]] [[just intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10- This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a heptatonic scale that demonstrates [[Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:7-limit]]
[[Category:Tritone]]
[[Category:Over-7]]</text>
      <sha1>salrc1nar2m3if715b8iks8tkkiea16</sha1>
    </revision>
    <revision>
      <id>97993</id>
      <parentid>86290</parentid>
      <timestamp>2022-10-25T11:59:56Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97993</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1261" sha1="c7nkeflvpej7o5nofga831lzj9wco5w" xml:space="preserve">{{Infobox Interval
| Name = high tritone, greater septimal tritone, Euler's tritone
| Color name = ry4, ruyo 4th
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}
{{Wikipedia|Septimal tritone}}

In [[7-limit]] [[just intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10- This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a heptatonic scale that demonstrates [[Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Tritone]]
[[Category:Over-7]]</text>
      <sha1>c7nkeflvpej7o5nofga831lzj9wco5w</sha1>
    </revision>
    <revision>
      <id>157033</id>
      <parentid>97993</parentid>
      <timestamp>2024-09-30T08:42:32Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Recategorize</comment>
      <origin>157033</origin>
      <model>wikitext</model>
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      <text bytes="1271" sha1="0dzp5pfjjfc82qtlxv8g0trpix4wwt8" xml:space="preserve">{{Infobox Interval
| Name = high tritone, greater septimal tritone, Euler's tritone
| Color name = ry4, ruyo 4th
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}
{{Wikipedia|Septimal tritone}}

In [[7-limit]] [[just intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10- This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a heptatonic scale that demonstrates [[Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Tritone]]
[[Category:Over-7 intervals]]</text>
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    <revision>
      <id>159180</id>
      <parentid>157033</parentid>
      <timestamp>2024-10-08T22:00:52Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Add line break</comment>
      <origin>159180</origin>
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      <text bytes="1273" sha1="5xsardsp5xtnid3h9eb6x3r9ztp947b" xml:space="preserve">{{Infobox Interval
| Name = high tritone, greater septimal tritone, Euler's tritone
| Color name = ry4, ruyo 4th
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}
{{Wikipedia|Septimal tritone}}

In [[7-limit]] [[just intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10- This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. 

While in the context of the harmonic seventh chord, it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth- an analysis that is required in cases where this interval occurs in a heptatonic scale that demonstrates [[Rothenberg propriety]].

== See also ==

* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Tritone]]
[[Category:Over-7 intervals]]</text>
      <sha1>5xsardsp5xtnid3h9eb6x3r9ztp947b</sha1>
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    <revision>
      <id>159236</id>
      <parentid>159180</parentid>
      <timestamp>2024-10-09T09:34:35Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Heptatonic -&gt; diatonic and misc. linking and style improvements</comment>
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      <text bytes="1290" sha1="r5prp4k7y1qrbv9t8thc4p6jp4i1dar" xml:space="preserve">{{Infobox Interval
| Name = high tritone, greater septimal tritone, Euler's tritone
| Color name = ry4, ruyo 4th
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}
{{Wikipedia|Septimal tritone}}

In [[7-limit]] [[just intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10 – This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. 

While in the context of the [[harmonic seventh chord]], it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth – an analysis that is required in cases where this interval occurs in a [[5L 2s|diatonic scale]] that demonstrates [[Rothenberg propriety]].

== See also ==
* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Tritone]]
[[Category:Over-7 intervals]]</text>
      <sha1>r5prp4k7y1qrbv9t8thc4p6jp4i1dar</sha1>
    </revision>
    <revision>
      <id>215852</id>
      <parentid>159236</parentid>
      <timestamp>2025-11-03T07:28:17Z</timestamp>
      <contributor>
        <username>Pailiaq</username>
        <id>507</id>
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      <text bytes="1348" sha1="8toa1qiikrlyjww6lzwqo8hrolh03av" xml:space="preserve">{{Infobox Interval
| Name = high tritone, greater septimal tritone, Euler's tritone
| Color name = ry4, ruyo 4th
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}
{{Wikipedia|Septimal tritone}}

In [[7-limit]] [[just intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10 – This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. 

While in the context of the [[harmonic seventh chord]], it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth – an analysis that is required in cases where this interval occurs in a [[5L 2s|diatonic scale]] that demonstrates [[Rothenberg propriety]].
== Approximation ==
 {{Interval_Edo_Approximation | 10/7}}
== See also ==
* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Tritone]]
[[Category:Over-7 intervals]]</text>
      <sha1>8toa1qiikrlyjww6lzwqo8hrolh03av</sha1>
    </revision>
    <revision>
      <id>216082</id>
      <parentid>215852</parentid>
      <timestamp>2025-11-03T13:13:55Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"</comment>
      <origin>216082</origin>
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      <format>text/x-wiki</format>
      <text bytes="1345" sha1="c7hmirgmc0dsqnno45hg4t0g23nl8u7" xml:space="preserve">{{Infobox Interval
| Name = high tritone, greater septimal tritone, Euler's tritone
| Color name = ry4, ruyo 4th
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}
{{Wikipedia|Septimal tritone}}

In [[7-limit]] [[just intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10 – This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. 

While in the context of the [[harmonic seventh chord]], it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth – an analysis that is required in cases where this interval occurs in a [[5L 2s|diatonic scale]] that demonstrates [[Rothenberg propriety]].
== Approximation ==
{{Interval edo approximation|10/7}}
== See also ==
* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Tritone]]
[[Category:Over-7 intervals]]</text>
      <sha1>c7hmirgmc0dsqnno45hg4t0g23nl8u7</sha1>
    </revision>
    <revision>
      <id>218871</id>
      <parentid>216082</parentid>
      <timestamp>2025-12-12T15:32:37Z</timestamp>
      <contributor>
        <username>Eufalesio</username>
        <id>581</id>
      </contributor>
      <comment>Hemifamity approximation</comment>
      <origin>218871</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1445" sha1="2h1a225vftauvajjke8jui7n7v175ek" xml:space="preserve">{{Infobox Interval
| Name = high tritone, greater septimal tritone, Euler's tritone
| Color name = ry4, ruyo 4th
| Sound = jid_10_7_pluck_adu_dr220.mp3
}}
{{Wikipedia|Septimal tritone}}

In [[7-limit]] [[just intonation]], '''10/7''' is a '''high [[tritone]]''' (or '''Euler's tritone''') measuring about 617.5¢. It has a similar sound to its inversion, [[7/5]], but may sound a little edgier, less relaxed. Nonetheless, it is considered a septimal consonance. It appears in chords where a major third ([[5/4]]) appears above the harmonic seventh ([[7/4]]), such as 4:6:7:10 – This particular chord is well-approximated in [[88cET]], which has a good approximation of 10/7, but no 7/5. It's well approximated by the Pythagorean augmented fourth [[729/512]], differing by [[5120/5103]]. 

While in the context of the [[harmonic seventh chord]], it is rightly recognized as a type of augmented fourth, it can also be argued on the basis of the fact that 10/7 interval is larger than 600 cents that it acts more as a type of diminished fifth than an augmented fourth – an analysis that is required in cases where this interval occurs in a [[5L 2s|diatonic scale]] that demonstrates [[Rothenberg propriety]].
== Approximation ==
{{Interval edo approximation|10/7}}
== See also ==
* [[7/5]] – its [[octave complement]]
* [[21/20]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Tritone]]
[[Category:Over-7 intervals]]</text>
      <sha1>2h1a225vftauvajjke8jui7n7v175ek</sha1>
    </revision>
  </page>
  <page>
    <title>10 7</title>
    <ns>0</ns>
    <id>34</id>
    <redirect title="10/7" />
    <revision>
      <id>34</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>34</origin>
      <model>wikitext</model>
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      <text bytes="18" sha1="im0ki909vlumq26o8flpzl9ehnxfuox" xml:space="preserve">#REDIRECT [[10/7]]</text>
      <sha1>im0ki909vlumq26o8flpzl9ehnxfuox</sha1>
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  </page>
  <page>
    <title>10/9</title>
    <ns>0</ns>
    <id>35</id>
    <revision>
      <id>35</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>35</origin>
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      <text bytes="1382" sha1="hmfuqtvbmxqf8wcnduvyxk6vxrs01xf" xml:space="preserve">'''10/9'''
|1 -2 1&amp;gt;

182.40371 cents

[[File:jid_10_9_pluck_adu_dr220.mp3]] [[:File:jid_10_9_pluck_adu_dr220.mp3|sound sample]]

In [[5-limit|5-limit]] [[Just_intonation|Just Intonation]], 10/9 is a small whole tone of about 182.4¢. It is a [[superparticular|superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8|9/8]] (about 203.9¢), which is [[81/80|81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo|12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 -- move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo|19edo]], [[31edo|31edo]] and other [[Meantone|meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]      [[Category:5-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:listen]]
[[Category:ratio]]
[[Category:second]]
[[Category:whole_tone]]</text>
      <sha1>hmfuqtvbmxqf8wcnduvyxk6vxrs01xf</sha1>
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    <revision>
      <id>4096</id>
      <timestamp>2016-09-13T10:31:58Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 591877702 - Original comment: **</comment>
      <origin>4096</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5447" sha1="221s2gmge5npjke8rxzwy7mco3qsssx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-09-13 10:31:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;591877702&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**10/9**
|1 -2 1&amp;gt;
182.40371 cents
[[media type="file" key="jid_10_9_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_10_9_pluck_adu_dr220.mp3|sound sample]]

In [[5-limit]] [[Just Intonation]], 10/9 is a small whole tone of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9_8|9/8]] (about 203.9¢), which is [[81_80|81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 -- move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;10/9&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|1 -2 1&amp;amp;gt;&amp;lt;br /&amp;gt;
182.40371 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_10_9_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_10_9_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_10_9_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_10_9_pluck_adu_dr220.mp3/513184856/jid_10_9_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_10_9_pluck_adu_dr220.mp3/513184856/jid_10_9_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 10/9 is a small whole tone of about 182.4¢. It is a &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt; (about 203.9¢), which is &amp;lt;a class="wiki_link" href="/81_80"&amp;gt;81/80&amp;lt;/a&amp;gt; (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 -- move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt; and other &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>221s2gmge5npjke8rxzwy7mco3qsssx</sha1>
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    <revision>
      <id>4097</id>
      <timestamp>2014-06-07T16:46:10Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513197650 - Original comment: **</comment>
      <origin>4097</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5259" sha1="ftk80u8xxvtye5n2wjhrg4jnsvfng7e" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-07 16:46:10 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513197650&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**10/9**
|1 -2 1&amp;gt;
182.40371 cents
[[media type="file" key="jid_10_9_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_10_9_pluck_adu_dr220.mp3|sound sample]]

In [[5-limit]] [[Just Intonation]], 10/9 is a small whole tone of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9_8|9/8]] (about 203.9¢), which is [[81_80|81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9. In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;10/9&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|1 -2 1&amp;amp;gt;&amp;lt;br /&amp;gt;
182.40371 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_10_9_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_10_9_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_10_9_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_10_9_pluck_adu_dr220.mp3/513184856/jid_10_9_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_10_9_pluck_adu_dr220.mp3/513184856/jid_10_9_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 10/9 is a small whole tone of about 182.4¢. It is a &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt; (about 203.9¢), which is &amp;lt;a class="wiki_link" href="/81_80"&amp;gt;81/80&amp;lt;/a&amp;gt; (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9. In systems where 81/80 is tempered out (in 12edo, &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt; and other &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    </revision>
    <revision>
      <id>4098</id>
      <timestamp>2014-06-07T12:34:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513184890 - Original comment: **</comment>
      <origin>4098</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4795" sha1="elb2u5i764ghy5ccnaef0887vpcygkw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-07 12:34:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513184890&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**10/9**
|1 -2 1&amp;gt;
182.40371 cents
[[media type="file" key="jid_10_9_pluck_adu_dr220.mp3"]]

In [[5-limit]] [[Just Intonation]], 10/9 is a small whole tone of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9_8|9/8]] (about 203.9¢), which is [[81_80|81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9. In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;10/9&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|1 -2 1&amp;amp;gt;&amp;lt;br /&amp;gt;
182.40371 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_10_9_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_10_9_pluck_adu_dr220.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_10_9_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 10/9 is a small whole tone of about 182.4¢. It is a &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt; (about 203.9¢), which is &amp;lt;a class="wiki_link" href="/81_80"&amp;gt;81/80&amp;lt;/a&amp;gt; (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9. In systems where 81/80 is tempered out (in 12edo, &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt; and other &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    </revision>
    <revision>
      <id>4099</id>
      <timestamp>2011-09-16T19:22:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 254921752 - Original comment: **</comment>
      <origin>4099</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3620" sha1="f2x31snjw730scw1y9dtqd0naeoudgb" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-09-16 19:22:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;254921752&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[5-limit]] [[Just Intonation]], 10/9 is a small whole tone of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9_8|9/8]] (about 203.9¢), which is [[81_80|81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9. In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 10/9 is a small whole tone of about 182.4¢. It is a &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt; (about 203.9¢), which is &amp;lt;a class="wiki_link" href="/81_80"&amp;gt;81/80&amp;lt;/a&amp;gt; (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9. In systems where 81/80 is tempered out (in 12edo, &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt; and other &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    </revision>
    <revision>
      <id>36585</id>
      <parentid>35</parentid>
      <timestamp>2018-10-11T22:25:03Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>typographical changes, links simplified, introduced infobox interval</comment>
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      <text bytes="1314" sha1="p0ho7zfg8170zqrhushcceud0spatdi" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 10/9
| Monzo = 1 -2 1
| Cents = 182.40371 
| Name = small whole tone
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}

In [[5-limit]] [[Just Intonation]], '''10/9''' is a small whole tone of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo|12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 -- move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

:''See also [[Gallery of Just Intervals]]''

[[Category:5-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:listen]]
[[Category:ratio]]
[[Category:second]]
[[Category:whole_tone]]</text>
      <sha1>p0ho7zfg8170zqrhushcceud0spatdi</sha1>
    </revision>
    <revision>
      <id>37581</id>
      <parentid>36585</parentid>
      <timestamp>2018-10-23T03:21:10Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <origin>37581</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1340" sha1="hmz7qf83fe0w84jwpv9kj34fecbt2ex" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 10/9
| Monzo = 1 -2 1
| Cents = 182.40371 
| Name = small whole tone
| Color name = y2, yo 2nd
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}

In [[5-limit]] [[Just Intonation]], '''10/9''' is a small whole tone of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo|12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 -- move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

:''See also [[Gallery of Just Intervals]]''

[[Category:5-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:listen]]
[[Category:ratio]]
[[Category:second]]
[[Category:whole_tone]]</text>
      <sha1>hmz7qf83fe0w84jwpv9kj34fecbt2ex</sha1>
    </revision>
    <revision>
      <id>38048</id>
      <parentid>37581</parentid>
      <timestamp>2018-10-28T13:19:07Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>secondary lemma bold</comment>
      <origin>38048</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1346" sha1="7gr38c6gyc8x5ys8uu3oe3j23ccdm7f" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 10/9
| Monzo = 1 -2 1
| Cents = 182.40371 
| Name = small whole tone
| Color name = y2, yo 2nd
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}

In [[5-limit]] [[Just Intonation]], '''10/9''' is a '''small whole tone''' of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo|12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 -- move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

:''See also [[Gallery of Just Intervals]]''

[[Category:5-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:listen]]
[[Category:ratio]]
[[Category:second]]
[[Category:whole_tone]]</text>
      <sha1>7gr38c6gyc8x5ys8uu3oe3j23ccdm7f</sha1>
    </revision>
    <revision>
      <id>46314</id>
      <parentid>38048</parentid>
      <timestamp>2020-06-10T07:11:17Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cats</comment>
      <origin>46314</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1333" sha1="eszawrjaxx2opww3th2diytofmw0ciy" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 10/9
| Monzo = 1 -2 1
| Cents = 182.40371 
| Name = small whole tone
| Color name = y2, yo 2nd
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}

In [[5-limit]] [[Just Intonation]], '''10/9''' is a '''small whole tone''' of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo|12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale -- 1/1, 9/8, 5/4 -- move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 -- move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

:''See also [[Gallery of Just Intervals]]''

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Listen]]
[[Category:Second]]
[[Category:Whole tone]]</text>
      <sha1>eszawrjaxx2opww3th2diytofmw0ciy</sha1>
    </revision>
    <revision>
      <id>49844</id>
      <parentid>46314</parentid>
      <timestamp>2020-09-19T13:00:38Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+FJS name; cleanup</comment>
      <origin>49844</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1526" sha1="2vzr322kslzirkwdd7i60u9mj5jx0t5" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 10/9
| Monzo = 1 -2 1
| Cents = 182.40371 
| Name = small whole tone
| Color name = y2, yo 2nd
| FJS name = M2&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}

In [[5-limit]] [[Just Intonation]], '''10/9''' is a '''small whole tone''' of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

== See also == 
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Major second|Major second - Wikipedia]]

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Listen]]
[[Category:Second]]
[[Category:Whole tone]]
[[Category:Superparticular]]</text>
      <sha1>2vzr322kslzirkwdd7i60u9mj5jx0t5</sha1>
    </revision>
    <revision>
      <id>64425</id>
      <parentid>49844</parentid>
      <timestamp>2021-03-27T03:52:42Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64425</origin>
      <model>wikitext</model>
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      <text bytes="1574" sha1="k4qo27j98iiapgltb2kopy2tc657feu" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 10/9
| Monzo = 1 -2 1
| Cents = 182.40371 
| Name = small whole tone
| Color name = y2, yo 2nd
| FJS name = M2&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}

In [[5-limit]] [[Just Intonation]], '''10/9''' is a '''small whole tone''' of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

== See also == 
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Major second|Major second - Wikipedia]]

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Listen]]
[[Category:Second]]
[[Category:Whole tone]]
[[Category:Superparticular]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>k4qo27j98iiapgltb2kopy2tc657feu</sha1>
    </revision>
    <revision>
      <id>72994</id>
      <parentid>64425</parentid>
      <timestamp>2021-06-23T17:41:03Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>72994</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1614" sha1="adgdbs3lx91gftxozw07qibweza4n0q" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 10/9
| Monzo = 1 -2 1
| Cents = 182.40371 
| Name = small whole tone
| Color name = y2, yo 2nd
| FJS name = M2&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}

In [[5-limit]] [[Just Intonation]], '''10/9''' is a '''small whole tone''' of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is tiny and hard to notice at first.

== See also == 
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Major second|Major second - Wikipedia]]

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Listen]]
[[Category:Second]]
[[Category:Whole tone]]
[[Category:Superparticular]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>adgdbs3lx91gftxozw07qibweza4n0q</sha1>
    </revision>
    <revision>
      <id>81002</id>
      <parentid>72994</parentid>
      <timestamp>2021-11-17T15:16:46Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>The difference is not necessarily tiny, however, it may still be easily overlooked for those who are used to meantone.</comment>
      <origin>81002</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1605" sha1="4xoalsmh5zjl07ixd885roaim0k1aoq" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 10/9
| Monzo = 1 -2 1
| Cents = 182.40371 
| Name = small whole tone
| Color name = y2, yo 2nd
| FJS name = M2&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}

In [[5-limit]] [[Just Intonation]], '''10/9''' is a '''small whole tone''' of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference is hard to notice at first.

== See also == 
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Major second|Major second - Wikipedia]]

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Listen]]
[[Category:Second]]
[[Category:Whole tone]]
[[Category:Superparticular]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>4xoalsmh5zjl07ixd885roaim0k1aoq</sha1>
    </revision>
    <revision>
      <id>81003</id>
      <parentid>81002</parentid>
      <timestamp>2021-11-17T15:16:57Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>81003</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1609" sha1="mjguefs2c918i1kc99xth77d4w2hlid" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 10/9
| Monzo = 1 -2 1
| Cents = 182.40371 
| Name = small whole tone
| Color name = y2, yo 2nd
| FJS name = M2&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}

In [[5-limit]] [[Just Intonation]], '''10/9''' is a '''small whole tone''' of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.

== See also == 
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Major second|Major second - Wikipedia]]

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Listen]]
[[Category:Second]]
[[Category:Whole tone]]
[[Category:Superparticular]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>mjguefs2c918i1kc99xth77d4w2hlid</sha1>
    </revision>
    <revision>
      <id>81210</id>
      <parentid>81003</parentid>
      <timestamp>2021-11-23T19:07:39Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* See also */</comment>
      <origin>81210</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1696" sha1="1uez140z1e5vwkqa9z8eiifhx80ap30" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 10/9
| Monzo = 1 -2 1
| Cents = 182.40371 
| Name = small whole tone
| Color name = y2, yo 2nd
| FJS name = M2&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}

In [[5-limit]] [[Just Intonation]], '''10/9''' is a '''small whole tone''' of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.

== See also == 
* [[Decaononic]] - temperament which sets the tone to this interval, instead of to 9/8
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Major second|Major second - Wikipedia]]

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Listen]]
[[Category:Second]]
[[Category:Whole tone]]
[[Category:Superparticular]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>1uez140z1e5vwkqa9z8eiifhx80ap30</sha1>
    </revision>
    <revision>
      <id>86060</id>
      <parentid>81210</parentid>
      <timestamp>2022-02-16T21:19:11Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Added Wikipedia box, misc. edits, categories</comment>
      <origin>86060</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1585" sha1="eadvnxqtr9qz175v6gpofzory0f1v7v" xml:space="preserve">{{Infobox Interval
| Ratio = 10/9
| Monzo = 1 -2 1
| Cents = 182.40371 
| Name = small whole tone
| Color name = y2, yo 2nd
| FJS name = M2&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major second}}

In [[5-limit]] [[just intonation]], '''10/9''' is a '''small whole tone''' of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.

== See also == 
* [[Decaononic]] - temperament which sets the tone to this interval, instead of to 9/8
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of just intervals]]

[[Category:5-limit]]
[[Category:Second]]
[[Category:Whole tone]]
[[Category:Superparticular]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>eadvnxqtr9qz175v6gpofzory0f1v7v</sha1>
    </revision>
    <revision>
      <id>86287</id>
      <parentid>86060</parentid>
      <timestamp>2022-02-22T23:46:38Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86287</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1537" sha1="ij32keto8oni4bddldne33e00fhepfd" xml:space="preserve">{{Infobox Interval
| Ratio = 10/9
| Monzo = 1 -2 1
| Cents = 182.40371 
| Name = small whole tone
| Color name = y2, yo 2nd
| FJS name = M2&lt;sup&gt;5&lt;/sup&gt;
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major second}}

In [[5-limit]] [[just intonation]], '''10/9''' is a '''small whole tone''' of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.

== See also == 
* [[Decaononic]] - temperament which sets the tone to this interval, instead of to 9/8
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of just intervals]]

[[Category:5-limit]]
[[Category:Second]]
[[Category:Whole tone]]
[[Category:Superparticular]]</text>
      <sha1>ij32keto8oni4bddldne33e00fhepfd</sha1>
    </revision>
    <revision>
      <id>98059</id>
      <parentid>86287</parentid>
      <timestamp>2022-10-25T12:49:45Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>98059</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1406" sha1="sbbu93z3zoaztqpt5xgx6zgvsx3egwi" xml:space="preserve">{{Infobox Interval
| Name = small whole tone
| Color name = y2, yo 2nd
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major second}}

In [[5-limit]] [[just intonation]], '''10/9''' is a '''small whole tone''' of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.

== See also == 
* [[Decaononic]] - temperament which sets the tone to this interval, instead of to 9/8
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of just intervals]]

[[Category:Second]]
[[Category:Whole tone]]</text>
      <sha1>sbbu93z3zoaztqpt5xgx6zgvsx3egwi</sha1>
    </revision>
    <revision>
      <id>101336</id>
      <parentid>98059</parentid>
      <timestamp>2023-01-12T12:35:18Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Names</comment>
      <origin>101336</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1559" sha1="o3322uzh1eww1frupinbcxsqczuuvke" xml:space="preserve">{{Infobox Interval
| Name = small whole tone, classic(al) whole tone, ptolemaic whole tone
| Color name = y2, yo 2nd
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major second}}

In [[5-limit]] [[just intonation]], '''10/9''' is the '''small''', '''classic(al)''', or '''ptolemaic whole tone'''&lt;ref&gt;For reference, see [[5/4]]. &lt;/ref&gt; of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.

== See also == 
* [[Decaononic]] - temperament which sets the tone to this interval, instead of to 9/8
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of just intervals]]

== Notes ==
&lt;references/&gt;

[[Category:Second]]
[[Category:Whole tone]]</text>
      <sha1>o3322uzh1eww1frupinbcxsqczuuvke</sha1>
    </revision>
    <revision>
      <id>112384</id>
      <parentid>101336</parentid>
      <timestamp>2023-05-07T01:02:01Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>112384</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1598" sha1="o3ecqdfotsvf357qtd18xizk6gdk6mx" xml:space="preserve">{{Infobox Interval
| Name = small whole tone, classic(al) whole tone, ptolemaic whole tone
| Color name = y2, yo 2nd
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major second}}

In [[5-limit]] [[just intonation]], '''10/9''' is the '''small''', '''classic(al)''', or '''ptolemaic whole tone'''&lt;ref&gt;For reference, see [[5/4]]. &lt;/ref&gt; of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.

== See also == 
* [[Decaononic]] - temperament which sets the tone to this interval, instead of to 9/8
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of just intervals]]

== Notes ==
&lt;references/&gt;

[[Category:Second]]
[[Category:Whole tone]]
[[Category:Tritave-reduced harmonics]]</text>
      <sha1>o3ecqdfotsvf357qtd18xizk6gdk6mx</sha1>
    </revision>
    <revision>
      <id>122060</id>
      <parentid>112384</parentid>
      <timestamp>2023-08-15T10:33:25Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Reference moved</comment>
      <origin>122060</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1602" sha1="ff88dmqh06wxtr5vbxu5td4a3iq47mi" xml:space="preserve">{{Infobox Interval
| Name = small whole tone, classic(al) whole tone, ptolemaic whole tone
| Color name = y2, yo 2nd
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major second}}

In [[5-limit]] [[just intonation]], '''10/9''' is the '''small''', '''classic(al)''', or '''ptolemaic whole tone'''&lt;ref&gt;For reference, see [[5-limit]]. &lt;/ref&gt; of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.

== See also == 
* [[Decaononic]] - temperament which sets the tone to this interval, instead of to 9/8
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of just intervals]]

== Notes ==
&lt;references/&gt;

[[Category:Second]]
[[Category:Whole tone]]
[[Category:Tritave-reduced harmonics]]</text>
      <sha1>ff88dmqh06wxtr5vbxu5td4a3iq47mi</sha1>
    </revision>
    <revision>
      <id>206205</id>
      <parentid>122060</parentid>
      <timestamp>2025-08-03T03:39:49Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Add temperaments (generated by this interval) section</comment>
      <origin>206205</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1750" sha1="phsztpdqji7qqk7tr07i9uwsy6ywwde" xml:space="preserve">{{Infobox Interval
| Name = small whole tone, classic(al) whole tone, ptolemaic whole tone
| Color name = y2, yo 2nd
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major second}}

In [[5-limit]] [[just intonation]], '''10/9''' is the '''small''', '''classic(al)''', or '''ptolemaic whole tone'''&lt;ref&gt;For reference, see [[5-limit]]. &lt;/ref&gt; of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.

== Temperaments ==
The following [[linear temperament]]s are [[generate]]d by a [[~]]10/9:
* [[Porcupine]]
* [[Minortone]]

{{todo|complete list}}

== See also == 
* [[Decaononic]] - temperament which sets the tone to this interval, instead of to 9/8
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of just intervals]]

== Notes ==
&lt;references/&gt;

[[Category:Second]]
[[Category:Whole tone]]
[[Category:Tritave-reduced harmonics]]</text>
      <sha1>phsztpdqji7qqk7tr07i9uwsy6ywwde</sha1>
    </revision>
    <revision>
      <id>217791</id>
      <parentid>206205</parentid>
      <timestamp>2025-11-27T00:46:30Z</timestamp>
      <contributor>
        <username>Pailiaq</username>
        <id>507</id>
      </contributor>
      <origin>217791</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1804" sha1="506g67d0we9kqfprxdb65dfps9efmv7" xml:space="preserve">{{Infobox Interval
| Name = small whole tone, classic(al) whole tone, ptolemaic whole tone
| Color name = y2, yo 2nd
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major second}}

In [[5-limit]] [[just intonation]], '''10/9''' is the '''small''', '''classic(al)''', or '''ptolemaic whole tone'''&lt;ref&gt;For reference, see [[5-limit]]. &lt;/ref&gt; of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.
== Approxcimations ==
{{interval edo approximation}}


== Temperaments ==
The following [[linear temperament]]s are [[generate]]d by a [[~]]10/9:
* [[Porcupine]]
* [[Minortone]]

{{todo|complete list}}

== See also == 
* [[Decaononic]] - temperament which sets the tone to this interval, instead of to 9/8
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of just intervals]]

== Notes ==
&lt;references/&gt;

[[Category:Second]]
[[Category:Whole tone]]
[[Category:Tritave-reduced harmonics]]</text>
      <sha1>506g67d0we9kqfprxdb65dfps9efmv7</sha1>
    </revision>
    <revision>
      <id>217792</id>
      <parentid>217791</parentid>
      <timestamp>2025-11-27T00:47:28Z</timestamp>
      <contributor>
        <username>Pailiaq</username>
        <id>507</id>
      </contributor>
      <origin>217792</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1802" sha1="qv5rln0az3b3at55hq6a7ptgwhvjl94" xml:space="preserve">{{Infobox Interval
| Name = small whole tone, classic(al) whole tone, ptolemaic whole tone
| Color name = y2, yo 2nd
| Sound = jid_10_9_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major second}}

In [[5-limit]] [[just intonation]], '''10/9''' is the '''small''', '''classic(al)''', or '''ptolemaic whole tone'''&lt;ref&gt;For reference, see [[5-limit]]. &lt;/ref&gt; of about 182.4¢. It is a [[superparticular]] interval, as you can find it in the harmonic series between the 9th and the 10th overtones. It is one of two essential whole tones in the 5-limit; the other one is [[9/8]] (about 203.9¢), which is [[81/80]] (about 21.5¢) higher than 10/9. 9/8 is an octave-reduced overtone, and it is closer to [[12edo]]'s single whole step of 200¢. Thus, 9/8 is more familiar and less difficult to tune by ear than 10/9.

The first three notes of a JI major scale – 1/1, 9/8, 5/4 – move by a step of 9/8 followed by a step of 10/9 (or alternatively 1/1, 10/9, 5/4 – move by a step of 10/9 followed by a step of 9/8). In systems where 81/80 is tempered out (in 12edo, [[19edo]], [[31edo]] and other [[meantone]] systems) that distinction is lost and this sounds like two equal-sized steps. In strict JI, the difference may be hard to notice at first.
== Approximations ==
{{interval edo approximation}}

== Temperaments ==
The following [[linear temperament]]s are [[generate]]d by a [[~]]10/9:
* [[Porcupine]]
* [[Minortone]]

{{todo|complete list}}

== See also == 
* [[Decaononic]] - temperament which sets the tone to this interval, instead of to 9/8
* [[9/5]] – its [[octave complement]]
* [[27/20]] – its [[fifth complement]]
* [[6/5]] – its [[fourth complement]]
* [[Gallery of just intervals]]

== Notes ==
&lt;references/&gt;

[[Category:Second]]
[[Category:Whole tone]]
[[Category:Tritave-reduced harmonics]]</text>
      <sha1>qv5rln0az3b3at55hq6a7ptgwhvjl94</sha1>
    </revision>
  </page>
  <page>
    <title>10ed5</title>
    <ns>0</ns>
    <id>37</id>
    <revision>
      <id>37</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>37</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1192" sha1="qgcxydtckgtm8vh1n33q29wq08jzo58" xml:space="preserve">=10 equal divisions of the 5th harmonic=

Half of [[20ed5|20ed5]] (obviously). But it has important characteristics of its own:

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

As 5ed5 is the simplest [[Hyperpyth|hyperpyth]] tuning, analogous to [[5edo|5edo]] and [[4edt|4edt]] in their own spheres, this, its double, can be compared, structurally, to, [[10edo|10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Adding octaves, strangely enough, relates this tuning to [[53edo|53edo]].

0: 1/1

1: 278.631 cents 13/11

2: 557.263 cents 7/5

3: 835.894 cents

4: 1114.525 cents "9/5"

5: 1393.157 cents 11/5

6: 1671.788 cents 13/5

7: 1950.420 cents

8: 2229.051 cents "17/5"

9: 2507.682 cents 21/5

10: 5/1

Music:

[http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky
[[Category:5th_harmonic]]
[[Category:ed5]]
[[Category:edonoi]]</text>
      <sha1>qgcxydtckgtm8vh1n33q29wq08jzo58</sha1>
    </revision>
    <revision>
      <id>4100</id>
      <timestamp>2012-01-01T04:44:18Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Kosmorsky</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 288953957 - Original comment: **</comment>
      <origin>4100</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4245" sha1="mqo74cn2sxuea0n7s0c9u1650c1aij6" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on &lt;tt&gt;2012-01-01 04:44:18 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288953957&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=10 equal divisions of the 5th harmonic= 

Half of [[20ed5]] (obviously). But it has important characteristics of its own:

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

As 5ed5 is the simplest [[hyperpyth]] tuning, analogous to [[5edo]] and [[4edt]] in their own spheres, this, its double, can be compared, structurally, to, [[10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Adding octaves, strangely enough, relates this tuning to [[53edo]].

0: 1/1
1: 278.631 cents 13/11
2: 557.263 cents 7/5
3: 835.894 cents
4: 1114.525 cents "9/5"
5: 1393.157 cents 11/5
6: 1671.788 cents 13/5
7: 1950.420 cents
8: 2229.051 cents "17/5"
9: 2507.682 cents 21/5
10: 5/1

Music:

[[http://www.youtube.com/watch?v=tjD7Es05zuI|Weird Blues]] -- Kosmorsky&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10ed5&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x10 equal divisions of the 5th harmonic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;10 equal divisions of the 5th harmonic&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
Half of &amp;lt;a class="wiki_link" href="/20ed5"&amp;gt;20ed5&amp;lt;/a&amp;gt; (obviously). But it has important characteristics of its own:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
As 5ed5 is the simplest &amp;lt;a class="wiki_link" href="/hyperpyth"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; tuning, analogous to &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/4edt"&amp;gt;4edt&amp;lt;/a&amp;gt; in their own spheres, this, its double, can be compared, structurally, to, &amp;lt;a class="wiki_link" href="/10edo"&amp;gt;10edo&amp;lt;/a&amp;gt;. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Adding octaves, strangely enough, relates this tuning to &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
0: 1/1&amp;lt;br /&amp;gt;
1: 278.631 cents 13/11&amp;lt;br /&amp;gt;
2: 557.263 cents 7/5&amp;lt;br /&amp;gt;
3: 835.894 cents&amp;lt;br /&amp;gt;
4: 1114.525 cents &amp;amp;quot;9/5&amp;amp;quot;&amp;lt;br /&amp;gt;
5: 1393.157 cents 11/5&amp;lt;br /&amp;gt;
6: 1671.788 cents 13/5&amp;lt;br /&amp;gt;
7: 1950.420 cents&amp;lt;br /&amp;gt;
8: 2229.051 cents &amp;amp;quot;17/5&amp;amp;quot;&amp;lt;br /&amp;gt;
9: 2507.682 cents 21/5&amp;lt;br /&amp;gt;
10: 5/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Music:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=tjD7Es05zuI" rel="nofollow"&amp;gt;Weird Blues&amp;lt;/a&amp;gt; -- Kosmorsky&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4101</id>
      <timestamp>2011-11-30T22:41:18Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Kosmorsky</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 280908706 - Original comment: **</comment>
      <origin>4101</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5014" sha1="jy579qz6vhzfsltc8y1m2kirattbkgx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on &lt;tt&gt;2011-11-30 22:41:18 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;280908706&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=10 equal divisions of the 5th harmonic= 

Half of [[20ed5]] (obviously). But it has important characteristics of its own:

Well, as [[17ed5|hyperpyth]] is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.

Furthermore, 5ed5 is the simplest hyperpyth tuning, analogous to [[5edo]] and [[4edt]] in their own spheres. So, while the approximation of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals. I think this is a relatively important tuning as well, it would certainly do well on a harmonica. Though it has a step size of around 273 cents, strange melodies may still be crafted around it, however the main feature is likely to be its variety of chords and harmonies.

Adding octaves, strangely enough, relates this tuning to [[53edo]].

0: 1/1
1: 278.631 cents 13/11
2: 557.263 cents 7/5
3: 835.894 cents
4: 1114.525 cents "9/5"
5: 1393.157 cents 11/5
6: 1671.788 cents 13/5
7: 1950.420 cents
8: 2229.051 cents "17/5"
9: 2507.682 cents 21/5
10: 5/1

Music:

[[http://www.youtube.com/watch?v=tjD7Es05zuI|Weird Blues]] -- Kosmorsky&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10ed5&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x10 equal divisions of the 5th harmonic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;10 equal divisions of the 5th harmonic&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
Half of &amp;lt;a class="wiki_link" href="/20ed5"&amp;gt;20ed5&amp;lt;/a&amp;gt; (obviously). But it has important characteristics of its own:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Well, as &amp;lt;a class="wiki_link" href="/17ed5"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Furthermore, 5ed5 is the simplest hyperpyth tuning, analogous to &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/4edt"&amp;gt;4edt&amp;lt;/a&amp;gt; in their own spheres. So, while the approximation of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals. I think this is a relatively important tuning as well, it would certainly do well on a harmonica. Though it has a step size of around 273 cents, strange melodies may still be crafted around it, however the main feature is likely to be its variety of chords and harmonies.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Adding octaves, strangely enough, relates this tuning to &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
0: 1/1&amp;lt;br /&amp;gt;
1: 278.631 cents 13/11&amp;lt;br /&amp;gt;
2: 557.263 cents 7/5&amp;lt;br /&amp;gt;
3: 835.894 cents&amp;lt;br /&amp;gt;
4: 1114.525 cents &amp;amp;quot;9/5&amp;amp;quot;&amp;lt;br /&amp;gt;
5: 1393.157 cents 11/5&amp;lt;br /&amp;gt;
6: 1671.788 cents 13/5&amp;lt;br /&amp;gt;
7: 1950.420 cents&amp;lt;br /&amp;gt;
8: 2229.051 cents &amp;amp;quot;17/5&amp;amp;quot;&amp;lt;br /&amp;gt;
9: 2507.682 cents 21/5&amp;lt;br /&amp;gt;
10: 5/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Music:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=tjD7Es05zuI" rel="nofollow"&amp;gt;Weird Blues&amp;lt;/a&amp;gt; -- Kosmorsky&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>jy579qz6vhzfsltc8y1m2kirattbkgx</sha1>
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    <revision>
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      <timestamp>2011-11-18T12:20:30Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Kosmorsky</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 276984936 - Original comment: **</comment>
      <origin>4102</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4941" sha1="rrvkrq5830epgrsgjobw42afliz4bbg" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on &lt;tt&gt;2011-11-18 12:20:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;276984936&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=10 equal divisions of the 5th harmonic= 

Half of [[20ed5]] (obviously). But it has important characteristics of its own:

Well, as [[17ed5|hyperpyth]] is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.

Furthermore, 5ed5 is the simplest hyperpyth tuning, analogous to [[5edo]] and [[4edt]] in their own spheres. So, while the approximation of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals. I think this is a relatively important tuning as well, it would certainly do well on a harmonica. Though it has a step size of around 273 cents, strange melodies may still be crafted around it, however the main feature is likely to be its variety of chords and harmonies. This would be the perfect tuning for blues from outer space (perhaps from a gas giant somewhere).

Adding octaves, strangely enough, relates this tuning to [[53edo]].

0: 1/1
1: 278.631 cents 13/11
2: 557.263 cents 7/5
3: 835.894 cents
4: 1114.525 cents "9/5"
5: 1393.157 cents 11/5
6: 1671.788 cents 13/5
7: 1950.420 cents
8: 2229.051 cents "17/5"
9: 2507.682 cents 21/5
10: 5/1&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10ed5&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x10 equal divisions of the 5th harmonic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;10 equal divisions of the 5th harmonic&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
Half of &amp;lt;a class="wiki_link" href="/20ed5"&amp;gt;20ed5&amp;lt;/a&amp;gt; (obviously). But it has important characteristics of its own:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Well, as &amp;lt;a class="wiki_link" href="/17ed5"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Furthermore, 5ed5 is the simplest hyperpyth tuning, analogous to &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/4edt"&amp;gt;4edt&amp;lt;/a&amp;gt; in their own spheres. So, while the approximation of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals. I think this is a relatively important tuning as well, it would certainly do well on a harmonica. Though it has a step size of around 273 cents, strange melodies may still be crafted around it, however the main feature is likely to be its variety of chords and harmonies. This would be the perfect tuning for blues from outer space (perhaps from a gas giant somewhere).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Adding octaves, strangely enough, relates this tuning to &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
0: 1/1&amp;lt;br /&amp;gt;
1: 278.631 cents 13/11&amp;lt;br /&amp;gt;
2: 557.263 cents 7/5&amp;lt;br /&amp;gt;
3: 835.894 cents&amp;lt;br /&amp;gt;
4: 1114.525 cents &amp;amp;quot;9/5&amp;amp;quot;&amp;lt;br /&amp;gt;
5: 1393.157 cents 11/5&amp;lt;br /&amp;gt;
6: 1671.788 cents 13/5&amp;lt;br /&amp;gt;
7: 1950.420 cents&amp;lt;br /&amp;gt;
8: 2229.051 cents &amp;amp;quot;17/5&amp;amp;quot;&amp;lt;br /&amp;gt;
9: 2507.682 cents 21/5&amp;lt;br /&amp;gt;
10: 5/1&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4103</id>
      <timestamp>2011-11-18T12:18:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Kosmorsky</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 276984104 - Original comment: **</comment>
      <origin>4103</origin>
      <model>wikitext</model>
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      <text bytes="4708" sha1="9gqx7hf40zwdb6pt3twhff8y1z513vo" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on &lt;tt&gt;2011-11-18 12:18:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;276984104&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=10 equal divisions of the 5th harmonic= 

Well, as [[17ed5|hyperpyth]] is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.

Furthermore, 5ed5 is the simplest hyperpyth tuning, analogous to [[5edo]] and [[4edt]] in their own spheres. So, while the approximation of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals. I think this is a relatively important tuning as well, it would certainly do well on a harmonica. Though it has a step size of around 273 cents, strange melodies may still be crafted around it, however the main feature is likely to be its variety of chords and harmonies. This would be the perfect tuning for blues from outer space (perhaps from a gas giant somewhere).

Adding octaves, strangely enough, relates this tuning to [[53edo]].

0: 1/1
1: 278.631 cents 13/11
2: 557.263 cents 7/5
3: 835.894 cents
4: 1114.525 cents "9/5"
5: 1393.157 cents 11/5
6: 1671.788 cents 13/5
7: 1950.420 cents
8: 2229.051 cents "17/5"
9: 2507.682 cents 21/5
10: 5/1&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10ed5&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x10 equal divisions of the 5th harmonic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;10 equal divisions of the 5th harmonic&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
Well, as &amp;lt;a class="wiki_link" href="/17ed5"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Furthermore, 5ed5 is the simplest hyperpyth tuning, analogous to &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/4edt"&amp;gt;4edt&amp;lt;/a&amp;gt; in their own spheres. So, while the approximation of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals. I think this is a relatively important tuning as well, it would certainly do well on a harmonica. Though it has a step size of around 273 cents, strange melodies may still be crafted around it, however the main feature is likely to be its variety of chords and harmonies. This would be the perfect tuning for blues from outer space (perhaps from a gas giant somewhere).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Adding octaves, strangely enough, relates this tuning to &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
0: 1/1&amp;lt;br /&amp;gt;
1: 278.631 cents 13/11&amp;lt;br /&amp;gt;
2: 557.263 cents 7/5&amp;lt;br /&amp;gt;
3: 835.894 cents&amp;lt;br /&amp;gt;
4: 1114.525 cents &amp;amp;quot;9/5&amp;amp;quot;&amp;lt;br /&amp;gt;
5: 1393.157 cents 11/5&amp;lt;br /&amp;gt;
6: 1671.788 cents 13/5&amp;lt;br /&amp;gt;
7: 1950.420 cents&amp;lt;br /&amp;gt;
8: 2229.051 cents &amp;amp;quot;17/5&amp;amp;quot;&amp;lt;br /&amp;gt;
9: 2507.682 cents 21/5&amp;lt;br /&amp;gt;
10: 5/1&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4104</id>
      <timestamp>2011-11-18T12:16:22Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Kosmorsky</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 276983050 - Original comment: **</comment>
      <origin>4104</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4500" sha1="snnw6r0ukut4n8lacrkwpsnrgnep5pc" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on &lt;tt&gt;2011-11-18 12:16:22 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;276983050&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=10 equal divisions of the 5th harmonic= 

Well, as [[17ed5|hyperpyth]] is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.

Furthermore, 5ed5 is the simplest hyperpyth tuning, analogous to [[5edo]] and [[4edt]] in their own spheres. So, while the approximation of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals. I think this is a relatively important tuning as well, it would certainly do well on a harmonica. Though it has a step size of around 273 cents, strange melodies may still be crafted around it, however the main feature is likely to be its variety of chords and harmonies. This would be the perfect tuning for blues from outer space (perhaps from a gas giant somewhere).

0: 1/1
1: 278.631 cents 13/11
2: 557.263 cents 7/5
3: 835.894 cents
4: 1114.525 cents "9/5"
5: 1393.157 cents 11/5
6: 1671.788 cents 13/5
7: 1950.420 cents
8: 2229.051 cents "17/5"
9: 2507.682 cents 21/5
10: 5/1 &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10ed5&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x10 equal divisions of the 5th harmonic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;10 equal divisions of the 5th harmonic&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
Well, as &amp;lt;a class="wiki_link" href="/17ed5"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Furthermore, 5ed5 is the simplest hyperpyth tuning, analogous to &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/4edt"&amp;gt;4edt&amp;lt;/a&amp;gt; in their own spheres. So, while the approximation of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals. I think this is a relatively important tuning as well, it would certainly do well on a harmonica. Though it has a step size of around 273 cents, strange melodies may still be crafted around it, however the main feature is likely to be its variety of chords and harmonies. This would be the perfect tuning for blues from outer space (perhaps from a gas giant somewhere).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
0: 1/1&amp;lt;br /&amp;gt;
1: 278.631 cents 13/11&amp;lt;br /&amp;gt;
2: 557.263 cents 7/5&amp;lt;br /&amp;gt;
3: 835.894 cents&amp;lt;br /&amp;gt;
4: 1114.525 cents &amp;amp;quot;9/5&amp;amp;quot;&amp;lt;br /&amp;gt;
5: 1393.157 cents 11/5&amp;lt;br /&amp;gt;
6: 1671.788 cents 13/5&amp;lt;br /&amp;gt;
7: 1950.420 cents&amp;lt;br /&amp;gt;
8: 2229.051 cents &amp;amp;quot;17/5&amp;amp;quot;&amp;lt;br /&amp;gt;
9: 2507.682 cents 21/5&amp;lt;br /&amp;gt;
10: 5/1&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4105</id>
      <timestamp>2011-11-01T15:16:55Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Kosmorsky</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 270764670 - Original comment: **</comment>
      <origin>4105</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2755" sha1="t6xb0fl9fl3uz6p1z584pigyudqh8md" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on &lt;tt&gt;2011-11-01 15:16:55 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;270764670&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=10 equal divisions of the 5th harmonic= 

Well, as [[17ed5|hyperpyth]] is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.

Though it has a step size of around 273 cents it has a weird musical sound.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10ed5&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x10 equal divisions of the 5th harmonic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;10 equal divisions of the 5th harmonic&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
Well, as &amp;lt;a class="wiki_link" href="/17ed5"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Though it has a step size of around 273 cents it has a weird musical sound.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>t6xb0fl9fl3uz6p1z584pigyudqh8md</sha1>
    </revision>
    <revision>
      <id>4106</id>
      <timestamp>2011-10-07T14:40:41Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Kosmorsky</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 262672144 - Original comment: **</comment>
      <origin>4106</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2945" sha1="tk00td0m3xgr9445u9856orzty0tlo3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on &lt;tt&gt;2011-10-07 14:40:41 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;262672144&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=10 equal divisions of the 5th harmonic= 

Well, as [[17ed5|hyperpyth]] is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.

Though it has a step size of around 273 cents it has a weird musical sound, and could be used musically. It might also be useful as a generator for an octave temperament.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10ed5&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x10 equal divisions of the 5th harmonic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;10 equal divisions of the 5th harmonic&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
Well, as &amp;lt;a class="wiki_link" href="/17ed5"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; is based on the chord 5:9:13:17:(21):25 there ought to be a companion system which emphasizes ratios of 7 and 11. 11/5 is ~30 cents away from the square root of five, so barring a relatively large and complex temperament with 60-80 cent intervals, the square root of five is an adequate approximation. 10ed5 approximates the 7/5 slightly sharp (merging it with 11/8) such that the 77/25 - an important orgone structural element, is within 3 cents of just. This is no coincidence.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Though it has a step size of around 273 cents it has a weird musical sound, and could be used musically. It might also be useful as a generator for an octave temperament.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>tk00td0m3xgr9445u9856orzty0tlo3</sha1>
    </revision>
    <revision>
      <id>82627</id>
      <parentid>37</parentid>
      <timestamp>2021-12-26T03:07:20Z</timestamp>
      <contributor>
        <username>KingHyperio</username>
        <id>213</id>
      </contributor>
      <origin>82627</origin>
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      <format>text/x-wiki</format>
      <text bytes="1186" sha1="evn0xyfl00l6qsydnhpjgmt3y0q16sk" xml:space="preserve">=10 equal divisions of the 5th harmonic=

Half of [[20ed5|20ed5]] (obviously). But it has important characteristics of its own:

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

As 5ed5 is the simplest [[Hyperpyth|hyperpyth]] tuning, analogous to [[5edo|5edo]] and [[4edt|4edt]] in their own spheres, this, its double, can be compared, structurally, to, [[10edo|10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Adding octaves, strangely enough, relates this tuning to [[43edo]].

0: 1/1

1: 278.631 cents 13/11

2: 557.263 cents 7/5

3: 835.894 cents

4: 1114.525 cents "9/5"

5: 1393.157 cents 11/5

6: 1671.788 cents 13/5

7: 1950.420 cents

8: 2229.051 cents "17/5"

9: 2507.682 cents 21/5

10: 5/1

Music:

[http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky
[[Category:5th_harmonic]]
[[Category:ed5]]
[[Category:edonoi]]</text>
      <sha1>evn0xyfl00l6qsydnhpjgmt3y0q16sk</sha1>
    </revision>
    <revision>
      <id>96772</id>
      <parentid>82627</parentid>
      <timestamp>2022-10-05T20:37:24Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
      <origin>96772</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1160" sha1="ifjk5fd6vgqw10lpw1ucp8sivmr3uy3" xml:space="preserve">{{Infobox ET}}

Half of [[20ed5|20ed5]] (obviously). But it has important characteristics of its own:

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

As 5ed5 is the simplest [[Hyperpyth|hyperpyth]] tuning, analogous to [[5edo|5edo]] and [[4edt|4edt]] in their own spheres, this, its double, can be compared, structurally, to, [[10edo|10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Adding octaves, strangely enough, relates this tuning to [[43edo]].

0: 1/1

1: 278.631 cents 13/11

2: 557.263 cents 7/5

3: 835.894 cents

4: 1114.525 cents "9/5"

5: 1393.157 cents 11/5

6: 1671.788 cents 13/5

7: 1950.420 cents

8: 2229.051 cents "17/5"

9: 2507.682 cents 21/5

10: 5/1

Music:

[http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky
[[Category:5th_harmonic]]
[[Category:ed5]]
[[Category:edonoi]]</text>
      <sha1>ifjk5fd6vgqw10lpw1ucp8sivmr3uy3</sha1>
    </revision>
    <revision>
      <id>131866</id>
      <parentid>96772</parentid>
      <timestamp>2024-01-15T09:44:18Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Changed headings to conform with the style of other wiki pages. Added table of harmonics.</comment>
      <origin>131866</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1230" sha1="he0t1inrqc8fz1o12rtvcj77sl4bprz" xml:space="preserve">{{Infobox ET}}

Half of [[20ed5|20ed5]] (obviously). But it has important characteristics of its own:

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

As 5ed5 is the simplest [[Hyperpyth|hyperpyth]] tuning, analogous to [[5edo|5edo]] and [[4edt|4edt]] in their own spheres, this, its double, can be compared, structurally, to, [[10edo|10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Adding octaves, strangely enough, relates this tuning to [[43edo]].

== Harmonics ==

{{harmonics in equal|10|5|1}}

== Intervals ==

0: 1/1

1: 278.631 cents 13/11

2: 557.263 cents 7/5

3: 835.894 cents

4: 1114.525 cents "9/5"

5: 1393.157 cents 11/5

6: 1671.788 cents 13/5

7: 1950.420 cents

8: 2229.051 cents "17/5"

9: 2507.682 cents 21/5

10: 5/1

== Music ==

[http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky
[[Category:5th_harmonic]]
[[Category:ed5]]
[[Category:edonoi]]</text>
      <sha1>he0t1inrqc8fz1o12rtvcj77sl4bprz</sha1>
    </revision>
    <revision>
      <id>131867</id>
      <parentid>131866</parentid>
      <timestamp>2024-01-15T09:44:38Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>131867</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1218" sha1="tqgav13ich5xilyiutfgn3d3r4512he" xml:space="preserve">{{Infobox ET}}

Half of [[20ed5|20ed5]]. But it has important characteristics of its own:

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

As 5ed5 is the simplest [[Hyperpyth|hyperpyth]] tuning, analogous to [[5edo|5edo]] and [[4edt|4edt]] in their own spheres, this, its double, can be compared, structurally, to, [[10edo|10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Adding octaves, strangely enough, relates this tuning to [[43edo]].

== Harmonics ==

{{harmonics in equal|10|5|1}}

== Intervals ==

0: 1/1

1: 278.631 cents 13/11

2: 557.263 cents 7/5

3: 835.894 cents

4: 1114.525 cents "9/5"

5: 1393.157 cents 11/5

6: 1671.788 cents 13/5

7: 1950.420 cents

8: 2229.051 cents "17/5"

9: 2507.682 cents 21/5

10: 5/1

== Music ==

[http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky
[[Category:5th_harmonic]]
[[Category:ed5]]
[[Category:edonoi]]</text>
      <sha1>tqgav13ich5xilyiutfgn3d3r4512he</sha1>
    </revision>
    <revision>
      <id>158737</id>
      <parentid>131867</parentid>
      <timestamp>2024-10-06T01:11:41Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Harmonics */</comment>
      <origin>158737</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1334" sha1="h5wlpzepz7r1k0697gttpo91vxtcvci" xml:space="preserve">{{Infobox ET}}

Half of [[20ed5|20ed5]]. But it has important characteristics of its own:

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

As 5ed5 is the simplest [[Hyperpyth|hyperpyth]] tuning, analogous to [[5edo|5edo]] and [[4edt|4edt]] in their own spheres, this, its double, can be compared, structurally, to, [[10edo|10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Adding octaves, strangely enough, relates this tuning to [[43edo]].

== Harmonics ==
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}

== Intervals ==

0: 1/1

1: 278.631 cents 13/11

2: 557.263 cents 7/5

3: 835.894 cents

4: 1114.525 cents "9/5"

5: 1393.157 cents 11/5

6: 1671.788 cents 13/5

7: 1950.420 cents

8: 2229.051 cents "17/5"

9: 2507.682 cents 21/5

10: 5/1

== Music ==

[http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky
[[Category:5th_harmonic]]
[[Category:ed5]]
[[Category:edonoi]]</text>
      <sha1>h5wlpzepz7r1k0697gttpo91vxtcvci</sha1>
    </revision>
    <revision>
      <id>205814</id>
      <parentid>158737</parentid>
      <timestamp>2025-08-01T18:39:02Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Ed5]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>205814</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1317" sha1="ljvi5r9zuq6z5nhivei90ktite611mp" xml:space="preserve">{{Infobox ET}}

Half of [[20ed5|20ed5]]. But it has important characteristics of its own:

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

As 5ed5 is the simplest [[Hyperpyth|hyperpyth]] tuning, analogous to [[5edo|5edo]] and [[4edt|4edt]] in their own spheres, this, its double, can be compared, structurally, to, [[10edo|10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Adding octaves, strangely enough, relates this tuning to [[43edo]].

== Harmonics ==
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}

== Intervals ==

0: 1/1

1: 278.631 cents 13/11

2: 557.263 cents 7/5

3: 835.894 cents

4: 1114.525 cents "9/5"

5: 1393.157 cents 11/5

6: 1671.788 cents 13/5

7: 1950.420 cents

8: 2229.051 cents "17/5"

9: 2507.682 cents 21/5

10: 5/1

== Music ==

[http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky
[[Category:5th_harmonic]]
[[Category:edonoi]]</text>
      <sha1>ljvi5r9zuq6z5nhivei90ktite611mp</sha1>
    </revision>
    <revision>
      <id>205930</id>
      <parentid>205814</parentid>
      <timestamp>2025-08-01T19:20:34Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Edonoi]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>205930</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1297" sha1="9xofg42wnmcb4wyjsvwgxvg40foj208" xml:space="preserve">{{Infobox ET}}

Half of [[20ed5|20ed5]]. But it has important characteristics of its own:

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

As 5ed5 is the simplest [[Hyperpyth|hyperpyth]] tuning, analogous to [[5edo|5edo]] and [[4edt|4edt]] in their own spheres, this, its double, can be compared, structurally, to, [[10edo|10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Adding octaves, strangely enough, relates this tuning to [[43edo]].

== Harmonics ==
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}

== Intervals ==

0: 1/1

1: 278.631 cents 13/11

2: 557.263 cents 7/5

3: 835.894 cents

4: 1114.525 cents "9/5"

5: 1393.157 cents 11/5

6: 1671.788 cents 13/5

7: 1950.420 cents

8: 2229.051 cents "17/5"

9: 2507.682 cents 21/5

10: 5/1

== Music ==

[http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky
[[Category:5th_harmonic]]</text>
      <sha1>9xofg42wnmcb4wyjsvwgxvg40foj208</sha1>
    </revision>
    <revision>
      <id>225599</id>
      <parentid>205930</parentid>
      <timestamp>2026-03-12T02:50:24Z</timestamp>
      <contributor>
        <username>Squib</username>
        <id>562</id>
      </contributor>
      <comment>partial cleanup, +note about structure</comment>
      <origin>225599</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1371" sha1="6zf205tbvbkxrdh637d83hae142teat" xml:space="preserve">{{Infobox ET}}

Half of [[20ed5|20ed5]]. But it has important characteristics of its own:

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

As 5ed5 is the simplest [[hyperpyth]] tuning (analogous to [[5edo]] and [[4edt]] in their own spheres) this, its double, can be compared structurally to [[10edo|10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals. It is especially important as a structural framework for the [[5.7.11.13 subgroup]].

Adding octaves relates this tuning to [[43edo]], which divides the step in ten.

== Harmonics ==
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}

== Intervals ==

0: 1/1

1: 278.631 cents 13/11

2: 557.263 cents 7/5

3: 835.894 cents

4: 1114.525 cents "9/5"

5: 1393.157 cents 11/5

6: 1671.788 cents 13/5

7: 1950.420 cents

8: 2229.051 cents "17/5"

9: 2507.682 cents 21/5

10: 5/1

== Music ==

[http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky
[[Category:5th_harmonic]]</text>
      <sha1>6zf205tbvbkxrdh637d83hae142teat</sha1>
    </revision>
    <revision>
      <id>227556</id>
      <parentid>225599</parentid>
      <timestamp>2026-04-10T01:59:00Z</timestamp>
      <contributor>
        <username>Squib</username>
        <id>562</id>
      </contributor>
      <comment>interval table, reorganization</comment>
      <origin>227556</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1624" sha1="rg8leygf8spwgspal46ahwo05e73ej1" xml:space="preserve">{{Infobox ET}}

In general, 10ed5 is simply a smashing tuning. The relatively large steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

It is especially important as a structural framework for the [[5.7.11.13 subgroup]].

== Harmonics ==
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}

== Intervals ==
{| class="wikitable"
|+
!Degree
!Cents
!5.7.11.13 intervals
|-
|0
|0.000
|1/1
|-
|1
|278.631
|13/11, 55/49
|-
|2
|557.263
|7/5
|-
|3
|835.894
|11/7
|-
|4
|1114.525
|13/7, 25/13
|-
|5
|1393.157
|11/5, 25/11
|-
|6
|1671.788
|13/5, 35/13
|-
|7
|1950.420
|35/11
|-
|8
|2229.051
|49/13
|-
|9
|2507.682
|49/11
|-
|10
|2786.314
|5/1
|}

== Subsets and supersets ==
Half of [[20ed5]].

As 5ed5 is the simplest [[hyperpyth]] tuning (analogous to [[5edo]] and [[4edt]] in their own spheres) this, its double, can be compared structurally to [[10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Adding octaves relates this tuning to 13edo, which divides the step in three, although the octaves are 7 cents sharp. If octaves are instead made just, everything else (especially prime 7) becomes flatter. Alternatively, the step can be divided in 10 to get 43edo.

== Music ==

[http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky
[[Category:5th_harmonic]]</text>
      <sha1>rg8leygf8spwgspal46ahwo05e73ej1</sha1>
    </revision>
    <revision>
      <id>227557</id>
      <parentid>227556</parentid>
      <timestamp>2026-04-10T01:59:19Z</timestamp>
      <contributor>
        <username>Squib</username>
        <id>562</id>
      </contributor>
      <comment>interval table, reorganization</comment>
      <origin>227557</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1447" sha1="6pyu5tsnqrip6pgkfwue20xgo76jxi6" xml:space="preserve">{{Infobox ET}}

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

It is especially important as a structural framework for the [[5.7.11.13 subgroup]].

== Harmonics ==
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}

== Intervals ==
{| class="wikitable"
|+
!Degree
!Cents
!5.7.11.13 intervals
|-
|0
|0.000
|1/1
|-
|1
|278.631
|13/11, 55/49
|-
|2
|557.263
|7/5
|-
|3
|835.894
|11/7
|-
|4
|1114.525
|13/7, 25/13
|-
|5
|1393.157
|11/5, 25/11
|-
|6
|1671.788
|13/5, 35/13
|-
|7
|1950.420
|35/11
|-
|8
|2229.051
|49/13
|-
|9
|2507.682
|49/11
|-
|10
|2786.314
|5/1
|}

== Subsets and supersets ==
Half of [[20ed5]].

As 5ed5 is the simplest [[hyperpyth]] tuning (analogous to [[5edo]] and [[4edt]] in their own spheres) this, its double, can be compared structurally to [[10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Adding octaves relates this tuning to [[13edo]], which divides the step in three.

== Music ==

[http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky
[[Category:5th_harmonic]]</text>
      <sha1>6pyu5tsnqrip6pgkfwue20xgo76jxi6</sha1>
    </revision>
    <revision>
      <id>227558</id>
      <parentid>227557</parentid>
      <timestamp>2026-04-10T02:04:49Z</timestamp>
      <contributor>
        <username>Squib</username>
        <id>562</id>
      </contributor>
      <origin>227558</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1625" sha1="3gz0ttdn8rja7why9ce11x59ikjz8dx" xml:space="preserve">{{Infobox ET}}

In general, 10ed5 is simply a smashing tuning. The relatively large small steps, about the size of a minor third or an orwell generator, actually work for melodies, and it's harmonies while strange have no lack of impact. It can be used such that the fifth harmonic is equivalent, but of course, doesn't have to.

It is especially important as a structural framework for the [[5.7.11.13 subgroup]].

== Harmonics ==
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 10
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}

== Intervals ==
{| class="wikitable"
|+
!Degree
!Cents
!5.7.11.13 intervals
|-
|0
|0.000
|1/1
|-
|1
|278.631
|13/11, 55/49
|-
|2
|557.263
|7/5
|-
|3
|835.894
|11/7
|-
|4
|1114.525
|13/7, 25/13
|-
|5
|1393.157
|11/5, 25/11
|-
|6
|1671.788
|13/5, 35/13
|-
|7
|1950.420
|35/11
|-
|8
|2229.051
|49/13
|-
|9
|2507.682
|49/11
|-
|10
|2786.314
|5/1
|}

== Subsets and supersets ==
Half of [[20ed5]].

As 5ed5 is the simplest [[hyperpyth]] tuning (analogous to [[5edo]] and [[4edt]] in their own spheres) this, its double, can be compared structurally to [[10edo]]. While its approximations of 9/5, 17/5 and 21/5 are quite far off, these are still categorically important intervals.

Octaves can be added by dividing the step in three to get [[13edo]] with octaves 7 cents sharp. If octaves are instead made just, prime 7 becomes very flat, as well as prime 5 to a lesser extent. Alternatively, the step can be divided in ten to get [[43edo]].

== Music ==

[http://www.youtube.com/watch?v=tjD7Es05zuI Weird Blues] -- Kosmorsky
[[Category:5th_harmonic]]</text>
      <sha1>3gz0ttdn8rja7why9ce11x59ikjz8dx</sha1>
    </revision>
  </page>
  <page>
    <title>10edf</title>
    <ns>0</ns>
    <id>38</id>
    <revision>
      <id>38</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>38</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="137" sha1="6pmcba7nrzrc6m09arr7a4gu75r89ki" xml:space="preserve">=10 ED 3/2=

Lookalikes: [[17edo|17edo]]

Music:

[http://www.archive.org/details/10Edf http://www.archive.org/details/10Edf] - Kosmorsky</text>
      <sha1>6pmcba7nrzrc6m09arr7a4gu75r89ki</sha1>
    </revision>
    <revision>
      <id>4107</id>
      <timestamp>2011-12-26T16:11:47Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Kosmorsky</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 288515854 - Original comment: **</comment>
      <origin>4107</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1776" sha1="446u1aya4wtd80fu9gvg2amigh9muix" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on &lt;tt&gt;2011-12-26 16:11:47 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288515854&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=10 ED 3/2= 

Lookalikes: [[17edo]]

Music:
[[http://www.archive.org/details/10Edf]] - Kosmorsky&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edf&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x10 ED 3/2"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;10 ED 3/2&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
Lookalikes: &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Music:&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.archive.org/details/10Edf" rel="nofollow"&amp;gt;http://www.archive.org/details/10Edf&amp;lt;/a&amp;gt; - Kosmorsky&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>446u1aya4wtd80fu9gvg2amigh9muix</sha1>
    </revision>
    <revision>
      <id>38064</id>
      <parentid>38</parentid>
      <timestamp>2018-10-28T14:07:49Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>+cat, some styling, still much to do</comment>
      <origin>38064</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="301" sha1="epm8y48b558g13lu759ttfexm6pxk65" xml:space="preserve">'''10edf''' is the 10-part equal division of the fifth ([[3/2]]).

Lookalikes: [[17edo]]

== Music ==
* [http://www.archive.org/details/10Edf http://www.archive.org/details/10Edf] by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>epm8y48b558g13lu759ttfexm6pxk65</sha1>
    </revision>
    <revision>
      <id>40041</id>
      <parentid>38064</parentid>
      <timestamp>2019-02-05T22:47:35Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>40041</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="662" sha1="esclsagaff5jwi9v4j03hc4id89lprh" xml:space="preserve">'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

== Music ==
* [http://www.archive.org/details/10Edf http://www.archive.org/details/10Edf] by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>esclsagaff5jwi9v4j03hc4id89lprh</sha1>
    </revision>
    <revision>
      <id>40405</id>
      <parentid>40041</parentid>
      <timestamp>2019-02-20T21:31:49Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>40405</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="968" sha1="p57cgxkk0pcrtuyp3qzadmng4fdet2l" xml:space="preserve">'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
!degree
!cents value
|-
|0
|0
|-
|1
|70.1955
|-
|2
|140.391
|-
|3
|210.5865
|-
|4
|280.782
|-
|5
|350.9775
|-
|6
|421.173
|-
|7
|491.3685
|-
|8
|561.564
|-
|9
|631.7595
|-
|10
|701.955
|-
|11
|772.1505
|-
|12
|842.346
|-
|13
|912.5415
|-
|14
|982.737
|-
|15
|1052.9325
|-
|16
|1123.128
|-
|17
|1193.3235
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>p57cgxkk0pcrtuyp3qzadmng4fdet2l</sha1>
    </revision>
    <revision>
      <id>40720</id>
      <parentid>40405</parentid>
      <timestamp>2019-03-08T20:49:16Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>40720</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1019" sha1="1vf5d6qe93neaukkjiyr25zy10c0lj0" xml:space="preserve">'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
!degree
!cents value
|-
|0
|0
|-
|1
|70.1955
|-
|2
|140.391
|-
|3
|210.5865
|-
|4
|280.782
|-
|5
|350.9775
|-
|6
|421.173
|-
|7
|491.3685
|-
|8
|561.564
|-
|9
|631.7595
|-
|10
|701.955
|-
|11
|772.1505
|-
|12
|842.346
|-
|13
|912.5415
|-
|14
|982.737
|-
|15
|1052.9325
|-
|16
|1123.128
|-
|17
|1193.3235
|-
|18
|1263.519
|-
|19
|1333.7145
|-
|20
|1403.91
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>1vf5d6qe93neaukkjiyr25zy10c0lj0</sha1>
    </revision>
    <revision>
      <id>41991</id>
      <parentid>40720</parentid>
      <timestamp>2019-06-24T02:39:36Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>41991</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3277" sha1="muodyilxsairt05f5edzdoelz6gbhao" xml:space="preserve">'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-3.5¢''
! rowspan="2" |ed31\54
! rowspan="2" |11/9-et (~ed11\19)
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+3.5¢''
|-
!(~ed17\29)
!(~ed10\17)
|-
| colspan="10" |0
|-
|1
|''66.2214-66.5714''
|68.8889
|69.4816
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.35''
|-
|2
|''132.4429-133.1429''
|137.7778
|138.9632
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-148.7''
|-
|3
|''198.6643-199.7143''
|206.6667
|208.4448
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.05''
|-
|4
|''264.8857-266.2857''
|275.5556
|277.92635
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-297.4''
|-
|5
|''331.1071-332.8571''
|344.4444
|347.4079
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-371.75''
|-
|6
|''397.3286-399.4286''
|413.3333
|416.8895
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-446.1''
|-
|7
|''463.55-466''
|482.2222
|486.3711
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-520.45''
|-
|8
|''529.7714-532.5714''
|551.1111
|555.8527
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-594.8''
|-
|9
|''595.9929-599.1429''
|620
|625.3343
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-669.15''
|-
|10
|''662.2143-665.7143''
|688.8889
|694.8159
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-743.5''
|-
|11
|''728.4357-732.2857''
|757.7778
|764.2974
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-817.85''
|-
|12
|''794.6571-798.8571''
|826.6667
|833.7791
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-892.2''
|-
|13
|''860.8786-865.4286''
|895.5556
|903.26065
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-966.55''
|-
|14
|''927.1-932''
|964.4444
|972.7422
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1040.9''
|-
|15
|''993.3214-998.5714''
|1033.3333
|1042.2238
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1115.25''
|-
|16
|''1059.5429-1065.1429''
|1102.2222
|1111.7054
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1189.6''
|-
|17
|''1125.7643-1131.7146''
|1171.1111
|1181.187
|1193.3235
|1196.18125
|1199.9954
|1205.4102
|1216.8451
|''1258-1263.95''
|-
|18
|''1191.9857-1198.2857''
|1240
|1250.6686
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1338.3''
|-
|19
|''1258.2071-1263.8571''
|1308.8889
|1320.1502
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1412.65''
|-
|20
|''1324.4286-1331.4286''
|1377.7778
|1389.6318
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1487''
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>muodyilxsairt05f5edzdoelz6gbhao</sha1>
    </revision>
    <revision>
      <id>41997</id>
      <parentid>41991</parentid>
      <timestamp>2019-06-25T19:45:25Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>41997</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3277" sha1="r57th9rvvpz2yc4gt0l6sovdppc0n1f" xml:space="preserve">'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-3.5¢''
! rowspan="2" |ed31\54
! rowspan="2" |11/9-et (~ed11\19)
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+3.5¢''
|-
!(~ed17\29)
!(~ed10\17)
|-
| colspan="10" |0
|-
|1
|''66.2214-66.5714''
|68.8889
|69.4816
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.35''
|-
|2
|''132.4429-133.1429''
|137.7778
|138.9632
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-148.7''
|-
|3
|''198.6643-199.7143''
|206.6667
|208.4448
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.05''
|-
|4
|''264.8857-266.2857''
|275.5556
|277.92635
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-297.4''
|-
|5
|''331.1071-332.8571''
|344.4444
|347.4079
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-371.75''
|-
|6
|''397.3286-399.4286''
|413.3333
|416.8895
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-446.1''
|-
|7
|''463.55-466''
|482.2222
|486.3711
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-520.45''
|-
|8
|''529.7714-532.5714''
|551.1111
|555.8527
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-594.8''
|-
|9
|''595.9929-599.1429''
|620
|625.3343
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-669.15''
|-
|10
|''662.2143-665.7143''
|688.8889
|694.8159
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-743.5''
|-
|11
|''728.4357-732.2857''
|757.7778
|764.2974
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-817.85''
|-
|12
|''794.6571-798.8571''
|826.6667
|833.7791
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-892.2''
|-
|13
|''860.8786-865.4286''
|895.5556
|903.26065
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-966.55''
|-
|14
|''927.1-932''
|964.4444
|972.7422
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1040.9''
|-
|15
|''993.3214-998.5714''
|1033.3333
|1042.2238
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1115.25''
|-
|16
|''1059.5429-1065.1429''
|1102.2222
|1111.7054
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1189.6''
|-
|17
|''1125.7643-1131.7143''
|1171.1111
|1181.187
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1263.95''
|-
|18
|''1191.9857-1198.2857''
|1240
|1250.6686
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1338.3''
|-
|19
|''1258.2071-1263.8571''
|1308.8889
|1320.1502
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1412.65''
|-
|20
|''1324.4286-1331.4286''
|1377.7778
|1389.6318
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1487''
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>r57th9rvvpz2yc4gt0l6sovdppc0n1f</sha1>
    </revision>
    <revision>
      <id>42243</id>
      <parentid>41997</parentid>
      <timestamp>2019-08-12T19:59:16Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>42243</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3243" sha1="ksk8y3t6bwoy5rmk2shxxw4tm9jy0lt" xml:space="preserve">'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
|-
!(~ed17\29)
!(~ed10\17)
|-
| colspan="10" |0
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|-
|7
|''462.5-466''
|482.2222
|486.3711
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|-
|14
|''925-932''
|964.4444
|972.7422
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>ksk8y3t6bwoy5rmk2shxxw4tm9jy0lt</sha1>
    </revision>
    <revision>
      <id>42264</id>
      <parentid>42243</parentid>
      <timestamp>2019-08-19T22:03:13Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>42264</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3648" sha1="strop5qorye9ih7brnp9nybalgo46jt" xml:space="preserve">'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
|-
!(~ed17\29)
!(~ed10\17)
|-
| colspan="12" |0
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>strop5qorye9ih7brnp9nybalgo46jt</sha1>
    </revision>
    <revision>
      <id>71018</id>
      <parentid>42264</parentid>
      <timestamp>2021-05-30T00:30:14Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>71018</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3648" sha1="dtwnrzuaeu13m9udvnqyztym4ughcrt" xml:space="preserve">'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>dtwnrzuaeu13m9udvnqyztym4ughcrt</sha1>
    </revision>
    <revision>
      <id>96596</id>
      <parentid>71018</parentid>
      <timestamp>2022-10-05T18:35:06Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
      <origin>96596</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3663" sha1="6geuesygx3no8zrev3gzahxm31ln2wk" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>6geuesygx3no8zrev3gzahxm31ln2wk</sha1>
    </revision>
    <revision>
      <id>103071</id>
      <parentid>96596</parentid>
      <timestamp>2023-02-17T09:09:44Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>103071</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3769" sha1="a9gekjx12qfmttkqpzwvzsjhv5n5vro" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
! rowspan="2" |2L 2s notation (soft)
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|A
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|A#
|
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|Bb
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|B
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|B#, Cb
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|C
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|C#
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|Db
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|D
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|D#, Ab
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|A
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>a9gekjx12qfmttkqpzwvzsjhv5n5vro</sha1>
    </revision>
    <revision>
      <id>103072</id>
      <parentid>103071</parentid>
      <timestamp>2023-02-17T09:10:00Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>103072</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3756" sha1="o81pq95wy4pulrg5j11169cijbala3u" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
! 2L 2s notation (soft)
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|A
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|A#
|
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|Bb
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|B
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|B#, Cb
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|C
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|C#
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|Db
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|D
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|D#, Ab
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|A
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>o81pq95wy4pulrg5j11169cijbala3u</sha1>
    </revision>
    <revision>
      <id>103073</id>
      <parentid>103072</parentid>
      <timestamp>2023-02-17T09:10:23Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>103073</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3770" sha1="lx9i9binrjir7sve1573rnqa67q0n5s" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
! rowspan="2" | 2L 2s notation (soft)
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|A
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|A#
|
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|Bb
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|B
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|B#, Cb
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|C
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|C#
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|Db
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|D
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|D#, Ab
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|A
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>lx9i9binrjir7sve1573rnqa67q0n5s</sha1>
    </revision>
    <revision>
      <id>103074</id>
      <parentid>103073</parentid>
      <timestamp>2023-02-17T09:11:26Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>103074</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3768" sha1="73ta60tk1zkxn2upugyiud644ek78d9" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
! rowspan="2" | 2L 2s notation (soft)
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|A
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|A#
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|Bb
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|B
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|B#, Cb
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|C
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|C#
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|Db
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|D
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|D#, Ab
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|A
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>73ta60tk1zkxn2upugyiud644ek78d9</sha1>
    </revision>
    <revision>
      <id>103076</id>
      <parentid>103074</parentid>
      <timestamp>2023-02-17T09:43:52Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>103076</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3773" sha1="kkjtssesqdfqjok9hn1mhhkcbjifzu8" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
! rowspan="2" | 2L 2s notation (soft)
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|C
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|C#
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|Db
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|D
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|D#, Eb
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|E
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|E#
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|Fb
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|F
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|F#, Cb
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|C
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|C#
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|Db
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|D
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>kkjtssesqdfqjok9hn1mhhkcbjifzu8</sha1>
    </revision>
    <revision>
      <id>104334</id>
      <parentid>103076</parentid>
      <timestamp>2023-03-04T04:31:31Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>104334</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3782" sha1="99pjs8vnrembm38aunbb0xsql3rl684" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
! rowspan="2" | 2L 2s notation (using 8\10edf)
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|C
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|C#
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|Db
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|D
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|D#, Eb
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|E
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|E#
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|Fb
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|F
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|F#, Cb
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|C
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|C#
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|Db
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|D
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>99pjs8vnrembm38aunbb0xsql3rl684</sha1>
    </revision>
    <revision>
      <id>104335</id>
      <parentid>104334</parentid>
      <timestamp>2023-03-04T04:31:51Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>104335</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3773" sha1="kkjtssesqdfqjok9hn1mhhkcbjifzu8" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
! rowspan="2" | 2L 2s notation (soft)
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|C
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|C#
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|Db
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|D
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|D#, Eb
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|E
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|E#
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|Fb
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|F
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|F#, Cb
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|C
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|C#
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|Db
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|D
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>kkjtssesqdfqjok9hn1mhhkcbjifzu8</sha1>
    </revision>
    <revision>
      <id>104339</id>
      <parentid>104335</parentid>
      <timestamp>2023-03-04T04:41:46Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>104339</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3836" sha1="70zsarfamxxwzd4ueb5otul33t3f81n" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
! rowspan="2" | [[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|C
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|^C, vDb
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|C#, Db
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|vD
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|D
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|^D, vE
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|E
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|^E, vF
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|F
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|F#, Cb
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|C
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|^C, vDb
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|C#, Db
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|vD
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>70zsarfamxxwzd4ueb5otul33t3f81n</sha1>
    </revision>
    <revision>
      <id>104347</id>
      <parentid>104339</parentid>
      <timestamp>2023-03-04T04:50:25Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>104347</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3836" sha1="8tq71rou4bjg8xz5ay8hzxntm649z2x" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
! rowspan="2" | [[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|C
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|^C, vDb
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|C#, Db
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|vD
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|D
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|^D, vE
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|E
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|^E, vF
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|F
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|^F, vC
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|C
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|^C, vDb
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|C#, Db
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|vD
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>8tq71rou4bjg8xz5ay8hzxntm649z2x</sha1>
    </revision>
    <revision>
      <id>104931</id>
      <parentid>104347</parentid>
      <timestamp>2023-03-07T07:25:16Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>104931</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5150" sha1="f1a30dj6hgbj7q2iie4ux32swipvt70" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
! rowspan="2" | [[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|C
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|^C, vDb
|-
|2
|''132.1429-133.1429''
|[[Tel:137.7778|137.7778]]
|[[Tel:138.9632|138.9632]]
|139.64
|''140''
|140.391
|[[Tel:140.7272|140.7272]]
|[[Tel:141.1711|141.1711]]
|141.813
|[[Tel:143.1579|143.1579]]
|''148-149''
|C#, Db
|-
|3
|''198.2143-199.7143''
|[[Tel:206.6667|206.6667]]
|[[Tel:208.4448|208.4448]]
|209.455
|''210''
|[[Tel:210.5865|210.5865]]
|[[Tel:211.0908|211.0908]]
|[[Tel:211.7566|211.7566]]
|[[Tel:212.7194|212.7194]]
|[[Tel:214.7368|214.7368]]
|''222-223.5''
|vD
|-
|4
|''264.2857-266.2857''
|[[Tel:275.5556|275.5556]]
|277.92635
|279.27
|''280''
|280.782
|[[Tel:281.4544|281.4544]]
|[[Tel:282.3422|282.3422]]
|[[Tel:283.6259|283.6259]]
|[[Tel:286.3158|286.3158]]
|''296-298''
|D
|-
|5
|''330.3571-332.8571''
|[[Tel:344.4444|344.4444]]
|[[Tel:347.4079|347.4079]]
|349.09
|''350''
|[[Tel:350.9775|350.9775]]
|351.818
|[[Tel:352.9277|352.9277]]
|[[Tel:354.5324|354.5324]]
|[[Tel:357.8947|357.8947]]
|''370-372.5''
|^D, vE
|-
|6
|''396.4286-399.4286''
|[[Tel:413.3333|413.3333]]
|[[Tel:416.8895|416.8895]]
|418.91
|''420''
|421.173
|[[Tel:422.1816|422.1816]]
|[[Tel:423.5133|423.5133]]
|[[Tel:425.4389|425.4389]]
|[[Tel:429.4737|429.4737]]
|''444-447''
|E
|-
|7
|''462.5-466''
|[[Tel:482.2222|482.2222]]
|[[Tel:486.3711|486.3711]]
|488.73
|''490''
|[[Tel:491.3685|491.3685]]
|[[Tel:492.5452|492.5452]]
|[[Tel:494.0988|494.0988]]
|[[Tel:496.3454|496.3454]]
|[[Tel:501.0526|501.0526]]
|''518-521.5''
|^E, vF
|-
|8
|''528.5714-532.5714''
|[[Tel:551.1111|551.1111]]
|[[Tel:555.8527|555.8527]]
|558.545
|''560''
|561.564
|[[Tel:562.9088|562.9088]]
|[[Tel:564.6843|564.6843]]
|[[Tel:567.2518|567.2518]]
|[[Tel:572.6316|572.6316]]
|''592-596''
|F
|-
|9
|''594.6429-599.1429''
|620
|[[Tel:625.3343|625.3343]]
|628.36
|''630''
|[[Tel:631.7595|631.7595]]
|[[Tel:633.2724|633.2724]]
|[[Tel:635.2699|635.2699]]
|[[Tel:638.1583|638.1583]]
|[[Tel:644.2105|644.2105]]
|''666-670.5''
|^F, vC
|-
|10
|''660.7143-665.714''3
|[[Tel:688.8889|688.8889]]
|[[Tel:694.8159|694.8159]]
|698.18
|''700''
|701.955
|703.636
|[[Tel:705.8555|705.8555]]
|[[Tel:709.0648|709.0648]]
|[[Tel:715.7895|715.7895]]
|''740-745''
|C
|-
|11
|''726.7857-732.2857''
|[[Tel:757.7778|757.7778]]
|[[Tel:764.2974|764.2974]]
|768
|''770''
|[[Tel:772.1505|772.1505]]
|[[Tel:773.9996|773.9996]]
|776.441
|[[Tel:779.9713|779.9713]]
|[[Tel:787.3684|787.3684]]
|''814-819.5''
|^C, vDb
|-
|12
|''792.8571-798.8571''
|[[Tel:826.6667|826.6667]]
|[[Tel:833.7791|833.7791]]
|837.82
|''840''
|842.346
|[[Tel:844.3632|844.3632]]
|[[Tel:847.0265|847.0265]]
|[[Tel:850.8778|850.8778]]
|[[Tel:858.9474|858.9474]]
|''888-894''
|C#, Db
|-
|13
|''858.9286-865.4286''
|[[Tel:895.5556|895.5556]]
|903.26065
|907.64
|''910''
|[[Tel:912.5415|912.5415]]
|[[Tel:914.7268|914.7268]]
|[[Tel:917.6121|917.6121]]
|[[Tel:921.7842|921.7842]]
|[[Tel:930.5263|930.5263]]
|''962-968.5''
|vD
|-
|14
|''925-932''
|[[Tel:964.4444|964.4444]]
|[[Tel:972.7422|972.7422]]
|977.455
|''980''
|982.737
|[[Tel:985.0904|985.0904]]
|[[Tel:988.1976|988.1976]]
|[[Tel:992.6907|992.6907]]
|1002.1053
|''1036-1043''
|D
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|^D, vE
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|E
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|^E, vF
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|F
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|^F, vC
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|C
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>f1a30dj6hgbj7q2iie4ux32swipvt70</sha1>
    </revision>
    <revision>
      <id>104938</id>
      <parentid>104931</parentid>
      <timestamp>2023-03-07T07:44:05Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Fix telephone links</comment>
      <origin>104938</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3858" sha1="abwfbioi3pe5nka6xeca4q6dg3hh3aw" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
|-
! rowspan="2" |degree
! rowspan="2" |''ed233\420-5¢''
! rowspan="2" |ed31\54
! rowspan="2" |ed121/81 (~ed11\19)
! rowspan="2" |ed32\55
! rowspan="2" |ed700¢=''r¢''
! rowspan="2" |ed3/2
! colspan="2" |Pyrite
! rowspan="2" |ed122/81 (~ed13\22)
! rowspan="2" |ed34\57
! rowspan="2" |''ed37\60+5¢''
! rowspan="2" | [[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
|-
!(~ed17\29)
!(~ed10\17)
|-
! colspan="12" |0
|C
|-
|1
|''66.0714-66.5714''
|68.8889
|69.4816
|69.82
|''70''
|70.1955
|70.3636
|70.58555
|70.9065
|71.57895
|''74-74.5''
|^C, vDb
|-
|2
|''132.1429-133.1429''
|137.7778
|138.9632
|139.64
|''140''
|140.391
|140.7272
|141.1711
|141.813
|143.1579
|''148-149''
|C#, Db
|-
|3
|''198.2143-199.7143''
|206.6667
|208.4448
|209.455
|''210''
|210.5865
|211.0908
|211.7566
|212.7194
|214.7368
|''222-223.5''
|vD
|-
|4
|''264.2857-266.2857''
|275.5556
|277.92635
|279.27
|''280''
|280.782
|281.4544
|282.3422
|283.6259
|286.3158
|''296-298''
|D
|-
|5
|''330.3571-332.8571''
|344.4444
|347.4079
|349.09
|''350''
|350.9775
|351.818
|352.9277
|354.5324
|357.8947
|''370-372.5''
|^D, vE
|-
|6
|''396.4286-399.4286''
|413.3333
|416.8895
|418.91
|''420''
|421.173
|422.1816
|423.5133
|425.4389
|429.4737
|''444-447''
|E
|-
|7
|''462.5-466''
|482.2222
|486.3711
|488.73
|''490''
|491.3685
|492.5452
|494.0988
|496.3454
|501.0526
|''518-521.5''
|^E, vF
|-
|8
|''528.5714-532.5714''
|551.1111
|555.8527
|558.545
|''560''
|561.564
|562.9088
|564.6843
|567.2518
|572.6316
|''592-596''
|F
|-
|9
|''594.6429-599.1429''
|620
|625.3343
|628.36
|''630''
|631.7595
|633.2724
|635.2699
|638.1583
|644.2105
|''666-670.5''
|^F, vC
|-
|10
|''660.7143-665.714''3
|688.8889
|694.8159
|698.18
|''700''
|701.955
|703.636
|705.8555
|709.0648
|715.7895
|''740-745''
|C
|-
|11
|''726.7857-732.2857''
|757.7778
|764.2974
|768
|''770''
|772.1505
|773.9996
|776.441
|779.9713
|787.3684
|''814-819.5''
|^C, vDb
|-
|12
|''792.8571-798.8571''
|826.6667
|833.7791
|837.82
|''840''
|842.346
|844.3632
|847.0265
|850.8778
|858.9474
|''888-894''
|C#, Db
|-
|13
|''858.9286-865.4286''
|895.5556
|903.26065
|907.64
|''910''
|912.5415
|914.7268
|917.6121
|921.7842
|930.5263
|''962-968.5''
|vD
|-
|14
|''925-932''
|964.4444
|972.7422
|977.455
|''980''
|982.737
|985.0904
|988.1976
|992.6907
|1002.1053
|''1036-1043''
|D
|-
|15
|''991.0714-998.5714''
|1033.3333
|1042.2238
|1047.27
|''1050''
|1052.9325
|1055.45405
|1058.7832
|1063.5972
|1073.6842
|''1110-1117.5''
|^D, vE
|-
|16
|''1057.1429-1065.1429''
|1102.2222
|1111.7054
|1117.09
|''1120''
|1123.128
|1125.81765
|1129.3688
|1134.5037
|1145.2632
|''1184-1192''
|E
|-
|17
|''1123.2143-1131.7143''
|1171.1111
|1181.187
|1186.91
|''1190''
|1193.3235
|1196.18125
|1199.9543
|1205.4102
|1216.8451
|''1258-1268.5''
|^E, vF
|-
|18
|''1189.2857-1198.2857''
|1240
|1250.6686
|1256.73
|''1260''
|1263.519
|1266.5449
|1270.5398
|1276.3166
|1288.42105
|''1332-1341''
|F
|-
|19
|''1255.3571-1263.8571''
|1308.8889
|1320.1502
|1326.545
|''1330''
|1333.7145
|1336.9085
|1341.1254
|1347.2231
|1360
|''1406-1415.5''
|^F, vC
|-
|20
|''1321.4286-1331.4286''
|1377.7778
|1389.6318
|1396.36
|''1400''
|1403.91
|1407.272
|1411.7109
|1418.1296
|1431.57895
|''1480-1490''
|C
|}
== Music ==
* http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>abwfbioi3pe5nka6xeca4q6dg3hh3aw</sha1>
    </revision>
    <revision>
      <id>105006</id>
      <parentid>104938</parentid>
      <timestamp>2023-03-08T04:03:57Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>105006</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1312" sha1="ei688i60lq12hjwow1fw56strcobhqe" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
!degree
!
![[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
|-
! colspan="2" |0
|C
|-
| 1
|70.1955
|^C, vDb
|-
|2
|140.391
|C#, Db
|-
|3
|[[Tel:210.5865|210.5865]]
|vD
|-
|4
|280.782
|D
|-
|5
|[[Tel:350.9775|350.9775]]
|^D, vE
|-
|6
|421.173
|E
|-
|7
|[[Tel:491.3685|491.3685]]
|^E, vF
|-
| 8
|561.564
|F
|-
|9
|[[Tel:631.7595|631.7595]]
|^F, vC
|-
|10
|701.955
|C
|-
|11
|[[Tel:772.1505|772.1505]]
|^C, vDb
|-
|12
|842.346
|C#, Db
|-
|13
|[[Tel:912.5415|912.5415]]
|vD
|-
|14
|982.737
|D
|-
|15
|1052.9325
|^D, vE
|-
|16
|1123.128
|E
|-
|17
|1193.3235
|^E, vF
|-
|18
|1263.519
|F
|-
|19
|1333.7145
|^F, vC
|-
|20
|1403.91
|C
|}
==Music==
*http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>ei688i60lq12hjwow1fw56strcobhqe</sha1>
    </revision>
    <revision>
      <id>105071</id>
      <parentid>105006</parentid>
      <timestamp>2023-03-08T20:41:16Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>105071</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1211" sha1="63w6ldy2l2eluthwotyxa4eyvdzgire" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
!degree
!
![[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
|-
! colspan="2" |0
|C
|-
| 1
|70.1955
|^C, vDb
|-
|2
|140.391
|C#, Db
|-
|3
|210.5865
|vD
|-
|4
|280.782
|D
|-
|5
|350.9775
|^D, vE
|-
|6
|421.173
|E
|-
|7
|491.3685
|^E, vF
|-
| 8
|561.564
|F
|-
|9
|631.7595
|^F, vC
|-
|10
|701.955
|C
|-
|11
|772.1505
|^C, vDb
|-
|12
|842.346
|C#, Db
|-
|13
|912.5415
|vD
|-
|14
|982.737
|D
|-
|15
|1052.9325
|^D, vE
|-
|16
|1123.128
|E
|-
|17
|1193.3235
|^E, vF
|-
|18
|1263.519
|F
|-
|19
|1333.7145
|^F, vC
|-
|20
|1403.91
|C
|}

==Music==
*http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>63w6ldy2l2eluthwotyxa4eyvdzgire</sha1>
    </revision>
    <revision>
      <id>105691</id>
      <parentid>105071</parentid>
      <timestamp>2023-03-16T22:15:27Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>105691</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5550" sha1="eyrf4dnk9qgqiye109iuvcr67846ele" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
!degree
!
![[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
|-
! colspan="2" |0
|C
|-
| 1
|70.1955
|^C, vDb
|-
|2
|140.391
|C#, Db
|-
|3
|210.5865
|vD
|-
|4
|280.782
|D
|-
|5
|350.9775
|^D, vE
|-
|6
|421.173
|E
|-
|7
|491.3685
|^E, vF
|-
| 8
|561.564
|F
|-
|9
|631.7595
|^F, vC
|-
|10
|701.955
|C
|-
|11
|772.1505
|^C, vDb
|-
|12
|842.346
|C#, Db
|-
|13
|912.5415
|vD
|-
|14
|982.737
|D
|-
|15
|1052.9325
|^D, vE
|-
|16
|1123.128
|E
|-
|17
|1193.3235
|^E, vF
|-
|18
|1263.519
|F
|-
|19
|1333.7145
|^F, vC
|-
|20
|1403.91
|C
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 68.57143 cents (4\7/10 = 2\35) to 72 cents (3\5/10 = 3\50)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || || || || ||68.5714||
|-
| || || || || || ||27\47||68.9362||
|-
| || || || || ||23\40|| ||69.000||
|-
| || || || || || ||42\73||69.0411||
|-
| || || || ||19\33|| || ||69.0909||
|-
| || || || || || ||53\92||69.1304||
|-
| || || || || ||34\59|| ||69.1525||
|-
| || || || || || ||49\85||69.1765||
|-
| || || ||15\26|| || || ||69.2308||
|-
| || || || || || ||56\97||69.2784||
|-
| || || || || ||41\71|| ||69.2958||
|-
| || || || || || ||67\116||69.3103||
|-
| || || || ||26\45|| || ||69.3333||[[Flattone]] is in this region
|-
| || || || || || ||63\109||69.3578||
|-
| || || || || ||37\64|| ||69.3750||
|-
| || || || || || ||48\83||69.3976||
|-
| || ||11\19|| || || || ||69.4737||
|-
| || || || || || ||51\88||69.5455||
|-
| || || || || ||40\69|| ||69.5652||
|-
| || || || || || ||69\119||69.5798||
|-
| || || || ||29\50|| || ||69.6000||
|-
| || || || || || ||66\131||69.6183||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||69.6296||
|-
| || || || || || ||65\112||69.6429||
|-
| || || ||18\31|| || || ||69.6774||[[Meantone]] is in this region
|-
| || || || || || ||61\105||69.7143||
|-
| || || || || ||43\74|| ||69.7297||
|-
| || || || || || ||68\117||69.7436||
|-
| || || || ||25\43|| || ||69.7674||
|-
| || || || || || ||57\98||69.7959||
|-
| || || || || ||32\55|| ||69.8182||
|-
| || || || || || ||39\67||69.8507||
|-
| ||7\12|| || || || || ||70.000||
|-
| || || || || || ||38\65||70.1539||
|-
| || || || || ||31\53|| ||70.1887||The generator closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||70.2128||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||70.2409||
|-
| || || || || || ||65\111||70.2703||
|-
| || || || || ||41\70|| ||70.2857||
|-
| || || || || || ||58\99||70.3030||
|-
| || || ||17\29|| || || ||70.3448||
|-
| || || || || || ||61\104||70.3846||
|-
| || || || || ||44\75|| ||70.4000||
|-
| || || || || || ||71\121||70.4132||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||70.4348||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||70.5487||
|-
| || || || || ||37\63|| ||70.4762||
|-
| || || || || || ||47\80||70.5000||
|-
| || ||10\17|| || || || ||70.5882||The generator closest to a just [[25/24]] for EDOs less than 200
|-
| || || || || || ||43\73||70.6849||
|-
| || || || || ||33\56|| ||70.7143||
|-
| || || || || || ||56\95||70.7368||
|-
| || || || ||23\39|| || ||70.7692||
|-
| || || || || || ||59\100||70.8000||
|-
| || || || || ||36\61|| ||70.8197||
|-
| || || || || || ||49\83||70.8434||
|-
| || || ||13\22|| || || ||70.9091||[[Archy]] is in this region
|-
| || || || || || ||42\71||70.9859||
|-
| || || || || ||29\49|| ||71.0204||
|-
| || || || || || ||45\76||71.0526||
|-
| || || || ||16\27|| || ||71.1111||
|-
| || || || || || ||35\59||71.1864||
|-
| || || || || ||19\32|| ||71.2500||
|-
| || || || || || ||22\37||71.3514||
|-
|3\5|| || || || || || ||72.000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
==Music==
*http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>eyrf4dnk9qgqiye109iuvcr67846ele</sha1>
    </revision>
    <revision>
      <id>106037</id>
      <parentid>105691</parentid>
      <timestamp>2023-03-22T18:11:51Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>106037</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5714" sha1="9bj57jzzn31fxyra0z8bm5ho6mfbsws" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
!degree
!
![[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
![[Ed9/4|Neapolitan]] notation using 3/10edf
|-
! colspan="2" |0
|C
|F
|-
| 1
|70.1955
|^C, vDb
|F^, Gb
|-
|2
|140.391
|C#, Db
|F#, Gd
|-
|3
|210.5865
|vD
|G
|-
|4
|280.782
|D
|G^, Ab
|-
|5
|350.9775
|^D, vE
|G#, Ad
|-
|6
|421.173
|E
|A
|-
|7
|491.3685
|^E, vF
|A^, Hb
|-
| 8
|561.564
|F
|A#, Hd
|-
|9
|631.7595
|^F, vC
|H
|-
|10
|701.955
|C
|B
|-
|11
|772.1505
|^C, vDb
|B^, Cb
|-
|12
|842.346
|C#, Db
|B#, Cd
|-
|13
|912.5415
|vD
|C
|-
|14
|982.737
|D
|C^, Db
|-
|15
|1052.9325
|^D, vE
|C#, Dd
|-
|16
|1123.128
|E
|D
|-
|17
|1193.3235
|^E, vF
|D^, Eb
|-
|18
|1263.519
|F
|D#, Eb
|-
|19
|1333.7145
|^F, vC
|E
|-
|20
|1403.91
|C
|F
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 68.57143 cents (4\7/10 = 2\35) to 72 cents (3\5/10 = 3\50)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || || || || ||68.5714||
|-
| || || || || || ||27\47||68.9362||
|-
| || || || || ||23\40|| ||69.000||
|-
| || || || || || ||42\73||69.0411||
|-
| || || || ||19\33|| || ||69.0909||
|-
| || || || || || ||53\92||69.1304||
|-
| || || || || ||34\59|| ||69.1525||
|-
| || || || || || ||49\85||69.1765||
|-
| || || ||15\26|| || || ||69.2308||
|-
| || || || || || ||56\97||69.2784||
|-
| || || || || ||41\71|| ||69.2958||
|-
| || || || || || ||67\116||69.3103||
|-
| || || || ||26\45|| || ||69.3333||[[Flattone]] is in this region
|-
| || || || || || ||63\109||69.3578||
|-
| || || || || ||37\64|| ||69.3750||
|-
| || || || || || ||48\83||69.3976||
|-
| || ||11\19|| || || || ||69.4737||
|-
| || || || || || ||51\88||69.5455||
|-
| || || || || ||40\69|| ||69.5652||
|-
| || || || || || ||69\119||69.5798||
|-
| || || || ||29\50|| || ||69.6000||
|-
| || || || || || ||66\131||69.6183||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||69.6296||
|-
| || || || || || ||65\112||69.6429||
|-
| || || ||18\31|| || || ||69.6774||[[Meantone]] is in this region
|-
| || || || || || ||61\105||69.7143||
|-
| || || || || ||43\74|| ||69.7297||
|-
| || || || || || ||68\117||69.7436||
|-
| || || || ||25\43|| || ||69.7674||
|-
| || || || || || ||57\98||69.7959||
|-
| || || || || ||32\55|| ||69.8182||
|-
| || || || || || ||39\67||69.8507||
|-
| ||7\12|| || || || || ||70.000||
|-
| || || || || || ||38\65||70.1539||
|-
| || || || || ||31\53|| ||70.1887||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||70.2128||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||70.2409||
|-
| || || || || || ||65\111||70.2703||
|-
| || || || || ||41\70|| ||70.2857||
|-
| || || || || || ||58\99||70.3030||
|-
| || || ||17\29|| || || ||70.3448||
|-
| || || || || || ||61\104||70.3846||
|-
| || || || || ||44\75|| ||70.4000||
|-
| || || || || || ||71\121||70.4132||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||70.4348||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||70.5487||
|-
| || || || || ||37\63|| ||70.4762||
|-
| || || || || || ||47\80||70.5000||
|-
| || ||10\17|| || || || ||70.5882||The generator closest to a just [[25/24]] for EDOs less than 200
|-
| || || || || || ||43\73||70.6849||
|-
| || || || || ||33\56|| ||70.7143||
|-
| || || || || || ||56\95||70.7368||
|-
| || || || ||23\39|| || ||70.7692||
|-
| || || || || || ||59\100||70.8000||
|-
| || || || || ||36\61|| ||70.8197||
|-
| || || || || || ||49\83||70.8434||
|-
| || || ||13\22|| || || ||70.9091||[[Archy]] is in this region
|-
| || || || || || ||42\71||70.9859||
|-
| || || || || ||29\49|| ||71.0204||
|-
| || || || || || ||45\76||71.0526||
|-
| || || || ||16\27|| || ||71.1111||
|-
| || || || || || ||35\59||71.1864||
|-
| || || || || ||19\32|| ||71.2500||
|-
| || || || || || ||22\37||71.3514||
|-
|3\5|| || || || || || ||72.000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
==Music==
*http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>9bj57jzzn31fxyra0z8bm5ho6mfbsws</sha1>
    </revision>
    <revision>
      <id>107009</id>
      <parentid>106037</parentid>
      <timestamp>2023-04-05T03:41:41Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>107009</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5789" sha1="siqhsmkg0na8o83j2gnwdnqjcw3wh8x" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
!degree
!
![[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
![[Ed9/4|Neapolitan]] notation using 3/10edf
|-
! colspan="2" |0
|C
|F
|-
| 1
|70.1955
|^C, vDb
|F^, Gb
|-
|2
|140.391
|C#, Db
|F#, Gd
|-
|3
|210.5865
|vD
|G
|-
|4
|280.782
|D
|G^, Ab
|-
|5
|350.9775
|^D, vE
|G#, Ad
|-
|6
|421.173
|E
|A
|-
|7
|491.3685
|^E, vF
|A^, Hb
|-
| 8
|561.564
|F
|A#, Hd
|-
|9
|631.7595
|^F, vC
|H
|-
|10
|701.955
|C
|B
|-
|11
|772.1505
|^C, vDb
|B^, Cb
|-
|12
|842.346
|C#, Db
|B#, Cd
|-
|13
|912.5415
|vD
|C
|-
|14
|982.737
|D
|C^, Db
|-
|15
|1052.9325
|^D, vE
|C#, Dd
|-
|16
|1123.128
|E
|D
|-
|17
|1193.3235
|^E, vF
|D^, Eb
|-
|18
|1263.519
|F
|D#, Eb
|-
|19
|1333.7145
|^F, vC
|E
|-
|20
|1403.91
|C
|F
|}
==Scale tree==
EDF scales can be approximated in [[EDO]]s by subdividing diatonic fifths. If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 68.57143 cents (4\7/10 = 2\35) to 72 cents (3\5/10 = 3\50)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || || || || ||68.5714||
|-
| || || || || || ||27\47||68.9362||
|-
| || || || || ||23\40|| ||69.000||
|-
| || || || || || ||42\73||69.0411||
|-
| || || || ||19\33|| || ||69.0909||
|-
| || || || || || ||53\92||69.1304||
|-
| || || || || ||34\59|| ||69.1525||
|-
| || || || || || ||49\85||69.1765||
|-
| || || ||15\26|| || || ||69.2308||
|-
| || || || || || ||56\97||69.2784||
|-
| || || || || ||41\71|| ||69.2958||
|-
| || || || || || ||67\116||69.3103||
|-
| || || || ||26\45|| || ||69.3333||[[Flattone]] is in this region
|-
| || || || || || ||63\109||69.3578||
|-
| || || || || ||37\64|| ||69.3750||
|-
| || || || || || ||48\83||69.3976||
|-
| || ||11\19|| || || || ||69.4737||
|-
| || || || || || ||51\88||69.5455||
|-
| || || || || ||40\69|| ||69.5652||
|-
| || || || || || ||69\119||69.5798||
|-
| || || || ||29\50|| || ||69.6000||
|-
| || || || || || ||66\131||69.6183||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||69.6296||
|-
| || || || || || ||65\112||69.6429||
|-
| || || ||18\31|| || || ||69.6774||[[Meantone]] is in this region
|-
| || || || || || ||61\105||69.7143||
|-
| || || || || ||43\74|| ||69.7297||
|-
| || || || || || ||68\117||69.7436||
|-
| || || || ||25\43|| || ||69.7674||
|-
| || || || || || ||57\98||69.7959||
|-
| || || || || ||32\55|| ||69.8182||
|-
| || || || || || ||39\67||69.8507||
|-
| ||7\12|| || || || || ||70.000||
|-
| || || || || || ||38\65||70.1539||
|-
| || || || || ||31\53|| ||70.1887||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||70.2128||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||70.2409||
|-
| || || || || || ||65\111||70.2703||
|-
| || || || || ||41\70|| ||70.2857||
|-
| || || || || || ||58\99||70.3030||
|-
| || || ||17\29|| || || ||70.3448||
|-
| || || || || || ||61\104||70.3846||
|-
| || || || || ||44\75|| ||70.4000||
|-
| || || || || || ||71\121||70.4132||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||70.4348||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||70.5487||
|-
| || || || || ||37\63|| ||70.4762||
|-
| || || || || || ||47\80||70.5000||
|-
| || ||10\17|| || || || ||70.5882||The generator closest to a just [[25/24]] for EDOs less than 200
|-
| || || || || || ||43\73||70.6849||
|-
| || || || || ||33\56|| ||70.7143||
|-
| || || || || || ||56\95||70.7368||
|-
| || || || ||23\39|| || ||70.7692||
|-
| || || || || || ||59\100||70.8000||
|-
| || || || || ||36\61|| ||70.8197||
|-
| || || || || || ||49\83||70.8434||
|-
| || || ||13\22|| || || ||70.9091||[[Archy]] is in this region
|-
| || || || || || ||42\71||70.9859||
|-
| || || || || ||29\49|| ||71.0204||
|-
| || || || || || ||45\76||71.0526||
|-
| || || || ||16\27|| || ||71.1111||
|-
| || || || || || ||35\59||71.1864||
|-
| || || || || ||19\32|| ||71.2500||
|-
| || || || || || ||22\37||71.3514||
|-
|3\5|| || || || || || ||72.000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
==Music==
*http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>siqhsmkg0na8o83j2gnwdnqjcw3wh8x</sha1>
    </revision>
    <revision>
      <id>142530</id>
      <parentid>107009</parentid>
      <timestamp>2024-05-07T13:20:19Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>-irrelevant shit</comment>
      <origin>142530</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1379" sha1="38i451n2x6al91vmoolfkx4hacua23t" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
!degree
!
![[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
![[Ed9/4|Neapolitan]] notation using 3/10edf
|-
! colspan="2" |0
|C
|F
|-
| 1
|70.1955
|^C, vDb
|F^, Gb
|-
|2
|140.391
|C#, Db
|F#, Gd
|-
|3
|210.5865
|vD
|G
|-
|4
|280.782
|D
|G^, Ab
|-
|5
|350.9775
|^D, vE
|G#, Ad
|-
|6
|421.173
|E
|A
|-
|7
|491.3685
|^E, vF
|A^, Hb
|-
| 8
|561.564
|F
|A#, Hd
|-
|9
|631.7595
|^F, vC
|H
|-
|10
|701.955
|C
|B
|-
|11
|772.1505
|^C, vDb
|B^, Cb
|-
|12
|842.346
|C#, Db
|B#, Cd
|-
|13
|912.5415
|vD
|C
|-
|14
|982.737
|D
|C^, Db
|-
|15
|1052.9325
|^D, vE
|C#, Dd
|-
|16
|1123.128
|E
|D
|-
|17
|1193.3235
|^E, vF
|D^, Eb
|-
|18
|1263.519
|F
|D#, Eb
|-
|19
|1333.7145
|^F, vC
|E
|-
|20
|1403.91
|C
|F
|}

==Music==
*http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>38i451n2x6al91vmoolfkx4hacua23t</sha1>
    </revision>
    <revision>
      <id>157492</id>
      <parentid>142530</parentid>
      <timestamp>2024-10-02T10:08:08Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Harmonics</comment>
      <origin>157492</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1543" sha1="3etn7qorms9v2xwhxhjey9e87sniacy" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]

==Intervals==
{| class="wikitable"
!degree
!
![[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
![[Ed9/4|Neapolitan]] notation using 3/10edf
|-
! colspan="2" |0
|C
|F
|-
| 1
|70.1955
|^C, vDb
|F^, Gb
|-
|2
|140.391
|C#, Db
|F#, Gd
|-
|3
|210.5865
|vD
|G
|-
|4
|280.782
|D
|G^, Ab
|-
|5
|350.9775
|^D, vE
|G#, Ad
|-
|6
|421.173
|E
|A
|-
|7
|491.3685
|^E, vF
|A^, Hb
|-
| 8
|561.564
|F
|A#, Hd
|-
|9
|631.7595
|^F, vC
|H
|-
|10
|701.955
|C
|B
|-
|11
|772.1505
|^C, vDb
|B^, Cb
|-
|12
|842.346
|C#, Db
|B#, Cd
|-
|13
|912.5415
|vD
|C
|-
|14
|982.737
|D
|C^, Db
|-
|15
|1052.9325
|^D, vE
|C#, Dd
|-
|16
|1123.128
|E
|D
|-
|17
|1193.3235
|^E, vF
|D^, Eb
|-
|18
|1263.519
|F
|D#, Eb
|-
|19
|1333.7145
|^F, vC
|E
|-
|20
|1403.91
|C
|F
|}

== Harmonics ==
{{Harmonics in equal
| steps = 10
| num = 3
| denom = 2
}}
{{Harmonics in equal
| steps = 10
| num = 3
| denom = 2
| start = 12
| collapsed = 1
}}

==Music==
*http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>3etn7qorms9v2xwhxhjey9e87sniacy</sha1>
    </revision>
    <revision>
      <id>171802</id>
      <parentid>157492</parentid>
      <timestamp>2024-12-19T08:58:19Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>{{todo|improve synopsis|text=Make it longer, describe more about the tuning.}}</comment>
      <origin>171802</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1622" sha1="1lz93g1bwwoczw0nyul1odkyx5z2k0k" xml:space="preserve">{{Infobox ET}}
'''[[EDF|Division of the just perfect fifth]] into 10 equal parts''' (10EDF) is related to [[17edo|17 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 6.6765 cents compressed and the step size is about 70.1955 cents. It is consistent to the [[7-odd-limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the [[3-odd-limit|4-integer-limit]].

Lookalikes: [[17edo]], [[27edt]]
{{todo|improve synopsis|text=Make it longer, describe more about the tuning.}}

==Intervals==
{| class="wikitable"
!degree
!
![[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
![[Ed9/4|Neapolitan]] notation using 3/10edf
|-
! colspan="2" |0
|C
|F
|-
| 1
|70.1955
|^C, vDb
|F^, Gb
|-
|2
|140.391
|C#, Db
|F#, Gd
|-
|3
|210.5865
|vD
|G
|-
|4
|280.782
|D
|G^, Ab
|-
|5
|350.9775
|^D, vE
|G#, Ad
|-
|6
|421.173
|E
|A
|-
|7
|491.3685
|^E, vF
|A^, Hb
|-
| 8
|561.564
|F
|A#, Hd
|-
|9
|631.7595
|^F, vC
|H
|-
|10
|701.955
|C
|B
|-
|11
|772.1505
|^C, vDb
|B^, Cb
|-
|12
|842.346
|C#, Db
|B#, Cd
|-
|13
|912.5415
|vD
|C
|-
|14
|982.737
|D
|C^, Db
|-
|15
|1052.9325
|^D, vE
|C#, Dd
|-
|16
|1123.128
|E
|D
|-
|17
|1193.3235
|^E, vF
|D^, Eb
|-
|18
|1263.519
|F
|D#, Eb
|-
|19
|1333.7145
|^F, vC
|E
|-
|20
|1403.91
|C
|F
|}

== Harmonics ==
{{Harmonics in equal
| steps = 10
| num = 3
| denom = 2
}}
{{Harmonics in equal
| steps = 10
| num = 3
| denom = 2
| start = 12
| collapsed = 1
}}

==Music==
*http://www.archive.org/details/10Edf by [[Peter Kosmorsky]]

[[Category:Edf]]
[[Category:Listen]]

[[Category:todo:expand]]
[[Category:todo:improve synopsis]]</text>
      <sha1>1lz93g1bwwoczw0nyul1odkyx5z2k0k</sha1>
    </revision>
    <revision>
      <id>197258</id>
      <parentid>171802</parentid>
      <timestamp>2025-05-21T14:26:52Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup (1/)</comment>
      <origin>197258</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1539" sha1="6f44byvv7m49cmk61acesg3xir5jcw9" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edf is related to [[17edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1|octave]] being just. The octave is about 6.68 cents compressed. 10edf is [[consistent]] to the [[integer limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the 4-integer-limit.

=== Harmonics ===
{{Harmonics in equal|10|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|10|3|2|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 10edf (continued)}}

==Intervals==
{| class="wikitable"
!degree
!
![[1L 3s (fifth-equivalent)|Neptunian]] notation using 8\10edf
![[Ed9/4|Neapolitan]] notation using 3/10edf
|-
! colspan="2" |0
|C
|F
|-
| 1
|70.1955
|^C, vDb
|F^, Gb
|-
|2
|140.391
|C#, Db
|F#, Gd
|-
|3
|210.5865
|vD
|G
|-
|4
|280.782
|D
|G^, Ab
|-
|5
|350.9775
|^D, vE
|G#, Ad
|-
|6
|421.173
|E
|A
|-
|7
|491.3685
|^E, vF
|A^, Hb
|-
| 8
|561.564
|F
|A#, Hd
|-
|9
|631.7595
|^F, vC
|H
|-
|10
|701.955
|C
|B
|-
|11
|772.1505
|^C, vDb
|B^, Cb
|-
|12
|842.346
|C#, Db
|B#, Cd
|-
|13
|912.5415
|vD
|C
|-
|14
|982.737
|D
|C^, Db
|-
|15
|1052.9325
|^D, vE
|C#, Dd
|-
|16
|1123.128
|E
|D
|-
|17
|1193.3235
|^E, vF
|D^, Eb
|-
|18
|1263.519
|F
|D#, Eb
|-
|19
|1333.7145
|^F, vC
|E
|-
|20
|1403.91
|C
|F
|}

== Music ==
; [[Peter Kosmorsky]]
* [https://www.archive.org/details/10Edf ''10 edf''] (archived 2011)

== See also ==
* [[17edo]] – relative edo
* [[27edt]] – relative edt
* [[44ed6]] – relative ed6

[[Category:Listen]]</text>
      <sha1>6f44byvv7m49cmk61acesg3xir5jcw9</sha1>
    </revision>
    <revision>
      <id>197259</id>
      <parentid>197258</parentid>
      <timestamp>2025-05-21T14:30:50Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup (2/2)</comment>
      <origin>197259</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1594" sha1="h1nfdtu5f2x1y8s0eui4c23ayixkmq7" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edf is related to [[17edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1|octave]] being just. The octave is about 6.68 cents compressed. 10edf is [[consistent]] to the [[integer limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the 4-integer-limit.

=== Harmonics ===
{{Harmonics in equal|10|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|10|3|2|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 10edf (continued)}}

== Intervals ==
{| class="wikitable center-all right-2"
! #
! Cents
! [[1L 3s (fifth-equivalent)|Neptunian]] notation&lt;br&gt;using 8\10edf
! [[Ed9/4|Neapolitan]] notation&lt;br&gt;using 3/10edf
|-
| 0
| 0.0
| C
| F
|-
| 1
| 70.2
| ^C, vDb
| F^, Gb
|-
| 2
| 140.4
| C#, Db
| F#, Gd
|-
| 3
| 210.6
| vD
| G
|-
| 4
| 280.8
| D
| G^, Ab
|-
| 5
| 351.0
| ^D, vE
| G#, Ad
|-
| 6
| 421.2
| E
| A
|-
| 7
| 491.4
| ^E, vF
| A^, Hb
|-
| 8
| 561.6
| F
| A#, Hd
|-
| 9
| 631.8
| ^F, vC
| H
|-
| 10
| 702.0
| C
| B
|-
| 11
| 772.2
| ^C, vDb
| B^, Cb
|-
| 12
| 842.3
| C#, Db
| B#, Cd
|-
| 13
| 912.5
| vD
| C
|-
| 14
| 982.7
| D
| C^, Db
|-
| 15
| 1052.9
| ^D, vE
| C#, Dd
|-
| 16
| 1123.1
| E
| D
|-
| 17
| 1193.3
| ^E, vF
| D^, Eb
|-
| 18
| 1263.5
| F
| D#, Eb
|-
| 19
| 1333.7
| ^F, vC
| E
|-
| 20
| 1403.9
| C
| F
|}

== Music ==
; [[Peter Kosmorsky]]
* [https://www.archive.org/details/10Edf ''10 edf''] (archived 2011)

== See also ==
* [[17edo]] – relative edo
* [[27edt]] – relative edt
* [[44ed6]] – relative ed6

[[Category:Listen]]</text>
      <sha1>h1nfdtu5f2x1y8s0eui4c23ayixkmq7</sha1>
    </revision>
    <revision>
      <id>197260</id>
      <parentid>197259</parentid>
      <timestamp>2025-05-21T14:32:19Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ +subsets and supersets</comment>
      <origin>197260</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1731" sha1="7c33qa4dis9astkusgjod8d4etol2j3" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edf is related to [[17edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1|octave]] being just. The octave is about 6.68 cents compressed. 10edf is [[consistent]] to the [[integer limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the 4-integer-limit.

=== Harmonics ===
{{Harmonics in equal|10|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|10|3|2|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 10edf (continued)}}

=== Subsets and supersets ===
Since 10 factors into primes as {{nowrap| 2 × 5 }}, 10edf contains [[2edf]] and [[5edf]] as subset edfs.

== Intervals ==
{| class="wikitable center-all right-2"
! #
! Cents
! [[1L 3s (fifth-equivalent)|Neptunian]] notation&lt;br&gt;using 8\10edf
! [[Ed9/4|Neapolitan]] notation&lt;br&gt;using 3/10edf
|-
| 0
| 0.0
| C
| F
|-
| 1
| 70.2
| ^C, vDb
| F^, Gb
|-
| 2
| 140.4
| C#, Db
| F#, Gd
|-
| 3
| 210.6
| vD
| G
|-
| 4
| 280.8
| D
| G^, Ab
|-
| 5
| 351.0
| ^D, vE
| G#, Ad
|-
| 6
| 421.2
| E
| A
|-
| 7
| 491.4
| ^E, vF
| A^, Hb
|-
| 8
| 561.6
| F
| A#, Hd
|-
| 9
| 631.8
| ^F, vC
| H
|-
| 10
| 702.0
| C
| B
|-
| 11
| 772.2
| ^C, vDb
| B^, Cb
|-
| 12
| 842.3
| C#, Db
| B#, Cd
|-
| 13
| 912.5
| vD
| C
|-
| 14
| 982.7
| D
| C^, Db
|-
| 15
| 1052.9
| ^D, vE
| C#, Dd
|-
| 16
| 1123.1
| E
| D
|-
| 17
| 1193.3
| ^E, vF
| D^, Eb
|-
| 18
| 1263.5
| F
| D#, Eb
|-
| 19
| 1333.7
| ^F, vC
| E
|-
| 20
| 1403.9
| C
| F
|}

== Music ==
; [[Peter Kosmorsky]]
* [https://www.archive.org/details/10Edf ''10 edf''] (archived 2011)

== See also ==
* [[17edo]] – relative edo
* [[27edt]] – relative edt
* [[44ed6]] – relative ed6

[[Category:Listen]]</text>
      <sha1>7c33qa4dis9astkusgjod8d4etol2j3</sha1>
    </revision>
    <revision>
      <id>197261</id>
      <parentid>197260</parentid>
      <timestamp>2025-05-21T14:37:09Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ + an evaluation on the amount of octave compression</comment>
      <origin>197261</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1889" sha1="8sdep2forluhl1pkffzm1ve5pgtpeo1" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edf is related to [[17edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1|octave]] being just. The octave is about 6.68 cents compressed. 10edf is [[consistent]] to the [[integer limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the 4-integer-limit. This makes 10edf a suitable tuning perhaps in the [[5-limit]], but overcompressed in any other limits, as well as the no-5 13-limit, where 17edo is best at. 

=== Harmonics ===
{{Harmonics in equal|10|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|10|3|2|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 10edf (continued)}}

=== Subsets and supersets ===
Since 10 factors into primes as {{nowrap| 2 × 5 }}, 10edf contains [[2edf]] and [[5edf]] as subset edfs.

== Intervals ==
{| class="wikitable center-all right-2"
! #
! Cents
! [[1L 3s (fifth-equivalent)|Neptunian]] notation&lt;br&gt;using 8\10edf
! [[Ed9/4|Neapolitan]] notation&lt;br&gt;using 3/10edf
|-
| 0
| 0.0
| C
| F
|-
| 1
| 70.2
| ^C, vDb
| F^, Gb
|-
| 2
| 140.4
| C#, Db
| F#, Gd
|-
| 3
| 210.6
| vD
| G
|-
| 4
| 280.8
| D
| G^, Ab
|-
| 5
| 351.0
| ^D, vE
| G#, Ad
|-
| 6
| 421.2
| E
| A
|-
| 7
| 491.4
| ^E, vF
| A^, Hb
|-
| 8
| 561.6
| F
| A#, Hd
|-
| 9
| 631.8
| ^F, vC
| H
|-
| 10
| 702.0
| C
| B
|-
| 11
| 772.2
| ^C, vDb
| B^, Cb
|-
| 12
| 842.3
| C#, Db
| B#, Cd
|-
| 13
| 912.5
| vD
| C
|-
| 14
| 982.7
| D
| C^, Db
|-
| 15
| 1052.9
| ^D, vE
| C#, Dd
|-
| 16
| 1123.1
| E
| D
|-
| 17
| 1193.3
| ^E, vF
| D^, Eb
|-
| 18
| 1263.5
| F
| D#, Eb
|-
| 19
| 1333.7
| ^F, vC
| E
|-
| 20
| 1403.9
| C
| F
|}

== Music ==
; [[Peter Kosmorsky]]
* [https://www.archive.org/details/10Edf ''10 edf''] (archived 2011)

== See also ==
* [[17edo]] – relative edo
* [[27edt]] – relative edt
* [[44ed6]] – relative ed6

[[Category:Listen]]</text>
      <sha1>8sdep2forluhl1pkffzm1ve5pgtpeo1</sha1>
    </revision>
    <revision>
      <id>200023</id>
      <parentid>197261</parentid>
      <timestamp>2025-06-19T15:29:56Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>200023</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1929" sha1="dp06vdk4mpca2dixbdwahkhpp9xn3m3" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edf is related to [[17edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1|octave]] being just. The octave is compressed by about 6.68{{c}}, a small but significant deviation. 10edf is [[consistent]] to the [[integer limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the 4-integer-limit. This makes 10edf a suitable tuning perhaps in the [[5-limit]], but overcompressed in any other limits, as well as the no-5 13-limit, where 17edo is best at. 

=== Harmonics ===
{{Harmonics in equal|10|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|10|3|2|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 10edf (continued)}}

=== Subsets and supersets ===
Since 10 factors into primes as {{nowrap| 2 × 5 }}, 10edf contains [[2edf]] and [[5edf]] as subset edfs.

== Intervals ==
{| class="wikitable center-all right-2"
|-
! #
! Cents
! [[1L 3s (fifth-equivalent)|Neptunian]] notation&lt;br&gt;using 8\10edf
! [[Ed9/4|Neapolitan]] notation&lt;br&gt;using 3/10edf
|-
| 0
| 0.0
| C
| F
|-
| 1
| 70.2
| ^C, vDb
| F^, Gb
|-
| 2
| 140.4
| C#, Db
| F#, Gd
|-
| 3
| 210.6
| vD
| G
|-
| 4
| 280.8
| D
| G^, Ab
|-
| 5
| 351.0
| ^D, vE
| G#, Ad
|-
| 6
| 421.2
| E
| A
|-
| 7
| 491.4
| ^E, vF
| A^, Hb
|-
| 8
| 561.6
| F
| A#, Hd
|-
| 9
| 631.8
| ^F, vC
| H
|-
| 10
| 702.0
| C
| B
|-
| 11
| 772.2
| ^C, vDb
| B^, Cb
|-
| 12
| 842.3
| C#, Db
| B#, Cd
|-
| 13
| 912.5
| vD
| C
|-
| 14
| 982.7
| D
| C^, Db
|-
| 15
| 1052.9
| ^D, vE
| C#, Dd
|-
| 16
| 1123.1
| E
| D
|-
| 17
| 1193.3
| ^E, vF
| D^, Eb
|-
| 18
| 1263.5
| F
| D#, Eb
|-
| 19
| 1333.7
| ^F, vC
| E
|-
| 20
| 1403.9
| C
| F
|}

== Music ==
; [[Peter Kosmorsky]]
* [https://www.archive.org/details/10Edf ''10 edf''] (archived 2011)

== See also ==
* [[17edo]] – relative edo
* [[27edt]] – relative edt
* [[44ed6]] – relative ed6

[[Category:Listen]]</text>
      <sha1>dp06vdk4mpca2dixbdwahkhpp9xn3m3</sha1>
    </revision>
    <revision>
      <id>233384</id>
      <parentid>200023</parentid>
      <timestamp>2026-07-08T06:03:21Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Music */ Add Bryan Deister's ''9ed(3/2) + 10ed(3/2) + 11ed(3/2) locked at 6:8:11 improv'' (2026)</comment>
      <origin>233384</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2146" sha1="prsxmlwv2zf6pd2pv6ppk1o7hcqk0mi" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edf is related to [[17edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1|octave]] being just. The octave is compressed by about 6.68{{c}}, a small but significant deviation. 10edf is [[consistent]] to the [[integer limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the 4-integer-limit. This makes 10edf a suitable tuning perhaps in the [[5-limit]], but overcompressed in any other limits, as well as the no-5 13-limit, where 17edo is best at. 

=== Harmonics ===
{{Harmonics in equal|10|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|10|3|2|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 10edf (continued)}}

=== Subsets and supersets ===
Since 10 factors into primes as {{nowrap| 2 × 5 }}, 10edf contains [[2edf]] and [[5edf]] as subset edfs.

== Intervals ==
{| class="wikitable center-all right-2"
|-
! #
! Cents
! [[1L 3s (fifth-equivalent)|Neptunian]] notation&lt;br&gt;using 8\10edf
! [[Ed9/4|Neapolitan]] notation&lt;br&gt;using 3/10edf
|-
| 0
| 0.0
| C
| F
|-
| 1
| 70.2
| ^C, vDb
| F^, Gb
|-
| 2
| 140.4
| C#, Db
| F#, Gd
|-
| 3
| 210.6
| vD
| G
|-
| 4
| 280.8
| D
| G^, Ab
|-
| 5
| 351.0
| ^D, vE
| G#, Ad
|-
| 6
| 421.2
| E
| A
|-
| 7
| 491.4
| ^E, vF
| A^, Hb
|-
| 8
| 561.6
| F
| A#, Hd
|-
| 9
| 631.8
| ^F, vC
| H
|-
| 10
| 702.0
| C
| B
|-
| 11
| 772.2
| ^C, vDb
| B^, Cb
|-
| 12
| 842.3
| C#, Db
| B#, Cd
|-
| 13
| 912.5
| vD
| C
|-
| 14
| 982.7
| D
| C^, Db
|-
| 15
| 1052.9
| ^D, vE
| C#, Dd
|-
| 16
| 1123.1
| E
| D
|-
| 17
| 1193.3
| ^E, vF
| D^, Eb
|-
| 18
| 1263.5
| F
| D#, Eb
|-
| 19
| 1333.7
| ^F, vC
| E
|-
| 20
| 1403.9
| C
| F
|}

== Music ==
=== 21st century ===
; [[Peter Kosmorsky]]
* [https://www.archive.org/details/10Edf ''10 edf''] (archived 2011)

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/U-XewMZiBC0 ''9ed(3/2) + 10ed(3/2) + 11ed(3/2) locked at 6:8:11 improv''] (2026, polymicrotonal, playing in 3 tuning systems simultaneously)

== See also ==
* [[17edo]] – relative edo
* [[27edt]] – relative edt
* [[44ed6]] – relative ed6

[[Category:Listen]]</text>
      <sha1>prsxmlwv2zf6pd2pv6ppk1o7hcqk0mi</sha1>
    </revision>
    <revision>
      <id>233827</id>
      <parentid>233384</parentid>
      <timestamp>2026-07-14T14:15:27Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[2/1|octave]]" to "[[octave]]"</comment>
      <origin>233827</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2142" sha1="kck01tvlkik66q36q7tcwjjven4i9ur" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edf is related to [[17edo]], but with the [[3/2|perfect fifth]] rather than the [[octave]] being just. The octave is compressed by about 6.68{{c}}, a small but significant deviation. 10edf is [[consistent]] to the [[integer limit|7-integer-limit]], but not to the 8-integer-limit. In comparison, 17edo is only consistent up to the 4-integer-limit. This makes 10edf a suitable tuning perhaps in the [[5-limit]], but overcompressed in any other limits, as well as the no-5 13-limit, where 17edo is best at. 

=== Harmonics ===
{{Harmonics in equal|10|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|10|3|2|intervals=integer|columns=12|start=12|collapsed=true|Approximation of harmonics in 10edf (continued)}}

=== Subsets and supersets ===
Since 10 factors into primes as {{nowrap| 2 × 5 }}, 10edf contains [[2edf]] and [[5edf]] as subset edfs.

== Intervals ==
{| class="wikitable center-all right-2"
|-
! #
! Cents
! [[1L 3s (fifth-equivalent)|Neptunian]] notation&lt;br&gt;using 8\10edf
! [[Ed9/4|Neapolitan]] notation&lt;br&gt;using 3/10edf
|-
| 0
| 0.0
| C
| F
|-
| 1
| 70.2
| ^C, vDb
| F^, Gb
|-
| 2
| 140.4
| C#, Db
| F#, Gd
|-
| 3
| 210.6
| vD
| G
|-
| 4
| 280.8
| D
| G^, Ab
|-
| 5
| 351.0
| ^D, vE
| G#, Ad
|-
| 6
| 421.2
| E
| A
|-
| 7
| 491.4
| ^E, vF
| A^, Hb
|-
| 8
| 561.6
| F
| A#, Hd
|-
| 9
| 631.8
| ^F, vC
| H
|-
| 10
| 702.0
| C
| B
|-
| 11
| 772.2
| ^C, vDb
| B^, Cb
|-
| 12
| 842.3
| C#, Db
| B#, Cd
|-
| 13
| 912.5
| vD
| C
|-
| 14
| 982.7
| D
| C^, Db
|-
| 15
| 1052.9
| ^D, vE
| C#, Dd
|-
| 16
| 1123.1
| E
| D
|-
| 17
| 1193.3
| ^E, vF
| D^, Eb
|-
| 18
| 1263.5
| F
| D#, Eb
|-
| 19
| 1333.7
| ^F, vC
| E
|-
| 20
| 1403.9
| C
| F
|}

== Music ==
=== 21st century ===
; [[Peter Kosmorsky]]
* [https://www.archive.org/details/10Edf ''10 edf''] (archived 2011)

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/U-XewMZiBC0 ''9ed(3/2) + 10ed(3/2) + 11ed(3/2) locked at 6:8:11 improv''] (2026, polymicrotonal, playing in 3 tuning systems simultaneously)

== See also ==
* [[17edo]] – relative edo
* [[27edt]] – relative edt
* [[44ed6]] – relative ed6

[[Category:Listen]]</text>
      <sha1>kck01tvlkik66q36q7tcwjjven4i9ur</sha1>
    </revision>
  </page>
  <page>
    <title>10edo</title>
    <ns>0</ns>
    <id>39</id>
    <revision>
      <id>39</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>39</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14918" sha1="t1qzvy6jjmrb0h18pqiwp04v34njpjp" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[10平均律|日本語]]&lt;/span&gt;

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
| 
__FORCETOC__
-----

==

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| | 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2,m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3,P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5,m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6,m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| | | 8 -5 &amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| | 
|-
| style="text-align:center;" | 25/24
| | | -3 -1 2 &amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| | 
|-
| style="text-align:center;" | 16875/16384
| | | -14 3 4 &amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| | 
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 525/512
| | | -9 1 2 1 &amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| | 
|-
| style="text-align:center;" | 49/48
| | | -4 -1 0 2 &amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| | 
|-
| style="text-align:center;" | 686/675
| | | 1 -3 -2 3 &amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| | 
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 6772805/6751042
| | | 11 -10 -10 10 &amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 243/242
| | | -1 5 0 0 -2 &amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 385/384
| | | -7 -1 1 1 1 &amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 540/539
| | | 2 3 1 -2 -1 &amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 3025/3024
| | | -4 -3 2 -1 2 &amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| | 
|-
| style="text-align:center;" | 676/675
| | | 2 -3 -2 0 0 2 &amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" | 
| | 
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

[[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

[[:File:decexperiment.ogg|decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]''&lt;/span&gt; by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]''&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]''&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3 The Csx Freight at 1:20 am]'' by [[Zach_Curley|Zach Curley]]

[http://soundcloud.com/martinsj013/sirmdbidnud2 Sirmdbidnud2 by martinsj013 on SoundCloud]

[https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      [[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>t1qzvy6jjmrb0h18pqiwp04v34njpjp</sha1>
    </revision>
    <revision>
      <id>4108</id>
      <timestamp>2018-02-06T11:16:30Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Stephen_Weigel</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 626024947 - Original comment: **</comment>
      <origin>4108</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="54827" sha1="n2biol1pixj4orey9p4up6khq0a2z6u" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Stephen_Weigel|Stephen_Weigel]] and made on &lt;tt&gt;2018-02-06 11:16:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;626024947&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
||~ Degree ||~ Cents ||~ Approximate Ratios* ||~ Additional Ratios*
of 3, 5 and 9 ||~ Interval Names ||||||~ [[Ups and Downs Notation|ups and downs notation]] ||
|| 0 ||&amp;gt; 0 ||= 1/1 || 256/243, 50/49, 25/24 || unison || unison, min 2nd ||= P1, m2 ||= D, Eb ||
|| 1 ||&amp;gt; 120 ||= 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second, large minor second || mid 2nd ||= ~2 ||= D^, Ev ||
|| 2 ||&amp;gt; 240 ||= 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || maj 2nd, min 3rd ||= M2,m3 ||= E, F ||
|| 3 ||&amp;gt; 360 ||= 16/13 || 5/4 || large neutral third || mid 3rd ||= ~3 ||= F^, Gv ||
|| 4 ||&amp;gt; 480 ||= 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || maj 3rd, perf 4th ||= M3,P4 ||= F#, G ||
|| 5 ||&amp;gt; 600 ||= 91/64, 128/91, 169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th, down 5th ||= ^4,v5 ||= G^, Av ||
|| 6 ||&amp;gt; 720 ||= 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || perf 5th, min 6th ||= P5,m6 ||= A, Bb ||
|| 7 ||&amp;gt; 840 ||= 13/8 || 8/5 || neutral sixth || mid 6th ||= ~6 ||= A^, Bv ||
|| 8 ||&amp;gt; 960 ||= 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || maj 6th, min 7th ||= M6,m7 ||= B, C ||
|| 9 ||&amp;gt; 1080 ||= 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||= ~7 ||= C^, Dv ||
|| 10 ||&amp;gt; 1200 ||= 2/1 || 243/128, 49/25, 48/25 || octave || maj 7th, octave ||= M7, P8 ||= C#, D ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
*adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Rational ||~ Monzo ||~ Size (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[https://www.reverbnation.com/1029821/album/172734|ZIA Space by Elaine Walker]] "Who Loves You, Me?," "Champagne," and "Avatar"
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Approximate Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Additional Ratios*&amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="3"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;ups and downs notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison, min 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1, m2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Eb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second, large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 2nd, min 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2,m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E, F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F^, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 3rd, perf 4th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3,P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F#, G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/64, 128/91, 169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th, down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4,v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G^, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;perf 5th, min 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P5,m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A, Bb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, Bv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 6th, min 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6,m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B, C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C^, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 7th, octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M7, P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C#, D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
*adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic fifth-based) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic 3rd-based).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:632:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10ed2-001.svg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10ed2-001.svg&amp;amp;quot; title=&amp;amp;quot;File: 10ed2-001.svg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');"&amp;gt;&amp;lt;img src="http://c1.wikicdn.com/i/mime/32/empty.png" height="32" width="32" alt="10ed2-001.svg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');" class="filename" title="10ed2-001.svg"&amp;gt;10ed2-001.svg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10ed2-001.svg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;15 KB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:632 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:629:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:629 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Rational&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="https://www.reverbnation.com/1029821/album/172734" rel="nofollow"&amp;gt;ZIA Space by Elaine Walker&amp;lt;/a&amp;gt; &amp;amp;quot;Who Loves You, Me?,&amp;amp;quot; &amp;amp;quot;Champagne,&amp;amp;quot; and &amp;amp;quot;Avatar&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:633:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://c1.wikicdn.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:633 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:634:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://c1.wikicdn.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:634 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:630:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:630 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:631:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:631 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>n2biol1pixj4orey9p4up6khq0a2z6u</sha1>
    </revision>
    <revision>
      <id>4109</id>
      <timestamp>2017-01-04T23:25:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;TallKite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 603105952 - Original comment: **</comment>
      <origin>4109</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="54444" sha1="s7718zf4zpw9xywdozzvqp5an6zlbg0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:TallKite|TallKite]] and made on &lt;tt&gt;2017-01-04 23:25:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;603105952&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
||~ Degree ||~ Cents ||~ Approximate Ratios* ||~ Additional Ratios*
of 3, 5 and 9 ||~ Interval Names ||||||~ [[Ups and Downs Notation|ups and downs notation]] ||
|| 0 ||&amp;gt; 0 ||= 1/1 || 256/243, 50/49, 25/24 || unison || unison, min 2nd ||= P1, m2 ||= D, Eb ||
|| 1 ||&amp;gt; 120 ||= 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second, large minor second || mid 2nd ||= ~2 ||= D^, Ev ||
|| 2 ||&amp;gt; 240 ||= 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || maj 2nd, min 3rd ||= M2,m3 ||= E, F ||
|| 3 ||&amp;gt; 360 ||= 16/13 || 5/4 || large neutral third || mid 3rd ||= ~3 ||= F^, Gv ||
|| 4 ||&amp;gt; 480 ||= 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || maj 3rd, perf 4th ||= M3,P4 ||= F#, G ||
|| 5 ||&amp;gt; 600 ||= 91/64, 128/91, 169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th, down 5th ||= ^4,v5 ||= G^, Av ||
|| 6 ||&amp;gt; 720 ||= 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || perf 5th, min 6th ||= P5,m6 ||= A, Bb ||
|| 7 ||&amp;gt; 840 ||= 13/8 || 8/5 || neutral sixth || mid 6th ||= ~6 ||= A^, Bv ||
|| 8 ||&amp;gt; 960 ||= 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || maj 6th, min 7th ||= M6,m7 ||= B, C ||
|| 9 ||&amp;gt; 1080 ||= 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||= ~7 ||= C^, Dv ||
|| 10 ||&amp;gt; 1200 ||= 2/1 || 243/128, 49/25, 48/25 || octave || maj 7th, octave ||= M7, P8 ||= C#, D ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
*adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Rational ||~ Monzo ||~ Size (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Approximate Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Additional Ratios*&amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="3"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;ups and downs notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison, min 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1, m2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Eb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second, large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 2nd, min 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2,m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E, F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F^, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 3rd, perf 4th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3,P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F#, G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/64, 128/91, 169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th, down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4,v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G^, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;perf 5th, min 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P5,m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A, Bb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, Bv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 6th, min 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6,m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B, C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C^, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 7th, octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M7, P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C#, D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
*adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic fifth-based) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic 3rd-based).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:629:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:629 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Rational&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:633:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:633 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:634:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:634 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:630:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:630 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:631:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:631 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>s7718zf4zpw9xywdozzvqp5an6zlbg0</sha1>
    </revision>
    <revision>
      <id>4110</id>
      <timestamp>2017-01-03T20:34:07Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;diagonalia</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 603051974 - Original comment: **</comment>
      <origin>4110</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="54440" sha1="dzbjy8veedivnnahbm5bausnbpeyecf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:diagonalia|diagonalia]] and made on &lt;tt&gt;2017-01-03 20:34:07 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;603051974&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
||~ Degree ||~ Cents ||~ Approximate Ratios* ||~ Additional Ratios*
of 3, 5 and 9 ||~ Interval Names ||||||~ [[Ups and Downs Notation|ups and downs notation]] ||
|| 0 ||&amp;gt; 0 ||= 1/1 || 256/243, 50/49, 25/24 || unison || unison, min 2nd ||= P1, m2 ||= D. Eb ||
|| 1 ||&amp;gt; 120 ||= 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second, large minor second || mid 2nd ||= ~2 ||= D^, Ev ||
|| 2 ||&amp;gt; 240 ||= 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || maj 2nd, min 3rd ||= M2,m3 ||= E, F ||
|| 3 ||&amp;gt; 360 ||= 16/13 || 5/4 || large neutral third || mid 3rd ||= ~3 ||= F^, Gv ||
|| 4 ||&amp;gt; 480 ||= 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || maj 3rd, perf 4th ||= M3,P4 ||= F#, G ||
|| 5 ||&amp;gt; 600 ||= 91/64, 128/91, 169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th, down 5th ||= ^4,v5 ||= G^, Av ||
|| 6 ||&amp;gt; 720 ||= 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || perf 5th, min 6th ||= P5,m6 ||= A, Bb ||
|| 7 ||&amp;gt; 840 ||= 13/8 || 8/5 || neutral sixth || mid 6th ||= ~6 ||= A^, Bv ||
|| 8 ||&amp;gt; 960 ||= 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || maj 6th, min 7th ||= M6,m7 ||= B, C ||
|| 9 ||&amp;gt; 1080 ||= 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||= ~7 ||= C^, Dv ||
|| 10 ||&amp;gt; 1200 ||= 2/1 || 243/128, 49/25, 48/25 || octave || maj 7th, octave ||= P8 ||= C#, D ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
*adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Rational ||~ Monzo ||~ Size (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Approximate Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Additional Ratios*&amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="3"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;ups and downs notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison, min 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1, m2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D. Eb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second, large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 2nd, min 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2,m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E, F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F^, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 3rd, perf 4th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3,P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F#, G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/64, 128/91, 169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th, down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4,v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G^, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;perf 5th, min 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P5,m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A, Bb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, Bv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 6th, min 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6,m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B, C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C^, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 7th, octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C#, D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
*adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic fifth-based) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic 3rd-based).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:632:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10ed2-001.svg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10ed2-001.svg&amp;amp;quot; title=&amp;amp;quot;File: 10ed2-001.svg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10ed2-001.svg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');" class="filename" title="10ed2-001.svg"&amp;gt;10ed2-001.svg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10ed2-001.svg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;15 KB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:632 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:629:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:629 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Rational&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:633:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:633 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:634:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:634 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:630:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:630 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:631:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:631 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>dzbjy8veedivnnahbm5bausnbpeyecf</sha1>
    </revision>
    <revision>
      <id>4111</id>
      <timestamp>2016-12-29T17:05:37Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602900664 - Original comment: **</comment>
      <origin>4111</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="54395" sha1="somi448wgyw5x6bnozhhrmdhp4cs0ro" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2016-12-29 17:05:37 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602900664&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
||~ Degree ||~ Cents ||~ Approximate Ratios* ||~ Additional Ratios*
of 3, 5 and 9 ||~ Interval Names ||||||~ [[Ups and Downs Notation|ups and downs notation]] ||
||  0 ||&amp;gt;    0 ||= 1/1 || 256/243, 50/49, 25/24 || unison || unison ||= P1 ||= D ||
||  1 ||&amp;gt;  120 ||= 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second, large minor second || mid 2nd ||= ~2 ||= D^, Ev ||
||  2 ||&amp;gt;  240 ||= 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || maj 2nd, min 3rd ||= M2,m3 ||= E, F ||
||  3 ||&amp;gt;  360 ||= 16/13 || 5/4 || large neutral third || mid 3rd ||= ~3 ||= F^, Gv ||
||  4 ||&amp;gt;  480 ||= 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || maj 3rd, perf 4th ||= M3,P4 ||= F#, G ||
||  5 ||&amp;gt;  600 ||= 91/64, 128/91, 169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th, down 5th ||= ^4,v5 ||= G^, Av ||
||  6 ||&amp;gt;  720 ||= 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || perf 5th, min 6th ||= P5,m6 ||= A, Bb ||
||  7 ||&amp;gt;  840 ||= 13/8 || 8/5 || neutral sixth || mid 6th ||= ~6 ||= A^, Bv ||
||  8 ||&amp;gt;  960 ||= 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || maj 6th, min 7th ||= M6,m7 ||= B, C ||
||  9 ||&amp;gt; 1080 ||= 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||= ~7 ||= C^, Dv ||
|| 10 ||&amp;gt; 1200 ||= 2/1 || 243/128, 49/25, 48/25 || octave || octave ||= P8 ||= D ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
*adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Rational ||~ Monzo ||~ Size (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Approximate Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Additional Ratios*&amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="3"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;ups and downs notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second, large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 2nd, min 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2,m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E, F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F^, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 3rd, perf 4th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3,P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F#, G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/64, 128/91, 169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th, down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4,v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G^, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;perf 5th, min 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P5,m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A, Bb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, Bv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 6th, min 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6,m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B, C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C^, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
*adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic fifth-based) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic 3rd-based).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:629:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:629 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Rational&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:633:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:633 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:634:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:634 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:630:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:630 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:631:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:631 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>somi448wgyw5x6bnozhhrmdhp4cs0ro</sha1>
    </revision>
    <revision>
      <id>4112</id>
      <timestamp>2016-12-29T17:02:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602900608 - Original comment: **</comment>
      <origin>4112</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="54304" sha1="9wpptv1bma8uy18tb97l5t3cv9qwf48" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2016-12-29 17:02:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602900608&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
||~ Degree ||~ Cents ||~ Approximate Ratios* ||~ Additional Ratios*
of 3, 5 and 9 ||~ Interval Names ||||||~ [[Ups and Downs Notation|ups and downs notation]] ||
||  0 ||&amp;gt;    0 ||= 1/1 || 256/243, 50/49, 25/24 || unison || unison ||= P1 ||= D ||
||  1 ||&amp;gt;  120 ||= 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second, large minor second || mid 2nd ||= ~2 ||= D^, Ev ||
||  2 ||&amp;gt;  240 ||= 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || maj 2nd, min 3rd ||= M2,m3 ||= E, F ||
||  3 ||&amp;gt;  360 ||= 16/13 || 5/4 || large neutral third || mid 3rd ||= ~3 ||= F^, Gv ||
||  4 ||&amp;gt;  480 ||= 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || maj 3rd, perf 4th ||= M3,P4 ||= F#, G ||
||  5 ||&amp;gt;  600 ||= 91/64, 128/91, 169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th, down 5th ||= ^4,v5 ||= G^, Av ||
||  6 ||&amp;gt;  720 ||= 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || perf 5th, min 6th ||= P5,m6 ||= A, Bb ||
||  7 ||&amp;gt;  840 ||= 13/8 || 8/5 || neutral sixth || mid 6th ||= ~6 ||= A^, Bv ||
||  8 ||&amp;gt;  960 ||= 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || maj 6th, min 7th ||= M6,m7 ||= B, C ||
||  9 ||&amp;gt; 1080 ||= 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||= ~7 ||= C^, Dv ||
|| 10 ||&amp;gt; 1200 ||= 2/1 || 243/128, 49/25, 48/25 || octave || octave ||= P8 ||= D ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
*adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||=   || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||=   || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||=   || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Approximate Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Additional Ratios*&amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="3"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;ups and downs notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second, large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 2nd, min 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2,m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E, F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F^, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 3rd, perf 4th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3,P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F#, G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/64, 128/91, 169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th, down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4,v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G^, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;perf 5th, min 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P5,m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A, Bb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, Bv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 6th, min 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6,m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B, C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C^, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
*adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic fifth-based) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic 3rd-based).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:633:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:633 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:634:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:634 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:630:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:630 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:631:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:631 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9wpptv1bma8uy18tb97l5t3cv9qwf48</sha1>
    </revision>
    <revision>
      <id>4113</id>
      <timestamp>2016-12-26T00:55:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;TallKite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602810518 - Original comment: **</comment>
      <origin>4113</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="53861" sha1="b70dymv1rm49ww8o7earld802mbs08q" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:TallKite|TallKite]] and made on &lt;tt&gt;2016-12-26 00:55:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602810518&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cents || Approximate
Ratios* || Additional Ratios
of 3, 5 and 9 || Interval Names ||||||= [[Ups and Downs Notation|ups and downs]]
[[Ups and Downs Notation|notation]] ||
|| 0 || 0 || 1/1 || 256/243, 50/49, 25/24 || unison || unison ||= P1 ||= D ||
|| 1 || 120 || 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second,
large minor second || mid 2nd ||= ~2 ||= D^, Ev ||
|| 2 || 240 || 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || maj 2nd,
min 3rd ||= M2,m3 ||= E, F ||
|| 3 || 360 || 16/13 || 5/4 || large neutral third || mid 3rd ||= ~3 ||= F^, Gv ||
|| 4 || 480 || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || maj 3rd,
perf 4th ||= M3,P4 ||= F#, G ||
|| 5 || 600 || 91/64, 128/91,
169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th,
down 5th ||= ^4,v5 ||= G^, Av ||
|| 6 || 720 || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || perf 5th,
min 6th ||= P5,m6 ||= A, Bb ||
|| 7 || 840 || 13/8 || 8/5 || neutral sixth || mid 6th ||= ~6 ||= A^, Bv ||
|| 8 || 960 || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || maj 6th,
min 7th ||= M6,m7 ||= B, C ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||= ~7 ||= C^, Dv ||
|| 10 || 1200 || 2/1 || 243/128, 49/25, 48/25 || octave || octave ||= P8 ||= D ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37/1 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||=   || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||=   || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||=   || | 11 -10 -10 10/1 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios&amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td colspan="3" style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;ups and downs&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second,&amp;lt;br /&amp;gt;
large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 2nd,&amp;lt;br /&amp;gt;
min 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2,m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E, F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F^, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 3rd,&amp;lt;br /&amp;gt;
perf 4th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3,P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F#, G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 128/91,&amp;lt;br /&amp;gt;
169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th,&amp;lt;br /&amp;gt;
down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4,v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G^, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;perf 5th,&amp;lt;br /&amp;gt;
min 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P5,m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A, Bb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, Bv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 6th,&amp;lt;br /&amp;gt;
min 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6,m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B, C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C^, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic fifth-based) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic 3rd-based).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37/1 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10/1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:633:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:633 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:634:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:634 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:630:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:630 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:631:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:631 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>b70dymv1rm49ww8o7earld802mbs08q</sha1>
    </revision>
    <revision>
      <id>4114</id>
      <timestamp>2016-12-22T03:52:26Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;TallKite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602738066 - Original comment: **</comment>
      <origin>4114</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="53765" sha1="tmbrkin3ni97alvcvbdisqhl15g13uf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:TallKite|TallKite]] and made on &lt;tt&gt;2016-12-22 03:52:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602738066&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cents || Approximate
Ratios* || Additional Ratios
of 3, 5 and 9 || Interval Names ||||||= [[Ups and Downs Notation|ups and downs]]
[[Ups and Downs Notation|notation]] ||
|| 0 || 0 || 1/1 || 256/243, 50/49, 25/24 || unison || unison ||= P1 ||= D ||
|| 1 || 120 || 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second,
large minor second || mid 2nd ||= ~2 ||= D^, Ev ||
|| 2 || 240 || 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || maj 2nd,
min 3rd ||= M2,m3 ||= E, F ||
|| 3 || 360 || 16/13 || 5/4 || large neutral third || mid 3rd ||= ~3 ||= F^, Gv ||
|| 4 || 480 || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || maj 3rd,
perf 4th ||= M3,P4 ||= F#, G ||
|| 5 || 600 || 91/64, 128/91,
169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th,
down 5th ||= ^4,v5 ||= G^, Av ||
|| 6 || 720 || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || perf 5th,
min 6th ||= P5,m6 ||= A, Bb ||
|| 7 || 840 || 13/8 || 8/5 || neutral sixth || mid 6th ||= ~6 ||= A^, Bv ||
|| 8 || 960 || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || maj 6th,
min 7th ||= M6,m7 ||= B, C ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||= ~7 ||= C^, Dv ||
|| 10 || 1200 || 2/1 || 243/128, 49/25, 48/25 || octave || octave ||= P8 ||= D ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37/1 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||=   || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||=   || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||=   || | 11 -10 -10 10/1 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios&amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td colspan="3" style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;ups and downs&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second,&amp;lt;br /&amp;gt;
large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 2nd,&amp;lt;br /&amp;gt;
min 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2,m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E, F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F^, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 3rd,&amp;lt;br /&amp;gt;
perf 4th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3,P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F#, G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 128/91,&amp;lt;br /&amp;gt;
169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th,&amp;lt;br /&amp;gt;
down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4,v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G^, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;perf 5th,&amp;lt;br /&amp;gt;
min 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P5,m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A, Bb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, Bv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 6th,&amp;lt;br /&amp;gt;
min 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6,m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B, C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C^, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:632:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10ed2-001.svg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10ed2-001.svg&amp;amp;quot; title=&amp;amp;quot;File: 10ed2-001.svg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10ed2-001.svg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');" class="filename" title="10ed2-001.svg"&amp;gt;10ed2-001.svg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10ed2-001.svg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;15 KB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:632 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:629:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:629 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37/1 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10/1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:633:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:633 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:634:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:634 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:630:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:630 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:631:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:631 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>tmbrkin3ni97alvcvbdisqhl15g13uf</sha1>
    </revision>
    <revision>
      <id>4115</id>
      <timestamp>2016-12-04T10:25:43Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 601330544 - Original comment: **</comment>
      <origin>4115</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="54099" sha1="htpjetlzr19f3gqlo5ya806j5gy0z4h" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-12-04 10:25:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;601330544&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cents (coarse and fine)
DMS || Approximate
Ratios* || Additional Ratios
of 3, 5 and 9 || Interval Names ||||||= [[Ups and Downs Notation|ups and downs]]
[[Ups and Downs Notation|notation]] ||
|| 0 || 0, 0° || 1/1 || 256/243, 50/49, 25/24 || unison || unison ||= P1 ||= D ||
|| 1 || 120, 144
36° || 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second,
large minor second || mid 2nd ||= ~2 ||= D^, Ev ||
|| 2 || 240, 288, 72° || 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || maj 2nd,
min 3rd ||= M2,m3 ||= E, F ||
|| 3 || 360, 432, 108° || 16/13 || 5/4 || large neutral third || mid 3rd ||= ~3 ||= F^, Gv ||
|| 4 || 480, 576, 144° || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || maj 3rd,
perf 4th ||= M3,P4 ||= F#, G ||
|| 5 || 600, 720
180° || 91/64, 128/91,
169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th,
down 5th ||= ^4,v5 ||= G^, Av ||
|| 6 || 720, 864, 216° || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || perf 5th,
min 6th ||= P5,m6 ||= A, Bb ||
|| 7 || 840, 1008, 252° || 13/8 || 8/5 || neutral sixth || mid 6th ||= ~6 ||= A^, Bv ||
|| 8 || 960, 1152, 288° || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || maj 6th,
min 7th ||= M6,m7 ||= B, C ||
|| 9 || 1080, 1296, 324° || 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||= ~7 ||= C^, Dv ||
|| 10 || 1200, 1440, 360° || 2/1 || 243/128, 49/25, 48/25 || octave || octave ||= P8 ||= D ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37/1 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||=   || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||=   || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||=   || | 11 -10 -10 10/1 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents (coarse and fine)&amp;lt;br /&amp;gt;
DMS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios&amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td colspan="3" style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;ups and downs&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0, 0°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120, 144&amp;lt;br /&amp;gt;
36°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second,&amp;lt;br /&amp;gt;
large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240, 288, 72°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 2nd,&amp;lt;br /&amp;gt;
min 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2,m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E, F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360, 432, 108°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F^, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480, 576, 144°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 3rd,&amp;lt;br /&amp;gt;
perf 4th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3,P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F#, G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600, 720&amp;lt;br /&amp;gt;
180°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 128/91,&amp;lt;br /&amp;gt;
169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th,&amp;lt;br /&amp;gt;
down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4,v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G^, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720, 864, 216°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;perf 5th,&amp;lt;br /&amp;gt;
min 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P5,m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A, Bb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840, 1008, 252°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, Bv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960, 1152, 288°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 6th,&amp;lt;br /&amp;gt;
min 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6,m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B, C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080, 1296, 324°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C^, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200, 1440, 360°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:629:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:629 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37/1 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10/1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:633:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:633 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:634:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:634 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:630:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:630 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:631:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:631 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>htpjetlzr19f3gqlo5ya806j5gy0z4h</sha1>
    </revision>
    <revision>
      <id>4116</id>
      <timestamp>2016-12-04T10:13:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 601329922 - Original comment: **</comment>
      <origin>4116</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="54207" sha1="4dfz6jajbk42jrlysvvflr1ywr1x2p0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-12-04 10:13:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;601329922&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cents (coarse and fine)
DMS || Approximate
Ratios* || Additional Ratios
of 3, 5 and 9 || Interval Names ||||||= [[Ups and Downs Notation|ups and downs]]
[[Ups and Downs Notation|notation]] ||
|| 0 || 0, 0° || 1/1 || 256/243, 50/49, 25/24 || unison || unison ||= P1 ||= D ||
|| 1 || 120, 144
36° || 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second,
large minor second || mid 2nd ||= ~2 ||= D^, Ev ||
|| 2 || 240, 288, 72° || 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || maj 2nd,
min 3rd ||= M2,m3 ||= E, F ||
|| 3 || 360, 432, 108° || 16/13 || 5/4 || large neutral third || mid 3rd ||= ~3 ||= F^, Gv ||
|| 4 || 480, 576, 144° || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || maj 3rd,
perf 4th ||= M3,P4 ||= F#, G ||
|| 5 || 600, 720
180° || 91/64, 128/91,
169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th,
down 5th ||= ^4,v5 ||= G^, Av ||
|| 6 || 720, 864, 216° || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || perf 5th,
min 6th ||= P5,m6 ||= A, Bb ||
|| 7 || 840, 1008, 252° || 13/8 || 8/5 || neutral sixth || mid 6th ||= ~6 ||= A^, Bv ||
|| 8 || 960, 1152, 288° || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || maj 6th,
min 7th ||= M6,m7 ||= B, C ||
|| 9 || 1080, 1296, 324° || 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||= ~7 ||= C^, Dv ||
|| 10 || 1200, 1440, 360° || 2/1 || 243/128, 49/25, 48/25 || octave || octave ||= P8 ||= D ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37/1 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||=   || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||=   || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||=   || | [[tel/11 -10 -10 10|11 -10 -10 10]] &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents (coarse and fine)&amp;lt;br /&amp;gt;
DMS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios&amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td colspan="3" style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;ups and downs&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0, 0°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120, 144&amp;lt;br /&amp;gt;
36°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second,&amp;lt;br /&amp;gt;
large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240, 288, 72°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 2nd,&amp;lt;br /&amp;gt;
min 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2,m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E, F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360, 432, 108°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F^, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480, 576, 144°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 3rd,&amp;lt;br /&amp;gt;
perf 4th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3,P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F#, G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600, 720&amp;lt;br /&amp;gt;
180°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 128/91,&amp;lt;br /&amp;gt;
169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th,&amp;lt;br /&amp;gt;
down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4,v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G^, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720, 864, 216°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;perf 5th,&amp;lt;br /&amp;gt;
min 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P5,m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A, Bb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840, 1008, 252°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, Bv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960, 1152, 288°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 6th,&amp;lt;br /&amp;gt;
min 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6,m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B, C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080, 1296, 324°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C^, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200, 1440, 360°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:629:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:629 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37/1 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| &amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/11%20-10%20-10%2010"&amp;gt;11 -10 -10 10&amp;lt;/a&amp;gt; &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:633:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:633 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:634:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:634 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:630:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:630 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:631:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:631 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>4dfz6jajbk42jrlysvvflr1ywr1x2p0</sha1>
    </revision>
    <revision>
      <id>4117</id>
      <timestamp>2016-12-04T10:11:18Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 601329828 - Original comment: **</comment>
      <origin>4117</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="54327" sha1="577vekoqi01oiexokn83guu83tbhjsj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-12-04 10:11:18 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;601329828&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cents (coarse and fine)
DMS || Approximate
Ratios* || Additional Ratios
of 3, 5 and 9 || Interval Names ||||||= [[Ups and Downs Notation|ups and downs]]
[[Ups and Downs Notation|notation]] ||
|| 0 || 0, 0° || 1/1 || 256/243, 50/49, 25/24 || unison || unison ||= P1 ||= D ||
|| 1 || 120, 144
36° || 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second,
large minor second || mid 2nd ||= ~2 ||= D^, Ev ||
|| 2 || 240, 288, 72° || 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || maj 2nd,
min 3rd ||= M2,m3 ||= E, F ||
|| 3 || 360, 432, 108° || 16/13 || 5/4 || large neutral third || mid 3rd ||= ~3 ||= F^, Gv ||
|| 4 || 480, 576, 144° || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || maj 3rd,
perf 4th ||= M3,P4 ||= F#, G ||
|| 5 || 600, 720
180° || 91/64, 128/91,
169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th,
down 5th ||= ^4,v5 ||= G^, Av ||
|| 6 || 720, 864, 216° || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || perf 5th,
min 6th ||= P5,m6 ||= A, Bb ||
|| 7 || 840, 1008, 252° || 13/8 || 8/5 || neutral sixth || mid 6th ||= ~6 ||= A^, Bv ||
|| 8 || 960, 1152, 288° || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || maj 6th,
min 7th ||= M6,m7 ||= B, C ||
|| 9 || 1080, 1296, 324° || 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||= ~7 ||= C^, Dv ||
|| 10 || 1200, 1440, 360° || 2/1 || 243/128, 49/25, 48/25 || octave || octave ||= P8 ||= D ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; [[tel:10 16 23 28 35 37|10 16 23 28 35 37]] |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||=   || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||=   || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||=   || | [[tel:11 -10 -10 10|11 -10 -10 10]] &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents (coarse and fine)&amp;lt;br /&amp;gt;
DMS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios&amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td colspan="3" style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;ups and downs&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0, 0°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120, 144&amp;lt;br /&amp;gt;
36°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second,&amp;lt;br /&amp;gt;
large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240, 288, 72°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 2nd,&amp;lt;br /&amp;gt;
min 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2,m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E, F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360, 432, 108°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F^, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480, 576, 144°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 3rd,&amp;lt;br /&amp;gt;
perf 4th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3,P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F#, G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600, 720&amp;lt;br /&amp;gt;
180°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 128/91,&amp;lt;br /&amp;gt;
169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th,&amp;lt;br /&amp;gt;
down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4,v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G^, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720, 864, 216°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;perf 5th,&amp;lt;br /&amp;gt;
min 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P5,m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A, Bb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840, 1008, 252°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, Bv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960, 1152, 288°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 6th,&amp;lt;br /&amp;gt;
min 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6,m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B, C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080, 1296, 324°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C^, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200, 1440, 360°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:632:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10ed2-001.svg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10ed2-001.svg&amp;amp;quot; title=&amp;amp;quot;File: 10ed2-001.svg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10ed2-001.svg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');" class="filename" title="10ed2-001.svg"&amp;gt;10ed2-001.svg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10ed2-001.svg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;15 KB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:632 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:629:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:629 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; &amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/10%2016%2023%2028%2035%2037"&amp;gt;10 16 23 28 35 37&amp;lt;/a&amp;gt; |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| &amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/11%20-10%20-10%2010"&amp;gt;11 -10 -10 10&amp;lt;/a&amp;gt; &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:633:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:633 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:634:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:634 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:630:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:630 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:631:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:631 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>577vekoqi01oiexokn83guu83tbhjsj</sha1>
    </revision>
    <revision>
      <id>4118</id>
      <timestamp>2016-11-21T07:25:59Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;TallKite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 599962808 - Original comment: **</comment>
      <origin>4118</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="54174" sha1="cwz7ag76szzwc9cfo44n6xqiqgi426a" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:TallKite|TallKite]] and made on &lt;tt&gt;2016-11-21 07:25:59 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;599962808&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cents (coarse and fine)
DMS || Approximate
Ratios* || Additional Ratios
of 3, 5 and 9 || Interval Names ||||||= [[Ups and Downs Notation|ups and downs]]
[[Ups and Downs Notation|notation]] ||
|| 0 || 0, 0° || 1/1 || 256/243, 50/49, 25/24 || unison || unison ||= P1 ||= D ||
|| 1 || 120, 144
36° || 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second,
large minor second || mid 2nd ||= ~2 ||= D^, Ev ||
|| 2 || 240, 288, 72° || 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || maj 2nd,
min 3rd ||= M2,m3 ||= E, F ||
|| 3 || 360, 432, 108° || 16/13 || 5/4 || large neutral third || mid 3rd ||= ~3 ||= F^, Gv ||
|| 4 || 480, 576, 144° || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || maj 3rd,
perf 4th ||= M3,P4 ||= F#, G ||
|| 5 || 600, 720
180° || 91/64, 128/91,
169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th,
down 5th ||= ^4,v5 ||= G^, Av ||
|| 6 || 720, 864, 216° || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || perf 5th,
min 6th ||= P5,m6 ||= A, Bb ||
|| 7 || 840, 1008, 252° || 13/8 || 8/5 || neutral sixth || mid 6th ||= ~6 ||= A^, Bv ||
|| 8 || 960, 1152, 288° || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || maj 6th,
min 7th ||= M6,m7 ||= B, C ||
|| 9 || 1080, 1296, 324° || 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||= ~7 ||= C^, Dv ||
|| 10 || 1200, 1440, 360° || 2/1 || 243/128, 49/25, 48/25 || octave || octave ||= P8 ||= D ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents (coarse and fine)&amp;lt;br /&amp;gt;
DMS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios&amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td colspan="3" style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;ups and downs&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0, 0°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120, 144&amp;lt;br /&amp;gt;
36°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second,&amp;lt;br /&amp;gt;
large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240, 288, 72°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 2nd,&amp;lt;br /&amp;gt;
min 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2,m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E, F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360, 432, 108°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F^, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480, 576, 144°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 3rd,&amp;lt;br /&amp;gt;
perf 4th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3,P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F#, G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600, 720&amp;lt;br /&amp;gt;
180°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 128/91,&amp;lt;br /&amp;gt;
169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th,&amp;lt;br /&amp;gt;
down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4,v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G^, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720, 864, 216°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;perf 5th,&amp;lt;br /&amp;gt;
min 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P5,m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A, Bb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840, 1008, 252°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, Bv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960, 1152, 288°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;maj 6th,&amp;lt;br /&amp;gt;
min 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6,m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B, C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080, 1296, 324°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C^, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200, 1440, 360°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:632:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10ed2-001.svg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10ed2-001.svg&amp;amp;quot; title=&amp;amp;quot;File: 10ed2-001.svg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10ed2-001.svg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');" class="filename" title="10ed2-001.svg"&amp;gt;10ed2-001.svg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10ed2-001.svg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;15 KB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:632 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:629:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:629 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:633:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:633 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:634:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:634 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:630:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:630 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:631:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:631 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>cwz7ag76szzwc9cfo44n6xqiqgi426a</sha1>
    </revision>
    <revision>
      <id>4119</id>
      <timestamp>2016-11-21T07:21:53Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;TallKite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 599962014 - Original comment: **</comment>
      <origin>4119</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="53124" sha1="41q05hz06svrae3q21soxtvz4mam5cs" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:TallKite|TallKite]] and made on &lt;tt&gt;2016-11-21 07:21:53 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;599962014&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cents (coarse and fine)
DMS || Approximate
Ratios* || Additional Ratios
of 3, 5 and 9 || Interval Names ||||= [[Ups and Downs Notation|ups and downs]]
[[Ups and Downs Notation|notation]] ||
|| 0 || 0, 0° || 1/1 || 256/243, 50/49, 25/24 || unison || unison ||= D ||
|| 1 || 120, 144
36° || 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second,
large minor second || mid 2nd ||= D^, Ev ||
|| 2 || 240, 288, 72° || 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || M2, m3 ||= E, F ||
|| 3 || 360, 432, 108° || 16/13 || 5/4 || large neutral third || mid 3rd ||= F^, Gv ||
|| 4 || 480, 576, 144° || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || M3, P4 ||= F#, G ||
|| 5 || 600, 720
180° || 91/64, 128/91,
169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th,
down 5th ||= G^, Av ||
|| 6 || 720, 864, 216° || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || P5, m6 ||= A, Bb ||
|| 7 || 840, 1008, 252° || 13/8 || 8/5 || neutral sixth || mid 6th ||= A^, Bv ||
|| 8 || 960, 1152, 288° || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || M6, m7 ||= B, C ||
|| 9 || 1080, 1296, 324° || 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||= C^, Dv ||
|| 10 || 1200, 1440, 360° || 2/1 || 243/128, 49/25, 48/25 || octave || 8ve ||= D ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents (coarse and fine)&amp;lt;br /&amp;gt;
DMS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios&amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td colspan="2" style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;ups and downs&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0, 0°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120, 144&amp;lt;br /&amp;gt;
36°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second,&amp;lt;br /&amp;gt;
large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240, 288, 72°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;M2, m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E, F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360, 432, 108°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F^, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480, 576, 144°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;M3, P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F#, G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600, 720&amp;lt;br /&amp;gt;
180°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 128/91,&amp;lt;br /&amp;gt;
169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th,&amp;lt;br /&amp;gt;
down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G^, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720, 864, 216°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P5, m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A, Bb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840, 1008, 252°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, Bv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960, 1152, 288°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;M6, m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B, C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080, 1296, 324°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C^, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200, 1440, 360°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8ve&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:607:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:607 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:611:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:611 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:612:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:612 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:608:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:608 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:609:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:609 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>41q05hz06svrae3q21soxtvz4mam5cs</sha1>
    </revision>
    <revision>
      <id>4120</id>
      <timestamp>2016-11-21T05:14:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;TallKite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 599950462 - Original comment: **</comment>
      <origin>4120</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="52137" sha1="p3mre2nsw0zgfev92uxeaffzwqrksyt" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:TallKite|TallKite]] and made on &lt;tt&gt;2016-11-21 05:14:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;599950462&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cents (coarse and fine)
DMS || Approximate
Ratios* || Additional Ratios 
of 3, 5 and 9 || Interval Names || [[Ups and Downs Notation|up/down]]
[[Ups and Downs Notation|notation]] ||
|| 0 || 0, 0° || 1/1 || 256/243, 50/49, 25/24 || unison || unison ||
|| 1 || 120, 144
36° || 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second,
large minor second || mid 2nd ||
|| 2 || 240, 288, 72° || 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third || M2, m3 ||
|| 3 || 360, 432, 108° || 16/13 || 5/4 || large neutral third || mid 3rd ||
|| 4 || 480, 576, 144° || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth || M3, P4 ||
|| 5 || 600, 720
180° || 91/64, 128/91, 
169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone || up 4th,
down 5th ||
|| 6 || 720, 864, 216° || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth || P5, m6 ||
|| 7 || 840, 1008, 252° || 13/8 || 8/5 || neutral sixth || mid 6th ||
|| 8 || 960, 1152, 288° || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh || M6, m7 ||
|| 9 || 1080, 1296, 324° || 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th || mid 7th ||
|| 10 || 1200, 1440, 360° || 2/1 || 243/128, 49/25, 48/25 || octave || 8ve ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents (coarse and fine)&amp;lt;br /&amp;gt;
DMS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios &amp;lt;br /&amp;gt;
of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Interval Names&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;up/down&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0, 0°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120, 144&amp;lt;br /&amp;gt;
36°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second,&amp;lt;br /&amp;gt;
large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240, 288, 72°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;M2, m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360, 432, 108°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 3rd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480, 576, 144°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;M3, P4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600, 720&amp;lt;br /&amp;gt;
180°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 128/91, &amp;lt;br /&amp;gt;
169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up 4th,&amp;lt;br /&amp;gt;
down 5th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720, 864, 216°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P5, m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840, 1008, 252°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 6th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960, 1152, 288°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;M6, m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080, 1296, 324°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mid 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200, 1440, 360°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8ve&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Pentatonic%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Pentatonic&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:588:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10ed2-001.svg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10ed2-001.svg&amp;amp;quot; title=&amp;amp;quot;File: 10ed2-001.svg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10ed2-001.svg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');" class="filename" title="10ed2-001.svg"&amp;gt;10ed2-001.svg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10ed2-001.svg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;15 KB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:588 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:585:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:585 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:589:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:589 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:590:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:590 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:586:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:586 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:587:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:587 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>p3mre2nsw0zgfev92uxeaffzwqrksyt</sha1>
    </revision>
    <revision>
      <id>4121</id>
      <timestamp>2016-10-30T07:03:59Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 597475228 - Original comment: **</comment>
      <origin>4121</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="50418" sha1="qkpx0b6vhgi52w7gfm37okhni0ke4x8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-30 07:03:59 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;597475228&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cents (coarse and fine)
DMS || Approximate
Ratios* || Additional Ratios of 3, 5 and 9 || Name ||
|| 0 || 0, 0° || 1/1 || 256/243, 50/49, 25/24 || unison ||
|| 1 || 120, 144
36° || 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second/large minor second ||
|| 2 || 240, 288, 72° || 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third ||
|| 3 || 360, 432, 108° || 16/13 || 5/4 || large neutral third ||
|| 4 || 480, 576, 144° || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth ||
|| 5 || 600, 720
180° || 91/64, 128/91, 169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone ||
|| 6 || 720, 864, 216° || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth ||
|| 7 || 840, 1008, 252° || 13/8 || 8/5 || neutral sixth ||
|| 8 || 960, 1152, 288° || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh ||
|| 9 || 1080, 1296, 324° || 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th ||
|| 10 || 1200, 1440, 360° || 2/1 || 243/128, 49/25, 48/25 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
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 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents (coarse and fine)&amp;lt;br /&amp;gt;
DMS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0, 0°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120, 144&amp;lt;br /&amp;gt;
36°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240, 288, 72°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360, 432, 108°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480, 576, 144°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600, 720&amp;lt;br /&amp;gt;
180°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 128/91, 169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720, 864, 216°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840, 1008, 252°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960, 1152, 288°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080, 1296, 324°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200, 1440, 360°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:565:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:565 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:566:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:566 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:562:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:562 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:563:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:563 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>qkpx0b6vhgi52w7gfm37okhni0ke4x8</sha1>
    </revision>
    <revision>
      <id>4122</id>
      <timestamp>2016-09-09T14:10:07Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 591535472 - Original comment: **</comment>
      <origin>4122</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="50374" sha1="flfhx439ti3ofge9uivetos5uq8hqso" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-09-09 14:10:07 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;591535472&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cents
DMS || Approximate
Ratios* || Additional Ratios of 3, 5 and 9 || Name ||
|| 0 || 0
0° || 1/1 || 256/243, 50/49, 25/24 || unison ||
|| 1 || 120
36° || 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second/large minor second ||
|| 2 || 240
72° || 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third ||
|| 3 || 360
108° || 16/13 || 5/4 || large neutral third ||
|| 4 || 480
144° || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth ||
|| 5 || 600
180° || 91/64, 128/91, 169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone ||
|| 6 || 720
216° || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth ||
|| 7 || 840
252° || 13/8 || 8/5 || neutral sixth ||
|| 8 || 960
288° || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh ||
|| 9 || 1080
324° || 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th ||
|| 10 || 1200
360° || 2/1 || 243/128, 49/25, 48/25 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents&amp;lt;br /&amp;gt;
DMS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
0°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
36°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
72°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
108°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
144°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
180°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 128/91, 169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
216°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
252°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
288°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
324°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
360°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25, 48/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:565:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:565 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:566:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:566 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:562:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:562 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:563:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:563 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>flfhx439ti3ofge9uivetos5uq8hqso</sha1>
    </revision>
    <revision>
      <id>4123</id>
      <timestamp>2016-09-09T14:09:38Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 591535428 - Original comment: **</comment>
      <origin>4123</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="50360" sha1="8y4cq4h3cdq814fw3w89pujwddbrigz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-09-09 14:09:38 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;591535428&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cents
DMS || Approximate
Ratios* || Additional Ratios of 3, 5 and 9 || Name ||
|| 0 || 0
0° || 1/1 || 256/243, 50/49, 25/24 || unison ||
|| 1 || 120
36° || 16/15, 15/14, 13/14 || 10/9, 13/12, 81/80 || small neutral second/large minor second ||
|| 2 || 240
72° || 8/7, 15/13, 144/125 || 9/8, 7/6 || second/third ||
|| 3 || 360
108° || 16/13 || 5/4 || large neutral third ||
|| 4 || 480
144° || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth ||
|| 5 || 600
180° || 91/64, 128/91, 169/120, 240/169 || 7/5, 10/7, 13/9, 18/13 || tritone ||
|| 6 || 720
216° || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth ||
|| 7 || 840
252° || 13/8 || 8/5 || neutral sixth ||
|| 8 || 960
288° || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh ||
|| 9 || 1080
324° || 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th ||
|| 10 || 1200
360° || 2/1 || 243/128, 49/25 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 25/24 || | -3 -1 2 &amp;gt; ||&amp;gt; 70.67 ||= 5-limit large semitone ||= 5-limit chromatic semitone ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
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&amp;lt;br /&amp;gt;
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&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents&amp;lt;br /&amp;gt;
DMS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
0°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49, 25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
36°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/15, 15/14, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
72°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 15/13, 144/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
108°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
144°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
180°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 128/91, 169/120, 240/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
216°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
252°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
288°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
324°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
360°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
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&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;25/24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;70.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit large semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5-limit chromatic semitone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:565:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:565 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:566:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:566 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:562:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:562 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:563:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:563 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>8y4cq4h3cdq814fw3w89pujwddbrigz</sha1>
    </revision>
    <revision>
      <id>4124</id>
      <timestamp>2016-08-21T13:30:42Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 589879786 - Original comment: **</comment>
      <origin>4124</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="49711" sha1="djoqopjchhs2e2ad9sahmjrstxvan8d" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-08-21 13:30:42 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;589879786&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cents
DMS || Approximate
Ratios* || Additional Ratios of 3, 5 and 9 || Name ||
|| 0 || 0
0° || 1/1 || 256/243, 50/49 || unison ||
|| 1 || 120
36° || 15/14, 16/15, 13/14 || 10/9, 13/12, 81/80 || small neutral second/large minor second ||
|| 2 || 240
72° || 15/13, 8/7 || 9/8, 7/6 || second/third ||
|| 3 || 360
108° || 16/13 || 5/4 || large neutral third ||
|| 4 || 480
144° || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth ||
|| 5 || 600
180° || 91/64, 169/120 || 7/5, 10/7, 13/9, 18/13 || tritone ||
|| 6 || 720
216° || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth ||
|| 7 || 840
252° || 13/8 || 8/5 || neutral sixth ||
|| 8 || 960
288° || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh ||
|| 9 || 1080
324° || 15/8, 28/15, 13/7 || 9/5, 24/13, 160/81 || small major 7th ||
|| 10 || 1200
360° || 2/1 || 243/128, 49/25 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
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 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents&amp;lt;br /&amp;gt;
DMS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
0°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;256/243, 50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
36°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12, 81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
72°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
108°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
144°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
180°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
216°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
252°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
288°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
324°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13, 160/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
360°&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;243/128, 49/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:550:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10ed2-001.svg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10ed2-001.svg&amp;amp;quot; title=&amp;amp;quot;File: 10ed2-001.svg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10ed2-001.svg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');" class="filename" title="10ed2-001.svg"&amp;gt;10ed2-001.svg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10ed2-001.svg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;15 KB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:550 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:547:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:547 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:551:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:551 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:552:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:552 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:548:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:548 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:549:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:549 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>djoqopjchhs2e2ad9sahmjrstxvan8d</sha1>
    </revision>
    <revision>
      <id>4125</id>
      <timestamp>2016-08-13T10:12:49Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 589279364 - Original comment: **</comment>
      <origin>4125</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="49375" sha1="7a7nmgkl5s8mjbeq2igws5n3r505m26" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-08-13 10:12:49 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;589279364&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Additional Ratios of 3, 5 and 9 || Name ||
|| 0 || 0 || 1/1 ||   || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || 10/9, 13/12 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || 9/8, 7/6 || second/third ||
|| 3 || 360 || 16/13 || 5/4 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || 7/5, 10/7, 13/9, 18/13 || tritone ||
|| 6 || 720 || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth ||
|| 7 || 840 || 13/8 || 8/5 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || 9/5, 24/13 || small major 7th ||
|| 10 || 1200 || 2/1 ||   || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicennma's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:551:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:551 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:552:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:552 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:548:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:548 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:549:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:549 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>7a7nmgkl5s8mjbeq2igws5n3r505m26</sha1>
    </revision>
    <revision>
      <id>4126</id>
      <timestamp>2016-07-02T11:59:20Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenjacob</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 586508935 - Original comment: **</comment>
      <origin>4126</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="49359" sha1="01ob25u3kiw6sppflunvh1mwt5db06u" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenjacob|xenjacob]] and made on &lt;tt&gt;2016-07-02 11:59:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;586508935&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Additional Ratios of 3, 5 and 9 || Name ||
|| 0 || 0 || 1/1 ||   || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || 10/9, 13/12 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || 9/8, 7/6 || second/third ||
|| 3 || 360 || 16/13 || 5/4 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || 7/5, 10/7, 13/9, 18/13 || tritone ||
|| 6 || 720 || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth ||
|| 7 || 840 || 13/8 || 8/5 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || 9/5, 24/13 || small major 7th ||
|| 10 || 1200 || 2/1 ||   || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]
[[@https://soundcloud.com/clem-fortuna/10tone|10tone demo]] by [[@http://clemfortuna.com|Clem Fortuna]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
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 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:547:&amp;amp;lt;img src=&amp;amp;quot;/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 285px; width: 280px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/10edo%20wheel.png/516974684/280x285/10edo%20wheel.png" alt="10edo wheel.png" title="10edo wheel.png" style="height: 285px; width: 280px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:547 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:551:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:551 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:552:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:552 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/clem-fortuna/10tone" rel="nofollow" target="_blank"&amp;gt;10tone demo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://clemfortuna.com" rel="nofollow" target="_blank"&amp;gt;Clem Fortuna&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:548:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:548 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:549:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:549 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>01ob25u3kiw6sppflunvh1mwt5db06u</sha1>
    </revision>
    <revision>
      <id>4127</id>
      <timestamp>2015-08-04T04:34:10Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 556152179 - Original comment: **</comment>
      <origin>4127</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="48989" sha1="3x1xhmm40512pn3xcbwvx7d55a4yat6" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2015-08-04 04:34:10 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;556152179&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[10平均律|日本語]]
&amp;lt;/span&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Additional Ratios of 3, 5 and 9 || Name ||
|| 0 || 0 || 1/1 ||   || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || 10/9, 13/12 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || 9/8, 7/6 || second/third ||
|| 3 || 360 || 16/13 || 5/4 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || 7/5, 10/7, 13/9, 18/13 || tritone ||
|| 6 || 720 || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth ||
|| 7 || 840 || 13/8 || 8/5 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || 9/5, 24/13 || small major 7th ||
|| 10 || 1200 || 2/1 ||   || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/10%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
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 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:550:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10ed2-001.svg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10ed2-001.svg&amp;amp;quot; title=&amp;amp;quot;File: 10ed2-001.svg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10ed2-001.svg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');" class="filename" title="10ed2-001.svg"&amp;gt;10ed2-001.svg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10ed2-001.svg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;15 KB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:550 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:551:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:551 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:552:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:552 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:548:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:548 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:549:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:549 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>3x1xhmm40512pn3xcbwvx7d55a4yat6</sha1>
    </revision>
    <revision>
      <id>4128</id>
      <timestamp>2014-07-24T11:44:06Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 516974806 - Original comment: **</comment>
      <origin>4128</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="48736" sha1="pewgbc64k9tckx8nsl3tpcorsr9sa1t" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2014-07-24 11:44:06 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;516974806&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Additional Ratios of 3, 5 and 9 || Name ||
|| 0 || 0 || 1/1 ||   || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || 10/9, 13/12 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || 9/8, 7/6 || second/third ||
|| 3 || 360 || 16/13 || 5/4 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || 7/5, 10/7, 13/9, 18/13 || tritone ||
|| 6 || 720 || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth ||
|| 7 || 840 || 13/8 || 8/5 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || 9/5, 24/13 || small major 7th ||
|| 10 || 1200 || 2/1 ||   || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Images= 

[[image:10edo wheel.png width="280" height="285"]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:17:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Images"&amp;gt;Images&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
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 &amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Images"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Images&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
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&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:551:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:551 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:552:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:552 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:548:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:548 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:549:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:549 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>pewgbc64k9tckx8nsl3tpcorsr9sa1t</sha1>
    </revision>
    <revision>
      <id>4129</id>
      <timestamp>2014-01-05T12:50:07Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 480703054 - Original comment: **</comment>
      <origin>4129</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47824" sha1="ovt1v49czibhndvg8sb6my3m69g5vnp" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-01-05 12:50:07 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;480703054&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Additional Ratios of 3, 5 and 9 || Name ||
|| 0 || 0 || 1/1 ||   || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || 10/9, 13/12 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || 9/8, 7/6 || second/third ||
|| 3 || 360 || 16/13 || 5/4 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || 7/5, 10/7, 13/9, 18/13 || tritone ||
|| 6 || 720 || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth ||
|| 7 || 840 || 13/8 || 8/5 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || 9/5, 24/13 || small major 7th ||
|| 10 || 1200 || 2/1 ||   || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

[[media type="custom" key="24802496"]]

[[file:10ed2-001.svg]]

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:15:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802496?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802496&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/10ed2-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:546:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10ed2-001.svg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10ed2-001.svg&amp;amp;quot; title=&amp;amp;quot;File: 10ed2-001.svg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10ed2-001.svg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg" onclick="ws.common.trackFileLink('/file/view/10ed2-001.svg/480702570/10ed2-001.svg');" class="filename" title="10ed2-001.svg"&amp;gt;10ed2-001.svg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10ed2-001.svg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10ed2-001.svg/480702570/10ed2-001.svg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;15 KB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:546 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:547:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:547 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:548:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:548 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:544:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:544 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:545:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:545 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:2:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:2 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ovt1v49czibhndvg8sb6my3m69g5vnp</sha1>
    </revision>
    <revision>
      <id>4130</id>
      <timestamp>2013-06-01T09:05:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;seraph57</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 435681956 - Original comment: **</comment>
      <origin>4130</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45901" sha1="pnyfbiiufe7q3289smicy9jd5rohw10" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:seraph57|seraph57]] and made on &lt;tt&gt;2013-06-01 09:05:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;435681956&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Additional Ratios of 3, 5 and 9 || Name ||
|| 0 || 0 || 1/1 ||   || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || 10/9, 13/12 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || 9/8, 7/6 || second/third ||
|| 3 || 360 || 16/13 || 5/4 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || 7/5, 10/7, 13/9, 18/13 || tritone ||
|| 6 || 720 || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth ||
|| 7 || 840 || 13/8 || 8/5 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || 9/5, 24/13 || small major 7th ||
|| 10 || 1200 || 2/1 ||   || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
//[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])
//[[http://www.seraph.it/dep/det/Smooth10.mp3|Smooth 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Smooth10.mp3" rel="nofollow"&amp;gt;Smooth 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:545:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:545 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:546:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:546 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:543:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:543 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:544:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:544 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>pnyfbiiufe7q3289smicy9jd5rohw10</sha1>
    </revision>
    <revision>
      <id>4131</id>
      <timestamp>2013-04-30T18:17:13Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;seraph57</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 427813294 - Original comment: **</comment>
      <origin>4131</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45095" sha1="1qt1asqi4j8bkdlg8kgj85668q78an9" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:seraph57|seraph57]] and made on &lt;tt&gt;2013-04-30 18:17:13 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;427813294&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Additional Ratios of 3, 5 and 9 || Name ||
|| 0 || 0 || 1/1 ||   || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || 10/9, 13/12 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || 9/8, 7/6 || second/third ||
|| 3 || 360 || 16/13 || 5/4 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || 7/5, 10/7, 13/9, 18/13 || tritone ||
|| 6 || 720 || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth ||
|| 7 || 840 || 13/8 || 8/5 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || 9/5, 24/13 || small major 7th ||
|| 10 || 1200 || 2/1 ||   || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
////[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]//// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])
////[[http://www.seraph.it/dep/det/Waltz10.mp3|Waltz 10]]//// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]////&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]////&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]////&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
////[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]//// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
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&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios of 3, 5 and 9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Waltz10.mp3" rel="nofollow"&amp;gt;Waltz 10&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:545:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:545 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:546:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:546 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:543:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:543 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:544:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:544 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>1qt1asqi4j8bkdlg8kgj85668q78an9</sha1>
    </revision>
    <revision>
      <id>4132</id>
      <timestamp>2013-04-14T12:55:20Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 422958116 - Original comment: **</comment>
      <origin>4132</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="44854" sha1="318x0uul7dbpijbxygw160hn2cmaxhx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2013-04-14 12:55:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;422958116&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Additional Ratios of 3, 5 and 9** || Name ||
|| 0 || 0 || 1/1 ||   || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || 10/9, 13/12 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || 9/8, 7/6 || second/third ||
|| 3 || 360 || 16/13 || 5/4 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || 4/3, 9/7, 13/10 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || 7/5, 10/7, 13/9, 18/13 || tritone ||
|| 6 || 720 || 49/32, 256/169 || 3/2, 14/9, 20/13 || bigger fifth ||
|| 7 || 840 || 13/8 || 8/5 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || 16/9, 12/7 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || 9/5, 24/13 || small major 7th ||
|| 10 || 1200 || 2/1 ||   || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
**adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
//[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]]// by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]//&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]//&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]//&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]]// by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Additional Ratios of 3, 5 and 9&amp;lt;strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/9, 13/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8, 7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3, 9/7, 13/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/5, 10/7, 13/9, 18/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2, 14/9, 20/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/9, 12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/5, 24/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;/strong&amp;gt;adding the ratios of 3, 5 and 9 introduces greater &amp;lt;a class="wiki_link" href="/error"&amp;gt;error&amp;lt;/a&amp;gt; while giving several more harmonic identities to the 10-EDO intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:545:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:545 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:546:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:546 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:543:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:543 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:544:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:544 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>318x0uul7dbpijbxygw160hn2cmaxhx</sha1>
    </revision>
    <revision>
      <id>4133</id>
      <timestamp>2013-04-04T15:55:24Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;seraph57</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 420211624 - Original comment: **</comment>
      <origin>4133</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="43306" sha1="saiuu9sm5kztyo01e4yogutcfkxfwjf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:seraph57|seraph57]] and made on &lt;tt&gt;2013-04-04 15:55:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;420211624&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[http://www.seraph.it/dep/det/Blues10.mp3|Blues 10]] by [[Carlo Serafini]] ([[http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html|blog entry]])

[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]] by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/dep/det/Blues10.mp3" rel="nofollow"&amp;gt;Blues 10&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html" rel="nofollow"&amp;gt;blog entry&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:521:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:521 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:522:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:522 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:519:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:519 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:520:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:520 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>saiuu9sm5kztyo01e4yogutcfkxfwjf</sha1>
    </revision>
    <revision>
      <id>4134</id>
      <timestamp>2012-12-13T12:57:40Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;phylingual</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 392031448 - Original comment: **</comment>
      <origin>4134</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="42774" sha1="m55u25dg6cuoxcdn8xeanltwa2j4n84" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:phylingual|phylingual]] and made on &lt;tt&gt;2012-12-13 12:57:40 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;392031448&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]] by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Igliashon Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:521:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:521 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:522:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:522 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:519:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:519 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:520:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:520 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>m55u25dg6cuoxcdn8xeanltwa2j4n84</sha1>
    </revision>
    <revision>
      <id>4135</id>
      <timestamp>2012-12-12T23:03:50Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;phylingual</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 391841406 - Original comment: **</comment>
      <origin>4135</origin>
      <model>wikitext</model>
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      <text bytes="42841" sha1="3fdsttjpa4bayzep98pa6npgvynl43l" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:phylingual|phylingual]] and made on &lt;tt&gt;2012-12-12 23:03:50 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;391841406&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]] by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:521:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:521 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:522:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:522 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:519:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:519 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:520:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:520 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>3fdsttjpa4bayzep98pa6npgvynl43l</sha1>
    </revision>
    <revision>
      <id>4136</id>
      <timestamp>2012-12-05T16:17:40Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 389547274 - Original comment: added on more piece**</comment>
      <origin>4136</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="43282" sha1="det9uh9hmqx71cxjkztgv1iycgftk5f" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2012-12-05 16:17:40 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;389547274&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;added on more piece&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
[[http://soundcloud.com/cityoftheasleep/domination|Domination]] and [[http://soundcloud.com/cityoftheasleep/anglebath|Anglebath]] by Igliashon Jones
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]] by [[Zach Curley]]
[[http://soundcloud.com/martinsj013/sirmdbidnud2|Sirmdbidnud2 by martinsj013 on SoundCloud]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/cityoftheasleep/domination" rel="nofollow"&amp;gt;Domination&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/cityoftheasleep/anglebath" rel="nofollow"&amp;gt;Anglebath&amp;lt;/a&amp;gt; by Igliashon Jones&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:521:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:521 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:522:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:522 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/martinsj013/sirmdbidnud2" rel="nofollow"&amp;gt;Sirmdbidnud2 by martinsj013 on SoundCloud&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:519:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:519 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:520:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:520 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>det9uh9hmqx71cxjkztgv1iycgftk5f</sha1>
    </revision>
    <revision>
      <id>4137</id>
      <timestamp>2012-07-05T18:49:20Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 350519804 - Original comment: **</comment>
      <origin>4137</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="42988" sha1="b1hhoou9cc3wc8romokoqz4mujt41zf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2012-07-05 18:49:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;350519804&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
[[http://soundcloud.com/cityoftheasleep/domination|Domination]] and [[http://soundcloud.com/cityoftheasleep/anglebath|Anglebath]] by Igliashon Jones
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]] by [[Zach Curley]]
=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/cityoftheasleep/domination" rel="nofollow"&amp;gt;Domination&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/cityoftheasleep/anglebath" rel="nofollow"&amp;gt;Anglebath&amp;lt;/a&amp;gt; by Igliashon Jones&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:521:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:521 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:522:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:522 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:519:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:519 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:520:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:520 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>b1hhoou9cc3wc8romokoqz4mujt41zf</sha1>
    </revision>
    <revision>
      <id>4138</id>
      <timestamp>2012-05-02T20:12:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 329012798 - Original comment: **</comment>
      <origin>4138</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="42576" sha1="o0s53x7j89k0ui67ag313478gg722hi" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-05-02 20:12:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;329012798&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3|The Csx Freight at 1:20 am]] by [[Zach Curley]]
=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
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&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:521:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:521 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:522:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:522 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3" rel="nofollow"&amp;gt;The Csx Freight at 1:20 am&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Zach%20Curley"&amp;gt;Zach Curley&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:519:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:519 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:520:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:520 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>o0s53x7j89k0ui67ag313478gg722hi</sha1>
    </revision>
    <revision>
      <id>4139</id>
      <timestamp>2012-03-15T22:31:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Kosmorsky</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 311594238 - Original comment: **</comment>
      <origin>4139</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="42120" sha1="k6iqup85kyl152gzye2od2fsacwvo0f" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on &lt;tt&gt;2012-03-15 22:31:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;311594238&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
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&amp;lt;br /&amp;gt;
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&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:521:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:521 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:522:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:522 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from &amp;lt;a class="wiki_link" href="/20edo"&amp;gt;20edo&amp;lt;/a&amp;gt; which approximates the 11th harmonic with relative accuracy, among other features.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:519:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:519 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:520:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:520 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>k6iqup85kyl152gzye2od2fsacwvo0f</sha1>
    </revision>
    <revision>
      <id>4140</id>
      <timestamp>2012-03-10T07:57:50Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 309620666 - Original comment: **</comment>
      <origin>4140</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="41327" sha1="iusnl1fkpz7ozoqo7ts3e9p0n9dynws" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2012-03-10 07:57:50 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;309620666&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]
[[@http://andrewheathwaite.bandcamp.com/track/shimmerwing|Shimmerwing]] by [[Andrew Heathwaite]] and [[Chris Vaisvil]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
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&amp;lt;br /&amp;gt;
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&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:521:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:521 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:522:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:522 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://andrewheathwaite.bandcamp.com/track/shimmerwing" rel="nofollow" target="_blank"&amp;gt;Shimmerwing&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:519:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:519 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:520:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:520 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>iusnl1fkpz7ozoqo7ts3e9p0n9dynws</sha1>
    </revision>
    <revision>
      <id>4141</id>
      <timestamp>2012-01-14T14:43:13Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 292232351 - Original comment: **</comment>
      <origin>4141</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="40663" sha1="gsrttur0urm1nkdjko4r1sea6qzc2mh" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2012-01-14 14:43:13 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;292232351&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]]&amp;lt;/span&amp;gt; by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
[[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3|The Dramatic Squirrel Overture]] by [[Chris Vaisvil]] [[@http://chrisvaisvil.com/?p=1368|Details]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:521:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:521 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:522:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:522 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3" rel="nofollow"&amp;gt;The Dramatic Squirrel Overture&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1368" rel="nofollow" target="_blank"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:519:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:519 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:520:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:520 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>gsrttur0urm1nkdjko4r1sea6qzc2mh</sha1>
    </revision>
    <revision>
      <id>4142</id>
      <timestamp>2011-11-23T07:07:24Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 278433142 - Original comment: **</comment>
      <origin>4142</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="37881" sha1="nxfwbrge3nuxxpkg55be7ulck4z0w2u" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-11-23 07:07:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;278433142&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]] by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3|Comets Over Flatland 9]] by [[Randy Winchester]]

=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:521:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:521 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:522:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:522 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 9&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:519:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:519 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:520:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:520 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>nxfwbrge3nuxxpkg55be7ulck4z0w2u</sha1>
    </revision>
    <revision>
      <id>4143</id>
      <timestamp>2011-10-16T10:53:57Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 265197380 - Original comment: **</comment>
      <origin>4143</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="37440" sha1="0lufxrx3q8dhy4dimtnhfwg2mj1led1" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2011-10-16 10:53:57 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;265197380&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=[[media type="custom" key="10886138"]]= 
=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
[[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3|10 Earwigs Invasive]] by [[@http://chrisvaisvil.com/?p=1397|Chris Vaisvil]]
=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:14:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10886138?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10886138&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:521:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:521 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:522:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:522 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3" rel="nofollow"&amp;gt;10 Earwigs Invasive&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1397" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:519:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:519 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:520:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:520 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>0lufxrx3q8dhy4dimtnhfwg2mj1led1</sha1>
    </revision>
    <revision>
      <id>4144</id>
      <timestamp>2011-09-20T19:39:35Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;igliashon</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 256314300 - Original comment: **</comment>
      <origin>4144</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="36215" sha1="k2d1bd9z7x5ukp64g9xgemq5aqtd70p" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:igliashon|igliashon]] and made on &lt;tt&gt;2011-09-20 19:39:35 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;256314300&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
=**Instruments**= 

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:11:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:11 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:12: --&amp;gt;&amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:12 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:13: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:13 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:14: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:517:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:517 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:518:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:518 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an &amp;amp;quot;E&amp;amp;quot; chord would be 0-2-2-1-0-0 (low to high), an &amp;amp;quot;A&amp;amp;quot; chord would be 0-0-2-2-1-0, and a &amp;amp;quot;D&amp;amp;quot; chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:515:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:515 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:516:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:516 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>k2d1bd9z7x5ukp64g9xgemq5aqtd70p</sha1>
    </revision>
    <revision>
      <id>4145</id>
      <timestamp>2011-09-18T15:43:53Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 255331488 - Original comment: **</comment>
      <origin>4145</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="34891" sha1="r7wqtipyfbngkb8248l2cjqjaibi7cg" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2011-09-18 15:43:53 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;255331488&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth 33/25 ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth 50/33 ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
=Linear temperaments= 
||~ Periods
per octave ||~ Generator ||~ Temperament(s) ||
|| 1 || 1\10 || Messed-up [[Negri|negri]] (or [[Miracle|miracle]]) ||
|| 1 || 3\10 || [[Dicot]]/[[Beatles|beatles]]/neutral thirds scale ||
|| 2 || 1\10 || Messed-up [[pajara]] ||
|| 2 || 2\10 || [[Decimal]] / messed-up [[Lemba|lemba]] ||
|| 5 || 1\10 || [[Blackwood]]/[[blacksmith]] ||
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
=**Instruments**= 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:11:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:11 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:12: --&amp;gt;&amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:12 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:13: --&amp;gt; | &amp;lt;a href="#Linear temperaments"&amp;gt;Linear temperaments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:13 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:14: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:15: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:15 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:16: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:16 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:17: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:17 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth 33/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth 50/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Linear temperaments&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperament(s)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/Negri"&amp;gt;negri&amp;lt;/a&amp;gt; (or &amp;lt;a class="wiki_link" href="/Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Dicot"&amp;gt;Dicot&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/Beatles"&amp;gt;beatles&amp;lt;/a&amp;gt;/neutral thirds scale&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Messed-up &amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Decimal"&amp;gt;Decimal&amp;lt;/a&amp;gt; / messed-up &amp;lt;a class="wiki_link" href="/Lemba"&amp;gt;lemba&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Blackwood"&amp;gt;Blackwood&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/blacksmith"&amp;gt;blacksmith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:517:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:517 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:518:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:518 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:515:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:515 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:516:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:516 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>r7wqtipyfbngkb8248l2cjqjaibi7cg</sha1>
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    <revision>
      <id>4146</id>
      <timestamp>2011-09-06T15:25:35Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Cenobyte</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 251285706 - Original comment: **</comment>
      <origin>4146</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="32482" sha1="ewd0gvsa5sx3iz2wm4r9ina5uu7lq1i" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Cenobyte|Cenobyte]] and made on &lt;tt&gt;2011-09-06 15:25:35 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;251285706&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth 33/25 ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth 50/33 ||
|| 7 || 840 || 13/8 || neutral sixth ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
=**Instruments**= 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:9:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:9 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:10: --&amp;gt;&amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:10 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:11: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:11 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:12: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:12 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:13: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:13 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:14: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth 33/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth 50/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;neutral sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:464:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:464 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:465:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:465 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:462:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:462 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:463:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:463 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ewd0gvsa5sx3iz2wm4r9ina5uu7lq1i</sha1>
    </revision>
    <revision>
      <id>4147</id>
      <timestamp>2011-09-06T02:01:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Cenobyte</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 251070706 - Original comment: **</comment>
      <origin>4147</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="32458" sha1="bjvz0rtlfrbvgqptt6hr7b4vbdl8k8u" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Cenobyte|Cenobyte]] and made on &lt;tt&gt;2011-09-06 02:01:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;251070706&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 || small neutral second/large minor second ||
|| 2 || 240 || 15/13, 8/7 || second/third ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 || smaller fourth 33/25 ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 || bigger fifth 50/33 ||
|| 7 || 840 || 13/8 || ? ||
|| 8 || 960 || 7/4, 26/15, 125/72 || sixth/seventh ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
=**Instruments**= 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[image:decaphonic-uke.JPG width="526" height="406"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:9:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:9 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:10: --&amp;gt;&amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:10 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:11: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:11 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:12: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:12 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:13: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:13 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:14: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small neutral second/large minor second&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;second/third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;smaller fourth 33/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;bigger fifth 50/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;?&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15, 125/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sixth/seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:464:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:464 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:465:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:465 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:462:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:462 --&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:463:&amp;amp;lt;img src=&amp;amp;quot;/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 406px; width: 526px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/decaphonic-uke.JPG/251070060/526x406/decaphonic-uke.JPG" alt="decaphonic-uke.JPG" title="decaphonic-uke.JPG" style="height: 406px; width: 526px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:463 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>bjvz0rtlfrbvgqptt6hr7b4vbdl8k8u</sha1>
    </revision>
    <revision>
      <id>4148</id>
      <timestamp>2011-09-05T01:04:38Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Cenobyte</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 250759958 - Original comment: **</comment>
      <origin>4148</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="31696" sha1="mibjxz3wlc0u8psku04zbujqqxabp9x" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Cenobyte|Cenobyte]] and made on &lt;tt&gt;2011-09-05 01:04:38 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;250759958&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* || Name ||
|| 0 || 0 || 1/1 || unison ||
|| 1 || 120 || 15/14, 16/15, 13/14 ||   ||
|| 2 || 240 || 15/13, 8/7 ||   ||
|| 3 || 360 || 16/13 || large neutral third ||
|| 4 || 480 || 64/49, 169/128 ||   ||
|| 5 || 600 || 91/64, 169/120 || tritone ||
|| 6 || 720 || 49/32, 256/169 ||   ||
|| 7 || 840 || 13/8 ||   ||
|| 8 || 960 || 7/4, 26/15 ||   ||
|| 9 || 1080 || 15/8, 28/15, 13/7 || small major 7th ||
|| 10 || 1200 || 2/1 || octave ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
=**Instruments**= 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:9:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:9 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:10: --&amp;gt;&amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:10 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:11: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:11 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:12: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:12 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:13: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:13 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:14: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;unison&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;large neutral third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;tritone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;small major 7th&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;octave&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:463:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:463 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:464:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:464 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:462:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:462 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>mibjxz3wlc0u8psku04zbujqqxabp9x</sha1>
    </revision>
    <revision>
      <id>4149</id>
      <timestamp>2011-08-19T11:39:49Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Cenobyte</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 246972179 - Original comment: **</comment>
      <origin>4149</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="31004" sha1="a7lshets253g5tobovtykwsyohdav6g" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Cenobyte|Cenobyte]] and made on &lt;tt&gt;2011-08-19 11:39:49 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;246972179&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* ||
|| 0 || 0 || 1/1 ||
|| 1 || 120 || 15/14, 16/15, 13/14 ||
|| 2 || 240 || 15/13, 8/7 ||
|| 3 || 360 || 16/13 ||
|| 4 || 480 || 64/49, 169/128 ||
|| 5 || 600 || 91/64, 169/120 ||
|| 6 || 720 || 49/32, 256/169 ||
|| 7 || 840 || 13/8 ||
|| 8 || 960 || 7/4, 26/15 ||
|| 9 || 1080 || 15/8, 28/15, 13/7 ||
|| 10 || 1200 || 2/1 ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
[[file:decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
=**Instruments**= 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:9:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:9 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:10: --&amp;gt;&amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:10 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:11: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:11 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:12: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:12 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:13: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:13 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:14: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:439:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:439 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:440:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/decexperiment.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@decexperiment.ogg&amp;amp;quot; title=&amp;amp;quot;File: decexperiment.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="decexperiment.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg" onclick="ws.common.trackFileLink('/file/view/decexperiment.ogg/246971619/decexperiment.ogg');" class="filename" title="decexperiment.ogg"&amp;gt;decexperiment.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/decexperiment.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/decexperiment.ogg/246971619/decexperiment.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;2 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:440 --&amp;gt;3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:438:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:438 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>a7lshets253g5tobovtykwsyohdav6g</sha1>
    </revision>
    <revision>
      <id>4150</id>
      <timestamp>2011-08-19T04:59:55Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 246904167 - Original comment: **</comment>
      <origin>4150</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="29543" sha1="ndzk9u3h353ld4tqv0jyreaqdpb5uvf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2011-08-19 04:59:55 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;246904167&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

[[toc|flat]]
----

=Intervals= 
|| Degree || Cent
Values || Approximate
Ratios* ||
|| 0 || 0 || 1/1 ||
|| 1 || 120 || 15/14, 16/15, 13/14 ||
|| 2 || 240 || 15/13, 8/7 ||
|| 3 || 360 || 16/13 ||
|| 4 || 480 || 64/49, 169/128 ||
|| 5 || 600 || 91/64, 169/120 ||
|| 6 || 720 || 49/32, 256/169 ||
|| 7 || 840 || 13/8 ||
|| 8 || 960 || 7/4, 26/15 ||
|| 9 || 1080 || 15/8, 28/15, 13/7 ||
|| 10 || 1200 || 2/1 ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
=Commas= 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
=Music= 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
=**Instruments**= 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextTocRule:9:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:9 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:10: --&amp;gt;&amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:10 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:11: --&amp;gt; | &amp;lt;a href="#Commas"&amp;gt;Commas&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:11 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:12: --&amp;gt; | &amp;lt;a href="#Music"&amp;gt;Music&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:12 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:13: --&amp;gt; | &amp;lt;a href="#Instruments"&amp;gt;Instruments&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:13 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:14: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:14 --&amp;gt;&amp;lt;hr /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Commas&amp;lt;/h1&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:439:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:439 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:438:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:438 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ndzk9u3h353ld4tqv0jyreaqdpb5uvf</sha1>
    </revision>
    <revision>
      <id>4151</id>
      <timestamp>2011-08-19T00:35:54Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Cenobyte</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 246873735 - Original comment: **</comment>
      <origin>4151</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28760" sha1="g5ygeuo9vd2jkp2a8wy9u7ncr8z9a2r" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Cenobyte|Cenobyte]] and made on &lt;tt&gt;2011-08-19 00:35:54 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;246873735&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
=== === 
===Intervals=== 
|| Degree || Cent
Values || Approximate
Ratios* ||
|| 0 || 0 || 1/1 ||
|| 1 || 120 || 15/14, 16/15, 13/14 ||
|| 2 || 240 || 15/13, 8/7 ||
|| 3 || 360 || 16/13 ||
|| 4 || 480 || 64/49, 169/128 ||
|| 5 || 600 || 91/64, 169/120 ||
|| 6 || 720 || 49/32, 256/169 ||
|| 7 || 840 || 13/8 ||
|| 8 || 960 || 7/4, 26/15 ||
|| 9 || 1080 || 15/8, 28/15, 13/7 ||
|| 10 || 1200 || 2/1 ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
[[file:10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
===**Instruments**=== 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:434:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/10preview.ogg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@10preview.ogg&amp;amp;quot; title=&amp;amp;quot;File: 10preview.ogg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="10preview.ogg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg" onclick="ws.common.trackFileLink('/file/view/10preview.ogg/246873297/10preview.ogg');" class="filename" title="10preview.ogg"&amp;gt;10preview.ogg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/10preview.ogg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/10preview.ogg/246873297/10preview.ogg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;3 MB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:434 --&amp;gt; A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x-Commas-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:433:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:433 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
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      <sha1>g5ygeuo9vd2jkp2a8wy9u7ncr8z9a2r</sha1>
    </revision>
    <revision>
      <id>4152</id>
      <timestamp>2011-07-20T15:59:57Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 242157653 - Original comment: **</comment>
      <origin>4152</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="27329" sha1="25xi90uc0gen9xwmlpuo8ii3kavozs7" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2011-07-20 15:59:57 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;242157653&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
=== === 
===Intervals=== 
|| Degree || Cent
Values || Approximate
Ratios* ||
|| 0 || 0 || 1/1 ||
|| 1 || 120 || 15/14, 16/15, 13/14 ||
|| 2 || 240 || 15/13, 8/7 ||
|| 3 || 360 || 16/13 ||
|| 4 || 480 || 64/49, 169/128 ||
|| 5 || 600 || 91/64, 169/120 ||
|| 6 || 720 || 49/32, 256/169 ||
|| 7 || 840 || 13/8 ||
|| 8 || 960 || 7/4, 26/15 ||
|| 9 || 1080 || 15/8, 28/15, 13/7 ||
|| 10 || 1200 || 2/1 ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
===**Instruments**=== 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x-Commas-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
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      <sha1>25xi90uc0gen9xwmlpuo8ii3kavozs7</sha1>
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    <revision>
      <id>4153</id>
      <timestamp>2011-07-20T15:57:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 242157353 - Original comment: **</comment>
      <origin>4153</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="27329" sha1="jzstmeibdasqlfr1xlfz74ju6sledsy" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2011-07-20 15:57:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;242157353&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 33614/32768 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
=== === 
===Intervals=== 
|| Degree || Cent
Values || Approximate
Ratios* ||
|| 0 || 0 || 1/1 ||
|| 1 || 120 || 15/14, 16/15, 13/14 ||
|| 2 || 240 || 15/13, 8/7 ||
|| 3 || 360 || 16/13 ||
|| 4 || 480 || 64/49, 169/128 ||
|| 5 || 600 || 91/64, 169/120 ||
|| 6 || 720 || 49/32, 256/169 ||
|| 7 || 840 || 13/8 ||
|| 8 || 960 || 7/4, 26/15 ||
|| 9 || 1080 || 15/8, 28/15, 13/7 ||
|| 10 || 1200 || 2/1 ||
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.
==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]]&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]]&amp;lt;/span&amp;gt; by [[Aaron Krister Johnson]] (synth monody)
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]]&amp;lt;/span&amp;gt; and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;[[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]]&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]]&amp;lt;/span&amp;gt; by [[Rick McGowan]] (Rhino synthesizer)
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]]&amp;lt;/span&amp;gt; by [[Iglashion Jones]] (synth)
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]]&amp;lt;/span&amp;gt; by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
===**Instruments**=== 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 33614/32768 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cent&amp;lt;br /&amp;gt;
Values&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 13/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/13, 8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64/49, 169/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91/64, 169/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/32, 256/169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 26/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 28/15, 13/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-0b41d890e4b2e2ea729c8a1c199295e9"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-86d97882546354d45034d3c272cbb56c"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-9ccb949290bc6c1cc7c1a4be8065871d"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-edd919cb4d239805e1b5cba4375a1e93"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a5c7622ba1020de4aa13d58769a6c357"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aa7c2b6de113578643d9a38ca7aa70e4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-08eecd4b04b03702c59dba2eb80c1428"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a9a8a238b39d19122886e0c933d01cca"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-1824c99256ed13ed49cba489058a8493"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x-Commas-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:433:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:433 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
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    <revision>
      <id>4154</id>
      <timestamp>2011-07-18T21:52:26Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 241868657 - Original comment: **</comment>
      <origin>4154</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23919" sha1="coqsb6e2r9iebl7kha6e4gejkhgqhhn" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2011-07-18 21:52:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;241868657&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak edo]].
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]] by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]] by [[Aaron Krister Johnson]] (synth monody)
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]] by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] [[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] [[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by [[Rick McGowan]] (Rhino synthesizer)
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]] by [[Iglashion Jones]] (synth)
[[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3|For two violas and gongs]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil (website)]] [[@http://chrisvaisvil.com/?p=1039|more composition information]]
===**Instruments**=== 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]][[media type="custom" key="10021077"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3" rel="nofollow"&amp;gt;For two violas and gongs&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil (website)&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1039" rel="nofollow" target="_blank"&amp;gt;more composition information&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x-Commas-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:403:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:403 --&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10021077?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10021077&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>coqsb6e2r9iebl7kha6e4gejkhgqhhn</sha1>
    </revision>
    <revision>
      <id>4155</id>
      <timestamp>2011-06-26T15:48:10Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 238828877 - Original comment: **</comment>
      <origin>4155</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22701" sha1="2ai2jikm28zlvyvucr7ce6rxd3jpj81" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-26 15:48:10 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;238828877&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak edo]].
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]] by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]] by [[Aaron Krister Johnson]] (synth monody)
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]] by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] [[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] [[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by [[Rick McGowan]] (Rhino synthesizer)
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]] by [[Iglashion Jones]] (synth)

===**Instruments**=== 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;). While not an integral or gap edo, it is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x-Commas-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:402:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:402 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>2ai2jikm28zlvyvucr7ce6rxd3jpj81</sha1>
    </revision>
    <revision>
      <id>4156</id>
      <timestamp>2011-06-10T17:08:01Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 235848360 - Original comment: **</comment>
      <origin>4156</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22424" sha1="sqhc58535fu09ji0zpxpeae3jymolu1" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-10 17:08:01 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;235848360&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]).
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]] by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]] by [[Aaron Krister Johnson]] (synth monody)
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]] by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] [[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] [[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by [[Rick McGowan]] (Rhino synthesizer)
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3|Ideas on the Waterfall of Expression]] by [[Iglashion Jones]] (synth)

===**Instruments**=== 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;).&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x-Commas-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:402:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:402 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>sqhc58535fu09ji0zpxpeae3jymolu1</sha1>
    </revision>
    <revision>
      <id>4157</id>
      <timestamp>2011-06-09T16:39:53Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 235574162 - Original comment: **</comment>
      <origin>4157</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22210" sha1="53bydcsz1ye8r3ops4vcqnm0kv15z04" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-09 16:39:53 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;235574162&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]).
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]] by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]] by [[Aaron Krister Johnson]] (synth monody)
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3|Prelude in 10ET]] by [[Aaron Andrew Hunt]]
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] [[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] [[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by [[Rick McGowan]] (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by [[Iglashion Jones]] (synth)

===**Instruments**=== 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;).&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3" rel="nofollow"&amp;gt;Prelude in 10ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x-Commas-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:402:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:402 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>53bydcsz1ye8r3ops4vcqnm0kv15z04</sha1>
    </revision>
    <revision>
      <id>4158</id>
      <timestamp>2011-06-07T17:26:36Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 234971950 - Original comment: **</comment>
      <origin>4158</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21788" sha1="gezweplece03rft3xbepvhq84gcorvl" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-07 17:26:36 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234971950&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]).
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]] by [[Bill Sethares]] (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]] by [[Aaron Krister Johnson]] (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] [[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] [[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by [[Rick McGowan]] (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by [[Iglashion Jones]] (synth)

===**Instruments**=== 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;).&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt; (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Rick%20McGowan"&amp;gt;Rick McGowan&amp;lt;/a&amp;gt; (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Iglashion%20Jones"&amp;gt;Iglashion Jones&amp;lt;/a&amp;gt; (synth)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x-Commas-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:402:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:402 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>gezweplece03rft3xbepvhq84gcorvl</sha1>
    </revision>
    <revision>
      <id>4159</id>
      <timestamp>2011-06-07T03:16:12Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 234780430 - Original comment: **</comment>
      <origin>4159</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21507" sha1="b79npv3iq4cd8l9tlw18pojs156su2d" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2011-06-07 03:16:12 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234780430&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]).
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] [[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] [[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)

===**Instruments**=== 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1 (&amp;lt;a class="wiki_link" href="/3L%204s"&amp;gt;3L 4s - mosh&amp;lt;/a&amp;gt;).&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x-Commas-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:402:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:402 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>b79npv3iq4cd8l9tlw18pojs156su2d</sha1>
    </revision>
    <revision>
      <id>4160</id>
      <timestamp>2011-06-06T23:24:38Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 234752280 - Original comment: **</comment>
      <origin>4160</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21426" sha1="tinpr7y8llc45tsnrq6vz9lsw4iqe51" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2011-06-06 23:24:38 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234752280&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1.
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] [[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] [[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)

===**Instruments**=== 

[[image:Decaphonic_Classic_Guitar.png caption="A Decaphonic (10-EDO) Classical Guitar"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x-Commas-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;&amp;lt;strong&amp;gt;Instruments&amp;lt;/strong&amp;gt;&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:402:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png&amp;amp;quot; alt=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; title=&amp;amp;quot;A Decaphonic (10-EDO) Classical Guitar&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/Decaphonic_Classic_Guitar.png/234752078/Decaphonic_Classic_Guitar.png" alt="Decaphonic_Classic_Guitar.png" title="Decaphonic_Classic_Guitar.png" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;A Decaphonic (10-EDO) Classical Guitar&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:402 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>tinpr7y8llc45tsnrq6vz9lsw4iqe51</sha1>
    </revision>
    <revision>
      <id>4161</id>
      <timestamp>2011-06-06T21:08:48Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 234728058 - Original comment: **</comment>
      <origin>4161</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="30407" sha1="mgf6tnw6npkx6g7rs4fbz3wqmbafzx1" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-06 21:08:48 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234728058&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1.
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] [[http://ziaspace.com/ZIA/mp3s/Future.mp3|play]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] [[http://ziaspace.com/ZIA/mp3s/Sol.mp3|play]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)


||~   ||~   ||~   ||~   ||~   ||~   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
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||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||
||=   ||   ||&amp;gt;   ||=   ||=   ||   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Future.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://ziaspace.com/ZIA/mp3s/Sol.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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      <sha1>mgf6tnw6npkx6g7rs4fbz3wqmbafzx1</sha1>
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    <revision>
      <id>4162</id>
      <timestamp>2011-06-06T21:03:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 234727268 - Original comment: **</comment>
      <origin>4162</origin>
      <model>wikitext</model>
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      <text bytes="30099" sha1="iaqhkwpzmir69n9odtiovdazcxfli7b" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-06 21:03:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234727268&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1.
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3|play]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)


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&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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      <sha1>iaqhkwpzmir69n9odtiovdazcxfli7b</sha1>
    </revision>
    <revision>
      <id>4163</id>
      <timestamp>2011-06-06T20:46:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 234724346 - Original comment: **</comment>
      <origin>4163</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="29856" sha1="e886maz7mv0a4fmrenfq82ouzzjti9i" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-06 20:46:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234724346&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1.
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3|play]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3|play]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)


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&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
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 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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      <sha1>e886maz7mv0a4fmrenfq82ouzzjti9i</sha1>
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    <revision>
      <id>4164</id>
      <timestamp>2011-05-31T21:25:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jdfreivald</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 233336834 - Original comment: **</comment>
      <origin>4164</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="29330" sha1="1y0vp49l9sc8bjxffcz2t6rvt6nsf51" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2011-05-31 21:25:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;233336834&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1.
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 0 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 0 0 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 0 0 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||
===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)


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&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1.&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 0 0 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 0 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x-Commas-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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      <sha1>1y0vp49l9sc8bjxffcz2t6rvt6nsf51</sha1>
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    <revision>
      <id>4165</id>
      <timestamp>2011-05-31T17:43:11Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jdfreivald</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 233289894 - Original comment: **</comment>
      <origin>4165</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19233" sha1="kuc8ah8t173szxk3p377by4djm3ekn3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2011-05-31 17:43:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;233289894&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1.
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||


===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 256/243 || | 8 -5 &amp;gt; ||&amp;gt; 90.22 ||= Limma ||= Pythagorean Minor 2nd ||   ||
||= 16875/16384 || | -14 3 4 &amp;gt; ||&amp;gt; 51.12 ||= Negri Comma ||= Double Augmentation Diesis ||   ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||   ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||   ||
||= 525/512 || | -9 1 2 1 &amp;gt; ||&amp;gt; 43.41 ||= Avicennma ||= Avicenna's Enharmonic Diesis ||   ||
||= 49/48 || | -4 -1 2 &amp;gt; ||&amp;gt; 35.70 ||= Slendro Diesis ||=   ||   ||
||= 50/49 || | 1 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||   ||
||= 686/675 || | 1 -3 -2 3 &amp;gt; ||&amp;gt; 27.99 ||= Senga ||=   ||   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma || Leipziger Komma ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||   ||
||= 16875/16807 || | 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||   ||
||= 6772805/6751042 || | 11 -10 -10 10 &amp;gt; ||&amp;gt; 5.57 ||= Linus ||=   ||   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||   ||
||= 243/242 || | -1 5 -2 &amp;gt; ||&amp;gt; 7.14 ||= Rastma ||=   ||   ||
||= 385/384 || | -7 -1 1 1 1 &amp;gt; ||&amp;gt; 4.50 ||= Keenanisma ||=   ||   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||   ||
||= 540/539 || | 2 3 1 -2 -1 &amp;gt; ||&amp;gt; 3.21 ||= Swetisma ||=   ||   ||
||= 3025/3024 || | -4 -3 2 -1 2 &amp;gt; ||&amp;gt; 0.57 ||= Lehmerisma ||=   ||   ||
||= 91/90 || | -1 -2 -1 1 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||   ||
||= 676/675 || | 2 -3 -2 2 &amp;gt; ||&amp;gt; 2.56 ||= Parizeksma ||=   ||   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x--Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc3"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>kuc8ah8t173szxk3p377by4djm3ekn3</sha1>
    </revision>
    <revision>
      <id>4166</id>
      <timestamp>2011-05-31T17:42:06Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jdfreivald</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 233289618 - Original comment: Added comma table.**</comment>
      <origin>4166</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16649" sha1="s78s9fpbsbk3er9pruzxjvr8zt5rx02" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2011-05-31 17:42:06 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;233289618&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;Added comma table.&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1.
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||


===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)

==Commas== 
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

|| Comma || Monzo || Value (Cents) || Name 1 || Name 2 || Name 3 ||
|| 256/243 || | 8 -5 &amp;gt; || 90.22 || Limma || Pythagorean Minor 2nd ||   ||
|| 16875/16384 || | -14 3 4 &amp;gt; || 51.12 || Negri Comma || Double Augmentation Diesis ||   ||
|| 9931568/9752117 || | -25 7 6 &amp;gt; || 31.57 || Ampersand's Comma ||   ||   ||
|| 2048/2025 || | 11 -4 -2 &amp;gt; || 19.55 || Diaschisma ||   ||   ||
|| 525/512 || | -9 1 2 1 &amp;gt; || 43.41 || Avicennma || Avicenna's Enharmonic Diesis ||   ||
|| 49/48 || | -4 -1 2 &amp;gt; || 35.70 || Slendro Diesis ||   ||   ||
|| 50/49 || | 1 2 -2 &amp;gt; || 34.98 || Tritonic Diesis || Jubilisma ||   ||
|| 686/675 || | 1 -3 -2 3 &amp;gt; || 27.99 || Senga ||   ||   ||
|| 64/63 || | 6 -2 0 -1 &amp;gt; || 27.26 || Septimal Comma || Archytas' Comma || Leipziger Komma ||
|| 9859966/9733137 || | -10 7 8 -7 &amp;gt; || 22.41 || Blackjackisma ||   ||   ||
|| 1029/1024 || | -10 1 0 3 &amp;gt; || 8.43 || Gamelisma ||   ||   ||
|| 225/224 || | -5 2 2 -1 &amp;gt; || 7.71 || Septimal Kleisma || Marvel Comma ||   ||
|| 16875/16807 || | 3 4 -5 &amp;gt; || 6.99 || Mirkwai ||   ||   ||
|| 6772805/6751042 || | 11 -10 -10 10 &amp;gt; || 5.57 || Linus ||   ||   ||
|| 2401/2400 || | -5 -1 -2 4 &amp;gt; || 0.72 || Breedsma ||   ||   ||
|| 243/242 || | -1 5 -2 &amp;gt; || 7.14 || Rastma ||   ||   ||
|| 385/384 || | -7 -1 1 1 1 &amp;gt; || 4.50 || Keenanisma ||   ||   ||
|| 441/440 || | -3 2 -1 2 -1 &amp;gt; || 3.93 || Werckisma ||   ||   ||
|| 540/539 || | 2 3 1 -2 -1 &amp;gt; || 3.21 || Swetisma ||   ||   ||
|| 3025/3024 || | -4 -3 2 -1 2 &amp;gt; || 0.57 || Lehmerisma ||   ||   ||
|| 91/90 || | -1 -2 -1 1 1 &amp;gt; || 19.13 || Superleap ||   ||   ||
|| 676/675 || | 2 -3 -2 2 &amp;gt; || 2.56 || Parizeksma ||   ||   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x--Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc3"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 10 EDO tempers out the following commas. (Note: This assumes the val &amp;amp;lt; 10 16 23 28 35 37 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;256/243&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 8 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;90.22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Pythagorean Minor 2nd&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16875/16384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -14 3 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;51.12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Negri Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Double Augmentation Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;525/512&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 1 2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Avicennma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Avicenna's Enharmonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;49/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;35.70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Slendro Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;686/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 -3 -2 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Senga&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6772805/6751042&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -10 -10 10 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Linus&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;243/242&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 5 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Rastma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;385/384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 -1 1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4.50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Keenanisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;540/539&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 3 1 -2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3.21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Swetisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3025/3024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 -3 2 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Lehmerisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;676/675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -3 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2.56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Parizeksma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>s78s9fpbsbk3er9pruzxjvr8zt5rx02</sha1>
    </revision>
    <revision>
      <id>4167</id>
      <timestamp>2011-03-31T04:02:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 215743600 - Original comment: **</comment>
      <origin>4167</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6695" sha1="dil7h3rlh596fjfdwvwtfzldcy39r2m" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-03-31 04:02:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;215743600&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of [[13_8|13/8]] and its inversion [[16_13|16/13]]; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1.
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||


===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of &amp;lt;a class="wiki_link" href="/13_8"&amp;gt;13/8&amp;lt;/a&amp;gt; and its inversion &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x--Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>dil7h3rlh596fjfdwvwtfzldcy39r2m</sha1>
    </revision>
    <revision>
      <id>4168</id>
      <timestamp>2011-03-31T04:01:32Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 215743416 - Original comment: **</comment>
      <origin>4168</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6575" sha1="5ts996ekj6kz5b6mgnsgc47actpu6kv" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-03-31 04:01:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;215743416&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s. It can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of 13/8 and its inversion 16/13; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1.
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||


===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 10 equal parts of exactly 120 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents (or 5 circles of &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;). It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of 13/8 and its inversion 16/13; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt; &amp;lt;/h3&amp;gt;
 &amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x--Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h3&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;600&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;720&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;960&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1080&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x--Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>5ts996ekj6kz5b6mgnsgc47actpu6kv</sha1>
    </revision>
    <revision>
      <id>4169</id>
      <timestamp>2011-03-30T14:49:37Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 215554736 - Original comment: **</comment>
      <origin>4169</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6377" sha1="gyhglwx28uas3jxxc3dpip28vzlouhx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-30 14:49:37 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;215554736&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the octave into 10 equal parts of exactly 120 cents. It can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of 13/8 and its inversion 16/13; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1.
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||


===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] by ZIA (synths and voice in 10)
[[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x--Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4170</id>
      <timestamp>2011-02-22T17:03:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 204127330 - Original comment: **</comment>
      <origin>4170</origin>
      <model>wikitext</model>
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      <text bytes="6409" sha1="l6q3imf6z5kpvra9ifzcutgk22fiuyo" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-02-22 17:03:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;204127330&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;10edo, or 10-tone equal temperament, is a tuning system which divides the octave into 10 equal parts of exactly 120 cents. It can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of 13/8 and its inversion 16/13; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1.
=== === 
===Intervals=== 
|| degrees of 10edo || cents value ||
|| 0 || 0 ||
|| 1 || 120 ||
|| 2 || 240 ||
|| 3 || 360 ||
|| 4 || 480 ||
|| 5 || 600 ||
|| 6 || 720 ||
|| 7 || 840 ||
|| 8 || 960 ||
|| 9 || 1080 ||


===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] by ZIA (synths and voice in 10)
[[http://rm-and-jo.laughingsquid.org/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;10edo, or 10-tone equal temperament, is a tuning system which divides the octave into 10 equal parts of exactly 120 cents. It can be thought of as two circles of &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt; separated by 120 cents. It adds to 5edo a small neutral second and its inversion a large neutral seventh; an excellent approximation of 13/8 and its inversion 16/13; and the droll 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment of symmetry scale&amp;lt;/a&amp;gt; of the form 1 2 1 2 1 2 1.&amp;lt;br /&amp;gt;
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&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;degrees of 10edo&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x--Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Music&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://rm-and-jo.laughingsquid.org/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>l6q3imf6z5kpvra9ifzcutgk22fiuyo</sha1>
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    <revision>
      <id>4171</id>
      <timestamp>2008-05-08T12:38:05Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 23813239 - Original comment: **</comment>
      <origin>4171</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3195" sha1="c0bvr608ferni0hayyjrrfapq98487w" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2008-05-08 12:38:05 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;23813239&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;===Music=== 
[[http://eceserv0.ece.wisc.edu/~sethares/xentone.html|Ten Fingers]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/~sethares/xentone.html|Circle of Thirds]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] by ZIA (synths and voice in 10)
[[http://rm-and-jo.laughingsquid.org/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)
[[http://soundclick.com/share?songid=5683815|Ideas on the Waterfall of Expression]] by Iglashion Jones (synth)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/~sethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/~sethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://rm-and-jo.laughingsquid.org/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=5683815" rel="nofollow"&amp;gt;Ideas on the Waterfall of Expression&amp;lt;/a&amp;gt; by Iglashion Jones (synth)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4172</id>
      <timestamp>2008-04-15T06:18:32Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 21797419 - Original comment: **</comment>
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      <text bytes="2910" sha1="ra0a87sox6bxx6980i6ithka9dhadgm" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2008-04-15 06:18:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;21797419&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] by ZIA (synths and voice in 10)
[[http://rm-and-jo.laughingsquid.org/Music/RhinoPrelude-2003-02-22.mp3|Prelude]] by Rick McGowan (Rhino synthesizer)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://rm-and-jo.laughingsquid.org/Music/RhinoPrelude-2003-02-22.mp3" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Rick McGowan (Rhino synthesizer)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2007-05-26T10:35:37Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenjacob</username>
        <id>0</id>
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      <comment>**Imported revision 4629619 - Original comment: **</comment>
      <origin>4173</origin>
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      <text bytes="2606" sha1="erl908ydjgje4654ye4t2bbfwp5j5jt" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenjacob|xenjacob]] and made on &lt;tt&gt;2007-05-26 10:35:37 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;4629619&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;===Music=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Ten Fingers]] by Bill Sethares (synth guitar)
[[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html|Circle of Thirds]] by Bill Sethares (synth ens.)
[[http://www.akjmusic.com/works.html|10_fantasy]] by Aaron K. Johnson (synth monody)
[[http://www.ziaspace.com/ZIA/mp3s/Future.html|Future]] and [[http://www.ziaspace.com/ZIA/mp3s/Sol.html|Sol]] by ZIA (synths and voice in 10)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Ten Fingers&amp;lt;/a&amp;gt; by Bill Sethares (synth guitar)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html" rel="nofollow"&amp;gt;Circle of Thirds&amp;lt;/a&amp;gt; by Bill Sethares (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/works.html" rel="nofollow"&amp;gt;10_fantasy&amp;lt;/a&amp;gt; by Aaron K. Johnson (synth monody)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Future.html" rel="nofollow"&amp;gt;Future&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link_ext" href="http://www.ziaspace.com/ZIA/mp3s/Sol.html" rel="nofollow"&amp;gt;Sol&amp;lt;/a&amp;gt; by ZIA (synths and voice in 10)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <id>31067</id>
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      <timestamp>2018-08-19T19:42:07Z</timestamp>
      <contributor>
        <username>PreImportHandFixes&gt;Battaglia01</username>
        <id>0</id>
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      <comment>Mike - cleaned TOC</comment>
      <origin>31067</origin>
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      <text bytes="14873" sha1="60vyj0iqrwk05q9aej8lk180gzqjuxm" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[10平均律|日本語]]&lt;/span&gt;

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
__FORCETOC__
-----

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| | 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2,m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3,P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5,m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6,m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| | | 8 -5 &amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| | | -3 -1 2 &amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| | | -14 3 4 &amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| | | -9 1 2 1 &amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| | | -4 -1 0 2 &amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| | | 1 -3 -2 3 &amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| | | 11 -10 -10 10 &amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| | | -1 5 0 0 -2 &amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| | | -7 -1 1 1 1 &amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| | | 2 3 1 -2 -1 &amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| | | -4 -3 2 -1 2 &amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| | | 2 -3 -2 0 0 2 &amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

[[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

[[:File:decexperiment.ogg|decexperiment.ogg]]3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]''&lt;/span&gt; by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]''&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]''&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3 The Csx Freight at 1:20 am]'' by [[Zach_Curley|Zach Curley]]

[http://soundcloud.com/martinsj013/sirmdbidnud2 Sirmdbidnud2 by martinsj013 on SoundCloud]

[https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      [[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>60vyj0iqrwk05q9aej8lk180gzqjuxm</sha1>
    </revision>
    <revision>
      <id>33809</id>
      <parentid>31067</parentid>
      <timestamp>2018-09-27T21:20:49Z</timestamp>
      <contributor>
        <username>Martins</username>
        <id>39</id>
      </contributor>
      <minor/>
      <comment>/* Music */</comment>
      <origin>33809</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14858" sha1="f78i13r4vsugku2354lfp8iq103vb96" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[10平均律|日本語]]&lt;/span&gt;

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
__FORCETOC__
-----

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| | 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2,m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3,P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5,m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6,m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| | | 8 -5 &amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| | | -3 -1 2 &amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| | | -14 3 4 &amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| | | -9 1 2 1 &amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| | | -4 -1 0 2 &amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| | | 1 -3 -2 3 &amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| | | 11 -10 -10 10 &amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| | | -1 5 0 0 -2 &amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| | | -7 -1 1 1 1 &amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| | | 2 3 1 -2 -1 &amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| | | -4 -3 2 -1 2 &amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| | | 2 -3 -2 0 0 2 &amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3 The Csx Freight at 1:20 am]'' by [[Zach_Curley|Zach Curley]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      [[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>f78i13r4vsugku2354lfp8iq103vb96</sha1>
    </revision>
    <revision>
      <id>36638</id>
      <parentid>33809</parentid>
      <timestamp>2018-10-12T11:59:10Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>introduced [[Template:Japanese version page]]</comment>
      <origin>36638</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14830" sha1="a4jjadxpgb71d84yddy0mp1pflp62uf" xml:space="preserve">{{Japanese version page| [[10平均律|日本語]] }}

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
__FORCETOC__
-----

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| | 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2,m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3,P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5,m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6,m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| | | 8 -5 &amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| | | -3 -1 2 &amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| | | -14 3 4 &amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| | | -9 1 2 1 &amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| | | -4 -1 0 2 &amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| | | 1 -3 -2 3 &amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| | | 11 -10 -10 10 &amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| | | -1 5 0 0 -2 &amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| | | -7 -1 1 1 1 &amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| | | 2 3 1 -2 -1 &amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| | | -4 -3 2 -1 2 &amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| | | 2 -3 -2 0 0 2 &amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3 The Csx Freight at 1:20 am]'' by [[Zach_Curley|Zach Curley]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      [[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>a4jjadxpgb71d84yddy0mp1pflp62uf</sha1>
    </revision>
    <revision>
      <id>36735</id>
      <parentid>36638</parentid>
      <timestamp>2018-10-12T22:21:25Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>replaced interwiki solution</comment>
      <origin>36735</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14834" sha1="mchhossh87pvubfi0p1omh1650jndsm" xml:space="preserve">{{interwiki
| de = 
| en = 
| es = 
| ja = 10平均律
}}

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
__FORCETOC__
-----

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| | 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2,m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3,P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5,m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6,m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| | | 8 -5 &amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| | | -3 -1 2 &amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| | | -14 3 4 &amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| | | -9 1 2 1 &amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| | | -4 -1 0 2 &amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| | | 1 -3 -2 3 &amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| | | 11 -10 -10 10 &amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| | | -1 5 0 0 -2 &amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| | | -7 -1 1 1 1 &amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| | | 2 3 1 -2 -1 &amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| | | -4 -3 2 -1 2 &amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| | | 2 -3 -2 0 0 2 &amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3 The Csx Freight at 1:20 am]'' by [[Zach_Curley|Zach Curley]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      [[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>mchhossh87pvubfi0p1omh1650jndsm</sha1>
    </revision>
    <revision>
      <id>36743</id>
      <parentid>36735</parentid>
      <timestamp>2018-10-12T22:37:39Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>unified interwiki section for all related pages possible</comment>
      <origin>36743</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14840" sha1="sgdxv52a8aalrctql5xrqwvla6m39ju" xml:space="preserve">{{interwiki
| de = 
| en = 10edo 
| es = 
| ja = 10平均律
}}

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
__FORCETOC__
-----

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| | 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2,m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3,P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5,m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6,m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| | | 8 -5 &amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| | | -3 -1 2 &amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| | | -14 3 4 &amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| | | -9 1 2 1 &amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| | | -4 -1 0 2 &amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| | | 1 -3 -2 3 &amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| | | 11 -10 -10 10 &amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| | | -1 5 0 0 -2 &amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| | | -7 -1 1 1 1 &amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| | | 2 3 1 -2 -1 &amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| | | -4 -3 2 -1 2 &amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| | | 2 -3 -2 0 0 2 &amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3 The Csx Freight at 1:20 am]'' by [[Zach_Curley|Zach Curley]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      [[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>sgdxv52a8aalrctql5xrqwvla6m39ju</sha1>
    </revision>
    <revision>
      <id>36796</id>
      <parentid>36743</parentid>
      <timestamp>2018-10-13T01:05:56Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>added interwiki de</comment>
      <origin>36796</origin>
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      <text bytes="14845" sha1="77woydalxu67731kigrfeituvapvqv1" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
__FORCETOC__
-----

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| | 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2,m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3,P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5,m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6,m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| | | 8 -5 &amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| | | -3 -1 2 &amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| | | -14 3 4 &amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| | | -9 1 2 1 &amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| | | -4 -1 0 2 &amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| | | 1 -3 -2 3 &amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| | | 11 -10 -10 10 &amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| | | -1 5 0 0 -2 &amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| | | -7 -1 1 1 1 &amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| | | 2 3 1 -2 -1 &amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| | | -4 -3 2 -1 2 &amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| | | 2 -3 -2 0 0 2 &amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3 The Csx Freight at 1:20 am]'' by [[Zach_Curley|Zach Curley]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      [[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>77woydalxu67731kigrfeituvapvqv1</sha1>
    </revision>
    <revision>
      <id>37024</id>
      <parentid>36796</parentid>
      <timestamp>2018-10-14T22:42:41Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>removed newline at article start</comment>
      <origin>37024</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14844" sha1="8uz996p14scqs7uivky0v4ofu66wxfz" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.
__FORCETOC__
-----

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| | 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2,m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3,P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5,m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6,m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| | | 8 -5 &amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| | | -3 -1 2 &amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| | | -14 3 4 &amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| | | -9 1 2 1 &amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| | | -4 -1 0 2 &amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| | | 1 -3 -2 3 &amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| | | 11 -10 -10 10 &amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| | | -1 5 0 0 -2 &amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| | | -7 -1 1 1 1 &amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| | | 2 3 1 -2 -1 &amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| | | -4 -3 2 -1 2 &amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| | | 2 -3 -2 0 0 2 &amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3 The Csx Freight at 1:20 am]'' by [[Zach_Curley|Zach Curley]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      [[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>8uz996p14scqs7uivky0v4ofu66wxfz</sha1>
    </revision>
    <revision>
      <id>41202</id>
      <parentid>37024</parentid>
      <timestamp>2019-03-26T20:46:04Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>41202</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15609" sha1="2d2fly0vbow7263o53n7quag9lyvtym" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
-----

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
!pions
!7mus
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| | 0
| colspan="3" style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
|127.2
|153.6 (99.A&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
|254.4
|307.2 (133.3&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
|381.6
|460.8 (1CC.D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
|508.8
|614.4 (266.6&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
|636
|768 (300&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
|763.2
|921.6 (399.A&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
|890.4
|875.2 (433.3&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
|1017.6
|1228.8 (4CC.D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
|1144.8
|1382.4 (566.6&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
|1272
|1536 (600&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3 The Csx Freight at 1:20 am]'' by [[Zach_Curley|Zach Curley]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      
[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>2d2fly0vbow7263o53n7quag9lyvtym</sha1>
    </revision>
    <revision>
      <id>41231</id>
      <parentid>41202</parentid>
      <timestamp>2019-03-28T02:05:47Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>41231</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15610" sha1="dblmjfjh4mntlr4y5cfzk8qhrp6crud" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
-----

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
!pions
!7mus
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| | 0
| colspan="3" style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
|127.2
|153.6 (99.A&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
|254.4
|307.2 (133.3&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
|381.6
|460.8 (1CC.D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
|508.8
|614.4 (266.6&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
|636
|768 (300&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
|763.2
|921.6 (399.A&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
|890.4
|1095.2 (433.3&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
|1017.6
|1228.8 (4CC.D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
|1144.8
|1382.4 (566.6&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
|1272
|1536 (600&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3 The Csx Freight at 1:20 am]'' by [[Zach_Curley|Zach Curley]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a convention keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      
[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>dblmjfjh4mntlr4y5cfzk8qhrp6crud</sha1>
    </revision>
    <revision>
      <id>41278</id>
      <parentid>41231</parentid>
      <timestamp>2019-03-31T21:10:57Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Instruments */</comment>
      <origin>41278</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15612" sha1="8b4ozl7743tnhzd24flpjn3w49fsuyn" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
-----

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
!pions
!7mus
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| | 0
| colspan="3" style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
|127.2
|153.6 (99.A&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
|254.4
|307.2 (133.3&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
|381.6
|460.8 (1CC.D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
|508.8
|614.4 (266.6&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
|636
|768 (300&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
|763.2
|921.6 (399.A&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
|890.4
|1095.2 (433.3&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
|1017.6
|1228.8 (4CC.D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
|1144.8
|1382.4 (566.6&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
|1272
|1536 (600&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3 The Csx Freight at 1:20 am]'' by [[Zach_Curley|Zach Curley]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      
[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>8b4ozl7743tnhzd24flpjn3w49fsuyn</sha1>
    </revision>
    <revision>
      <id>41735</id>
      <parentid>41278</parentid>
      <timestamp>2019-05-18T19:34:01Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>41735</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15580" sha1="rowq90hhcob0i5mm5io1kuqrcziv8q8" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
-----

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
!pions
!7mus
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
|  colspan="4"| 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
|127.2
|153.6 (99.A&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
|254.4
|307.2 (133.3&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
|381.6
|460.8 (1CC.D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
|508.8
|614.4 (266.6&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
|636
|768 (300&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
|763.2
|921.6 (399.A&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
|890.4
|1095.2 (433.3&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
|1017.6
|1228.8 (4CC.D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
|1144.8
|1382.4 (566.6&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
|1272
|1536 (600&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Curley/Zach%20Curley%20-%20The%20Csx%20Freight%20at%201-20%20am.mp3 The Csx Freight at 1:20 am]'' by [[Zach_Curley|Zach Curley]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      
[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>rowq90hhcob0i5mm5io1kuqrcziv8q8</sha1>
    </revision>
    <revision>
      <id>42522</id>
      <parentid>41735</parentid>
      <timestamp>2019-11-23T15:04:09Z</timestamp>
      <contributor>
        <username>Zach Curley</username>
        <id>166</id>
      </contributor>
      <comment>I removed another one of my tracks that I wasn't proud of.</comment>
      <origin>42522</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15397" sha1="arsitzqxqw29afvqkr84sso2zgjssif" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
-----

=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
!pions
!7mus
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| colspan="4" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
|127.2
|153.6 (99.A&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | D^, Ev
|-
| | 2
| style="text-align:right;" | 240
|254.4
|307.2 (133.3&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
|381.6
|460.8 (1CC.D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | F^, Gv
|-
| | 4
| style="text-align:right;" | 480
|508.8
|614.4 (266.6&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
|636
|768 (300&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | G^, Av
|-
| | 6
| style="text-align:right;" | 720
|763.2
|921.6 (399.A&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
|890.4
|1095.2 (433.3&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | A^, Bv
|-
| | 8
| style="text-align:right;" | 960
|1017.6
|1228.8 (4CC.D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
|1144.8
|1382.4 (566.6&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | C^, Dv
|-
| | 10
| style="text-align:right;" | 1200
|1272
|1536 (600&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Images=

[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

='''Instruments'''=

10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]      
[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>arsitzqxqw29afvqkr84sso2zgjssif</sha1>
    </revision>
    <revision>
      <id>42591</id>
      <parentid>42522</parentid>
      <timestamp>2019-12-12T10:58:11Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <origin>42591</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15037" sha1="mu2p9zwn2x04va7b5b156z8v97cmnbv" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D, vE
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ^F, vG
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^G, vA
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^A, vB
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ^C, vD
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Pentatonic" EDOs|Ups and Downs Notation -"Pentatonic" EDOs]] (pentatonic fifth-based) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic 3rd-based).

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Images=
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Instruments=
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]
=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>mu2p9zwn2x04va7b5b156z8v97cmnbv</sha1>
    </revision>
    <revision>
      <id>42751</id>
      <parentid>42591</parentid>
      <timestamp>2019-12-13T01:32:56Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>added a section about alternative notations</comment>
      <origin>42751</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15948" sha1="7dlxyzytc9gta2x6lx2prkmv7gts0o5" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D, vE
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ^F, vG
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^G, vA
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^A, vB
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ^C, vD
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
| |
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
| |
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
| |
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
| |
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| |
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" | Senga
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| | Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| |
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" | Linus
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" | Rastma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" |
| |
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
| |
|}

=Images=
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Instruments=
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]
=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>7dlxyzytc9gta2x6lx2prkmv7gts0o5</sha1>
    </revision>
    <revision>
      <id>42758</id>
      <parentid>42751</parentid>
      <timestamp>2019-12-13T06:00:03Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>/* Commas */ added color names</comment>
      <origin>42758</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16856" sha1="pxystw992l1rejrf9uekikhiahlvlv3" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D, vE
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ^F, vG
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^G, vA
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^A, vB
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ^C, vD
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | [[Ratios|Ratio]]
! | [[Monzo]]
! | [[Cent|Cents]]
! |[[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
!Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" |Sawa
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
|
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" |Yoyo
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
|
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" |Laquadyo
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
|
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" |Lazoyoyo
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
|
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" |Zozo
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
|
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| style="text-align:center;" | Senga
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
|Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
|
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" |Saquinbizogu
| style="text-align:center;" | Linus
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" |Lulu
| style="text-align:center;" | Rastma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" |Lururuyo
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" |Loloruyoyo
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" |Bithogu
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
|
|}

=Images=
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Instruments=
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]
=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>pxystw992l1rejrf9uekikhiahlvlv3</sha1>
    </revision>
    <revision>
      <id>42821</id>
      <parentid>42758</parentid>
      <timestamp>2019-12-15T07:26:37Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>more cleanup</comment>
      <origin>42821</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16868" sha1="b10qxq3qvex3vlhw0lu6x72q9hzbd0p" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

= Theory =
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D, vE
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ^F, vG
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^G, vA
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^A, vB
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ^C, vD
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | [[Ratios|Ratio]]
! | [[Monzo]]
! | [[Cent|Cents]]
! |[[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
!Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" |Sawa
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
|
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" |Yoyo
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
|
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" |Laquadyo
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
|
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" |Lazoyoyo
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
|
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" |Zozo
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
|
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| style="text-align:center;" | Senga
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
|Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
|
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" |Saquinbizogu
| style="text-align:center;" | Linus
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" |Lulu
| style="text-align:center;" | Rastma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" |Lururuyo
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" |Loloruyoyo
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" |Bithogu
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
|
|}

=Images=
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Instruments=
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]
=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>b10qxq3qvex3vlhw0lu6x72q9hzbd0p</sha1>
    </revision>
    <revision>
      <id>42904</id>
      <parentid>42821</parentid>
      <timestamp>2019-12-26T04:38:42Z</timestamp>
      <contributor>
        <username>Stephen Weigel</username>
        <id>121</id>
      </contributor>
      <minor/>
      <comment>/* Music */</comment>
      <origin>42904</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17011" sha1="i5kw9d51p3xdf4b2k79zjpp6rk2sls9" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

= Theory =
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D, vE
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ^F, vG
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^G, vA
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^A, vB
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ^C, vD
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Linear temperaments=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | [[Ratios|Ratio]]
! | [[Monzo]]
! | [[Cent|Cents]]
! |[[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
!Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" |Sawa
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
|
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" |Yoyo
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
|
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" |Laquadyo
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
|
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" |Lazoyoyo
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
|
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" |Zozo
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
|
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| style="text-align:center;" | Senga
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
|Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
|
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" |Saquinbizogu
| style="text-align:center;" | Linus
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" |Lulu
| style="text-align:center;" | Rastma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" |Lururuyo
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" |Loloruyoyo
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" |Bithogu
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
|
|}

=Images=
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Instruments=
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]
=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>i5kw9d51p3xdf4b2k79zjpp6rk2sls9</sha1>
    </revision>
    <revision>
      <id>43993</id>
      <parentid>42904</parentid>
      <timestamp>2020-04-24T03:18:58Z</timestamp>
      <contributor>
        <username>Stephen Weigel</username>
        <id>121</id>
      </contributor>
      <comment>/* Images */</comment>
      <origin>43993</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17106" sha1="0ylgdvmxkn6r1echzjl4628ttliqcew" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

= Theory =
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D, vE
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ^F, vG
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^G, vA
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^A, vB
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ^C, vD
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Linear temperaments (with images for MOS horagrams)=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|1060x1060px]]

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | [[Ratios|Ratio]]
! | [[Monzo]]
! | [[Cent|Cents]]
! |[[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
!Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" |Sawa
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
|
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" |Yoyo
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
|
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" |Laquadyo
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
|
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" |Lazoyoyo
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
|
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" |Zozo
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
|
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| style="text-align:center;" | Senga
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
|Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
|
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" |Saquinbizogu
| style="text-align:center;" | Linus
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" |Lulu
| style="text-align:center;" | Rastma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" |Lururuyo
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" |Loloruyoyo
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" |Bithogu
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
|
|}

=Images=
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Instruments=
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]
=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>0ylgdvmxkn6r1echzjl4628ttliqcew</sha1>
    </revision>
    <revision>
      <id>43995</id>
      <parentid>43993</parentid>
      <timestamp>2020-04-24T03:30:47Z</timestamp>
      <contributor>
        <username>Stephen Weigel</username>
        <id>121</id>
      </contributor>
      <minor/>
      <comment>Added horagrams for MOS scales so people without Scala or temperament knowledge can find the modes immediately</comment>
      <origin>43995</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17316" sha1="b19qfncxgqgi8x5er1406neqdiuhl52" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

= Theory =
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
=Intervals=

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D, vE
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ^F, vG
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^G, vA
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^A, vB
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ^C, vD
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

=Linear temperaments (with images for MOS horagrams)=

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=Commas=
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | [[Ratios|Ratio]]
! | [[Monzo]]
! | [[Cent|Cents]]
! |[[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
!Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" |Sawa
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
|
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" |Yoyo
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
|
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" |Laquadyo
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
|
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" |Lazoyoyo
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
|
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" |Zozo
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
|
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| style="text-align:center;" | Senga
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
|Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
|
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" |Saquinbizogu
| style="text-align:center;" | Linus
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" |Lulu
| style="text-align:center;" | Rastma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" |Lururuyo
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" |Loloruyoyo
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" |Bithogu
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
|
|}

=Images=
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

=Instruments=
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]
=Music=
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>b19qfncxgqgi8x5er1406neqdiuhl52</sha1>
    </revision>
    <revision>
      <id>44925</id>
      <parentid>43995</parentid>
      <timestamp>2020-05-25T12:00:07Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>heading shifting</comment>
      <origin>44925</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17343" sha1="fhxx5rf36z3vmw010168ogyjeow837x" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

== Theory  ==
10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
== Intervals ==

{| class="wikitable"
|-
! | Degree
! | Cents
! | Approximate Ratios*
! | Additional Ratios*

of 3, 5 and 9
! | Interval Names
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs notation]]
|-
| 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| | 256/243, 50/49, 25/24
| | unison
| | unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| | 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| | 10/9, 13/12, 81/80
| | small neutral second, large minor second
| | mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D, vE
|-
| | 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| | 9/8, 7/6
| | second/third
| | maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| | 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| | 5/4
| | large neutral third
| | mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ^F, vG
|-
| | 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| | 4/3, 9/7, 13/10
| | smaller fourth
| | maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| | 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| | 7/5, 10/7, 13/9, 18/13
| | tritone
| | up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^G, vA
|-
| | 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| | 3/2, 14/9, 20/13
| | bigger fifth
| | perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| | 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| | 8/5
| | neutral sixth
| | mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^A, vB
|-
| | 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| | 16/9, 12/7
| | sixth/seventh
| | maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| | 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| | 9/5, 24/13, 160/81
| | small major 7th
| | mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ^C, vD
|-
| | 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| | 243/128, 49/25, 48/25
| | octave
| | maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}
*based on treating 10-EDO as a 2.7.13.15 subgroup temperament.

*adding the ratios of 3, 5 and 9 introduces greater [[error|error]] while giving several more harmonic identities to the 10-EDO intervals.

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperament(s)
|-
| | 1
| | 1\10
| | Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| | 1
| | 3\10
| | [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| | 2
| | 1\10
| | Messed-up [[pajara|pajara]]
|-
| | 2
| | 2\10
| | [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| | 5
| | 1\10
| | [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! | [[Ratios|Ratio]]
! | [[Monzo]]
! | [[Cent|Cents]]
! |[[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
!Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" |Sawa
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
|
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" |Yoyo
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
|
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" |Laquadyo
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
|
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" |Lazoyoyo
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
|
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" |Zozo
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
|
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| style="text-align:center;" | Senga
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
|Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
|
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" |Saquinbizogu
| style="text-align:center;" | Linus
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" |Lulu
| style="text-align:center;" | Rastma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" |Lururuyo
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" |Loloruyoyo
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" |Bithogu
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
|
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]
== Music ==
[https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

[https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (synth guitar)

[http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]''&lt;/span&gt;&lt;/span&gt; by Bill Sethares (synth ens.)

[http://www.akjmusic.com/works.html 10_fantasy] &lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Krister_Johnson|Aaron Krister Johnson]] (synth monody)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]''&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]]

[http://www.ziaspace.com/ZIA/mp3s/Future.html Future] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]''&lt;/span&gt;&lt;/span&gt; and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] &lt;span style=""&gt;&lt;span style=""&gt;''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]''&lt;/span&gt;&lt;/span&gt; by ZIA (synths and voice in 10)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]''&lt;/span&gt;&lt;/span&gt; by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]''&lt;/span&gt;&lt;/span&gt; by Igliashon Jones (synth)

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]''&lt;/span&gt;&lt;/span&gt; by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
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      <id>44926</id>
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| de = 10edo
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== Theory  ==

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
== Intervals ==

{| class="wikitable"
! Degree
! Cents
! Approximate Ratios &lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
|-
| 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D, vE
|-
| 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| 5/4
| large neutral third
| mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ^F, vG
|-
| 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^G, vA
|-
| 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| 8/5
| neutral sixth
| mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^A, vB
|-
| 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ^C, vD
|-
| 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}

&lt;references/&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| 1
| 3\10
| [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara|pajara]]
|-
| 2
| 2\10
| [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| 5
| 1\10
| [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! [[Ratios|Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" |Sawa
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
|
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" |Yoyo
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
|
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" |Laquadyo
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
|
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" |Lazoyoyo
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
|
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" |Zozo
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
|
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| style="text-align:center;" | Senga
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
|Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
|
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" |Saquinbizogu
| style="text-align:center;" | Linus
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" |Lulu
| style="text-align:center;" | Rastma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" |Lururuyo
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" |Loloruyoyo
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" |Bithogu
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
|
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"

* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]

* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]'' by [[Bill Sethares]] (synth guitar)

* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]'' by Bill Sethares (synth ens.)

* [http://www.akjmusic.com/works.html 10_fantasy] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]'' by [[Aaron Krister Johnson]] (synth monody)

* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]'' by [[Aaron Andrew Hunt]]

* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] ''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]'' and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] ''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]'' by ZIA (synths and voice in 10)

* ''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]'' by [[Rick_McGowan|Rick McGowan]] (Rhino synthesizer)

* ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]'' by Igliashon Jones (synth)

* ''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]'' by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]

* ''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])

* ''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])

* ''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])

* ''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)

* ''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)

* ''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]

* ''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy_Winchester|Randy Winchester]]

* ''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris_Vaisvil|Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]

* ''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew_Heathwaite|Andrew Heathwaite]] and [[Chris_Vaisvil|Chris Vaisvil]]

* ''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve_Martin|Steve Martin]] on SoundCloud

* ''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>a9b0k2nh74u8r4ruly2uokglu5zxm52</sha1>
    </revision>
    <revision>
      <id>44927</id>
      <parentid>44926</parentid>
      <timestamp>2020-05-25T12:25:13Z</timestamp>
      <contributor>
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        <id>6</id>
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      <text bytes="16718" sha1="6sped5kpo967ialgpc4ofcsecg5xgik" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

== Theory  ==

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

__FORCETOC__
== Intervals ==

{| class="wikitable"
! Degree
! Cents
! Approximate Ratios &lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
|-
| 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D, vE
|-
| 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| 5/4
| large neutral third
| mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ^F, vG
|-
| 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^G, vA
|-
| 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| 8/5
| neutral sixth
| mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^A, vB
|-
| 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ^C, vD
|-
| 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}

&lt;references/&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| 1
| 3\10
| [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara|pajara]]
|-
| 2
| 2\10
| [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| 5
| 1\10
| [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! [[Ratios|Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" |Sawa
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
|
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" |Yoyo
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
|
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" |Laquadyo
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
|
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" |Lazoyoyo
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
|
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" |Zozo
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
|
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| style="text-align:center;" | Senga
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
|Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
|
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" |Saquinbizogu
| style="text-align:center;" | Linus
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" |Lulu
| style="text-align:center;" | Rastma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" |Lururuyo
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" |Loloruyoyo
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" |Bithogu
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
|
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]'' by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]'' by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]'' by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]'' by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] ''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]'' and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] ''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]'' by ZIA (synths and voice in 10)
* ''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]'' by [[Rick McGowan]] (Rhino synthesizer)
* ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]'' by [[Igliashon Jones]] (synth)
* ''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]'' by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]
* ''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])
* ''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])
* ''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])
* ''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
* ''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* ''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]
* ''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy Winchester]]
* ''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]
* ''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* ''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve Martin]] on SoundCloud
* ''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>6sped5kpo967ialgpc4ofcsecg5xgik</sha1>
    </revision>
    <revision>
      <id>44928</id>
      <parentid>44927</parentid>
      <timestamp>2020-05-25T12:26:28Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>toc is automatically shown for articles with more than 3 headings</comment>
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      <text bytes="16705" sha1="pc3vqsc2tqsqp9wombmbnwzpyaz7pmz" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

== Theory  ==

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable"
! Degree
! Cents
! Approximate Ratios &lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
|-
| 0
| style="text-align:right;" | 0
| style="text-align:center;" | 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| style="text-align:center;" | P1, m2
| style="text-align:center;" | D, Eb
|-
| 1
| style="text-align:right;" | 120
| style="text-align:center;" | 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| style="text-align:center;" | ~2
| style="text-align:center;" | ^D, vE
|-
| 2
| style="text-align:right;" | 240
| style="text-align:center;" | 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| style="text-align:center;" | M2, m3
| style="text-align:center;" | E, F
|-
| 3
| style="text-align:right;" | 360
| style="text-align:center;" | 16/13
| 5/4
| large neutral third
| mid 3rd
| style="text-align:center;" | ~3
| style="text-align:center;" | ^F, vG
|-
| 4
| style="text-align:right;" | 480
| style="text-align:center;" | 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| style="text-align:center;" | M3, P4
| style="text-align:center;" | F#, G
|-
| 5
| style="text-align:right;" | 600
| style="text-align:center;" | 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| style="text-align:center;" | ^4,v5
| style="text-align:center;" | ^G, vA
|-
| 6
| style="text-align:right;" | 720
| style="text-align:center;" | 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| style="text-align:center;" | P5, m6
| style="text-align:center;" | A, Bb
|-
| 7
| style="text-align:right;" | 840
| style="text-align:center;" | 13/8
| 8/5
| neutral sixth
| mid 6th
| style="text-align:center;" | ~6
| style="text-align:center;" | ^A, vB
|-
| 8
| style="text-align:right;" | 960
| style="text-align:center;" | 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| style="text-align:center;" | M6, m7
| style="text-align:center;" | B, C
|-
| 9
| style="text-align:right;" | 1080
| style="text-align:center;" | 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| style="text-align:center;" | ~7
| style="text-align:center;" | ^C, vD
|-
| 10
| style="text-align:right;" | 1200
| style="text-align:center;" | 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| style="text-align:center;" | M7, P8
| style="text-align:center;" | C#, D
|}

&lt;references/&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| 1
| 3\10
| [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara|pajara]]
|-
| 2
| 2\10
| [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| 5
| 1\10
| [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! [[Ratios|Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" |Sawa
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
|
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" |Yoyo
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
|
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" |Laquadyo
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
|
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" |Lazoyoyo
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
|
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" |Zozo
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
|
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| style="text-align:center;" | Senga
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
|Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
|
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" |Saquinbizogu
| style="text-align:center;" | Linus
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" |Lulu
| style="text-align:center;" | Rastma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" |Lururuyo
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" |Loloruyoyo
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" |Bithogu
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
|
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]'' by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]'' by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]'' by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]'' by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] ''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]'' and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] ''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]'' by ZIA (synths and voice in 10)
* ''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]'' by [[Rick McGowan]] (Rhino synthesizer)
* ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]'' by [[Igliashon Jones]] (synth)
* ''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]'' by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]
* ''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])
* ''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])
* ''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])
* ''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
* ''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* ''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]
* ''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy Winchester]]
* ''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]
* ''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* ''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve Martin]] on SoundCloud
* ''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>pc3vqsc2tqsqp9wombmbnwzpyaz7pmz</sha1>
    </revision>
    <revision>
      <id>46052</id>
      <parentid>44928</parentid>
      <timestamp>2020-06-08T08:16:25Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Intervals */ replaced inline styling for text-alignment in table cells</comment>
      <origin>46052</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15473" sha1="hsidgr6ggxw66c2ejms9r316rar4x4c" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

== Theory  ==

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| 16/13
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| 13/8
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references/&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| 1
| 3\10
| [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara|pajara]]
|-
| 2
| 2\10
| [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| 5
| 1\10
| [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! [[Ratios|Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" |Sawa
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
|
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" |Yoyo
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
|
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" |Laquadyo
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
|
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" |Lazoyoyo
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
|
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" |Zozo
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
|
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| style="text-align:center;" | Senga
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
|Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
|
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" |Saquinbizogu
| style="text-align:center;" | Linus
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" |Lulu
| style="text-align:center;" | Rastma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" |Lururuyo
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" |Loloruyoyo
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" |Bithogu
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
|
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]'' by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]'' by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]'' by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]'' by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] ''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]'' and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] ''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]'' by ZIA (synths and voice in 10)
* ''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]'' by [[Rick McGowan]] (Rhino synthesizer)
* ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]'' by [[Igliashon Jones]] (synth)
* ''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]'' by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]
* ''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])
* ''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])
* ''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])
* ''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
* ''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* ''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]
* ''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy Winchester]]
* ''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]
* ''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* ''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve Martin]] on SoundCloud
* ''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>hsidgr6ggxw66c2ejms9r316rar4x4c</sha1>
    </revision>
    <revision>
      <id>46053</id>
      <parentid>46052</parentid>
      <timestamp>2020-06-08T08:19:33Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Linear temperaments (with images for MOS horagrams) */ table cell formatting centered, because texts are short and of same length per column</comment>
      <origin>46053</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15491" sha1="c4sqslqv2dom2if020y6jed55ipb4dl" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

== Theory  ==

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| 16/13
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| 13/8
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references/&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| 1
| 3\10
| [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara|pajara]]
|-
| 2
| 2\10
| [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| 5
| 1\10
| [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable"
|-
! [[Ratios|Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | 256/243
| |&lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 90.22
| style="text-align:center;" |Sawa
| style="text-align:center;" | Limma
| style="text-align:center;" | Pythagorean Minor 2nd
|
|-
| style="text-align:center;" | 25/24
| |&lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 70.67
| style="text-align:center;" |Yoyo
| style="text-align:center;" | 5-limit large semitone
| style="text-align:center;" | 5-limit chromatic semitone
|
|-
| style="text-align:center;" | 16875/16384
| |&lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 51.12
| style="text-align:center;" |Laquadyo
| style="text-align:center;" | Negri Comma
| style="text-align:center;" | Double Augmentation Diesis
|
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2048/2025
| |&lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 525/512
| |&lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 43.41
| style="text-align:center;" |Lazoyoyo
| style="text-align:center;" | Avicennma
| style="text-align:center;" | Avicennma's Enharmonic Diesis
|
|-
| style="text-align:center;" | 49/48
| |&lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 35.70
| style="text-align:center;" |Zozo
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 50/49
| |&lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
|
|-
| style="text-align:center;" | 686/675
| |&lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.99
| style="text-align:center;" |Trizo-agugu
| style="text-align:center;" | Senga
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 64/63
| |&lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
|Leipziger Komma
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 225/224
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
|
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 6772805/6751042
| |&lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 5.57
| style="text-align:center;" |Saquinbizogu
| style="text-align:center;" | Linus
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 243/242
| |&lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.14
| style="text-align:center;" |Lulu
| style="text-align:center;" | Rastma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 385/384
| |&lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 4.50
| style="text-align:center;" |Lozoyo
| style="text-align:center;" | Keenanisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 441/440
| |&lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 540/539
| |&lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 3.21
| style="text-align:center;" |Lururuyo
| style="text-align:center;" | Swetisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 3025/3024
| |&lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.57
| style="text-align:center;" |Loloruyoyo
| style="text-align:center;" | Lehmerisma
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 91/90
| |&lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" |
|
|-
| style="text-align:center;" | 676/675
| |&lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.56
| style="text-align:center;" |Bithogu
| style="text-align:center;" | Parizeksma
| style="text-align:center;" |
|
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]'' by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]'' by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]'' by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]'' by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] ''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]'' and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] ''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]'' by ZIA (synths and voice in 10)
* ''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]'' by [[Rick McGowan]] (Rhino synthesizer)
* ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]'' by [[Igliashon Jones]] (synth)
* ''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]'' by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]
* ''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])
* ''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])
* ''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])
* ''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
* ''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* ''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]
* ''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy Winchester]]
* ''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]
* ''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* ''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve Martin]] on SoundCloud
* ''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>c4sqslqv2dom2if020y6jed55ipb4dl</sha1>
    </revision>
    <revision>
      <id>46054</id>
      <parentid>46053</parentid>
      <timestamp>2020-06-08T08:31:32Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Commas */ replaced inline styling by alignment classes for columns</comment>
      <origin>46054</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12205" sha1="ijouf2mo9fvap55gjjecvi9qb7mbxg2" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

== Theory  ==

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| 16/13
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| 13/8
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references/&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| 1
| 3\10
| [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara|pajara]]
|-
| 2
| 2\10
| [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| 5
| 1\10
| [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 right-3 center-4"
|-
! [[Ratios|Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma
| Pythagorean Minor 2nd
|
|-
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone
| 5-limit chromatic semitone
|
|-
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri Comma
| Double Augmentation Diesis
|
|-
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's Comma
|
|
|-
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|
|
|-
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma
| Avicennma's Enharmonic Diesis
|
|-
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro Diesis
|
|
|-
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic Diesis
| Jubilisma
|
|-
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|
|
|-
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal Comma
| Archytas' Comma
| Leipziger Komma
|-
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|
|
|-
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|
|
|-
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal Kleisma
| Marvel Comma
|
|-
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|
|
|-
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|
|
|-
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|
|
|-
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|
|
|-
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|
|
|-
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|
|
|-
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|
|
|-
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|
|
|-
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|
|
|-
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|
|
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space by Elaine Walker] "Who Loves You, Me?," "Champagne," and "Avatar"
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]'' by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]'' by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]'' by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]'' by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] ''[http://ziaspace.com/ZIA/mp3s/Future.mp3 play]'' and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] ''[http://ziaspace.com/ZIA/mp3s/Sol.mp3 play]'' by ZIA (synths and voice in 10)
* ''[http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]'' by [[Rick McGowan]] (Rhino synthesizer)
* ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]'' by [[Igliashon Jones]] (synth)
* ''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs]'' by [http://www.chrisvaisvil.com Chris Vaisvil (website)] [http://chrisvaisvil.com/?p=1039 more composition information]
* ''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])
* ''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])
* ''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo_Serafini|Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])
* ''[[:File:10preview.ogg|10preview.ogg]]'' A sample of orchestral possibilities made using ZynAddSubFx under Linux (cenobyte)
* ''[[:File:decexperiment.ogg|decexperiment.ogg]]'' 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* ''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' by [http://chrisvaisvil.com/?p=1397 Chris Vaisvil]
* ''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]'' by [[Randy Winchester]]
* ''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1368 Details]
* ''[http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]'' by [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* ''[http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' by [[Steve Martin]] on SoundCloud
* ''[https://soundcloud.com/clem-fortuna/10tone 10tone demo]'' by [http://clemfortuna.com Clem Fortuna]

[[Category:10-tone]]
[[Category:10edo]]
[[Category:edo]]
[[Category:instruments]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:scale]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>ijouf2mo9fvap55gjjecvi9qb7mbxg2</sha1>
    </revision>
    <revision>
      <id>46093</id>
      <parentid>46054</parentid>
      <timestamp>2020-06-08T21:24:21Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Music */ simplified wiki markup, reduced links in music section, reduced cats</comment>
      <origin>46093</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11957" sha1="k85yy3n9olkgj5cx1iz144cxcznnsip" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

== Theory  ==

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| 16/13
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| 13/8
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references/&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| 1
| 3\10
| [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara|pajara]]
|-
| 2
| 2\10
| [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| 5
| 1\10
| [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 right-3 center-4"
|-
! [[Ratios|Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma
| Pythagorean Minor 2nd
|
|-
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone
| 5-limit chromatic semitone
|
|-
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri Comma
| Double Augmentation Diesis
|
|-
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's Comma
|
|
|-
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|
|
|-
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma
| Avicennma's Enharmonic Diesis
|
|-
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro Diesis
|
|
|-
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic Diesis
| Jubilisma
|
|-
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|
|
|-
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal Comma
| Archytas' Comma
| Leipziger Komma
|-
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|
|
|-
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|
|
|-
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal Kleisma
| Marvel Comma
|
|-
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|
|
|-
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|
|
|-
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|
|
|-
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|
|
|-
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|
|
|-
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|
|
|-
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|
|
|-
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|
|
|-
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|
|
|-
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|
|
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]

[[Category:Edo]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]</text>
      <sha1>k85yy3n9olkgj5cx1iz144cxcznnsip</sha1>
    </revision>
    <revision>
      <id>46094</id>
      <parentid>46093</parentid>
      <timestamp>2020-06-09T02:38:57Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Comma table cleanup</comment>
      <origin>46094</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11958" sha1="fbtsjn720vyz7isto8t8emgw77qzbkb" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

== Theory  ==

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| 16/13
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| 13/8
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| 1
| 3\10
| [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara|pajara]]
|-
| 2
| 2\10
| [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| 5
| 1\10
| [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratios|Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
|3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
|5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
|"
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
|"
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
|"
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
|7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
|"
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
|"
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
|"
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
|"
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
|"
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
|"
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
|"
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
|"
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
|"
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
|"
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
|11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
|"
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
|"
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
|"
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
|"
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
|13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
|"
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]

[[Category:Edo]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]</text>
      <sha1>fbtsjn720vyz7isto8t8emgw77qzbkb</sha1>
    </revision>
    <revision>
      <id>47577</id>
      <parentid>46094</parentid>
      <timestamp>2020-07-11T21:22:51Z</timestamp>
      <contributor>
        <username>Дмитрий Баженов</username>
        <id>187</id>
      </contributor>
      <comment>/* Music */</comment>
      <origin>47577</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12069" sha1="f78hnq0h5lwacpwpgoyh3383nhup132" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

== Theory  ==

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| 16/13
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| 13/8
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:10ed2-001.svg|10ed2-001.svg]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| 1
| 3\10
| [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara|pajara]]
|-
| 2
| 2\10
| [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| 5
| 1\10
| [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}

[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratios|Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
|3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
|5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
|"
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
|"
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
|"
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
|7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
|"
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
|"
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
|"
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
|"
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
|"
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
|"
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
|"
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
|"
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
|"
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
|"
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
|11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
|"
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
|"
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
|"
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
|"
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
|13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
|"
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Edo]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]</text>
      <sha1>f78hnq0h5lwacpwpgoyh3383nhup132</sha1>
    </revision>
    <revision>
      <id>48021</id>
      <parentid>47577</parentid>
      <timestamp>2020-07-31T05:31:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+Error table</comment>
      <origin>48021</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12439" sha1="4kbdu116jyk2nvt4kpzjtqdhy3kig0s" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}

== Theory  ==

10edo, or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. It can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| 16/13
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| 13/8
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| 1
| 3\10
| [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara|pajara]]
|-
| 2
| 2\10
| [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| 5
| 1\10
| [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratios|Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
|3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
|5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
|"
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
|"
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
|"
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
|7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
|"
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
|"
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
|"
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
|"
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
|"
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
|"
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
|"
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
|"
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
|"
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
|"
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
|11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
|"
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
|"
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
|"
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
|"
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
|13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
|"
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Edo]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]</text>
      <sha1>4kbdu116jyk2nvt4kpzjtqdhy3kig0s</sha1>
    </revision>
    <revision>
      <id>48022</id>
      <parentid>48021</parentid>
      <timestamp>2020-07-31T05:32:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Intro</comment>
      <origin>48022</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12448" sha1="kxlq4s4yfpao0zg52u50o3k0puxzip1" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[Octave|octave]] into 10 equal parts of exactly 120 [[cent|cent]]s. 

== Theory  ==

10edo can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8|13/8]] and its inversion [[16/13|16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation|ups and downs notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| 16/13
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| 13/8
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[Negri|negri]] (or [[Miracle|miracle]])
|-
| 1
| 3\10
| [[Dicot|Dicot]]/[[Beatles|beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara|pajara]]
|-
| 2
| 2\10
| [[Decimal|Decimal]] / messed-up [[Lemba|lemba]]
|-
| 5
| 1\10
| [[Blackwood|Blackwood]]/[[blacksmith|blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratios|Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
|3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
|5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
|"
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
|"
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
|"
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
|7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
|"
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
|"
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
|"
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
|"
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
|"
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
|"
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
|"
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
|"
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
|"
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
|"
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
|11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
|"
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
|"
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
|"
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
|"
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
|13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
|"
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| | [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| | A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Edo]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]</text>
      <sha1>kxlq4s4yfpao0zg52u50o3k0puxzip1</sha1>
    </revision>
    <revision>
      <id>48023</id>
      <parentid>48022</parentid>
      <timestamp>2020-07-31T05:37:13Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Simplify internal links</comment>
      <origin>48023</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12340" sha1="9p0nn2vjedefxun7imbqf0t16mch32z" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent|cents]]. 

== Theory  ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| 16/13
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| 13/8
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Edo]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]</text>
      <sha1>9p0nn2vjedefxun7imbqf0t16mch32z</sha1>
    </revision>
    <revision>
      <id>48037</id>
      <parentid>48023</parentid>
      <timestamp>2020-08-01T06:03:47Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+Temperament measures</comment>
      <origin>48037</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13243" sha1="mfjaslyoo71u2onnbahexqws91pi3f5" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent|cents]]. 

== Theory  ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| 16/13
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| 13/8
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Edo]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]</text>
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      <text bytes="13404" sha1="shpt0y2wo0a1kk3kunvarn4nvx1pytb" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent|cents]]. 

== Theory  ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| 16/13
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| 13/8
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by Carlo Serafini ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by Carlo Serafini ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Edo]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]</text>
      <sha1>shpt0y2wo0a1kk3kunvarn4nvx1pytb</sha1>
    </revision>
    <revision>
      <id>48056</id>
      <parentid>48055</parentid>
      <timestamp>2020-08-02T11:50:40Z</timestamp>
      <contributor>
        <username>Seraph57</username>
        <id>20</id>
      </contributor>
      <origin>48056</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13416" sha1="oyh9oi5mk9ekhdqprzqarfz5cr9cllx" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent|cents]]. 

== Theory  ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| 16/13
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| 13/8
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Edo]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]</text>
      <sha1>oyh9oi5mk9ekhdqprzqarfz5cr9cllx</sha1>
    </revision>
    <revision>
      <id>51550</id>
      <parentid>48056</parentid>
      <timestamp>2020-10-17T17:50:37Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Intervals */ links added</comment>
      <origin>51550</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13424" sha1="5qh95rw3gg2hos9z0dj2pvmwim9z91b" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent|cents]]. 

== Theory  ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Edo]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]</text>
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      <timestamp>2020-10-21T21:14:05Z</timestamp>
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| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent|cents]]. 

== Theory  ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Edo]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]</text>
      <sha1>gee1xd94gphxggvkei2fm7fehq82g3y</sha1>
    </revision>
    <revision>
      <id>54958</id>
      <parentid>51701</parentid>
      <timestamp>2020-12-04T23:13:05Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54958</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13558" sha1="61gozy4nuzt6agry32rwal0mttc448m" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent|cents]]. 

== Theory  ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Equal divisions of the octave]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]
[[Category:Instruments]]</text>
      <sha1>61gozy4nuzt6agry32rwal0mttc448m</sha1>
    </revision>
    <revision>
      <id>55360</id>
      <parentid>54958</parentid>
      <timestamp>2020-12-04T23:52:08Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>not an instrument</comment>
      <origin>55360</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13533" sha1="lef7dwbkfta6w0aitisejyxyyfmn4l0" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent|cents]]. 

== Theory  ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Equal divisions of the octave]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
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      <text bytes="13811" sha1="f4m26d3x3ncz6hns2m2dredivrw27mz" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 * 5
| Subgroup = 2.7.13.15
| Step size = 120
| Fifth type = [[5edo]] 3\10
| Common uses = chromatic [[5edo]], neutral thirds MOS
| Important MOS = neutral thirds ([[beatles]]) 3L4s 3131311 (3\10, 1\1)
| Example composition = 
| Score = 
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent|cents]].

== Theory  ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Equal divisions of the octave]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>f4m26d3x3ncz6hns2m2dredivrw27mz</sha1>
    </revision>
    <revision>
      <id>55755</id>
      <parentid>55754</parentid>
      <timestamp>2020-12-06T03:38:54Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>55755</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13811" sha1="avf205ipnvn4rky73e6xlih8byoeqcl" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 * 5
| Subgroup = 2.7.13.15
| Step size = 120
| Fifth type = [[5edo]] 3\10
| Common uses = chromatic [[5edo]], neutral thirds MOS
| Important MOS = neutral thirds ([[beatles]]) 3L4s 2121211 (3\10, 1\1)
| Example composition = 
| Score = 
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent|cents]].

== Theory  ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOSScales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Equal divisions of the octave]]
[[Category:10edo]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>avf205ipnvn4rky73e6xlih8byoeqcl</sha1>
    </revision>
    <revision>
      <id>55801</id>
      <parentid>55755</parentid>
      <timestamp>2020-12-06T10:12:41Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cat sorting, links</comment>
      <origin>55801</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13831" sha1="kppnph8z8u96zs23hokt8gqv7vuluru" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 * 5
| Subgroup = 2.7.13.15
| Step size = 120
| Fifth type = [[5edo]] 3\10
| Common uses = chromatic [[5edo]], neutral thirds MOS
| Important MOS = neutral thirds ([[beatles]]) 3L4s 2121211 (3\10, 1\1)
| Example composition = 
| Score = 
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0.0
| +15.0
| -21.9
| -7.3
| +40.6
| -0.4
| +15.0
|}

==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>kppnph8z8u96zs23hokt8gqv7vuluru</sha1>
    </revision>
    <revision>
      <id>56358</id>
      <parentid>55801</parentid>
      <timestamp>2020-12-09T09:43:19Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>added M2, m2 and A1 to the template, moved the primes-error table up to the top</comment>
      <origin>56358</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13987" sha1="ptvjt4a16ciy55zydnum87vj723k1bu" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 * 5
| Subgroup = 2.7.13.15
| Step size = 120¢
| Fifth type = 6\10 = 720¢ = [[5edo]]
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
| Common uses = chromatic [[5edo]], neutral thirds MOS
| Important MOS = neutral thirds ([[beatles]]) 3L4s 2121211 (3\10, 1\1)
| Example composition = 
| Score = 
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
|  +18.0
|  -26.3
|  -8.8
|  +48.7
|  -0.5
|  +12.5
|-
![[Relative error|relative]] (%)
| 0
|  +15
|  -22
|  -7
|  +41
|  -0.4
|  +15
|-
! colspan="2" |[[nearest edomapping]]
|10
|6
|3
|8
|5
|7
|1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>ptvjt4a16ciy55zydnum87vj723k1bu</sha1>
    </revision>
    <revision>
      <id>56438</id>
      <parentid>56358</parentid>
      <timestamp>2020-12-09T22:35:56Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Music */ +1</comment>
      <origin>56438</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14092" sha1="c104gaxx1lm1dveaa2vrhx6hi6nyook" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 * 5
| Subgroup = 2.7.13.15
| Step size = 120¢
| Fifth type = 6\10 = 720¢ = [[5edo]]
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
| Common uses = chromatic [[5edo]], neutral thirds MOS
| Important MOS = neutral thirds ([[beatles]]) 3L4s 2121211 (3\10, 1\1)
| Example composition = 
| Score = 
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
|  +18.0
|  -26.3
|  -8.8
|  +48.7
|  -0.5
|  +12.5
|-
![[Relative error|relative]] (%)
| 0
|  +15
|  -22
|  -7
|  +41
|  -0.4
|  +15
|-
! colspan="2" |[[nearest edomapping]]
|10
|6
|3
|8
|5
|7
|1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>c104gaxx1lm1dveaa2vrhx6hi6nyook</sha1>
    </revision>
    <revision>
      <id>56898</id>
      <parentid>56438</parentid>
      <timestamp>2020-12-18T05:50:56Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>template cleanup</comment>
      <origin>56898</origin>
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      <text bytes="13906" sha1="3tq7zxgt3rhr3de1x3qcburqw2wnuk3" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 * 5
| Step size = 120¢
| Fifth type = 6\10 = 720¢ = [[5edo]]
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
|  +18.0
|  -26.3
|  -8.8
|  +48.7
|  -0.5
|  +12.5
|-
![[Relative error|relative]] (%)
| 0
|  +15
|  -22
|  -7
|  +41
|  -0.4
|  +15
|-
! colspan="2" |[[nearest edomapping]]
|10
|6
|3
|8
|5
|7
|1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>3tq7zxgt3rhr3de1x3qcburqw2wnuk3</sha1>
    </revision>
    <revision>
      <id>57001</id>
      <parentid>56898</parentid>
      <timestamp>2020-12-18T15:54:23Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>ET parameter name, cleanup</comment>
      <origin>57001</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13927" sha1="g9me7tm1vuspczht9z5o4i1k5x6bmui" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. (Note: This assumes the val &amp;lt; 10 16 23 28 35 37 |.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
!Prime&lt;br&gt;Limit
! [[Ratio]]
! [[Monzo]]
! [[Cent|Cents]]
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| 256/243
| &lt;nowiki&gt; | 8 -5 &lt;/nowiki&gt;&amp;gt;
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| 25/24
| &lt;nowiki&gt; | -3 -1 2 &lt;/nowiki&gt;&amp;gt;
| 70.67
| Yoyo
| 5-limit large semitone, 5-limit chromatic semitone
|-
| "
| 16875/16384
| &lt;nowiki&gt; | -14 3 4 &lt;/nowiki&gt;&amp;gt;
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| 9931568/9752117
| &lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| 31.57
| Lala-tribiyo
| Ampersand's comma
|-
| "
| 2048/2025
| &lt;nowiki&gt; | 11 -4 -2 &lt;/nowiki&gt;&amp;gt;
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| 525/512
| &lt;nowiki&gt; | -9 1 2 1 &lt;/nowiki&gt;&amp;gt;
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| 49/48
| &lt;nowiki&gt; | -4 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| 35.70
| Zozo
| Slendro diesis
|-
| "
| 50/49
| &lt;nowiki&gt; | 1 0 2 -2 &lt;/nowiki&gt;&amp;gt;
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| 686/675
| &lt;nowiki&gt; | 1 -3 -2 3 &lt;/nowiki&gt;&amp;gt;
| 27.99
| Trizo-agugu
| Senga
|-
| "
| 64/63
| &lt;nowiki&gt; | 6 -2 0 -1 &lt;/nowiki&gt;&amp;gt;
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| 9859966/9733137
| &lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| 1029/1024
| &lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| 8.43
| Latrizo
| Gamelisma
|-
| "
| 225/224
| &lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| 16875/16807
| &lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| 6772805/6751042
| &lt;nowiki&gt; | 11 -10 -10 10 &lt;/nowiki&gt;&amp;gt;
| 5.57
| Saquinbizogu
| Linus
|-
| "
| 2401/2400
| &lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| 243/242
| &lt;nowiki&gt; | -1 5 0 0 -2 &lt;/nowiki&gt;&amp;gt;
| 7.14
| Lulu
| Rastma
|-
| "
| 385/384
| &lt;nowiki&gt; | -7 -1 1 1 1 &lt;/nowiki&gt;&amp;gt;
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| 441/440
| &lt;nowiki&gt; | -3 2 -1 2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.93
| Luzozogu
| Werckisma
|-
| "
| 540/539
| &lt;nowiki&gt; | 2 3 1 -2 -1 &lt;/nowiki&gt;&amp;gt;
| 3.21
| Lururuyo
| Swetisma
|-
| "
| 3025/3024
| &lt;nowiki&gt; | -4 -3 2 -1 2 &lt;/nowiki&gt;&amp;gt;
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| 91/90
| &lt;nowiki&gt; | -1 -2 -1 1 0 1 &lt;/nowiki&gt;&amp;gt;
| 19.13
| Thozogu
| Superleap
|-
| "
| 676/675
| &lt;nowiki&gt; | 2 -3 -2 0 0 2 &lt;/nowiki&gt;&amp;gt;
| 2.56
| Bithogu
| Parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>g9me7tm1vuspczht9z5o4i1k5x6bmui</sha1>
    </revision>
    <revision>
      <id>57058</id>
      <parentid>57001</parentid>
      <timestamp>2020-12-19T08:23:57Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Commas */ Cleanup and corrections</comment>
      <origin>57058</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13754" sha1="7frfibcoidcua8ivz4zipr3fcbstbz5" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, classic chroma
|-
| "
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| "
| [[34171875/33554432|…]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| Ampersand, Ampersand's comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| "
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| "
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| "
| [[9859966/9733137]]
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| "
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| "
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| "
| [[6772805/6751042]]
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| Linus
|-
| "
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| "
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| "
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| "
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| "
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| "
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>7frfibcoidcua8ivz4zipr3fcbstbz5</sha1>
    </revision>
    <revision>
      <id>57074</id>
      <parentid>57058</parentid>
      <timestamp>2020-12-19T18:54:42Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* Commas */ de-sophisticated cell content</comment>
      <origin>57074</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13755" sha1="lniebwfo9l85pzrqb4dr40cj74hmwgn" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean minor 2nd
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| Ampersand, Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[9859966/9733137]]
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[6772805/6751042]]
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| Linus
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>lniebwfo9l85pzrqb4dr40cj74hmwgn</sha1>
    </revision>
    <revision>
      <id>57087</id>
      <parentid>57074</parentid>
      <timestamp>2020-12-20T07:11:28Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Commas */ -wrong ratios; hide long ratios</comment>
      <origin>57087</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13732" sha1="p6h03thvq4zq765qxz6ln6dgmdw8tnl" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|…]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| Ampersand, Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| …
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| Blackjackisma
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| …
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| Linus
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>p6h03thvq4zq765qxz6ln6dgmdw8tnl</sha1>
    </revision>
    <revision>
      <id>57089</id>
      <parentid>57087</parentid>
      <timestamp>2020-12-20T11:13:24Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Commas */ make abbreviation of ratios useful (actual values calculated by wolfram alpha)</comment>
      <origin>57089</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13926" sha1="9d3sbtr42zn2u0fthi4xz0t6sy0om1h" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are not included&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>9d3sbtr42zn2u0fthi4xz0t6sy0om1h</sha1>
    </revision>
    <revision>
      <id>57090</id>
      <parentid>57089</parentid>
      <timestamp>2020-12-20T11:15:29Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Commas */ give (hopefully) better explanation</comment>
      <origin>57090</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13962" sha1="sjhfczejvlx1g7dhhybch3aee1ppls7" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color notation/Temperament Names|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>sjhfczejvlx1g7dhhybch3aee1ppls7</sha1>
    </revision>
    <revision>
      <id>57143</id>
      <parentid>57090</parentid>
      <timestamp>2020-12-20T14:13:19Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Commas */ shorten [[color name]] header</comment>
      <origin>57143</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13929" sha1="8sjprneho1q6o87gc0pm0vymnol0ypx" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>8sjprneho1q6o87gc0pm0vymnol0ypx</sha1>
    </revision>
    <revision>
      <id>57148</id>
      <parentid>57143</parentid>
      <timestamp>2020-12-20T14:41:26Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Commas */ add class commatable</comment>
      <origin>57148</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13940" sha1="7e9kjvgc13k12t1uve1n1excg8s50h9" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 13/14
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>7e9kjvgc13k12t1uve1n1excg8s50h9</sha1>
    </revision>
    <revision>
      <id>58336</id>
      <parentid>57148</parentid>
      <timestamp>2021-01-06T16:36:49Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <minor/>
      <comment>Math error.</comment>
      <origin>58336</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13940" sha1="4dfbkjvmc4479l8sbsx6sypa34a7glo" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>4dfbkjvmc4479l8sbsx6sypa34a7glo</sha1>
    </revision>
    <revision>
      <id>62990</id>
      <parentid>58336</parentid>
      <timestamp>2021-03-13T21:42:22Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <comment>/* Music */</comment>
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      <text bytes="14001" sha1="h1myfoib26erm3cpnnkb92k4gl1wcx3" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/7PBQXcf0kZU Userminusone - Bit Crystals]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>h1myfoib26erm3cpnnkb92k4gl1wcx3</sha1>
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    <revision>
      <id>62991</id>
      <parentid>62990</parentid>
      <timestamp>2021-03-13T21:43:43Z</timestamp>
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        <username>Userminusone</username>
        <id>273</id>
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      <text bytes="14024" sha1="t6edrxw2322lopydygkrp9ly7wxnng6" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/7PBQXcf0kZU Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
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{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/7PBQXcf0kZU Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
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    </revision>
    <revision>
      <id>66185</id>
      <parentid>64844</parentid>
      <timestamp>2021-04-11T04:12:28Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <comment>reuploaded version</comment>
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      <text bytes="14062" sha1="iyxpaugud72p7cjf1nz85xngkcui5wl" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>iyxpaugud72p7cjf1nz85xngkcui5wl</sha1>
    </revision>
    <revision>
      <id>66595</id>
      <parentid>66185</parentid>
      <timestamp>2021-04-13T06:54:49Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>66595</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14024" sha1="nd1x34wl0g6tqubla8l7nmjx1lb718x" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>nd1x34wl0g6tqubla8l7nmjx1lb718x</sha1>
    </revision>
    <revision>
      <id>66954</id>
      <parentid>66595</parentid>
      <timestamp>2021-04-14T01:18:11Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
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      <text bytes="14113" sha1="2tidxvpikpahl8ku8i8pg8z9fdwg1hy" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

2 1 1 1 2 1 1 1 [[2L 6s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>2tidxvpikpahl8ku8i8pg8z9fdwg1hy</sha1>
    </revision>
    <revision>
      <id>66964</id>
      <parentid>66954</parentid>
      <timestamp>2021-04-14T01:36:16Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
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      <text bytes="14147" sha1="nls9xcxse29bi95alwhwmzodcvalxe7" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

2 1 1 1 2 1 1 1 [[2L 6s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>nls9xcxse29bi95alwhwmzodcvalxe7</sha1>
    </revision>
    <revision>
      <id>66975</id>
      <parentid>66964</parentid>
      <timestamp>2021-04-14T01:53:21Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Pathological Modes */</comment>
      <origin>66975</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14147" sha1="em2uagq44mba8amalndystxoxdodidn" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>em2uagq44mba8amalndystxoxdodidn</sha1>
    </revision>
    <revision>
      <id>67017</id>
      <parentid>66975</parentid>
      <timestamp>2021-04-14T05:51:17Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <comment>revert to before policy on march 27</comment>
      <origin>67017</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14024" sha1="t6edrxw2322lopydygkrp9ly7wxnng6" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/7PBQXcf0kZU Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>t6edrxw2322lopydygkrp9ly7wxnng6</sha1>
    </revision>
    <revision>
      <id>67036</id>
      <parentid>67017</parentid>
      <timestamp>2021-04-14T06:05:20Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <comment>restore changes by another users, no "policy" related changes happened</comment>
      <origin>67036</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14147" sha1="em2uagq44mba8amalndystxoxdodidn" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et has a lower relative error than any previous ETs in the 7- and 17-limit. The next ET that does better in these subgroups is 12 and 22, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ET that does better in these subgroups is 17, 19, 36 and 31, respectively. 

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>em2uagq44mba8amalndystxoxdodidn</sha1>
    </revision>
    <revision>
      <id>68898</id>
      <parentid>67036</parentid>
      <timestamp>2021-04-27T08:17:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Temperament measures */ update</comment>
      <origin>68898</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14120" sha1="fb2uoqc4q9xxlflaeheunh816mr1lim" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4,v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>fb2uoqc4q9xxlflaeheunh816mr1lim</sha1>
    </revision>
    <revision>
      <id>71672</id>
      <parentid>68898</parentid>
      <timestamp>2021-06-02T21:48:39Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>71672</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14477" sha1="h7uhyi9boe42uviam5i8pl8jfdrris2" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
unison, min 1st
| P1, m2
P0, m1
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
small neutral prime, large minor prime
| mid 2nd
mid 1st
| ~2
~1
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
prime/second
| maj 2nd, min 3rd
maj 1st, min 2nd
| M2, m3
M1, m2
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
large neutral second
| mid 3rd
mid 2nd
| ~3
~2
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
smaller third
| maj 3rd, perf 4th
maj 2nd, perf 3rd
| M3, P4
M2, P3
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
up 3rd, down 4th
| ^4, v5
^3, v4
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
bigger fourth
| perf 5th, min 6th
perf 4th, min 5th
| P5, m6
P4, m5
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
neutral fifth
| mid 6th
mid 5th
| ~6
~5
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
fifth/sixth
| maj 6th, min 7th
maj 5th, min 6th
| M6, m7
M6, m6
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
small major 6th
| mid 7th
mid 6th
| ~7
~6
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
7th
| maj 7th, octave
maj 6th, 7th
| M7, P8
M6, P7
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>h7uhyi9boe42uviam5i8pl8jfdrris2</sha1>
    </revision>
    <revision>
      <id>71683</id>
      <parentid>71672</parentid>
      <timestamp>2021-06-03T00:18:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ remove the attempt to "correct" degrees since it lacks context</comment>
      <origin>71683</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14124" sha1="jmdkp78zmxvr1h52p1qpybw1f567vlh" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| small neutral second, large minor second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| second/third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| large neutral third
| mid 3rd
| ~3
~2
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| smaller fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| bigger fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| sixth/seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| small major 7th
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>jmdkp78zmxvr1h52p1qpybw1f567vlh</sha1>
    </revision>
    <revision>
      <id>71684</id>
      <parentid>71683</parentid>
      <timestamp>2021-06-03T00:26:27Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ improve interval names to better comply to diatonic scheme</comment>
      <origin>71684</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14152" sha1="qh02mylrq58aqx9icrh35lstj4popi2" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>qh02mylrq58aqx9icrh35lstj4popi2</sha1>
    </revision>
    <revision>
      <id>72838</id>
      <parentid>71684</parentid>
      <timestamp>2021-06-20T14:53:11Z</timestamp>
      <contributor>
        <username>Cerdded41</username>
        <id>300</id>
      </contributor>
      <comment>Added: "Vidya" --&gt; 10-EDO music by Sevish</comment>
      <origin>72838</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14242" sha1="1kzpff9fsbsqsfda9v81v7hn6i942kz" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10edo''', or 10-tone equal temperament, is a tuning system which divides the [[octave]] into 10 equal parts of exactly 120 [[cent]]s.

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>1kzpff9fsbsqsfda9v81v7hn6i942kz</sha1>
    </revision>
    <revision>
      <id>73767</id>
      <parentid>72838</parentid>
      <timestamp>2021-07-06T14:22:13Z</timestamp>
      <contributor>
        <username>SupahstarSaga</username>
        <id>281</id>
      </contributor>
      <minor/>
      <comment>Rewrote header sentence</comment>
      <origin>73767</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14359" sha1="qnkt3iri3m3r912s91685smdxkaf7sr" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
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    </revision>
    <revision>
      <id>74366</id>
      <parentid>73767</parentid>
      <timestamp>2021-07-14T19:04:44Z</timestamp>
      <contributor>
        <username>Дмитрий Баженов</username>
        <id>187</id>
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      <text bytes="14435" sha1="s074gk4decwrrb75uvwwj95g497b0do" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")
* [https://www.youtube.com/watch?v=9KBdibXR3xg Strunka] by Dmitriy Bazhenov

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>s074gk4decwrrb75uvwwj95g497b0do</sha1>
    </revision>
    <revision>
      <id>74369</id>
      <parentid>74366</parentid>
      <timestamp>2021-07-14T19:15:52Z</timestamp>
      <contributor>
        <username>Дмитрий Баженов</username>
        <id>187</id>
      </contributor>
      <comment>/* Music */</comment>
      <origin>74369</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14435" sha1="3740sp9rtttq91rlo7hs3jxcvc1ao9p" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Strunka] by Dmitriy Bazhenov

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>3740sp9rtttq91rlo7hs3jxcvc1ao9p</sha1>
    </revision>
    <revision>
      <id>74387</id>
      <parentid>74369</parentid>
      <timestamp>2021-07-15T05:09:53Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* Music */ add Cyrillic spelling as well</comment>
      <origin>74387</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14500" sha1="3l052an5gi1i8ozv28dhia43tv1cbfu" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory  ==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
|-
! rowspan="2" |Error
! absolute (¢)
| 0.0
| +18.0
| -26.3
| -8.8
| +48.7
| -0.5
| +12.5
|-
! [[Relative error|relative]] (%)
| 0
| +15
| -22
| -7
| +41
| -0
| +15
|-
! colspan="2" | [[nearest edomapping]]
| 10
| 6
| 3
| 8
| 5
| 7
| 1
|}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10&amp;#45;РДО) &amp;#124; Strunka (10&amp;#45;EDO) &amp;#45; YouTube] by Dmitriy Bazhenov

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>3l052an5gi1i8ozv28dhia43tv1cbfu</sha1>
    </revision>
    <revision>
      <id>75194</id>
      <parentid>74387</parentid>
      <timestamp>2021-07-29T02:41:13Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
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      <text bytes="14150" sha1="cbl8i1h6nncb3mk1ust7atuztlmns9v" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory  ==
{{Odd harmonics in edo|edo=10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10&amp;#45;РДО) &amp;#124; Strunka (10&amp;#45;EDO) &amp;#45; YouTube] by Dmitriy Bazhenov

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>cbl8i1h6nncb3mk1ust7atuztlmns9v</sha1>
    </revision>
    <revision>
      <id>75454</id>
      <parentid>75194</parentid>
      <timestamp>2021-08-04T18:10:15Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
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      <text bytes="14406" sha1="2d9jga7xxud3chkkhn9z1069pqq6os1" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢
| Minor 2nd = 0\10 = 0¢
| Augmented 1sn = 2\10 = 240¢
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory  ==
{{Odd harmonics in edo|edo=10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of Just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==

{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

===Differences between distributionally-even scales and smaller edos===
{| class="wikitable"
|+
!N
!L-Nedo
! s-Nedo
|-
|3
|80¢
| -40¢
|-
|4
|60¢
|  -60¢
|-
| 6
|40¢
| -80¢
|-
|7
|68.571¢
| -51.429¢
|-
|8
|90¢
|30¢
|-
|9
|106.667¢
| -13.333¢
|}
== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" |Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10et is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10 EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10-EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12-TET, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10-EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==

* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10edo)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10&amp;#45;РДО) &amp;#124; Strunka (10&amp;#45;EDO) &amp;#45; YouTube] by Dmitriy Bazhenov

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>2d9jga7xxud3chkkhn9z1069pqq6os1</sha1>
    </revision>
    <revision>
      <id>79496</id>
      <parentid>75454</parentid>
      <timestamp>2021-10-13T23:01:01Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <minor/>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14448" sha1="qmojeml0ymmgjmty2bzlblvqrpmvau2" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2\10 : 0\10
}}
'''10 equal divisions of the octave''' ('''10EDO'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory ==
{{Odd harmonics in edo|edo=10}}

10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L_4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10-EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
!N
! L-Nedo
! s-Nedo
|-
| 3
| 80¢
| -40¢
|-
| 4
| 60¢
|  -60¢
|-
| 6
| 40¢
| -80¢
|-
| 7
| 68.571¢
| -51.429¢
|-
| 8
| 90¢
| 30¢
|-
| 9
| 106.667¢
| -13.333¢
|}
== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12EDO, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10EDO may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20EDO]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10&amp;#45;РДО) &amp;#124; Strunka (10&amp;#45;EDO) &amp;#45; YouTube] by Dmitriy Bazhenov

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
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| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2\10 : 0\10
}}
'''10 equal divisions of the octave''' ('''10EDO'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory ==
{{Odd harmonics in edo|edo=10}}

10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5EDO a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 3
| 80¢
| -40¢
|-
| 4
| 60¢
|  -60¢
|-
| 6
| 40¢
| -80¢
|-
| 7
| 68.571¢
| -51.429¢
|-
| 8
| 90¢
| 30¢
|-
| 9
| 106.667¢
| -13.333¢
|}
== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12EDO, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10EDO may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20EDO]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by Dmitriy Bazhenov

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
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      <text bytes="14469" sha1="jygu2sv75ty5cz7e1foxkb2bx02ghio" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2 : 0
| Consistency = 7
| Monotonicity = 9
}}
'''10 equal divisions of the octave''' ('''10EDO'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory ==
{{Odd harmonics in edo|edo=10}}

10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5EDO a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak EDO]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 3
| 80¢
| -40¢
|-
| 4
| 60¢
| -60¢
|-
| 6
| 40¢
| -80¢
|-
| 7
| 68.571¢
| -51.429¢
|-
| 8
| 90¢
| 30¢
|-
| 9
| 106.667¢
| -13.333¢
|}
== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12EDO, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10EDO may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20EDO]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by Dmitriy Bazhenov

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
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      <text bytes="14550" sha1="szyljrw9u1sn6s9g6tl4y4l2w3qwt1r" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2 : 0
| Consistency = 7
| Monotonicity = 9
}}
'''10 equal divisions of the octave''' ('''10EDO'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory ==
{{Odd harmonics in edo|edo=10}}

10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5EDO a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap EDO, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak EDO]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 3
| 80¢
| -40¢
|-
| 4
| 60¢
| -60¢
|-
| 6
| 40¢
| -80¢
|-
| 7
| 68.571¢
| -51.429¢
|-
| 8
| 90¢
| 30¢
|-
| 9
| 106.667¢
| -13.333¢
|}
== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12EDO, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10EDO may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20EDO]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://youtu.be/CPmlCcJZGZQ "Vidya"] by Sevish (from his 2017 album "Harmony Hacker")
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by Dmitriy Bazhenov

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
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| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2 : 0
| Consistency = 7
| Monotonicity = 9
}}
'''10 equal divisions of the octave''' ('''10EDO'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory ==
{{Odd harmonics in edo|edo=10}}

10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5EDO a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap EDO, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak EDO]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 3
| 80¢
| -40¢
|-
| 4
| 60¢
| -60¢
|-
| 6
| 40¢
| -80¢
|-
| 7
| 68.571¢
| -51.429¢
|-
| 8
| 90¢
| 30¢
|-
| 9
| 106.667¢
| -13.333¢
|}
== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12EDO, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10EDO may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20EDO]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by Dmitriy Bazhenov

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
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      <id>81281</id>
      <parentid>80010</parentid>
      <timestamp>2021-11-26T16:44:11Z</timestamp>
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        <username>NerisHarp</username>
        <id>325</id>
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      <text bytes="14773" sha1="lcyetvdk3nbfl4hm52bqn0f91ygzok2" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2 : 0
| Consistency = 7
| Monotonicity = 9
}}
'''10 equal divisions of the octave''' ('''10EDO'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory ==
{{Odd harmonics in edo|edo=10}}

10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5EDO a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap EDO, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak EDO]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 3
| 80¢
| -40¢
|-
| 4
| 60¢
| -60¢
|-
| 6
| 40¢
| -80¢
|-
| 7
| 68.571¢
| -51.429¢
|-
| 8
| 90¢
| 30¢
|-
| 9
| 106.667¢
| -13.333¢
|}
== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12EDO, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10EDO may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20EDO]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by Dmitriy Bazhenov
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>lcyetvdk3nbfl4hm52bqn0f91ygzok2</sha1>
    </revision>
    <revision>
      <id>81431</id>
      <parentid>81281</parentid>
      <timestamp>2021-11-30T16:23:38Z</timestamp>
      <contributor>
        <username>KingHyperio</username>
        <id>213</id>
      </contributor>
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      <model>wikitext</model>
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      <text bytes="14791" sha1="r49401sgm99d65h1rtlucaag4vkswp5" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2 : 0
| Consistency = 7
| Monotonicity = 9
}}
'''10 equal divisions of the octave''' ('''10EDO'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory ==
{{Odd harmonics in edo|edo=10}}

10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5EDO a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap EDO, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak EDO]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 3
| 80¢
| -40¢
|-
| 4
| 60¢
| -60¢
|-
| 6
| 40¢
| -80¢
|-
| 7
| 68.571¢
| -51.429¢
|-
| 8
| 90¢
| 30¢
|-
| 9
| 106.667¢
| -13.333¢
|}
== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12EDO, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10EDO may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20EDO]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by Dmitriy Bazhenov (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by Dmitriy Bazhenov
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>r49401sgm99d65h1rtlucaag4vkswp5</sha1>
    </revision>
    <revision>
      <id>83488</id>
      <parentid>81431</parentid>
      <timestamp>2022-01-08T21:55:08Z</timestamp>
      <contributor>
        <username>Дмитрий Баженов</username>
        <id>187</id>
      </contributor>
      <comment>/* Music */</comment>
      <origin>83488</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14867" sha1="9pvvyojo7k27q3mn0tidtgecruubhh1" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2 : 0
| Consistency = 7
| Monotonicity = 9
}}
'''10 equal divisions of the octave''' ('''10EDO'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory ==
{{Odd harmonics in edo|edo=10}}

10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5EDO a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap EDO, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak EDO]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 3
| 80¢
| -40¢
|-
| 4
| 60¢
| -60¢
|-
| 6
| 40¢
| -80¢
|-
| 7
| 68.571¢
| -51.429¢
|-
| 8
| 90¢
| 30¢
|-
| 9
| 106.667¢
| -13.333¢
|}
== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12EDO, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10EDO may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20EDO]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>9pvvyojo7k27q3mn0tidtgecruubhh1</sha1>
    </revision>
    <revision>
      <id>83754</id>
      <parentid>83488</parentid>
      <timestamp>2022-01-13T00:02:44Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>remove table "Differences between distributionally-even scales and smaller edos"</comment>
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      <text bytes="14596" sha1="auxft39rbm562m6i19rfj8sbo35z10c" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2 : 0
| Consistency = 7
| Monotonicity = 9
}}
'''10 equal divisions of the octave''' ('''10EDO'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into ten equal steps of exactly 120 [[cent|cents]].

== Theory ==
{{Odd harmonics in edo|edo=10}}

10EDO can be thought of as two circles of [[5edo|5EDO]] separated by 120 cents (or 5 circles of [[2edo|2EDO]]). It adds to 5EDO a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered EDO. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap EDO, it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak EDO]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10EDO as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-EDO intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10EDO tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10EDO lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12EDO, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all EDOs which are multiples of 5, but in 10-EDO it is particularly simple.

Retuning a conventional keyboard to 10EDO may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20EDO]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10EDO) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play] by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play] by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play] by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET] by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression] by [[Igliashon Jones]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9] by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:Equal divisions of the octave]]
[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>auxft39rbm562m6i19rfj8sbo35z10c</sha1>
    </revision>
    <revision>
      <id>84024</id>
      <parentid>83754</parentid>
      <timestamp>2022-01-15T01:43:43Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Used new primes template, lowercase edo, marked 11 dead links, misc. edits</comment>
      <origin>84024</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14738" sha1="m44xcwwywr0uzu3prwl0jk29ko3u7l5" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2 : 0
| Consistency = 7
| Monotonicity = 9
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{primes in equal|10}}

10edo can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10edo tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]{{dead link}} by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]{{dead link}} by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]{{dead link}} by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]{{dead link}} by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]{{dead link}} by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]{{dead link}} by [[City of the Asleep]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]{{dead link}}
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>m44xcwwywr0uzu3prwl0jk29ko3u7l5</sha1>
    </revision>
    <revision>
      <id>84111</id>
      <parentid>84024</parentid>
      <timestamp>2022-01-15T20:05:28Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>New template name</comment>
      <origin>84111</origin>
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      <format>text/x-wiki</format>
      <text bytes="14741" sha1="k2vb4d0yh5gt933nwdp4vg5lx55qbro" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2 : 0
| Consistency = 7
| Monotonicity = 9
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10edo tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]{{dead link}} by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]{{dead link}} by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]{{dead link}} by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]{{dead link}} by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]{{dead link}} by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]{{dead link}} by [[City of the Asleep]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]{{dead link}}
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>k2vb4d0yh5gt933nwdp4vg5lx55qbro</sha1>
    </revision>
    <revision>
      <id>84217</id>
      <parentid>84111</parentid>
      <timestamp>2022-01-18T22:21:37Z</timestamp>
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      <text bytes="14770" sha1="hznsmlaophxl10w3soactpqw9bilhc1" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
Relative Radian = 19.09859¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2 : 0
| Consistency = 7
| Monotonicity = 9
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10edo tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]{{dead link}} by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]{{dead link}} by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]{{dead link}} by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]{{dead link}} by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]{{dead link}} by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]{{dead link}} by [[City of the Asleep]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]{{dead link}}
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>hznsmlaophxl10w3soactpqw9bilhc1</sha1>
    </revision>
    <revision>
      <id>84232</id>
      <parentid>84217</parentid>
      <timestamp>2022-01-19T00:28:07Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Undo relative radian: this type of addition should be discussed in Template:Infobox ET</comment>
      <origin>84232</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14741" sha1="k2vb4d0yh5gt933nwdp4vg5lx55qbro" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2 : 0
| Consistency = 7
| Monotonicity = 9
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10edo tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]{{dead link}} by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]{{dead link}} by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]{{dead link}} by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]{{dead link}} by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude]{{dead link}} by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]{{dead link}} by [[City of the Asleep]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]{{dead link}}
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>k2vb4d0yh5gt933nwdp4vg5lx55qbro</sha1>
    </revision>
    <revision>
      <id>89142</id>
      <parentid>84232</parentid>
      <timestamp>2022-03-08T02:58:40Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Fixed 1 dead link</comment>
      <origin>89142</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14770" sha1="oxxbl3f323m96oq1jlljj7yw6jdtnas" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120¢
| Fifth = 6\10 = 720¢ (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 = 240¢ (&amp;rarr;1\5)
| Semitones = 2 : 0
| Consistency = 7
| Monotonicity = 9
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10-TET''', '''10ET''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo|5edo]] separated by 120 cents (or 5 circles of [[2edo|2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Just approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

== Linear temperaments (with images for MOS horagrams) ==
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 [[2L 6s]] MOS

3 1 1 1 1 1 1 1 [[1L 7s]] MOS  

2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10edo tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480*5=2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]{{dead link}} by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]{{dead link}} by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]{{dead link}} by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]{{dead link}} by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]{{dead link}} by [[City of the Asleep]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]{{dead link}}
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>oxxbl3f323m96oq1jlljj7yw6jdtnas</sha1>
    </revision>
    <revision>
      <id>90402</id>
      <parentid>89142</parentid>
      <timestamp>2022-03-26T12:55:49Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>-monotonicity; cleanup</comment>
      <origin>90402</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14757" sha1="89kyimbqoecmc5y6b3fza4rndnxr9uf" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120.000¢
| Fifth = 6\10 (720¢) (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 (240¢) (&amp;rarr;1\5)
| Semitones = 2:0 (240¢ : 0¢)
| Consistency = 7
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
The following table shows [[TE temperament measures]] (RMS normalized) of 10ET. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 2.3.7
! 2.3.7.13
! 2.3.7.13.17
! 5-limit
! 7-limit
! 2.3.5.7.13
! 2.3.5.7.13.17
|-
! colspan="2" | Octave stretch (¢)
| -5.69
| -2.77
| -2.05
| -2.37
| -0.09
| +0.72
| +0.60
| -0.11
|-
! rowspan="2" |Error
![[TE error|absolute]] (¢)
| 5.66
| 6.23
| 5.54
| 5.00
| 9.27
| 8.15
| 7.30
| 6.85
|-
![[TE simple badness|relative]] (%)
| 4.74
| 5.20
| 4.62
| 4.17
| 7.73
| 6.79
| 6.08
| 5.70
|}
* 10ET is lower in relative error than any previous ETs in the 7- and 17-limit. The next ETs better in those subgroups are 12 and 19eg, respectively. 
* 10ET is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next ETs better in those subgroups are 17, 19, 36 and 31, respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| Messed-up [[negri]] (or [[miracle]])
|-
| 1
| 3\10
| [[Dicot]]/[[beatles]]/neutral thirds scale
|-
| 2
| 1\10
| Messed-up [[pajara]]
|-
| 2
| 2\10
| [[Decimal]] / messed-up [[lemba]]
|-
| 5
| 1\10
| [[Blackwood]]/[[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]{{dead link}} by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]{{dead link}} by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]{{dead link}} by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]{{dead link}} by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]{{dead link}} by [[City of the Asleep]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]{{dead link}}
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>89kyimbqoecmc5y6b3fza4rndnxr9uf</sha1>
    </revision>
    <revision>
      <id>90409</id>
      <parentid>90402</parentid>
      <timestamp>2022-03-26T13:34:33Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ rework RTT tables</comment>
      <origin>90409</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14721" sha1="6wz294nr1ersbawkn77rq8288dlmyrn" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120.000¢
| Fifth = 6\10 (720¢) (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 (240¢) (&amp;rarr;1\5)
| Semitones = 2:0 (240¢ : 0¢)
| Consistency = 7
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Images ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]{{dead link}} by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]{{dead link}} by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]{{dead link}} by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]{{dead link}} by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]{{dead link}} by [[City of the Asleep]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]{{dead link}}
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>6wz294nr1ersbawkn77rq8288dlmyrn</sha1>
    </revision>
    <revision>
      <id>90410</id>
      <parentid>90409</parentid>
      <timestamp>2022-03-26T13:35:02Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Section title</comment>
      <origin>90410</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14723" sha1="axtlrso9n1w17qvl3frtu10eq9me235" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120.000¢
| Fifth = 6\10 (720¢) (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 (240¢) (&amp;rarr;1\5)
| Semitones = 2:0 (240¢ : 0¢)
| Consistency = 7
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]{{dead link}} by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]{{dead link}} by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]{{dead link}} by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]{{dead link}} by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]{{dead link}} by [[City of the Asleep]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]{{dead link}}
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>axtlrso9n1w17qvl3frtu10eq9me235</sha1>
    </revision>
    <revision>
      <id>91124</id>
      <parentid>90410</parentid>
      <timestamp>2022-04-11T07:12:08Z</timestamp>
      <contributor>
        <username>Cerdded41</username>
        <id>300</id>
      </contributor>
      <comment>Added audio files for the intervals.</comment>
      <origin>91124</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15250" sha1="6tu4jcoy8obweyszs6ogiu4a997onfj" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120.000¢
| Fifth = 6\10 (720¢) (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 (240¢) (&amp;rarr;1\5)
| Semitones = 2:0 (240¢ : 0¢)
| Consistency = 7
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]{{dead link}} by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]{{dead link}} by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]{{dead link}} by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]{{dead link}} by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]{{dead link}} by [[City of the Asleep]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]{{dead link}}
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
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      <text bytes="15277" sha1="g9pdvd9oa6tbxvbn073czvn4n1bqxfl" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120.000¢
| Fifth = 6\10 (720¢) (&amp;rarr;[[5edo|3\5]])
| Major 2nd = 2\10 (240¢) (&amp;rarr;1\5)
| Semitones = 2:0 (240¢ : 0¢)
| Consistency = 7
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]{{dead link}} by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]{{dead link}} by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]{{dead link}} by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]{{dead link}} by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]{{dead link}} by [[City of the Asleep]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]{{dead link}}
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>g9pdvd9oa6tbxvbn073czvn4n1bqxfl</sha1>
    </revision>
    <revision>
      <id>94746</id>
      <parentid>92573</parentid>
      <timestamp>2022-08-19T18:51:28Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>-major 2nd in infobox</comment>
      <origin>94746</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15238" sha1="a2wn65povntsts2ywlpiqrd22jgovf2" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120.000¢
| Fifth = 6\10 (720¢) (&amp;rarr;[[5edo|3\5]])
| Semitones = 2:0 (240¢ : 0¢)
| Consistency = 7
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the happy 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]{{dead link}} by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]{{dead link}} by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]{{dead link}} by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]{{dead link}} by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [http://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]{{dead link}} by [[City of the Asleep]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [http://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [http://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]{{dead link}}
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>a2wn65povntsts2ywlpiqrd22jgovf2</sha1>
    </revision>
    <revision>
      <id>94836</id>
      <parentid>94746</parentid>
      <timestamp>2022-08-21T03:42:22Z</timestamp>
      <contributor>
        <username>Morion</username>
        <id>322</id>
      </contributor>
      <comment>/* Theory */ "Familiar" is the proper word to use here.</comment>
      <origin>94836</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15244" sha1="joyficiylf20tomy3dh526vx2336b50" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120.000¢
| Fifth = 6\10 (720¢) (&amp;rarr;[[5edo|3\5]])
| Semitones = 2:0 (240¢ : 0¢)
| Consistency = 7
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}
* [https://www.reverbnation.com/1029821/album/172734 ZIA Space] "Who Loves You, Me?", "Champagne", and "Avatar" by [[Elaine Walker]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel]
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Ten Fingers]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Ten_Fingers.mp3 play]{{dead link}} by [[Bill Sethares]] (synth guitar)
* [http://eceserv0.ece.wisc.edu/%7Esethares/xentone.html Circle of Thirds]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Circle_of_Thirds.mp3 play]{{dead link}} by Bill Sethares (synth ens.)
* [http://www.akjmusic.com/works.html 10_fantasy]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/AKJ/10_fantasy.mp3 play]{{dead link}} by [[Aaron Krister Johnson]] (synth monody)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Hunt/Prelude%20in%2010ET.mp3 Prelude in 10ET]{{dead link}} by [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fugue-in-10et Fugue in 10ET &amp;#124; SoundCloud] by Aaron Andrew Hunt, 2015
* [http://www.ziaspace.com/ZIA/mp3s/Future.html Future] [http://ziaspace.com/ZIA/mp3s/Future.mp3 play] and [http://www.ziaspace.com/ZIA/mp3s/Sol.html Sol] [http://ziaspace.com/ZIA/mp3s/Sol.mp3 play] by ZIA (synths and voice in 10)
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Prelude] by [[Rick McGowan]] (Rhino synthesizer)
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20Ideas%20on%20the%20Waterfall%20of%20Expression.mp3 Ideas on the Waterfall of Expression]{{dead link}} by [[City of the Asleep]] (synth)
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For two violas and gongs] by [[Chris Vaisvil]] [http://chrisvaisvil.com/?p=1039 Details]
* [http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html blog entry])''
* [http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html blog entry])''
* [http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html blog entry])''
* [http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' by [[Carlo Serafini]] ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html blog entry])''
* [[:File:10preview.ogg|10preview.ogg]] A sample of orchestral possibilities made using ZynAddSubFx under Linux ([[cenobyte]])
* [[:File:decexperiment.ogg|decexperiment.ogg]] 3 tracks made in ZynAddSubFx simply mixed in Audacity (cenobyte)
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive] by Chris Vaisvil [http://chrisvaisvil.com/?p=1397 Details]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/09%20-%209.%2010%20octave.mp3 Comets Over Flatland 9]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture] by Chris Vaisvil [http://chrisvaisvil.com/?p=1368 Details]
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing] by [[Andrew Heathwaite]] and Chris Vaisvil
* [https://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner] by [[Steve Martin]] on SoundCloud
* [https://soundcloud.com/clem-fortuna/10tone 10-tone demo] by [http://clemfortuna.com Clem Fortuna]{{dead link}}
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!] by [[User:Дмитрий Баженов|Dmitry Bazhenov]] (second part in 10-edo)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike (10EDO)] by [[City of the Asleep]]
* [https://youtu.be/gZrD3gHUnnM Bit Crystals] by [[User:Userminusone|Userminusone]]
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка (10-РДО) &amp;#124; Strunka (10-EDO) &amp;#45; YouTube] by [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal moment musicale 2] by Evan Bennet
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three weeping tyrants] by Stephen Weigel

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>joyficiylf20tomy3dh526vx2336b50</sha1>
    </revision>
    <revision>
      <id>95647</id>
      <parentid>94836</parentid>
      <timestamp>2022-09-22T01:32:12Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>/* Music */ Sorted by artists, fix some broken links, a few new links</comment>
      <origin>95647</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16317" sha1="0kziq832z2xq5h82meb2h9tq4btuik7" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120.000¢
| Fifth = 6\10 (720¢) (&amp;rarr;[[5edo|3\5]])
| Semitones = 2:0 (240¢ : 0¢)
| Consistency = 7
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* ''[https://soundcloud.com/user-544568549/ey-ule-hey-ule Hey, ule!]'' (second part only) (2020)
* ''[https://www.youtube.com/watch?v=RU1Cpe1Szo8 Струнка]'' (''Strunka'') (2021)

'''Evan Bennet'''
* ''[https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 Microtonal Moment Musicale 2]'' (2018)

'''Cenobyte'''
* ''[[:File:10preview.ogg|10preview]]'' (2018)
* ''[[:File:decexperiment.ogg|decexperiment]]'' (2018)

'''[[City of the Asleep]]'''
* "[https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression Ideas on the Waterfall of Expression]", from ''[https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue Map of an Internal Landscape]'' (2007)
* "[https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo Sad Mike]", from ''[https://cityoftheasleep.bandcamp.com/album/cloiiids CLOIIIDS]'' (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* ''[https://soundcloud.com/clem-fortuna/10tone 10Tone]'' (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* "[https://andrewheathwaite.bandcamp.com/track/shimmerwing Shimmerwing]", from ''[https://andrewheathwaite.bandcamp.com/album/newbeams Newbeams]'' (2012)

'''[[Aaron Andrew Hunt]]'''
* "[https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et Prelude in 10ET]", from ''[https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard The Equal-Tempered Keyboard]'' (1999-2022)
* "[https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et Fugue a3 in 10ET]", from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''[[Aaron Krister Johnson]]'''
* ''[http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg 10 fantasy]'' (≤2005)

'''Steve Martin'''
* ''[https://soundcloud.com/martinsj013/sirmdbidnud2 Shall I Refuse My Dinner]'' (2012)

'''[[Rick McGowan]]'''
* ''[https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 Rhino Prelude]'' (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* ''[http://www.seraph.it/dep/det/Blues10.mp3 Blues 10]'' (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* ''[http://www.seraph.it/dep/det/Waltz10.mp3 Waltz 10]'' (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* ''[http://www.seraph.it/dep/det/Smooth10.mp3 Smooth 10]'' (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* ''[http://www.seraph.it/dep/det/10PRS.mp3 10 PRS]'' (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* "[https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 Ten Fingers]", from ''[https://sethares.engr.wisc.edu/xentone.html Xentonality]'' (1997)
* "[https://sethares.engr.wisc.edu/mp3s/circleofthirds.html Circle of Thirds]", from ''Xentonality'' (1997)

'''[[User:Userminusone|Userminusone]]'''
* ''[https://youtu.be/gZrD3gHUnnM Bit Crystals]'' (2021)

'''[[Chris Vaisvil]]'''
* ''[http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 For 2 Violas and Gongs in 10 edo]'' (2011) ([http://chrisvaisvil.com/?p=1039 details])
* ''[http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 The Dramatic Squirrel Overture]'' (2011) ([http://chrisvaisvil.com/?p=1368 details])
* ''[http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 10 Earwigs Invasive]'' (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* ''[https://soundcloud.com/overtoneshock/fiat-circadia-10-edo Fiat Circadia]'' (2018)
* ''[https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 Three Weeping Tyrants]'' (2020)

'''[[Randy Winchester]]'''
* "[https://archive.org/details/jamendo-005173/09.mp3 9. 10 / octave]", from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* "[https://ziaspace.bandcamp.com/track/sol-10edo Sol]", from ''[https://ziaspace.bandcamp.com/album/zia-15-1992 ZIA 1.5]'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* "[https://ziaspace.bandcamp.com/track/agribusiness-10edo Agribusiness]", from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* "[https://ziaspace.bandcamp.com/track/future-10edo Future]", from ''[https://ziaspace.bandcamp.com/album/shem-1996 SHEM]'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* "[https://ziaspace.bandcamp.com/track/river-10edo River]", from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* "[https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo Who Loves You.. Me?]", from ''[https://ziaspace.bandcamp.com/album/drum-n-space-2011 Drum 'n' Space]'' (2011)
* "[https://ziaspace.bandcamp.com/track/champagne-10edo Champagne]", from ''Drum 'n' Space'' (2011)
* "[https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo Love is the Catalyst]", from ''Drum 'n' Space'' (2011)
* "[https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo Decagon Dancefloor]", from ''[https://ziaspace.bandcamp.com/album/trapezoid-2019 Trapezoid]'' (2019)
* "[https://ziaspace.bandcamp.com/track/matter-over-mind-10edo Matter Over Mind]", from ''[https://ziaspace.bandcamp.com/album/trapezoid-2019 No Terrestrial Road]" (2020)

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>0kziq832z2xq5h82meb2h9tq4btuik7</sha1>
    </revision>
    <revision>
      <id>95741</id>
      <parentid>95647</parentid>
      <timestamp>2022-09-26T03:32:31Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
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      <text bytes="16386" sha1="ox4vfcw0bgb7n5l8d6t5bo38o7wa42x" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120.000¢
| Fifth = 6\10 (720¢) (&amp;rarr;[[5edo|3\5]])
| Semitones = 2:0 (240¢ : 0¢)
| Consistency = 7
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>ox4vfcw0bgb7n5l8d6t5bo38o7wa42x</sha1>
    </revision>
    <revision>
      <id>95742</id>
      <parentid>95741</parentid>
      <timestamp>2022-09-26T03:34:07Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>/* Music */ Added 1 track</comment>
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      <text bytes="16457" sha1="sdy66ydgfqxtm0l7nstek0e42v9kphd" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET
| Prime factorization = 2 × 5
| Step size = 120.000¢
| Fifth = 6\10 (720¢) (&amp;rarr;[[5edo|3\5]])
| Semitones = 2:0 (240¢ : 0¢)
| Consistency = 7
}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>sdy66ydgfqxtm0l7nstek0e42v9kphd</sha1>
    </revision>
    <revision>
      <id>96082</id>
      <parentid>95742</parentid>
      <timestamp>2022-10-04T16:01:54Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET now computes most parameters automatically</comment>
      <origin>96082</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16307" sha1="jteb284i0pfwnbuxyer8gvl3bj9pjis" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that 105/104, 225/224, and 16807/16384 are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>jteb284i0pfwnbuxyer8gvl3bj9pjis</sha1>
    </revision>
    <revision>
      <id>101067</id>
      <parentid>96082</parentid>
      <timestamp>2023-01-07T19:14:38Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
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      <comment>Add new album to music list</comment>
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      <text bytes="16446" sha1="b84qj7vqwi9i7yklrrmgcjenj05z7k9" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are 12 and 19eg, respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are 17, 19, 36 and 31, respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:10edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Macrotonal]]
[[Category:Zeta]]
[[Category:Listen]]</text>
      <sha1>b84qj7vqwi9i7yklrrmgcjenj05z7k9</sha1>
    </revision>
    <revision>
      <id>102390</id>
      <parentid>101067</parentid>
      <timestamp>2023-02-03T11:58:30Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Style; cleanup; -redundant categories</comment>
      <origin>102390</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16377" sha1="gd7paclfayje8nd48h9k39kgeujrfpu" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Commas ==
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
      <sha1>gd7paclfayje8nd48h9k39kgeujrfpu</sha1>
    </revision>
    <revision>
      <id>102391</id>
      <parentid>102390</parentid>
      <timestamp>2023-02-03T12:04:21Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Move commas to a subsection of RTT properties</comment>
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      <text bytes="16379" sha1="25r7kul4g1q7zpdxnr94m459g5zsz4q" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|1060x1060px|1\10 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 MOS with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

=== Pathological scales ===
* 2 1 1 1 2 1 1 1 [[2L 6s]] MOS
* 3 1 1 1 1 1 1 1 [[1L 7s]] MOS  
* 2 1 1 1 1 1 1 1 1 [[1L 8s]] MOS

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
      <sha1>25r7kul4g1q7zpdxnr94m459g5zsz4q</sha1>
    </revision>
    <revision>
      <id>102392</id>
      <parentid>102391</parentid>
      <timestamp>2023-02-03T12:06:35Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Scales */ -"pathological scales" (it's unclear why those are noted at all)</comment>
      <origin>102392</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16248" sha1="gdsxnuvylhxp104rdlcghmu440tmpk9" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
'''10 equal divisions of the octave''' ('''10edo'''), or '''10-tone equal temperament''' ('''10tet''', '''10et''') when viewed from a [[regular temperament]] perspective, is the [[tuning system]] that divides the [[octave]] into ten equal steps of exactly 120 [[cent]]s.

== Theory ==
{{Harmonics in equal|10}}

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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      <text bytes="16019" sha1="4m4810v9a3di8ki4yoplp3o0recfn5e" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
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{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave
| up 4th, down 5th
| ^4, v5
| ^G, vA
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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      <text bytes="16108" sha1="0g03b5xvwxqc6zrjskrstj3xw8i0hqg" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
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{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]). While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| C#, Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| D
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| E#, Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| F
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| G#, Ab
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| A
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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      <text bytes="16201" sha1="f4re581fb0yn4dfwdme623n7vsjuc3t" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
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{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| C#, Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| D
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| E#, Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| F
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| G#, Ab
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| A
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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        <id>381</id>
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      <text bytes="16292" sha1="d90c2t3jct8xpnu7vuzpbixnwiv5wsm" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
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{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| C#, Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| D
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| E#, Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| F
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| G#, Ab
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| A
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''5 hideya'''
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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    </revision>
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      <id>113704</id>
      <parentid>113700</parentid>
      <timestamp>2023-05-19T11:54:42Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Music */ Added music</comment>
      <origin>113704</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16367" sha1="o89s3bb2hvzcmzc6vc00jnnryo3ni3d" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| C#, Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| D
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| E#, Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| F
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| G#, Ab
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| A
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''5 hideya'''
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)
* [https://www.youtube.com/watch?v=lkVooFr1vmk ''Like Yagi-bushi''] (2021)

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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        <username>Francium</username>
        <id>381</id>
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      <text bytes="16448" sha1="jaiv86oyxfdu29tshdttjgdf4y0pnxy" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| C#, Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| D
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| E#, Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| F
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| G#, Ab
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| A
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''5 hideya'''
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)
* [https://www.youtube.com/watch?v=lkVooFr1vmk ''Like Yagi-bushi''] (2021)
* [https://www.youtube.com/watch?v=-L4VfJpYk-g ''Like Summer Creatures''] (2021)

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| C#, Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| D
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| E#, Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| F
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| G#, Ab
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| A
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

'''5 hideya'''
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)
* [https://www.youtube.com/watch?v=lkVooFr1vmk ''Like Yagi-bushi''] (2021)
* [https://www.youtube.com/watch?v=-L4VfJpYk-g ''Like Summer Creatures''] (2021)
* [https://www.youtube.com/watch?v=BCyhjlLVHl8 ''Like spring flowers''] (2023)

'''[[User:Дмитрий Баженов|Dmitry Bazhenov]]'''
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

'''Evan Bennet'''
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

'''Cenobyte'''
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

'''Clem Fortuna''' ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

'''[[Andrew Heathwaite]] and [[Chris Vaisvil]]'''
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

'''Jama'''
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

'''[[Aaron Krister Johnson]]'''
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

'''Steve Martin'''
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

'''[[Rick McGowan]]'''
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

'''[[Carlo Serafini]]'''
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

'''[[Sevish]]'''
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

'''[[User:Userminusone|Userminusone]]'''
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

'''[[Stephen Weigel]]'''
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[ZIA]]'''
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| C#, Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| D
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| E#, Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| F
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| G#, Ab
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| A
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

; [[5 hideya]]
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)
* [https://www.youtube.com/watch?v=lkVooFr1vmk ''Like Yagi-bushi''] (2021)
* [https://www.youtube.com/watch?v=-L4VfJpYk-g ''Like Summer Creatures''] (2021)
* [https://www.youtube.com/watch?v=BCyhjlLVHl8 ''Like spring flowers''] (2023)

; [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

; [[Bazil Müzik]]
* [https://www.youtube.com/watch?v=OolR2oYiuyQ ''Lost Keys''] (2019)

; [[Evan Bennet]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

; [[Cenobyte]]
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

; Clem Fortuna ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

; [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

; Jama
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

; [[Aaron Krister Johnson]]
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

; Steve Martin
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

; [[Rick McGowan]]
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

; [[Sevish]]
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

; [[User:Userminusone|Userminusone]]
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[ZIA]]
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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      <timestamp>2023-05-26T15:27:27Z</timestamp>
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| de = 10edo
| en = 10edo 
| es = 
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{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| C#, Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| D
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| E#, Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| F
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| G#, Ab
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| A
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

; [[5 hideya]]
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)
* [https://www.youtube.com/watch?v=lkVooFr1vmk ''Like Yagi-bushi''] (2021)
* [https://www.youtube.com/watch?v=-L4VfJpYk-g ''Like Summer Creatures''] (2021)
* [https://www.youtube.com/watch?v=BCyhjlLVHl8 ''Like spring flowers''] (2023)

; [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

; [[Bazil Müzik]]
* [https://www.youtube.com/watch?v=OolR2oYiuyQ ''Lost Keys''] (2019)

; [[Evan Bennet]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

; [[Cenobyte]]
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

; Clem Fortuna ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

; [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

; Jama
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

; [[Aaron Krister Johnson]]
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

; Steve Martin
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

; [[Rick McGowan]]
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

; [[Sevish]]
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

; [[User:Userminusone|Userminusone]]
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Xotla]]
* [https://www.youtube.com/watch?v=DzKDz-CeI_E ''Phosphoresce''] (2017)

; [[ZIA]]
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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| de = 10edo
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{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| augmented unison, minor second
| C#, Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second
| D
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

; [[5 hideya]]
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)
* [https://www.youtube.com/watch?v=lkVooFr1vmk ''Like Yagi-bushi''] (2021)
* [https://www.youtube.com/watch?v=-L4VfJpYk-g ''Like Summer Creatures''] (2021)
* [https://www.youtube.com/watch?v=BCyhjlLVHl8 ''Like spring flowers''] (2023)

; [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

; [[Bazil Müzik]]
* [https://www.youtube.com/watch?v=OolR2oYiuyQ ''Lost Keys''] (2019)

; [[Evan Bennet]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

; [[Cenobyte]]
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

; Clem Fortuna ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

; [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

; Jama
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

; [[Aaron Krister Johnson]]
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

; Steve Martin
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

; [[Rick McGowan]]
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

; [[Sevish]]
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

; [[User:Userminusone|Userminusone]]
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Xotla]]
* [https://www.youtube.com/watch?v=DzKDz-CeI_E ''Phosphoresce''] (2017)

; [[ZIA]]
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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    </revision>
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      <id>115013</id>
      <parentid>115012</parentid>
      <timestamp>2023-05-29T03:21:46Z</timestamp>
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      <text bytes="16845" sha1="kkab1lbkvc4keapvx3wv37qoq2ff3do" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
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{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second
| D
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

; [[5 hideya]]
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)
* [https://www.youtube.com/watch?v=lkVooFr1vmk ''Like Yagi-bushi''] (2021)
* [https://www.youtube.com/watch?v=-L4VfJpYk-g ''Like Summer Creatures''] (2021)
* [https://www.youtube.com/watch?v=BCyhjlLVHl8 ''Like spring flowers''] (2023)

; [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

; [[Bazil Müzik]]
* [https://www.youtube.com/watch?v=OolR2oYiuyQ ''Lost Keys''] (2019)

; [[Evan Bennet]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

; [[Cenobyte]]
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

; Clem Fortuna ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

; [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

; Jama
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

; [[Aaron Krister Johnson]]
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

; Steve Martin
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

; [[Rick McGowan]]
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

; [[Sevish]]
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

; [[User:Userminusone|Userminusone]]
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Xotla]]
* [https://www.youtube.com/watch?v=DzKDz-CeI_E ''Phosphoresce''] (2017)

; [[ZIA]]
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
      <sha1>kkab1lbkvc4keapvx3wv37qoq2ff3do</sha1>
    </revision>
    <revision>
      <id>115014</id>
      <parentid>115013</parentid>
      <timestamp>2023-05-29T03:23:51Z</timestamp>
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      <text bytes="16867" sha1="6rc95fiiz1e90zvk4inma0qobgblfzy" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{See also|:Category:10edo tracks}}

; [[5 hideya]]
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)
* [https://www.youtube.com/watch?v=lkVooFr1vmk ''Like Yagi-bushi''] (2021)
* [https://www.youtube.com/watch?v=-L4VfJpYk-g ''Like Summer Creatures''] (2021)
* [https://www.youtube.com/watch?v=BCyhjlLVHl8 ''Like spring flowers''] (2023)

; [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

; [[Bazil Müzik]]
* [https://www.youtube.com/watch?v=OolR2oYiuyQ ''Lost Keys''] (2019)

; [[Evan Bennet]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

; [[Cenobyte]]
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

; Clem Fortuna ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

; [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

; Jama
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

; [[Aaron Krister Johnson]]
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

; Steve Martin
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

; [[Rick McGowan]]
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

; [[Sevish]]
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

; [[User:Userminusone|Userminusone]]
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Xotla]]
* [https://www.youtube.com/watch?v=DzKDz-CeI_E ''Phosphoresce''] (2017)

; [[ZIA]]
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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    <revision>
      <id>115632</id>
      <parentid>115014</parentid>
      <timestamp>2023-06-03T14:22:33Z</timestamp>
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      <text bytes="16855" sha1="g8t8ysawxvrs9wfwpgsbhg7ize6e81r" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Catrel|10edo tracks}}

; [[5 hideya]]
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)
* [https://www.youtube.com/watch?v=lkVooFr1vmk ''Like Yagi-bushi''] (2021)
* [https://www.youtube.com/watch?v=-L4VfJpYk-g ''Like Summer Creatures''] (2021)
* [https://www.youtube.com/watch?v=BCyhjlLVHl8 ''Like spring flowers''] (2023)

; [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

; [[Bazil Müzik]]
* [https://www.youtube.com/watch?v=OolR2oYiuyQ ''Lost Keys''] (2019)

; [[Evan Bennet]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

; [[Cenobyte]]
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

; Clem Fortuna ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

; [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

; Jama
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

; [[Aaron Krister Johnson]]
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

; Steve Martin
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

; [[Rick McGowan]]
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

; [[Sevish]]
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

; [[User:Userminusone|Userminusone]]
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Xotla]]
* [https://www.youtube.com/watch?v=DzKDz-CeI_E ''Phosphoresce''] (2017)

; [[ZIA]]
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
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    <revision>
      <id>117754</id>
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      <timestamp>2023-06-29T12:06:24Z</timestamp>
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        <id>381</id>
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      <text bytes="17023" sha1="761huguqyou22uadjstuef6uffyhakh" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Catrel|10edo tracks}}

; [[5 hideya]]
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)
* [https://www.youtube.com/watch?v=lkVooFr1vmk ''Like Yagi-bushi''] (2021)
* [https://www.youtube.com/watch?v=-L4VfJpYk-g ''Like Summer Creatures''] (2021)
* [https://www.youtube.com/watch?v=BCyhjlLVHl8 ''Like spring flowers''] (2023)

; [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

; [[Bazil Müzik]]
* [https://www.youtube.com/watch?v=OolR2oYiuyQ ''Lost Keys''] (2019)

; [[Evan Bennet]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

; [[Cenobyte]]
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

; Clem Fortuna ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

; [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

; Jama
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

; [[Aaron Krister Johnson]]
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

; Steve Martin
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

; [[Rick McGowan]]
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

; [[Sevish]]
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

; [[User:Userminusone|Userminusone]]
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Xotla]]
* [https://www.youtube.com/watch?v=DzKDz-CeI_E ''Phosphoresce''] (2017)

; Zewen Senpai
* [https://www.youtube.com/watch?v=3j7JLG-Iiuc ''Dreamscape No. 1''] (2022)
* [https://www.youtube.com/watch?v=ZVwDCluoWIc ''Dreamscape No. 2''] (2022)

; [[ZIA]]
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
      <sha1>761huguqyou22uadjstuef6uffyhakh</sha1>
    </revision>
    <revision>
      <id>117816</id>
      <parentid>117754</parentid>
      <timestamp>2023-06-30T14:32:11Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <minor/>
      <comment>/* Music */ Added music</comment>
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      <format>text/x-wiki</format>
      <text bytes="17135" sha1="2ul4ggagzpzz8kpt542dy7bkbzjq4b1" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Catrel|10edo tracks}}

; [[5 hideya]]
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)
* [https://www.youtube.com/watch?v=lkVooFr1vmk ''Like Yagi-bushi''] (2021)
* [https://www.youtube.com/watch?v=-L4VfJpYk-g ''Like Summer Creatures''] (2021)
* [https://www.youtube.com/watch?v=BCyhjlLVHl8 ''Like spring flowers''] (2023)

; [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

; [[Bazil Müzik]]
* [https://www.youtube.com/watch?v=OolR2oYiuyQ ''Lost Keys''] (2019)

; [[Evan Bennet]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

; [[Cenobyte]]
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

; Clem Fortuna ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

; [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

; Jama
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

; [[Aaron Krister Johnson]]
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

; Steve Martin
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

; [[Rick McGowan]]
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

; [[Sevish]]
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

; [[User:Userminusone|Userminusone]]
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Xotla]]
* [https://www.youtube.com/watch?v=DzKDz-CeI_E ''Phosphoresce''] (2017)

; Zewen Senpai
* [https://www.youtube.com/watch?v=3j7JLG-Iiuc ''Dreamscape No. 1''] (2022)
* [https://www.youtube.com/watch?v=ZVwDCluoWIc ''Dreamscape No. 2''] (2022)
* [https://www.youtube.com/watch?v=O-dSKT4JP64 ''Chromatic Etude for 10 EDO Piano No. 1 “Intervals”] (2023)

; [[ZIA]]
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
      <sha1>2ul4ggagzpzz8kpt542dy7bkbzjq4b1</sha1>
    </revision>
    <revision>
      <id>117817</id>
      <parentid>117816</parentid>
      <timestamp>2023-06-30T14:33:09Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
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      <text bytes="17137" sha1="0ab30ylta06qhkj2hyw2w2e0m46mlun" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Catrel|10edo tracks}}

; [[5 hideya]]
* [https://www.youtube.com/watch?v=ymLlBwO8XAY ''Like spring sea''] (2020)
* [https://www.youtube.com/watch?v=lkVooFr1vmk ''Like Yagi-bushi''] (2021)
* [https://www.youtube.com/watch?v=-L4VfJpYk-g ''Like Summer Creatures''] (2021)
* [https://www.youtube.com/watch?v=BCyhjlLVHl8 ''Like spring flowers''] (2023)

; [[User:Дмитрий Баженов|Dmitry Bazhenov]]
* [https://soundcloud.com/user-544568549/ey-ule-hey-ule ''Hey, ule!''] (second part only) (2020)
* [https://www.youtube.com/watch?v=RU1Cpe1Szo8 ''Струнка''] (''Strunka'') (2021)

; [[Bazil Müzik]]
* [https://www.youtube.com/watch?v=OolR2oYiuyQ ''Lost Keys''] (2019)

; [[Evan Bennet]]
* [https://www.youtube.com/watch?v=nbAsmGuPAuo&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=15 ''Microtonal Moment Musicale 2''] (2018)

; [[Cenobyte]]
* [[:File:10preview.ogg|10preview.ogg]] (2018)
* [[:File:decexperiment.ogg|decexperiment.ogg]] (2018)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/ideas-on-the-waterfall-of-expression "Ideas on the Waterfall of Expression"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)
* [https://cityoftheasleep.bandcamp.com/track/sad-mike-10edo "Sad Mike"], from [https://cityoftheasleep.bandcamp.com/album/cloiiids ''CLOIIIDS''] (2018)

; Clem Fortuna ([http://web.archive.org/web/20181106171816/http://clemfortuna.com/ website] (archived))
* [https://soundcloud.com/clem-fortuna/10tone ''10Tone''] (2015)

; [[Andrew Heathwaite]] and [[Chris Vaisvil]]
* [https://andrewheathwaite.bandcamp.com/track/shimmerwing "Shimmerwing"], from [https://andrewheathwaite.bandcamp.com/album/newbeams ''Newbeams''] (2012)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-10et "Prelude in 10ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/fugue-a3-in-10et "Fugue a3 in 10ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/fugue-in-10et SoundCloud])

; Jama
* [https://youtu.be/pKwbUL_bnpQ ''Zona del Ruido''] (2022)

; [[Aaron Krister Johnson]]
* [http://web.archive.org/web/20070613204725/http://www.akjmusic.com:80/audio/10_fantasy.ogg ''10 fantasy''] (≤2005)

; Steve Martin
* [https://soundcloud.com/martinsj013/sirmdbidnud2 ''Shall I Refuse My Dinner''] (2012)

; [[Rick McGowan]]
* [https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/RhinoPrelude-2003-02-22.mp3 ''Rhino Prelude''] (2003) ([https://web.archive.org/web/20161013092841/http://azuma-asobi.com/Music/index.html details])

; [[Carlo Serafini]]
* [http://www.seraph.it/dep/det/Blues10.mp3 ''Blues 10''] (2013) ([http://www.seraph.it/blog_files/dd7ee3a756851e812778e2fb222b5c47-168.html details])
* [http://www.seraph.it/dep/det/Waltz10.mp3 ''Waltz 10''] (2013) ([http://www.seraph.it/blog_files/875fbcf1467c4ed837ac1ba4b666d417-169.html details])
* [http://www.seraph.it/dep/det/Smooth10.mp3 ''Smooth 10''] (2013) ([http://www.seraph.it/blog_files/65405f5999f3960d5704b83c7318c673-170.html details])
* [http://www.seraph.it/dep/det/10PRS.mp3 ''10 PRS''] (2020) ([http://www.seraph.it/blog_files/be240f62b27fe8b738f44749e5f8c414-296.html details])

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/Ten_Fingers.mp3 "Ten Fingers"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)
* [https://sethares.engr.wisc.edu/mp3s/circleofthirds.html "Circle of Thirds"], from ''Xentonality'' (1997)

; [[Sevish]]
* "[[Vidya]]", from ''[[Harmony Hacker]]'' (2017)

; [[User:Userminusone|Userminusone]]
* [https://youtu.be/gZrD3gHUnnM ''Bit Crystals''] (2021)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/10edo/daily20110713_q49_10_viola_duet_and_gongs.mp3 ''For 2 Violas and Gongs in 10 edo''] (2011) ([http://chrisvaisvil.com/?p=1039 details])
* [http://micro.soonlabel.com/10edo/daily20110923-10edo_dramatic_squirrel_overture.mp3 ''The Dramatic Squirrel Overture''] (2011) ([http://chrisvaisvil.com/?p=1368 details])
* [http://micro.soonlabel.com/10edo/10_earwigs_invasive.mp3 ''10 Earwigs Invasive''] (2011) ([http://chrisvaisvil.com/?p=1397 details])

; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/fiat-circadia-10-edo ''Fiat Circadia''] (2018)
* [https://www.youtube.com/watch?v=GtoRNALtg3I&amp;list=PLdd4_BAP8pSsQHA8Sf3_wwTbVYSr9iJDy&amp;index=4 ''Three Weeping Tyrants''] (2020)

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/09.mp3 "9. 10 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Xotla]]
* [https://www.youtube.com/watch?v=DzKDz-CeI_E ''Phosphoresce''] (2017)

; Zewen Senpai
* [https://www.youtube.com/watch?v=3j7JLG-Iiuc ''Dreamscape No. 1''] (2022)
* [https://www.youtube.com/watch?v=ZVwDCluoWIc ''Dreamscape No. 2''] (2022)
* [https://www.youtube.com/watch?v=O-dSKT4JP64 ''Chromatic Etude for 10 EDO Piano No. 1 “Intervals”''] (2023)

; [[ZIA]]
* [https://ziaspace.bandcamp.com/track/sol-10edo "Sol"], from [https://ziaspace.bandcamp.com/album/zia-15-1992 ''ZIA 1.5''] (1992) ([http://www.ziaspace.com/ZIA/mp3s/Sol.html details])
* [https://ziaspace.bandcamp.com/track/agribusiness-10edo "Agribusiness"], from ''ZIA 1.5'' (1992) ([http://www.ziaspace.com/ZIA/mp3s/Agribusiness.html details])
* [https://ziaspace.bandcamp.com/track/future-10edo "Future"], from [https://ziaspace.bandcamp.com/album/shem-1996 ''SHEM''] (1996) ([http://www.ziaspace.com/ZIA/mp3s/Future.html details])
* [https://ziaspace.bandcamp.com/track/river-10edo "River"], from ''SHEM'' (1996) ([http://www.ziaspace.com/ZIA/mp3s/River.html details])
* [https://ziaspace.bandcamp.com/track/who-loves-you-me-10edo "Who Loves You.. Me?"], from [https://ziaspace.bandcamp.com/album/drum-n-space-2011 ''Drum 'n' Space''] (2011)
* [https://ziaspace.bandcamp.com/track/champagne-10edo "Champagne"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/love-is-the-catalyst-10edo "Love is the Catalyst"], from ''Drum 'n' Space'' (2011)
* [https://ziaspace.bandcamp.com/track/decagon-dancefloor-10edo "Decagon Dancefloor"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''Trapezoid''] (2019)
* [https://ziaspace.bandcamp.com/track/matter-over-mind-10edo "Matter Over Mind"], from [https://ziaspace.bandcamp.com/album/trapezoid-2019 ''No Terrestrial Road''] (2020)

'''[https://jonlervold.com/ Syzygy]'''

* [https://syzygybnr.bandcamp.com/album/escape "Escape"] (full album)

[[Category:Macrotonal]]
[[Category:Listen]]</text>
      <sha1>0ab30ylta06qhkj2hyw2w2e0m46mlun</sha1>
    </revision>
    <revision>
      <id>120477</id>
      <parentid>117817</parentid>
      <timestamp>2023-07-17T07:40:40Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
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      <text bytes="10591" sha1="qispzasd4yg0pv42qyg1rgpvi2dnjnc" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>qispzasd4yg0pv42qyg1rgpvi2dnjnc</sha1>
    </revision>
    <revision>
      <id>120896</id>
      <parentid>120477</parentid>
      <timestamp>2023-07-23T05:12:16Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Add 13-limit uniform maps</comment>
      <origin>120896</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10641" sha1="ah7qupdnssnc7xb3mjcbzcot7hy7ln5" xml:space="preserve">{{interwiki
| de = 10edo
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>ah7qupdnssnc7xb3mjcbzcot7hy7ln5</sha1>
    </revision>
    <revision>
      <id>124037</id>
      <parentid>120896</parentid>
      <timestamp>2023-09-13T02:47:49Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Match German wiki page title</comment>
      <origin>124037</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10642" sha1="np4xegdyu5zruwyrv2kg9hp86gy5o9h" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== JI approximation ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>np4xegdyu5zruwyrv2kg9hp86gy5o9h</sha1>
    </revision>
    <revision>
      <id>138869</id>
      <parentid>124037</parentid>
      <timestamp>2024-03-11T04:26:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* JI approximation */ -&gt; approximation to JI</comment>
      <origin>138869</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10645" sha1="lsw6eiqscmbfp5ag3wxpdiuq7onjdyu" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>lsw6eiqscmbfp5ag3wxpdiuq7onjdyu</sha1>
    </revision>
    <revision>
      <id>138990</id>
      <parentid>138869</parentid>
      <timestamp>2024-03-12T03:46:07Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <comment>/* Scales */</comment>
      <origin>138990</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11013" sha1="i4q0mfryc6aspleqj87pn5h7y707814" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Dicot[7]: 1 2 1 2 1 2 1
* Decimal/Lemba[8]: 2 1 1 1 2 1 1 1 1
* Negri[9]: 1 1 1 1 2 1 1 1 1

=== Other scales ===

* Marvel Augmented Hexatonic: 2 1 3 1 2 1 (subset of Negri[9])
* [[The Pinetone System#Pinetone%20octatonic%20scales|Pinetone major-harmonic/minor-harmonic octatonic]]: 1 2 1 1 1 1 2 1 (Decimal\Lemba[8] MODMOS)

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>i4q0mfryc6aspleqj87pn5h7y707814</sha1>
    </revision>
    <revision>
      <id>138991</id>
      <parentid>138990</parentid>
      <timestamp>2024-03-12T04:47:17Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <comment>/* Scales */Added Lemba[6]</comment>
      <origin>138991</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11045" sha1="ez0yr72nqpsxy4r9ttym2hesfyt2yoe" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6]: 2 2 1 2 2 1
* Dicot[7]: 1 2 1 2 1 2 1
* Decimal/Lemba[8]: 2 1 1 1 2 1 1 1 1
* Negri[9]: 1 1 1 1 2 1 1 1 1

=== Other scales ===

* Marvel Augmented Hexatonic: 2 1 3 1 2 1 (subset of Negri[9])
* [[The Pinetone System#Pinetone%20octatonic%20scales|Pinetone major-harmonic/minor-harmonic octatonic]]: 1 2 1 1 1 1 2 1 (Decimal\Lemba[8] MODMOS)

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>ez0yr72nqpsxy4r9ttym2hesfyt2yoe</sha1>
    </revision>
    <revision>
      <id>138992</id>
      <parentid>138991</parentid>
      <timestamp>2024-03-12T05:01:37Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <comment>/* Other scales */Added a scale</comment>
      <origin>138992</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11114" sha1="tfcktzrp7bo6npk5nbrz0pqssjpturj" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6]: 2 2 1 2 2 1
* Dicot[7]: 1 2 1 2 1 2 1
* Decimal/Lemba[8]: 2 1 1 1 2 1 1 1 1
* Negri[9]: 1 1 1 1 2 1 1 1 1

=== Other scales ===

* Enharmonic/chromatic trichord octave species: 3 1 2 3 1, 1 3 2 1 3
* Marvel Augmented Hexatonic: 2 1 3 1 2 1 (subset of Negri[9])
* [[The Pinetone System#Pinetone%20octatonic%20scales|Pinetone major-harmonic/minor-harmonic octatonic]]: 1 2 1 1 1 1 2 1 (Decimal\Lemba[8] MODMOS)

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>tfcktzrp7bo6npk5nbrz0pqssjpturj</sha1>
    </revision>
    <revision>
      <id>138995</id>
      <parentid>138992</parentid>
      <timestamp>2024-03-12T05:24:02Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <comment>/* Other scales */Added a scale</comment>
      <origin>138995</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11148" sha1="pl2t7rogf1zgxwsst74j2tjd6o7bxpg" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6]: 2 2 1 2 2 1
* Dicot[7]: 1 2 1 2 1 2 1
* Decimal/Lemba[8]: 2 1 1 1 2 1 1 1 1
* Negri[9]: 1 1 1 1 2 1 1 1 1

=== Other scales ===

* Enharmonic/chromatic trichord octave species: 3 1 2 3 1, 1 3 2 1 3
* Marvel Augmented Hexatonic: 2 1 3 1 2 1 (subset of Negri[9])
* 2 1 1 2 2 1 1 (Dicot[7] MODMOS)
* [[The Pinetone System#Pinetone%20octatonic%20scales|Pinetone major-harmonic/minor-harmonic octatonic]]: 1 2 1 1 1 1 2 1 (Decimal\Lemba[8] MODMOS)

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>pl2t7rogf1zgxwsst74j2tjd6o7bxpg</sha1>
    </revision>
    <revision>
      <id>139105</id>
      <parentid>138995</parentid>
      <timestamp>2024-03-13T07:42:20Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <comment>/* Other scales */</comment>
      <origin>139105</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11284" sha1="eu0j2jeya7x0smwrkjhrvvh1buhw026" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6]: 2 2 1 2 2 1
* Dicot[7]: 1 2 1 2 1 2 1
* Decimal/Lemba[8]: 2 1 1 1 2 1 1 1 1
* Negri[9]: 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]]: 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]]: 3 1 2 3 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1 
* Marvel Augmented Hexatonic: 2 1 3 1 2 1 (subset of Negri[9])
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1
* [[The Pinetone System#Pinetone%20octatonic%20scales|Pinetone major-harmonic/minor-harmonic octatonic]]: 1 2 1 1 1 1 2 1 (Decimal\Lemba[8] [[4M]])

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>eu0j2jeya7x0smwrkjhrvvh1buhw026</sha1>
    </revision>
    <revision>
      <id>139106</id>
      <parentid>139105</parentid>
      <timestamp>2024-03-13T07:45:00Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <minor/>
      <comment>/* Other scales */</comment>
      <origin>139106</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11326" sha1="bip0ey202h83y5omwldsf30mzngkxx1" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6]: 2 2 1 2 2 1
* Dicot[7]: 1 2 1 2 1 2 1
* Decimal/Lemba[8]: 2 1 1 1 2 1 1 1 1
* Negri[9]: 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1 
* Marvel Augmented Hexatonic (subset of Negri[9]): 2 1 3 1 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1
* [[The Pinetone System#Pinetone%20octatonic%20scales|Pinetone major-harmonic/minor-harmonic octatonic]] (Decimal\Lemba[8] [[4M]]): 1 2 1 1 1 1 2 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>bip0ey202h83y5omwldsf30mzngkxx1</sha1>
    </revision>
    <revision>
      <id>139161</id>
      <parentid>139106</parentid>
      <timestamp>2024-03-14T00:26:50Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <comment>/* Other scales */</comment>
      <origin>139161</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11408" sha1="9jml6o7du2kv3g33iiuidglsgrnwz7s" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6]: 2 2 1 2 2 1
* Dicot[7]: 1 2 1 2 1 2 1
* Decimal/Lemba[8]: 2 1 1 1 2 1 1 1 1
* Negri[9]: 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1 
* Marvel augmented hexatonic (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmomic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1
* [[The Pinetone System#Pinetone%20octatonic%20scales|Pinetone major-harmonic/minor-harmonic octatonic]] (Decimal\Lemba[8] [[4M]]): 1 2 1 1 1 1 2 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>9jml6o7du2kv3g33iiuidglsgrnwz7s</sha1>
    </revision>
    <revision>
      <id>139163</id>
      <parentid>139161</parentid>
      <timestamp>2024-03-14T00:49:21Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <comment>/* MOS scales */Added step signatures, got rid of fake mos</comment>
      <origin>139163</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11400" sha1="1tt0ffl9x2x04uaiyjjwcqkblhq33cj" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]]: 2 2 1 2 2 1
* Dicot[7] [[3L 4s]]: 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]]: 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1 
* Marvel augmented hexatonic (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmomic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1
* [[The Pinetone System#Pinetone%20octatonic%20scales|Pinetone major-harmonic/minor-harmonic octatonic]] (Decimal\Lemba[8] [[4M]]): 1 2 1 1 1 1 2 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>1tt0ffl9x2x04uaiyjjwcqkblhq33cj</sha1>
    </revision>
    <revision>
      <id>139164</id>
      <parentid>139163</parentid>
      <timestamp>2024-03-14T00:51:04Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <comment>/* Other scales */</comment>
      <origin>139164</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11374" sha1="oazudqxfpn86idtktsty87fhhr7xmpf" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]]: 2 2 1 2 2 1
* Dicot[7] [[3L 4s]]: 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]]: 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1 
* Marvel augmented hexatonic (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmomic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1
* [[The Pinetone System#Pinetone%20octatonic%20scales|Pinetone major-harmonic/minor-harmonic octatonic]]: 1 2 1 1 1 1 2 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>oazudqxfpn86idtktsty87fhhr7xmpf</sha1>
    </revision>
    <revision>
      <id>139171</id>
      <parentid>139164</parentid>
      <timestamp>2024-03-14T02:53:22Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <comment>/* Scales */ added gens to MOS scales, updated Other scales</comment>
      <origin>139171</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11327" sha1="b0dhe94re6a6d2fs92jgq0jgjcz7dx5" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1 
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>b0dhe94re6a6d2fs92jgq0jgjcz7dx5</sha1>
    </revision>
    <revision>
      <id>139172</id>
      <parentid>139171</parentid>
      <timestamp>2024-03-14T02:54:11Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <minor/>
      <comment>/* Other scales */ just reordered</comment>
      <origin>139172</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11326" sha1="qt9i2afhtdm7j7xe2k20qrvm1sz214l" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| [{{val| 10 16 23 }}]
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| [{{val| 10 16 23 28 }}]
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Limma, Pythagorean diatonic semitone
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Classic chromatic semitone, dicot comma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[15/14ths equal temperament|Linus]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>qt9i2afhtdm7j7xe2k20qrvm1sz214l</sha1>
    </revision>
    <revision>
      <id>165071</id>
      <parentid>139172</parentid>
      <timestamp>2024-11-12T12:40:32Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ style and comma names</comment>
      <origin>165071</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11323" sha1="8bw0ytpjh0oql0cffrc7zd78pa099cy" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.09
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.72
| 8.15
| 6.79
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>8bw0ytpjh0oql0cffrc7zd78pa099cy</sha1>
    </revision>
    <revision>
      <id>165076</id>
      <parentid>165071</parentid>
      <timestamp>2024-11-12T12:58:58Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ +2.3.5.7.13 subgroup</comment>
      <origin>165076</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11424" sha1="4pt6r9hak2z7dr2upl43nrcpm8rimlz" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]</text>
      <sha1>4pt6r9hak2z7dr2upl43nrcpm8rimlz</sha1>
    </revision>
    <revision>
      <id>166588</id>
      <parentid>165076</parentid>
      <timestamp>2024-11-17T01:51:57Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Todo</comment>
      <origin>166588</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11454" sha1="n275jsttw22nivc7c3cpmh2jop2o9o8" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form 1 2 1 2 1 2 1 ([[3L 4s|3L 4s - mosh]]), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]] and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>n275jsttw22nivc7c3cpmh2jop2o9o8</sha1>
    </revision>
    <revision>
      <id>166863</id>
      <parentid>166588</parentid>
      <timestamp>2024-11-18T15:08:00Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
      <origin>166863</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11653" sha1="9vu22i3zus41rglbwaq73rtsx6r6i2n" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

Rather surprisingly, 10edo is also the smallest EDO that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the harmonic series.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate Ratios&lt;ref&gt;based on treating 10edo as a 2.7.13.15 subgroup temperament&lt;/ref&gt;
! Additional Ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval Names
! colspan="3" | [[Ups and Downs Notation]]
! colspan="2" | [[3L 4s]] notation
!Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| unison
| C
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| minor second
| Db
|[[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| major second, diminished third
| D, Eb
|[[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| perfect/neutral third
| E
|[[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| minor fourth
| Fb
|[[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| major fourth, minor fifth
| F, Gb
|[[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| major fifth
| G
|[[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| perfect/neutral sixth
| A
|[[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| augmented sixth, minor seventh
| A#, Bb
|[[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| major seventh
| B
|[[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| octave
| C
|[[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>9vu22i3zus41rglbwaq73rtsx6r6i2n</sha1>
    </revision>
    <revision>
      <id>167153</id>
      <parentid>166863</parentid>
      <timestamp>2024-11-21T11:15:23Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Move mosh notation to its own section</comment>
      <origin>167153</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12149" sha1="mnius5bmx0scq5dyp1ncvkx75b4eif6" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==

10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. Taking the the 360 cent large neutral third as a generator produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most diatonic-like scale in 10edo excluding the 5edo degenerate diatonic scale. While not an integral or gap edo, it is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. One way to interpret it in terms of a temperament of just intonation is as a 2.7.13.15 subgroup, such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are tempered out. It can also be treated as a full 13-limit temperament, but it is a closer match to the aforementioned subgroup.

Rather surprisingly, 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the harmonic series.

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate ratios&lt;ref&gt;based on treating 10edo as a 2.15.7.13-subgroup temperament&lt;/ref&gt;
! Additional ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

== Notation ==
This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

=== 3L 4s (mosh) notation ===
The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>mnius5bmx0scq5dyp1ncvkx75b4eif6</sha1>
    </revision>
    <revision>
      <id>171563</id>
      <parentid>167153</parentid>
      <timestamp>2024-12-18T08:02:04Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ Add links &amp; line breaks. Reorder a sentence. Add the line “Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].”</comment>
      <origin>171563</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12278" sha1="n5qeq5qgyu756v4092a3n54g7w5wco1" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360 cent large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale. 

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate ratios&lt;ref&gt;based on treating 10edo as a 2.15.7.13-subgroup temperament&lt;/ref&gt;
! Additional ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

== Notation ==
This is the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

=== 3L 4s (mosh) notation ===
The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>n5qeq5qgyu756v4092a3n54g7w5wco1</sha1>
    </revision>
    <revision>
      <id>173622</id>
      <parentid>171563</parentid>
      <timestamp>2024-12-29T07:10:22Z</timestamp>
      <contributor>
        <username>Dave Keenan</username>
        <id>218</id>
      </contributor>
      <comment>/* Notation */ Added Sagittal notation</comment>
      <origin>173622</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13395" sha1="1r6zc9lua1o6dylbtway5gih7m5n7vz" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360 cent large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale. 

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate ratios&lt;ref&gt;based on treating 10edo as a 2.15.7.13-subgroup temperament&lt;/ref&gt;
! Additional ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

====Evo-SZ flavor====

&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

=== 3L 4s (mosh) notation ===
The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>1r6zc9lua1o6dylbtway5gih7m5n7vz</sha1>
    </revision>
    <revision>
      <id>173691</id>
      <parentid>173622</parentid>
      <timestamp>2024-12-29T13:18:43Z</timestamp>
      <contributor>
        <username>Dave Keenan</username>
        <id>218</id>
      </contributor>
      <comment>/* Sagittal notation */ added subsubheading "Evo and Revo flavors"</comment>
      <origin>173691</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13425" sha1="gg69my97a0ylhg1yxse23mj2cljbzji" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360 cent large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale. 

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate ratios&lt;ref&gt;based on treating 10edo as a 2.15.7.13-subgroup temperament&lt;/ref&gt;
! Additional ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated&lt;/u&gt;: D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic gencircles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic gencircles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...) (s = sub-, d = -oid)

'''&lt;u&gt;Heptatonic 3rd-generated&lt;/u&gt;: D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

====Evo and Revo flavors====

&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

====Evo-SZ flavor====

&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

=== 3L 4s (mosh) notation ===
The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>gg69my97a0ylhg1yxse23mj2cljbzji</sha1>
    </revision>
    <revision>
      <id>173953</id>
      <parentid>173691</parentid>
      <timestamp>2024-12-31T09:41:54Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>/* Intervals */ added EUs</comment>
      <origin>173953</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13683" sha1="fdlxslus6ouq0o2mh6qkgds34wg7h08" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360 cent large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale. 

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate ratios&lt;ref&gt;based on treating 10edo as a 2.15.7.13-subgroup temperament&lt;/ref&gt;
! Additional ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]
([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L 4s Mosh notation

=== 3L 4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

====Evo and Revo flavors====

&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

====Evo-SZ flavor====

&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]], Ampersand's comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>fdlxslus6ouq0o2mh6qkgds34wg7h08</sha1>
    </revision>
    <revision>
      <id>176889</id>
      <parentid>173953</parentid>
      <timestamp>2025-01-18T08:52:53Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ ampersand -&gt; ampersand comma to distinguish it from the temp</comment>
      <origin>176889</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13670" sha1="rvmh9gb62000a0r4f2b5ktt0x6780f6" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360 cent large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale. 

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate ratios&lt;ref&gt;based on treating 10edo as a 2.15.7.13-subgroup temperament&lt;/ref&gt;
! Additional ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]
([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L 4s Mosh notation

=== 3L 4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

====Evo and Revo flavors====

&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

====Evo-SZ flavor====

&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>rvmh9gb62000a0r4f2b5ktt0x6780f6</sha1>
    </revision>
    <revision>
      <id>176923</id>
      <parentid>176889</parentid>
      <timestamp>2025-01-18T20:20:02Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <origin>176923</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14066" sha1="9q5n6ltun63my2xyrbi9qz1kzrvg8ku" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360 cent large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale. 

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
! Degree
! Cents
! Approximate ratios&lt;ref&gt;based on treating 10edo as a 2.15.7.13-subgroup temperament&lt;/ref&gt;
! Additional ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]
([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L 4s Mosh notation

=== 3L 4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

====Evo and Revo flavors====

&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

====Evo-SZ flavor====

&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Zeta properties ==

=== Zeta peak index ===
{| class="wikitable"
! colspan="3" |Tuning
! colspan="3" |Strength
! colspan="2" |Closest EDO
! colspan="2" |Integer limit
|-
!ZPI
!Steps per octave
!Step size (cents)
!Height
!Integral
!Gap
!EDO
!Octave (cents)
!Consistent
!Distinct
|-
|[[26zpi]]
|10.0084563372591
|119.898609691954
|4.477141
|1.082282
|14.181485
|10edo
|1198.98609691954
|8
|5
|}
== Scales ==

=== MOS scales ===

* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===

* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>9q5n6ltun63my2xyrbi9qz1kzrvg8ku</sha1>
    </revision>
    <revision>
      <id>176996</id>
      <parentid>176923</parentid>
      <timestamp>2025-01-19T01:16:22Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Formatting</comment>
      <origin>176996</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14069" sha1="0kyitxxce93rgx4zhmf43p1ktszyyct" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360 cent large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale. 

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref&gt;based on treating 10edo as a 2.15.7.13-subgroup temperament&lt;/ref&gt;
! Additional ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L 4s Mosh notation

=== 3L 4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Zeta properties ==
=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[26zpi]]
| 10.0084563372591
| 119.898609691954
| 4.477141
| 1.082282
| 14.181485
| 10edo
| 1198.98609691954
| 8
| 5
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>0kyitxxce93rgx4zhmf43p1ktszyyct</sha1>
    </revision>
    <revision>
      <id>176997</id>
      <parentid>176996</parentid>
      <timestamp>2025-01-19T01:17:43Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176997</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14082" sha1="6wvslvqnyhwt8k1kk3ec95og8p999yw" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360 cent large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale. 

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref&gt;based on treating 10edo as a 2.15.7.13-subgroup temperament&lt;/ref&gt;
! Additional ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L 4s Mosh notation

=== 3L 4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Zeta properties ==
=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[26zpi]]
| 10.0084563372591
| 119.898609691954
| 4.477141
| 1.082282
| 14.181485
| 10edo
| 1198.98609691954
| 8
| 5
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>6wvslvqnyhwt8k1kk3ec95og8p999yw</sha1>
    </revision>
    <revision>
      <id>177088</id>
      <parentid>176997</parentid>
      <timestamp>2025-01-19T08:31:15Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Consolidate sections</comment>
      <origin>177088</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14071" sha1="r3nr0mv09bd9ip2ypfsc5n6kvd648at" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{EDO intro|10}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360 cent large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale. 

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref&gt;based on treating 10edo as a 2.15.7.13-subgroup temperament&lt;/ref&gt;
! Additional ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L 4s Mosh notation

=== 3L 4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[26zpi]]
| 10.0084563372591
| 119.898609691954
| 4.477141
| 1.082282
| 14.181485
| 10edo
| 1198.98609691954
| 8
| 5
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>r3nr0mv09bd9ip2ypfsc5n6kvd648at</sha1>
    </revision>
    <revision>
      <id>181313</id>
      <parentid>177088</parentid>
      <timestamp>2025-02-18T16:49:03Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181313</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14067" sha1="ciuzka7kgaljh71mns49qpnrb9iq9mj" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360 cent large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale. 

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref&gt;based on treating 10edo as a 2.15.7.13-subgroup temperament&lt;/ref&gt;
! Additional ratios &lt;br&gt; of 3, 5 and 9&lt;ref&gt;adding the ratios of 3, 5 and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

&lt;references /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L 4s Mosh notation

=== 3L 4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[26zpi]]
| 10.0084563372591
| 119.898609691954
| 4.477141
| 1.082282
| 14.181485
| 10edo
| 1198.98609691954
| 8
| 5
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L 2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L 4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L 8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 1L 7s, and 1L 8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L 1s, 1L 2s, 3L 1s, 3L 4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480 cents) exactly spans two octaves (480 × 5 = 2400), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be 0-2-2-1-0-0 (low to high), an "A" chord would be 0-0-2-2-1-0, and a "D" chord would be 1-0-0-2-2-1. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>ciuzka7kgaljh71mns49qpnrb9iq9mj</sha1>
    </revision>
    <revision>
      <id>181810</id>
      <parentid>181313</parentid>
      <timestamp>2025-02-20T15:34:21Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181810</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14183" sha1="k1371klt57sim17lox80z4pv9ezeydi" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale.

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s Mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[26zpi]]
| 10.0084563372591
| 119.898609691954
| 4.477141
| 1.082282
| 14.181485
| 10edo
| 1198.98609691954
| 8
| 5
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:Macrotonal]]
{{todo|add lumatone mapping}}</text>
      <sha1>k1371klt57sim17lox80z4pv9ezeydi</sha1>
    </revision>
    <revision>
      <id>182802</id>
      <parentid>181810</parentid>
      <timestamp>2025-02-23T13:29:13Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Recategorize</comment>
      <origin>182802</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14187" sha1="b5sb9r13r42t8tsfevkayy6zowxinob" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale.

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s Mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[26zpi]]
| 10.0084563372591
| 119.898609691954
| 4.477141
| 1.082282
| 14.181485
| 10edo
| 1198.98609691954
| 8
| 5
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]
{{Todo|add lumatone mapping}}</text>
      <sha1>b5sb9r13r42t8tsfevkayy6zowxinob</sha1>
    </revision>
    <revision>
      <id>187323</id>
      <parentid>182802</parentid>
      <timestamp>2025-03-20T15:15:38Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Zeta peak index */</comment>
      <origin>187323</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14032" sha1="copsfoz3ickczb9jylr3cjw2mtphvk6" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale.

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s Mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{{ZPI
| zpi = 26
| steps = 10.0084563372591
| step size = 119.898609691954
| tempered height = 4.477141
| pure height = 4.464866
| integral = 1.082282
| gap = 14.181485
| octave = 1198.98609691954
| consistent = 8
| distinct = 5
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|13|9.5|10.5}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]
{{Todo|add lumatone mapping}}</text>
      <sha1>copsfoz3ickczb9jylr3cjw2mtphvk6</sha1>
    </revision>
    <revision>
      <id>188766</id>
      <parentid>187323</parentid>
      <timestamp>2025-03-29T17:03:52Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>/* Uniform maps */ use new args</comment>
      <origin>188766</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14027" sha1="eyfvn7w1hupaszd7pcxtjyk852ybwyn" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale.

While not an integral or gap edo, 10edo is a [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s Mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{{ZPI
| zpi = 26
| steps = 10.0084563372591
| step size = 119.898609691954
| tempered height = 4.477141
| pure height = 4.464866
| integral = 1.082282
| gap = 14.181485
| octave = 1198.98609691954
| consistent = 8
| distinct = 5
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]
{{Todo|add lumatone mapping}}</text>
      <sha1>eyfvn7w1hupaszd7pcxtjyk852ybwyn</sha1>
    </revision>
    <revision>
      <id>190557</id>
      <parentid>188766</parentid>
      <timestamp>2025-04-10T13:01:12Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>190557</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14027" sha1="gtfr2bd72g82jt6vf27kkooe8sqt21w" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale.

While not an integral or gap edo, 10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s Mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{{ZPI
| zpi = 26
| steps = 10.0084563372591
| step size = 119.898609691954
| tempered height = 4.477141
| pure height = 4.464866
| integral = 1.082282
| gap = 14.181485
| octave = 1198.98609691954
| consistent = 8
| distinct = 5
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]
{{Todo|add lumatone mapping}}</text>
      <sha1>gtfr2bd72g82jt6vf27kkooe8sqt21w</sha1>
    </revision>
    <revision>
      <id>190929</id>
      <parentid>190557</parentid>
      <timestamp>2025-04-11T06:21:37Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>/* Rank-2 temperaments */ add line “  See [[regular temperament]] for information about what these are.”</comment>
      <origin>190929</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14094" sha1="4u8ei8xottl0v1sg6t1tnv6nzct1187" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale.

While not an integral or gap edo, 10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s Mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{{ZPI
| zpi = 26
| steps = 10.0084563372591
| step size = 119.898609691954
| tempered height = 4.477141
| pure height = 4.464866
| integral = 1.082282
| gap = 14.181485
| octave = 1198.98609691954
| consistent = 8
| distinct = 5
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
See [[regular temperament]] for information about what these are.

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]
{{Todo|add lumatone mapping}}</text>
      <sha1>4u8ei8xottl0v1sg6t1tnv6nzct1187</sha1>
    </revision>
    <revision>
      <id>191075</id>
      <parentid>190929</parentid>
      <timestamp>2025-04-11T10:00:58Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>No need to remind readers of what a regular temperament is everywhere</comment>
      <origin>191075</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14027" sha1="gtfr2bd72g82jt6vf27kkooe8sqt21w" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale.

While not an integral or gap edo, 10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s Mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{{ZPI
| zpi = 26
| steps = 10.0084563372591
| step size = 119.898609691954
| tempered height = 4.477141
| pure height = 4.464866
| integral = 1.082282
| gap = 14.181485
| octave = 1198.98609691954
| consistent = 8
| distinct = 5
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]
{{Todo|add lumatone mapping}}</text>
      <sha1>gtfr2bd72g82jt6vf27kkooe8sqt21w</sha1>
    </revision>
    <revision>
      <id>194612</id>
      <parentid>191075</parentid>
      <timestamp>2025-04-28T20:51:20Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Remove todo (Lumatone mapping page now exists)</comment>
      <origin>194612</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13997" sha1="6b2iwnrglbqhqg3kx099vdo26wt6c7g" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale.

While not an integral or gap edo, 10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s Mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{{ZPI
| zpi = 26
| steps = 10.0084563372591
| step size = 119.898609691954
| tempered height = 4.477141
| pure height = 4.464866
| integral = 1.082282
| gap = 14.181485
| octave = 1198.98609691954
| consistent = 8
| distinct = 5
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]</text>
      <sha1>6b2iwnrglbqhqg3kx099vdo26wt6c7g</sha1>
    </revision>
    <revision>
      <id>194764</id>
      <parentid>194612</parentid>
      <timestamp>2025-04-30T05:16:06Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Instruments */ link to Lumatone mapping</comment>
      <origin>194764</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14055" sha1="5grrarhgccn2qhia7pztbuoiz2irm1x" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents (or 5 circles of [[2edo]]). It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale.

While not an integral or gap edo, 10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]]. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

One way to interpret it in terms of a [[Temperament|temperament of just intonation]] is as a 2.7.13.15 [[subgroup]], such that [[105/104]], [[225/224]], [[43904/43875]], and [[16807/16384]] are [[tempered out]]. It can also be treated as a full [[13-limit]] temperament, but it is a closer match to the aforementioned subgroup.

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s Mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{{ZPI
| zpi = 26
| steps = 10.0084563372591
| step size = 119.898609691954
| tempered height = 4.477141
| pure height = 4.464866
| integral = 1.082282
| gap = 14.181485
| octave = 1198.98609691954
| consistent = 8
| distinct = 5
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]</text>
      <sha1>5grrarhgccn2qhia7pztbuoiz2irm1x</sha1>
    </revision>
    <revision>
      <id>198538</id>
      <parentid>194764</parentid>
      <timestamp>2025-06-02T22:39:39Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>198538</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15351" sha1="36x4fx4688kdrfbkik90sjy5uv3lctk" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale, and can be seen as a neutralized diatonic scale. 

It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all.  

This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360c as 11/9, we arrive at 11/8 = 600c (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.  

10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s Mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of mosh. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! Degree
! Cents
! Note
! Name
! Associated ratio
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for EDOs [[20edo#Sagittal notation|20]] and [[30edo#Sagittal notation|30]] and a superset of the notation for [[5edo#Sagittal notation|5-EDO]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{{ZPI
| zpi = 26
| steps = 10.0084563372591
| step size = 119.898609691954
| tempered height = 4.477141
| pure height = 4.464866
| integral = 1.082282
| gap = 14.181485
| octave = 1198.98609691954
| consistent = 8
| distinct = 5
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]</text>
      <sha1>36x4fx4688kdrfbkik90sjy5uv3lctk</sha1>
    </revision>
    <revision>
      <id>198571</id>
      <parentid>198538</parentid>
      <timestamp>2025-06-03T11:39:59Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Notation */ restore Neutral[7] since mosh has been given in the section title. Misc. cleanup</comment>
      <origin>198571</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15363" sha1="eqjgc37pfwhc22x2rh25c05tfgcoi27" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale, and can be seen as a neutralized diatonic scale. 

It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all.  

This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360c as 11/9, we arrive at 11/8 = 600c (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.  

10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{{ZPI
| zpi = 26
| steps = 10.0084563372591
| step size = 119.898609691954
| tempered height = 4.477141
| pure height = 4.464866
| integral = 1.082282
| gap = 14.181485
| octave = 1198.98609691954
| consistent = 8
| distinct = 5
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]</text>
      <sha1>eqjgc37pfwhc22x2rh25c05tfgcoi27</sha1>
    </revision>
    <revision>
      <id>199890</id>
      <parentid>198571</parentid>
      <timestamp>2025-06-18T00:10:21Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>199890</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15495" sha1="pqdgjuw489mipw43q1wh7e6umhheb4w" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale, and can be seen as a neutralized diatonic scale. 

It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all.  

This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360c as 11/9, we arrive at 11/8 = 600c (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.  

10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second, ''grade''
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third, ''lacuna''
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third, ''semitres''
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth, ''bilatus''
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone, ''median''
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth, ''trilatus''
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth, ''semisept''
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh, ''antilatus''
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh, ''degrade''
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave, ''duplance''
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{{ZPI
| zpi = 26
| steps = 10.0084563372591
| step size = 119.898609691954
| tempered height = 4.477141
| pure height = 4.464866
| integral = 1.082282
| gap = 14.181485
| octave = 1198.98609691954
| consistent = 8
| distinct = 5
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]</text>
      <sha1>pqdgjuw489mipw43q1wh7e6umhheb4w</sha1>
    </revision>
    <revision>
      <id>199893</id>
      <parentid>199890</parentid>
      <timestamp>2025-06-18T03:58:03Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Undo revision [[Special:Diff/199890|199890]] by [[Special:Contributions/VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]])</comment>
      <origin>199893</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15363" sha1="eqjgc37pfwhc22x2rh25c05tfgcoi27" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale, and can be seen as a neutralized diatonic scale. 

It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all.  

This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360c as 11/9, we arrive at 11/8 = 600c (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.  

10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

=== Zeta peak index ===
{{ZPI
| zpi = 26
| steps = 10.0084563372591
| step size = 119.898609691954
| tempered height = 4.477141
| pure height = 4.464866
| integral = 1.082282
| gap = 14.181485
| octave = 1198.98609691954
| consistent = 8
| distinct = 5
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]</text>
      <sha1>eqjgc37pfwhc22x2rh25c05tfgcoi27</sha1>
    </revision>
    <revision>
      <id>207036</id>
      <parentid>199893</parentid>
      <timestamp>2025-08-15T23:58:47Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>/* Approximation to JI */ -zeta peak index</comment>
      <origin>207036</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15106" sha1="nw7bqryr1avqg5n9yb57tba2uli2naj" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale, and can be seen as a neutralized diatonic scale. 

It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all.  

This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360c as 11/9, we arrive at 11/8 = 600c (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.  

10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br /&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]] / [[blacksmith]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]</text>
      <sha1>nw7bqryr1avqg5n9yb57tba2uli2naj</sha1>
    </revision>
    <revision>
      <id>209780</id>
      <parentid>207036</parentid>
      <timestamp>2025-09-11T13:26:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Septimal blackwood is now canon</comment>
      <origin>209780</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15063" sha1="rmkd8uza11xiu36u5oj6syr6zne0t2g" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scales|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale, and can be seen as a neutralized diatonic scale. 

It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all.  

This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360c as 11/9, we arrive at 11/8 = 600c (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.  

10edo is a [[The Riemann zeta function and tuning #Zeta edo lists|zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{Mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{Mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{Mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[Pinetone #Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[Pinetone #Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]</text>
      <sha1>rmkd8uza11xiu36u5oj6syr6zne0t2g</sha1>
    </revision>
    <revision>
      <id>209781</id>
      <parentid>209780</parentid>
      <timestamp>2025-09-11T13:27:26Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ duh</comment>
      <origin>209781</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15017" sha1="rvws4a4kiyjcufzdodlwixvlrto6mzp" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale, and can be seen as a neutralized diatonic scale. 

It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all.  

This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.  

10edo is a [[zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{Mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{Mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{Mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]]
|}

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[Pinetone #Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[Pinetone #Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]</text>
      <sha1>rvws4a4kiyjcufzdodlwixvlrto6mzp</sha1>
    </revision>
    <revision>
      <id>209986</id>
      <parentid>209781</parentid>
      <timestamp>2025-09-16T08:21:27Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Temporary until full version is done</comment>
      <origin>209986</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15391" sha1="gwrtyxlpxswlkobkp0zfskqze1dl182" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale, and can be seen as a neutralized diatonic scale. 

It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all.  

This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.  

10edo is a [[zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{Mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{Mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{Mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]]
|}

== Octave stretch or compression ==
If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from [[octave shrinking]]. [[zpi|26zpi]] and [[36ed12]] are compressed-octave versions of 10edo.

If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from [[octave stretching]]. [[ed7|28ed7]] is a stretched version of 10edo.

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[Pinetone #Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[Pinetone #Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== Notes ==
&lt;references group="note" /&gt;

[[Category:10-tone scales]]</text>
      <sha1>gwrtyxlpxswlkobkp0zfskqze1dl182</sha1>
    </revision>
    <revision>
      <id>212429</id>
      <parentid>209986</parentid>
      <timestamp>2025-10-09T10:22:34Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>Move notes to appear under the corresponding tables</comment>
      <origin>212429</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15378" sha1="hv7e5gzgibj7gnct1stp5t21lpetrzw" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale, and can be seen as a neutralized diatonic scale. 

It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all.  

This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.  

10edo is a [[zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}
&lt;references group="note" /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{Mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{Mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{Mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]]
|}

== Octave stretch or compression ==
If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from [[octave shrinking]]. [[zpi|26zpi]] and [[36ed12]] are compressed-octave versions of 10edo.

If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from [[octave stretching]]. [[ed7|28ed7]] is a stretched version of 10edo.

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[Pinetone #Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[Pinetone #Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:10-tone scales]]</text>
      <sha1>hv7e5gzgibj7gnct1stp5t21lpetrzw</sha1>
    </revision>
    <revision>
      <id>213190</id>
      <parentid>212429</parentid>
      <timestamp>2025-10-15T09:19:56Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Fix ref error</comment>
      <origin>213190</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15405" sha1="mrnwvl1ez3d2ck799cu6otxyp0yu0dc" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo can be thought of as two circles of [[5edo]] separated by 120 cents. It adds to 5edo a small neutral second (or large minor 2nd) and its inversion a large neutral seventh (or small major 7th); an excellent approximation of [[13/8]] and its inversion [[16/13]]; and the familiar 600-cent tritone that appears in every even-numbered edo. 

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|moment of symmetry scale]] of the form {{nowrap|1 2 1 2 1 2 1}} ([[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo degenerate diatonic scale, and can be seen as a neutralized diatonic scale. 

It shares [[5edo]]'s approximation quality in the 2.3.7 subgroup (though the detuned fifth could be seen as a bigger problem with the more fine division of steps), but expands on its accuracy in the full 7-limit, by including a better approximation of 5/4 at 360 cents, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, [[6/5]] is very poorly approximated, over 40 cents sharp, due to to the errors on 3/2 and 5/4 compounding. In fact, it is mapped to the exact same interval as 5/4, which results in the [[dicot]] exotemperament. So, if one wishes to represent JI with 10edo, it is best to use 5 carefully or not at all.  

This third also serves as an extremely accurate approximation of [[16/13]], making 10edo usable as a 2.3.5.7.13 temperament, in which, alongside 5edo's temperaments in 2.3.7, septimal supermajor intervals are equated with tridecimal ultramajor intervals (tempering out [[105/104]]), and 5-limit major and minor thirds are equated as mentioned before (tempering out [[25/24]]). Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and from this it can be seen that the syntonic comma is mapped to 120 cents. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate to the no-11 subgroup.  

10edo is a [[zeta peak edo]], due to its decent tuning of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}
&lt;references group="note" /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{Mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{Mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{Mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references group="note"/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]]
|}

== Octave stretch or compression ==
If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from [[octave shrinking]]. [[zpi|26zpi]] and [[36ed12]] are compressed-octave versions of 10edo.

If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from [[octave stretching]]. [[ed7|28ed7]] is a stretched version of 10edo.

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[Pinetone #Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[Pinetone #Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:10-tone scales]]</text>
      <sha1>mrnwvl1ez3d2ck799cu6otxyp0yu0dc</sha1>
    </revision>
    <revision>
      <id>223906</id>
      <parentid>213190</parentid>
      <timestamp>2026-02-13T01:49:07Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Theory */ Add archetype paragraph; overall rewrite</comment>
      <origin>223906</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17204" sha1="kaklo3mgiesilnogen1gk8auy8gkca9" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo contains all the intervals of [[5edo]], but also adds another copy of it separated by 120 [[cent]]s. The new intervals have sizes of 120{{c}}, 360{{c}}, 600{{c}}, 840{{c}}, and 1080{{c}}. The 120{{c}} interval can be treated a small neutral second or large minor 2nd, and its inversion a large neutral seventh or small major 7th, with the 120{{c}} and 1080{{c}} intervals being close (about 0.6{{c}} off) to [[15/14]] and [[28/15]] respectively. The 360{{c}} interval is a large neutral third, being about 0.5{{c}} sharp of [[16/13]], with its inversion being equally close to [[13/8]]. Finally, the 600{{c}} interva is the tritone that appears in every even-numbered edo, including [[12edo]].

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|moment of symmetry scale]] with step sizes {{nowrap|2 1 1 2 1 2 1}} (pattern [[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo [[collapsed]] diatonic scale, and can be seen as a [[neutralized]] diatonic scale. 

It shares [[5edo]]'s approximation quality in the [[2.3.7 subgroup]], though the detuned fifth could be seen as a bigger problem with the more fine division of steps. Compared to 5edo, 10edo attains more accuracy in the full [[7-limit]], by including a better approximation of [[5/4]] at 360{{c}}, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, the approximation to 5/4 is still over 25{{c}} flat, and this interval is also equated with [[6/5]] (which is even more inaccurate, at 44{{c}} sharp), tempering out [[25/24]] and resulting in the [[dicot]] exotemperament. Thus, if one wishes to represent JI with 10edo, it is best to use prime [[5/1|5]] carefully or not at all.

Even if 10edo isn't directly used to represent JI, it could still serve as a structural archetype for the 7-limit. The fact that 25/24 is tempered out means that the 5-limit major triad [[4:5:6]] and minor triad [[10:12:15|1/(6:5:4)]] are mapped to the same number of scale steps in the 10-form, a feature shared with [[7edo]] and the [[heptatonic]] system used in western music. 10edo additionally sends [[49/48]] to the unison, meaning the 7-limit triad [[4:6:7]] and its inverse [[14:21:24|1/(12:8:7)]] are the same number of scale steps in a decatonic system as well, and therefore also the [[4:5:6:7]] major and [[70:84:105:120|1/(12:10:8:7)]] minor tetrads as well. Tempering out 25/24 and 49/48 leads to the [[decimal]] exotemperament (which is named after 10edo). A more accurate temperament based off of the 10-form that doesn't temper out 25/24 or 49/48 is [[pajara]], which shares many desireable properties with diatonic&lt;ref&gt;Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]&lt;/ref&gt;.

Since the neutral third is very close to 16/13, 10edo is usable as a 2.3.5.7.13 temperament,  which includes 5edo's representation of 2.3.7; however, it is not without high damage. For one, all of [[9/7]], [[13/10]], [[21/16]], and [[4/3]] are equated to a flat fourth (or an extremely sharp supermajor third), tempering out [[28/27]], [[40/39]], [[49/48]], [[64/63]], [[91/90]], and [[105/104]]. Also, 5-limit major and minor thirds are equated as mentioned before (tempering out 25/24), and the third is also equated to 16/13, tempering out 40/39 and [[65/64]]. Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and since 3/2 is tuned sharp and 5/4 is tuned flat, the syntonic comma is exaggerated to a full step, or 120{{c}}. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate as a no-11 system.  

10edo is a [[zeta peak edo]], due to its relatively decent tunings of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}
&lt;references group="note" /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{Mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{Mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{Mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references group="note"/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] / [[neutral]]
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]]
|}

== Octave stretch or compression ==
If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from [[octave shrinking]]. [[zpi|26zpi]] and [[36ed12]] are compressed-octave versions of 10edo.

If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from [[octave stretching]]. [[ed7|28ed7]] is a stretched version of 10edo.

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[Pinetone #Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[Pinetone #Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:10-tone scales]]</text>
      <sha1>kaklo3mgiesilnogen1gk8auy8gkca9</sha1>
    </revision>
    <revision>
      <id>223907</id>
      <parentid>223906</parentid>
      <timestamp>2026-02-13T01:50:54Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Rank-2 temperaments */ these can be tuned much better in other systems</comment>
      <origin>223907</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17232" sha1="4j8jpj1dtcpd57ii6kr18rq00ugs176" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo contains all the intervals of [[5edo]], but also adds another copy of it separated by 120 [[cent]]s. The new intervals have sizes of 120{{c}}, 360{{c}}, 600{{c}}, 840{{c}}, and 1080{{c}}. The 120{{c}} interval can be treated a small neutral second or large minor 2nd, and its inversion a large neutral seventh or small major 7th, with the 120{{c}} and 1080{{c}} intervals being close (about 0.6{{c}} off) to [[15/14]] and [[28/15]] respectively. The 360{{c}} interval is a large neutral third, being about 0.5{{c}} sharp of [[16/13]], with its inversion being equally close to [[13/8]]. Finally, the 600{{c}} interva is the tritone that appears in every even-numbered edo, including [[12edo]].

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|moment of symmetry scale]] with step sizes {{nowrap|2 1 1 2 1 2 1}} (pattern [[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo [[collapsed]] diatonic scale, and can be seen as a [[neutralized]] diatonic scale. 

It shares [[5edo]]'s approximation quality in the [[2.3.7 subgroup]], though the detuned fifth could be seen as a bigger problem with the more fine division of steps. Compared to 5edo, 10edo attains more accuracy in the full [[7-limit]], by including a better approximation of [[5/4]] at 360{{c}}, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, the approximation to 5/4 is still over 25{{c}} flat, and this interval is also equated with [[6/5]] (which is even more inaccurate, at 44{{c}} sharp), tempering out [[25/24]] and resulting in the [[dicot]] exotemperament. Thus, if one wishes to represent JI with 10edo, it is best to use prime [[5/1|5]] carefully or not at all.

Even if 10edo isn't directly used to represent JI, it could still serve as a structural archetype for the 7-limit. The fact that 25/24 is tempered out means that the 5-limit major triad [[4:5:6]] and minor triad [[10:12:15|1/(6:5:4)]] are mapped to the same number of scale steps in the 10-form, a feature shared with [[7edo]] and the [[heptatonic]] system used in western music. 10edo additionally sends [[49/48]] to the unison, meaning the 7-limit triad [[4:6:7]] and its inverse [[14:21:24|1/(12:8:7)]] are the same number of scale steps in a decatonic system as well, and therefore also the [[4:5:6:7]] major and [[70:84:105:120|1/(12:10:8:7)]] minor tetrads as well. Tempering out 25/24 and 49/48 leads to the [[decimal]] exotemperament (which is named after 10edo). A more accurate temperament based off of the 10-form that doesn't temper out 25/24 or 49/48 is [[pajara]], which shares many desireable properties with diatonic&lt;ref&gt;Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]&lt;/ref&gt;.

Since the neutral third is very close to 16/13, 10edo is usable as a 2.3.5.7.13 temperament,  which includes 5edo's representation of 2.3.7; however, it is not without high damage. For one, all of [[9/7]], [[13/10]], [[21/16]], and [[4/3]] are equated to a flat fourth (or an extremely sharp supermajor third), tempering out [[28/27]], [[40/39]], [[49/48]], [[64/63]], [[91/90]], and [[105/104]]. Also, 5-limit major and minor thirds are equated as mentioned before (tempering out 25/24), and the third is also equated to 16/13, tempering out 40/39 and [[65/64]]. Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and since 3/2 is tuned sharp and 5/4 is tuned flat, the syntonic comma is exaggerated to a full step, or 120{{c}}. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate as a no-11 system.  

10edo is a [[zeta peak edo]], due to its relatively decent tunings of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}
&lt;references group="note" /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{Mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{Mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{Mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references group="note"/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] (out-of-tune) / [[neutral]] (out-of-tune)
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]]
|}

== Octave stretch or compression ==
If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from [[octave shrinking]]. [[zpi|26zpi]] and [[36ed12]] are compressed-octave versions of 10edo.

If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from [[octave stretching]]. [[ed7|28ed7]] is a stretched version of 10edo.

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[Pinetone #Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[Pinetone #Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

[[Category:10-tone scales]]</text>
      <sha1>4j8jpj1dtcpd57ii6kr18rq00ugs176</sha1>
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    <revision>
      <id>223908</id>
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      <timestamp>2026-02-13T01:51:43Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
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| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
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{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo contains all the intervals of [[5edo]], but also adds another copy of it separated by 120 [[cent]]s. The new intervals have sizes of 120{{c}}, 360{{c}}, 600{{c}}, 840{{c}}, and 1080{{c}}. The 120{{c}} interval can be treated a small neutral second or large minor 2nd, and its inversion a large neutral seventh or small major 7th, with the 120{{c}} and 1080{{c}} intervals being close (about 0.6{{c}} off) to [[15/14]] and [[28/15]] respectively. The 360{{c}} interval is a large neutral third, being about 0.5{{c}} sharp of [[16/13]], with its inversion being equally close to [[13/8]]. Finally, the 600{{c}} interva is the tritone that appears in every even-numbered edo, including [[12edo]].

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|moment of symmetry scale]] with step sizes {{nowrap|2 1 1 2 1 2 1}} (pattern [[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo [[collapsed]] diatonic scale, and can be seen as a [[neutralized]] diatonic scale. 

It shares [[5edo]]'s approximation quality in the [[2.3.7 subgroup]], though the detuned fifth could be seen as a bigger problem with the more fine division of steps. Compared to 5edo, 10edo attains more accuracy in the full [[7-limit]], by including a better approximation of [[5/4]] at 360{{c}}, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, the approximation to 5/4 is still over 25{{c}} flat, and this interval is also equated with [[6/5]] (which is even more inaccurate, at 44{{c}} sharp), tempering out [[25/24]] and resulting in the [[dicot]] exotemperament. Thus, if one wishes to represent JI with 10edo, it is best to use prime [[5/1|5]] carefully or not at all.

Even if 10edo isn't directly used to represent JI, it could still serve as a structural archetype for the 7-limit. The fact that 25/24 is tempered out means that the 5-limit major triad [[4:5:6]] and minor triad [[10:12:15|1/(6:5:4)]] are mapped to the same number of scale steps in the 10-form, a feature shared with [[7edo]] and the [[heptatonic]] system used in western music. 10edo additionally sends [[49/48]] to the unison, meaning the 7-limit triad [[4:6:7]] and its inverse [[14:21:24|1/(12:8:7)]] are the same number of scale steps in a decatonic system as well, and therefore also the [[4:5:6:7]] major and [[70:84:105:120|1/(12:10:8:7)]] minor tetrads as well. Tempering out 25/24 and 49/48 leads to the [[decimal]] exotemperament (which is named after 10edo). A more accurate temperament based off of the 10-form that doesn't temper out 25/24 or 49/48 is [[pajara]], which shares many desireable properties with diatonic&lt;ref&gt;Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]&lt;/ref&gt;.

Since the neutral third is very close to 16/13, 10edo is usable as a 2.3.5.7.13 temperament,  which includes 5edo's representation of 2.3.7; however, it is not without high damage. For one, all of [[9/7]], [[13/10]], [[21/16]], and [[4/3]] are equated to a flat fourth (or an extremely sharp supermajor third), tempering out [[28/27]], [[40/39]], [[49/48]], [[64/63]], [[91/90]], and [[105/104]]. Also, 5-limit major and minor thirds are equated as mentioned before (tempering out 25/24), and the third is also equated to 16/13, tempering out 40/39 and [[65/64]]. Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and since 3/2 is tuned sharp and 5/4 is tuned flat, the syntonic comma is exaggerated to a full step, or 120{{c}}. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate as a no-11 system.  

10edo is a [[zeta peak edo]], due to its relatively decent tunings of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}
&lt;references group="note" /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{Mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{Mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{Mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references group="note"/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] (out-of-tune) / [[neutral]] (out-of-tune)
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]]
|}

== Octave stretch or compression ==
If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from [[octave shrinking]]. [[zpi|26zpi]] and [[36ed12]] are compressed-octave versions of 10edo.

If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from [[octave stretching]]. [[ed7|28ed7]] is a stretched version of 10edo.

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[Pinetone #Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[Pinetone #Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== References ==
&lt;references/&gt;

[[Category:10-tone scales]]</text>
      <sha1>8jm1e9dkedd76nbsz3hkbj71gds2kio</sha1>
    </revision>
    <revision>
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      <contributor>
        <username>Squib</username>
        <id>562</id>
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      <text bytes="17280" sha1="rj6jdlr14ckpdpdrgcnqpngeb2qqfet" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo contains all the intervals of [[5edo]], but also adds another copy of it separated by 120 [[cent]]s. The new intervals have sizes of 120{{c}}, 360{{c}}, 600{{c}}, 840{{c}}, and 1080{{c}}. The 120{{c}} interval can be treated a small neutral second or large minor 2nd, and its inversion a large neutral seventh or small major 7th, with the 120{{c}} and 1080{{c}} intervals being close (about 0.6{{c}} off) to [[15/14]] and [[28/15]] respectively. The 360{{c}} interval is a large neutral third, being about 0.5{{c}} sharp of [[16/13]], with its inversion being equally close to [[13/8]]. Finally, the 600{{c}} interva is the tritone that appears in every even-numbered edo, including [[12edo]].

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|moment of symmetry scale]] with step sizes {{nowrap|2 1 1 2 1 2 1}} (pattern [[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo [[collapsed]] diatonic scale, and can be seen as a [[neutralized]] diatonic scale. 

It shares [[5edo]]'s approximation quality in the [[2.3.7 subgroup]], though the detuned fifth could be seen as a bigger problem with the more fine division of steps. Compared to 5edo, 10edo attains more accuracy in the full [[7-limit]], by including a better approximation of [[5/4]] at 360{{c}}, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, the approximation to 5/4 is still over 25{{c}} flat, and this interval is also equated with [[6/5]] (which is even more inaccurate, at 44{{c}} sharp), tempering out [[25/24]] and resulting in the [[dicot]] exotemperament. Thus, if one wishes to represent JI with 10edo, it is best to use prime [[5/1|5]] carefully or not at all.

Even if 10edo isn't directly used to represent JI, it could still serve as a structural archetype for the 7-limit. The fact that 25/24 is tempered out means that the 5-limit major triad [[4:5:6]] and minor triad [[10:12:15|1/(6:5:4)]] are mapped to the same number of scale steps in the 10-form, a feature shared with [[7edo]] and the [[heptatonic]] system used in western music. 10edo additionally sends [[49/48]] to the unison, meaning the 7-limit triad [[4:6:7]] and its inverse [[14:21:24|1/(12:8:7)]] are the same number of scale steps in a decatonic system as well, and therefore also the [[4:5:6:7]] major and [[70:84:105:120|1/(12:10:8:7)]] minor tetrads as well. Tempering out 25/24 and 49/48 leads to the [[decimal]] exotemperament (which is named after 10edo). A more accurate temperament based off of the 10-form that doesn't temper out 25/24 or 49/48 is [[pajara]], which shares many desireable properties with diatonic&lt;ref&gt;Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]&lt;/ref&gt;.

Since the neutral third is very close to 16/13, 10edo is usable as a 2.3.5.7.13 temperament,  which includes 5edo's representation of 2.3.7; however, it is not without high damage. For one, all of [[9/7]], [[13/10]], [[21/16]], and [[4/3]] are equated to a flat fourth (or an extremely sharp supermajor third), tempering out [[28/27]], [[40/39]], [[49/48]], [[64/63]], [[91/90]], and [[105/104]]. Also, 5-limit major and minor thirds are equated as mentioned before (tempering out 25/24), and the third is also equated to 16/13, tempering out 40/39 and [[65/64]]. Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and since 3/2 is tuned sharp and 5/4 is tuned flat, the syntonic comma is exaggerated to a full step, or 120{{c}}. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]] and [[243/242]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate as a no-11 system.  

10edo is a [[zeta peak edo]], due to its relatively decent tunings of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}
&lt;references group="note" /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{Mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{Mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{Mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references group="note"/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] (out-of-tune) / [[neutral]] (out-of-tune)
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]]
|}

== Octave stretch or compression ==
If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from [[octave shrinking]]. [[zpi|26zpi]] and [[36ed12]] are compressed-octave versions of 10edo.

If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from [[octave stretching]]. [[ed7|28ed7]] is a stretched version of 10edo.

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[Pinetone #Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[Pinetone #Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== References ==
&lt;references/&gt;

[[Category:10-tone scales]]</text>
      <sha1>rj6jdlr14ckpdpdrgcnqpngeb2qqfet</sha1>
    </revision>
    <revision>
      <id>225644</id>
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      <contributor>
        <username>MisterShafXen</username>
        <id>509</id>
      </contributor>
      <minor/>
      <comment>Fixed typo</comment>
      <origin>225644</origin>
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      <text bytes="17281" sha1="1qax1wwemc0v11bv6enzfj4p6ay4day" xml:space="preserve">{{interwiki
| de = 10-EDO
| en = 10edo 
| es = 
| ja = 10平均律
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
10edo contains all the intervals of [[5edo]], but also adds another copy of it separated by 120 [[cent]]s. The new intervals have sizes of 120{{c}}, 360{{c}}, 600{{c}}, 840{{c}}, and 1080{{c}}. The 120{{c}} interval can be treated a small neutral second or large minor 2nd, and its inversion a large neutral seventh or small major 7th, with the 120{{c}} and 1080{{c}} intervals being close (about 0.6{{c}} off) to [[15/14]] and [[28/15]] respectively. The 360{{c}} interval is a large neutral third, being about 0.5{{c}} sharp of [[16/13]], with its inversion being equally close to [[13/8]]. Finally, the 600{{c}} interval is the tritone that appears in every even-numbered edo, including [[12edo]].

Taking the the 360{{c}} large neutral third as a [[generator]] produces a heptatonic [[MOS scale|moment of symmetry scale]] with step sizes {{nowrap|2 1 1 2 1 2 1}} (pattern [[3L&amp;nbsp;4s]], or "mosh"), which is the most [[Diatonic scale|diatonic]]-like scale in 10edo excluding the 5edo [[collapsed]] diatonic scale, and can be seen as a [[neutralized]] diatonic scale. 

It shares [[5edo]]'s approximation quality in the [[2.3.7 subgroup]], though the detuned fifth could be seen as a bigger problem with the more fine division of steps. Compared to 5edo, 10edo attains more accuracy in the full [[7-limit]], by including a better approximation of [[5/4]] at 360{{c}}, resulting in the better tuning of various intervals including 5, such as [[16/15]] and [[7/5]]. However, the approximation to 5/4 is still over 25{{c}} flat, and this interval is also equated with [[6/5]] (which is even more inaccurate, at 44{{c}} sharp), tempering out [[25/24]] and resulting in the [[dicot]] exotemperament. Thus, if one wishes to represent JI with 10edo, it is best to use prime [[5/1|5]] carefully or not at all.

Even if 10edo isn't directly used to represent JI, it could still serve as a structural archetype for the 7-limit. The fact that 25/24 is tempered out means that the 5-limit major triad [[4:5:6]] and minor triad [[10:12:15|1/(6:5:4)]] are mapped to the same number of scale steps in the 10-form, a feature shared with [[7edo]] and the [[heptatonic]] system used in western music. 10edo additionally sends [[49/48]] to the unison, meaning the 7-limit triad [[4:6:7]] and its inverse [[14:21:24|1/(12:8:7)]] are the same number of scale steps in a decatonic system as well, and therefore also the [[4:5:6:7]] major and [[70:84:105:120|1/(12:10:8:7)]] minor tetrads as well. Tempering out 25/24 and 49/48 leads to the [[decimal]] exotemperament (which is named after 10edo). A more accurate temperament based off of the 10-form that doesn't temper out 25/24 or 49/48 is [[pajara]], which shares many desireable properties with diatonic&lt;ref&gt;Erlich, Paul. "Tuning, Tonality and 22-Tone Temperament." Xenharmonicon 17, 1998. [http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf http://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf]&lt;/ref&gt;.

Since the neutral third is very close to 16/13, 10edo is usable as a 2.3.5.7.13 temperament,  which includes 5edo's representation of 2.3.7; however, it is not without high damage. For one, all of [[9/7]], [[13/10]], [[21/16]], and [[4/3]] are equated to a flat fourth (or an extremely sharp supermajor third), tempering out [[28/27]], [[40/39]], [[49/48]], [[64/63]], [[91/90]], and [[105/104]]. Also, 5-limit major and minor thirds are equated as mentioned before (tempering out 25/24), and the third is also equated to 16/13, tempering out 40/39 and [[65/64]]. Additionally, 5-limit augmented and diminished intervals are equated with nearby septimal intervals (tempering out [[225/224]]), and since 3/2 is tuned sharp and 5/4 is tuned flat, the syntonic comma is exaggerated to a full step, or 120{{c}}. More accurately, it can be seen as a 2.7.13.15 temperament, restricting the 3.5 subgroup to powers of 15.  

By treating 360{{c}} as 11/9, we arrive at 11/8 = 600{{c}} (tempering out [[144/143]] and [[243/242]]), which allows 10edo to be treated as a full [[13-limit]] temperament. However, it is more accurate as a no-11 system.  

10edo is a [[zeta peak edo]], due to its relatively decent tunings of the harmonics 2, 3, 5, 7, 13, and 17. 10edo is also the smallest edo that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

Thanks to its sevenths, 10edo is an ideal tuning for its size for [[metallic harmony]].

=== Prime harmonics ===
{{Harmonics in equal|10}}

== Intervals ==
{| class="wikitable right-1 right-2 center-7 center-8"
|-
! Degree
! Cents
! Approximate ratios&lt;ref group="note"&gt;{{sg|limit=2.15.7.13-subgroup}}&lt;/ref&gt;
! Additional ratios&lt;br /&gt;of 3, 5, and 9&lt;ref group="note"&gt;Adding the ratios of 3, 5, and 9 introduces greater [[error]] while giving several more harmonic identities to the 10-edo intervals&lt;/ref&gt;
! Interval names
! colspan="3" | [[Ups and downs notation]]&lt;br /&gt;([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and m2)
! Audio
|-
| 0
| 0
| 1/1
| 256/243, 50/49, 25/24
| unison
| unison, min 2nd
| P1, m2
| D, Eb
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 120
| 16/15, 15/14, 14/13
| 10/9, 13/12, 81/80
| neutral second
| mid 2nd
| ~2
| ^D, vE
| [[File:0-120 neutral second (10-EDO).mp3|frameless]]
|-
| 2
| 240
| 8/7, 15/13, 144/125, 224/195
| 9/8, 7/6
| hemifourth, major second, minor third
| maj 2nd, min 3rd
| M2, m3
| E, F
| [[File:0-240 second, third (5-EDO).mp3|frameless]]
|-
| 3
| 360
| [[16/13]]
| 5/4
| neutral third
| mid 3rd
| ~3
| ^F, vG
| [[File:0-360 neutral third (10-EDO).mp3|frameless]]
|-
| 4
| 480
| 64/49, 169/128
| 4/3, 9/7, 13/10
| perfect fourth
| maj 3rd, perf 4th
| M3, P4
| F#, G
| [[File:0-480 fourth (5-EDO).mp3|frameless]]
|-
| 5
| 600
| 91/64, 128/91, 169/120, 240/169
| 7/5, 10/7, 13/9, 18/13
| hemioctave, tritone
| up 4th, down 5th
| ^4, v5
| ^G, vA
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 6
| 720
| 49/32, 256/169
| 3/2, 14/9, 20/13
| perfect fifth
| perf 5th, min 6th
| P5, m6
| A, Bb
| [[File:0-720 fifth (5-EDO).mp3|frameless]]
|-
| 7
| 840
| [[13/8]]
| 8/5
| neutral sixth
| mid 6th
| ~6
| ^A, vB
| [[File:0-840 neutral sixth (10-EDO).mp3|frameless]]
|-
| 8
| 960
| 7/4, 26/15, 125/72, 195/112
| 16/9, 12/7
| hemitwelfth, major sixth, minor seventh
| maj 6th, min 7th
| M6, m7
| B, C
| [[File:0-960 sixth, seventh (5-EDO).mp3|frameless]]
|-
| 9
| 1080
| 15/8, 28/15, 13/7
| 9/5, 24/13, 160/81
| neutral seventh
| mid 7th
| ~7
| ^C, vD
| [[File:0-1080 major seventh (10-EDO).mp3|frameless]]
|-
| 10
| 1200
| 2/1
| 243/128, 49/25, 48/25
| octave
| maj 7th, octave
| M7, P8
| C#, D
| [[File:0-1200 octave.mp3|frameless]]
|}
&lt;references group="note" /&gt;

== Notation ==
=== Ups and downs notation ===
The interval table above shows the diatonic notation, generated by 5ths (6\10, representing 3/2). Alternative notations include pentatonic fifth-generated and heptatonic 3rd-generated.

==== Pentatonic 5th-generated ====
'''D * E * G * A * C * D''' (generator = 3/2 = 6\10 = perfect 5thoid)

D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D

1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d (s = sub-, d = -oid)

pentatonic circles of fifths: ...D - A - E - C - G - D... and ...^D - ^A - ^E - ^C - ^G - ^D... (or equivalently ...vD - vA - vE - vC - vG - vD...)

pentatonic circles of fifths: ...1 - 5d - s3 - s7 - 4d - 1... and ...^1 - ^5d - ^s3 - ^s7 - ^4d - ^1... (or equivalently ...v1 - v5d - vs3 - vs7 - v4d - v1...)

(s- = sub-, -d = -oid, see [[5edo#Alternative%20notations|5edo notation]])

[[Enharmonic unison]]: vvs3

==== Heptatonic 3rd-generated ====
'''D E * F G * A B * C D''' (generator = 3\10 = perfect 3rd)

D - E - E#/Fb - F - G - G#/Ab - A - B - B#/Cb - C - D

P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8

genchain of 3rds: ...Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D#... ("Every good boy deserves fudge and candy")

genchain of 3rds: ...d8 - d3 - m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1...

[[Enharmonic unison]]: d2

See below: 3L&amp;nbsp;4s mosh notation

=== 3L&amp;nbsp;4s (mosh) notation ===
See above: Heptatonic 3rd-generated notation.

The notation of Neutral[7]. Notes are denoted as LsLssLs = CDEFGABC, and raising and lowering by a chroma (L − s), 1 step in this instance, is denoted by ♯ and ♭. 

{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed"
! #
! Cents
! Note
! Name
! Associated ratios
|-
| 0
| 0
| C
| Perf 1sn
| 1/1
|-
| 1
| 120
| Db
| Min 2nd
| 12/11
|-
| 2
| 240
| D, Eb
| Maj 2nd, dim 3rd
| 9/8, 32/27
|-
| 3
| 360
| E
| Perf 3rd
| 11/9, 27/22
|-
| 4
| 480
| Fb
| Min 4th
| 4/3
|-
| 5
| 600
| F, Gb
| Maj 4th, min 5th
| 11/8, 16/11
|-
| 6
| 720
| G
| Maj 5th
| 3/2
|-
| 7
| 840
| A
| Perf 6th
| 18/11, 44/27
|-
| 8
| 960
| A#, Bb
| Aug 6th, min 7th
| 16/9, 27/16
|-
| 9
| 1080
| B
| Maj 7th
| 11/6
|-
| 10
| 1200
| C
| Perf 8ve
| 2/1
|}

=== Sagittal notation ===
This notation is a subset of the notations for edos [[20edo #Sagittal notation|20]] and [[30edo #Sagittal notation|30]] and a superset of the notation for [[5edo #Sagittal notation|5edo]].

==== Evo and Revo flavors ====
&lt;imagemap&gt;
File:10-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 319 0 479 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 319 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Sagittal.svg]]
&lt;/imagemap&gt;

==== Evo-SZ flavor ====
&lt;imagemap&gt;
File:10-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 315 0 475 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 315 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:10-EDO_Evo-SZ_Sagittal.svg]]
&lt;/imagemap&gt;

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is identical to Stein–Zimmerman notation.

== Approximation to JI ==
=== Selected just intervals by error ===
==== Selected 13-limit intervals ====
[[File:10ed2-001.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 25/24, 256/243
| {{Mapping| 10 16 23 }}
| -0.089
| 9.27
| 7.73
|-
| 2.3.5.7
| 25/24, 28/27, 49/48
| {{Mapping| 10 16 23 28 }}
| +0.718
| 8.15
| 6.79
|-
| 2.3.5.7.13
| 25/24, 28/27, 40/39, 49/48
| {{Mapping| 10 16 23 28 37 }}
| +0.603
| 7.30
| 6.08
|}
* 10et is lower in relative error than any previous equal temperaments in the 7- and 17-limit. The next equal temperaments doing better in those subgroups are [[12edo|12]] and [[19edo|19eg]], respectively. 
* 10et is prominent in the 2.3.7.13, 2.3.5.7.13, 2.3.7.13.17, and 2.3.5.7.13.17 subgroup. The next equal temperaments doing better in those subgroups are [[17edo|17]], 19, [[36edo|36]] and [[31edo|31]], respectively.

=== Uniform maps ===
{{Uniform map|edo=10}}

=== Commas ===
10et tempers out the following commas. This assumes the val {{val| 10 16 23 28 35 37 }}. 

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[256/243]]
| {{Monzo| 8 -5 }}
| 90.22
| Sawa
| Blackwood comma, Pythagorean limma
|-
| 5
| [[25/24]]
| {{Monzo| -3 -1 2 }}
| 70.67
| Yoyo
| Dicot comma, classic chroma
|-
| 5
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma, double augmentation diesis
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 7
| [[525/512]]
| {{Monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma, Avicenna's enharmonic diesis
|-
| 7
| [[49/48]]
| {{Monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma, tritonic diesis
|-
| 7
| [[686/675]]
| {{Monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma, septimal kleisma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| &lt;abbr title="578509309952/576650390625"&gt;(24 digits)&lt;/abbr&gt;
| {{Monzo| 11 -10 -10 10 }}
| 5.57
| Saquinbizogu
| [[Linus comma]]
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[243/242]]
| {{Monzo| -1 5 0 0 -2 }}
| 7.14
| Lulu
| Rastma
|-
| 11
| [[385/384]]
| {{Monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[540/539]]
| {{Monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 11
| [[3025/3024]]
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[196/195]]
| {{Monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Thuzozogu
| Mynucuma
|-
| 13
| [[676/675]]
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma, parizeksma
|}
&lt;references group="note"/&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator
! Temperament(s)
|-
| 1
| 1\10
| [[Negri]], [[miracle]] (out-of-tune)
|-
| 1
| 3\10
| [[Dicot]] / [[beatles]] (out-of-tune) / [[neutral]] (out-of-tune)
|-
| 2
| 1\10
| [[Pajara]] (out-of-tune)
|-
| 2
| 2\10
| [[Decimal]], [[lemba]] (out-of-tune)
|-
| 5
| 1\10
| [[Blackwood]]
|}

== Octave stretch or compression ==
If one wishes to use 10edo as a no-5s, 19-or-lower-limit tuning, then it benefits from [[octave shrinking]]. [[zpi|26zpi]] and [[36ed12]] are compressed-octave versions of 10edo.

If one wishes to use 10edo as a no-3s, 13-or-lower-limit tuning, then it benefits from [[octave stretching]]. [[ed7|28ed7]] is a stretched version of 10edo.

== Scales ==
=== MOS scales ===
* Decimal/Lemba[6] [[4L&amp;nbsp;2s]] (period = 5\10, gen = 2\10): 2 2 1 2 2 1
* Dicot[7] [[3L&amp;nbsp;4s]] (gen = 3\10): 1 2 1 2 1 2 1
* Negri[9] [[1L&amp;nbsp;8s]] (gen = 1\10): 1 1 1 1 2 1 1 1 1

=== Other scales ===
* [[Pinetone #Pinetone pentatonic|Pinetone major pentatonic]] (subset of Dicot[7]): 2 1 3 1 3
* [[Pinetone #Pinetone pentatonic|Pinetone minor pentatonic]] (subset of Dicot[7]): 3 1 2 3 1
* [[Marvel hexatonic|Marvel augmented hexatonic]] (subset of Dicot[7]): 2 1 3 1 2 1
* Marvel double harmonic hexatonic (subset of Dicot[7]): 1 2 1 3 2 1, 1 2 3 1 2 1
* Decimal/Lemba[6] [[4M]]: 2 1 2 2 2 1
* Dicot[7] [[4M]]: 2 1 1 2 2 1 1

=== Horagrams ===
[[File:Screen Shot 2020-04-23 at 11.13.09 PM.png|alt=1\10 MOS|none|thumb|697x697px|1\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 1L&amp;nbsp;3s, 1L&amp;nbsp;4s, 1L&amp;nbsp;5s, 1L&amp;nbsp;6s, 1L&amp;nbsp;7s, and 1L&amp;nbsp;8s]]
[[File:Screen Shot 2020-04-23 at 11.13.35 PM.png|none|thumb|697x697px|3\10 mos with 1L&amp;nbsp;1s, 1L&amp;nbsp;2s, 3L&amp;nbsp;1s, 3L&amp;nbsp;4s]]

== Diagrams ==
[[File:10edo_wheel.png|alt=10edo wheel.png|280x285px|10edo wheel.png]]

== Instruments ==
10edo lends itself exceptionally well to guitar (and other fretted strings), on account of the fact that five of its flat 4ths (at 480{{c}}) exactly spans two octaves ({{nowrap|480 × 5 {{=}} 2400}}), meaning the open strings can be uniformly tuned in 4ths. This allows for greater uniformity in chord and scale fingering patterns than in 12edo, making it exceptionally easy to learn. For instance, the fingering for an "E" chord would be {{dash|0, 2, 2, 1, 0, 0}} (low to high), an "A" chord would be {{dash|0, 0, 2, 2, 1, 0}}, and a "D" chord would be {{nowrap|1, 0, 0, 2, 2, 1}}. This is also the case in all edos which are multiples of 5, but in 10-edo it is particularly simple.

Retuning a conventional keyboard to 10edo may be done in many ways, but neglecting or making redundant the Eb and Ab keys preserves the sLsLsLs scale on the white keys. Redundancy may make modulation easier, but another option is tuning the superfluous keys to selections from [[20edo|20edo]] which approximates the 11th harmonic with relative accuracy, among other features.

{| class="wikitable"
|-
| [[File:Decaphonic_Classic_Guitar.png|alt=Decaphonic_Classic_Guitar.png|Decaphonic_Classic_Guitar.png]]
|-
| A Decaphonic (10edo) Classical Guitar
|}
[[File:decaphonic-uke.JPG|alt=decaphonic-uke.JPG|526x406px|decaphonic-uke.JPG]]

=== Lumatone ===
''See [[Lumatone mapping for 10edo]]''.

== Music ==
{{Main| 10edo/Music }}
{{Catrel|10edo tracks}}

== References ==
&lt;references/&gt;

[[Category:10-tone scales]]</text>
      <sha1>1qax1wwemc0v11bv6enzfj4p6ay4day</sha1>
    </revision>
  </page>
  <page>
    <title>10edt</title>
    <ns>0</ns>
    <id>40</id>
    <revision>
      <id>40</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>40</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1207" sha1="sbogi62b8l3i05iicktzbtxu1tylca7" xml:space="preserve">&lt;span style="font-size: 18px; line-height: 27px;"&gt;'''10 Equal Divisions of the Tritave'''&lt;/span&gt;

{| class="wikitable"
|-
| | Degrees
| | Cents
| | Approximate Ratios
|-
| | 0
| | 0
| | &lt;span style="color: #660000;"&gt;[[1/1|1/1]]&lt;/span&gt;
|-
| | 1
| | 190.196
| | [[10/9|10/9]], [[28/25|28/25]]
|-
| | 2
| | 380.391
| | &lt;span style="color: #660000;"&gt;[[5/4|5/4]]&lt;/span&gt;
|-
| | 3
| | 570.587
| | [[7/5|7/5]]
|-
| | 4
| | 760.782
| | &lt;span style="color: #660000;"&gt;[[14/9|14/9]]&lt;/span&gt;
|-
| | 5
| | 950.978
| | [[19/11|19/11]]?
|-
| | 6
| | 1141.173
| | &lt;span style="color: #660000;"&gt;[[27/14|27/14]]&lt;/span&gt;
|-
| | 7
| | 1331.369
| | [[15/7|15/7]] ([[15/14|15/14]] plus an octave)
|-
| | 8
| | 1521.564
| | [[12/5|5/5]] (&lt;span style="color: #660000;"&gt;[[6/5|6/5]]&lt;/span&gt; plus an octave)
|-
| | 9
| | 1711.760
| | [[27/10|27/10]]
|-
| | 10
| | 1901.955
| | [[3/1|3/1]]
|}

10edt, like [[5edt|5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
[[Category:edt]]</text>
      <sha1>sbogi62b8l3i05iicktzbtxu1tylca7</sha1>
    </revision>
    <revision>
      <id>4174</id>
      <timestamp>2014-08-25T04:13:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 519523810 - Original comment: **</comment>
      <origin>4174</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6124" sha1="fzkeo2jyyqxlz35es13ukpdcpu11cnf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2014-08-25 04:13:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;519523810&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;
&amp;lt;span style="font-size: 18px; line-height: 27px;"&amp;gt;**10 Equal Divisions of the Tritave**&amp;lt;/span&amp;gt;

|| Degrees || Cents || Approximate Ratios ||
|| 0 || 0 || &amp;lt;span style="color: #660000;"&amp;gt;[[1_1|1/1]]&amp;lt;/span&amp;gt; ||
|| 1 || 190.196 || [[10_9|10/9]], [[28_25|28/25]] ||
|| 2 || 380.391 || &amp;lt;span style="color: #660000;"&amp;gt;[[5_4|5/4]]&amp;lt;/span&amp;gt; ||
|| 3 || 570.587 || [[7_5|7/5]] ||
|| 4 || 760.782 || &amp;lt;span style="color: #660000;"&amp;gt;[[14_9|14/9]]&amp;lt;/span&amp;gt; ||
|| 5 || 950.978 || [[19_11|19/11]]? ||
|| 6 || 1141.173 || &amp;lt;span style="color: #660000;"&amp;gt;[[27_14|27/14]]&amp;lt;/span&amp;gt; ||
|| 7 || 1331.369 || [[15_7|15/7]] ([[15_14|15/14]] plus an octave) ||
|| 8 || 1521.564 || [[12_5|5/5]] (&amp;lt;span style="color: #660000;"&amp;gt;[[6_5|6/5]]&amp;lt;/span&amp;gt; plus an octave) ||
|| 9 || 1711.760 || [[27_10|27/10]] ||
|| 10 || 1901.955 || [[3_1|3/1]] ||


10edt, like [[5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 18px; line-height: 27px;"&amp;gt;&amp;lt;strong&amp;gt;10 Equal Divisions of the Tritave&amp;lt;/strong&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degrees&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate Ratios&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/1_1"&amp;gt;1/1&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;190.196&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/10_9"&amp;gt;10/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_25"&amp;gt;28/25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;380.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;570.587&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;760.782&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/14_9"&amp;gt;14/9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;950.978&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/19_11"&amp;gt;19/11&amp;lt;/a&amp;gt;?&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1141.173&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/27_14"&amp;gt;27/14&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1331.369&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/15_7"&amp;gt;15/7&amp;lt;/a&amp;gt; (&amp;lt;a class="wiki_link" href="/15_14"&amp;gt;15/14&amp;lt;/a&amp;gt; plus an octave)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1521.564&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/12_5"&amp;gt;5/5&amp;lt;/a&amp;gt; (&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; plus an octave)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1711.760&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/27_10"&amp;gt;27/10&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1901.955&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/3_1"&amp;gt;3/1&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10edt, like &amp;lt;a class="wiki_link" href="/5edt"&amp;gt;5edt&amp;lt;/a&amp;gt;, has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of &amp;amp;quot;meantone&amp;amp;quot; temperament.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>fzkeo2jyyqxlz35es13ukpdcpu11cnf</sha1>
    </revision>
    <revision>
      <id>4175</id>
      <timestamp>2014-08-25T04:06:04Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 519523510 - Original comment: **</comment>
      <origin>4175</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5829" sha1="r68l8gxni6x198g1utiho1wopdzpa1v" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2014-08-25 04:06:04 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;519523510&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;
&amp;lt;span style="font-size: 18px; line-height: 27px;"&amp;gt;**10 Equal Divisions of the Tritave**&amp;lt;/span&amp;gt;

|| Degrees || Cents || Approximate Ratios ||
|| 0 || 0 || &amp;lt;span style="color: #660000;"&amp;gt;[[1_1|1/1]]&amp;lt;/span&amp;gt; ||
|| 1 || 190.196 || [[10_9|10/9]], [[28_25|28/25]] ||
|| 2 || 380.391 || &amp;lt;span style="color: #660000;"&amp;gt;[[5_4|5/4]]&amp;lt;/span&amp;gt; ||
|| 3 || 570.587 || [[7_5|7/5]] ||
|| 4 || 760.782 || &amp;lt;span style="color: #660000;"&amp;gt;[[14_9|14/9]]&amp;lt;/span&amp;gt; ||
|| 5 || 950.978 || 19/11? ||
|| 6 || 1141.173 || &amp;lt;span style="color: #660000;"&amp;gt;[[27_14|27/14]]&amp;lt;/span&amp;gt; ||
|| 7 || 1331.369 || 15/7 ([[15_14|15/14]] plus an octave) ||
|| 8 || 1521.564 || 12/5 (&amp;lt;span style="color: #660000;"&amp;gt;[[6_5|6/5]]&amp;lt;/span&amp;gt; plus an octave) ||
|| 9 || 1711.760 || 27/10 ||
|| 10 || 1901.955 || 3/1 ||


10edt, like [[5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 18px; line-height: 27px;"&amp;gt;&amp;lt;strong&amp;gt;10 Equal Divisions of the Tritave&amp;lt;/strong&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degrees&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate Ratios&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/1_1"&amp;gt;1/1&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;190.196&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/10_9"&amp;gt;10/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_25"&amp;gt;28/25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;380.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;570.587&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;760.782&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/14_9"&amp;gt;14/9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;950.978&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/11?&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1141.173&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/27_14"&amp;gt;27/14&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1331.369&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/7 (&amp;lt;a class="wiki_link" href="/15_14"&amp;gt;15/14&amp;lt;/a&amp;gt; plus an octave)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1521.564&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;12/5 (&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; plus an octave)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1711.760&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1901.955&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10edt, like &amp;lt;a class="wiki_link" href="/5edt"&amp;gt;5edt&amp;lt;/a&amp;gt;, has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of &amp;amp;quot;meantone&amp;amp;quot; temperament.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>r68l8gxni6x198g1utiho1wopdzpa1v</sha1>
    </revision>
    <revision>
      <id>4176</id>
      <timestamp>2011-08-16T13:35:12Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Sarzadoce</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 246268983 - Original comment: **</comment>
      <origin>4176</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5772" sha1="b6k9ipe1h0h1t4x2a0ppg0706p5vnfg" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on &lt;tt&gt;2011-08-16 13:35:12 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;246268983&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;
&amp;lt;span style="font-size: 18px; line-height: 27px;"&amp;gt;**10 Equal Divisions of the Tritave**&amp;lt;/span&amp;gt;

|| Degrees || Cents || Approximate Ratios ||
|| 0 || 0 || &amp;lt;span style="color: #660000;"&amp;gt;[[1_1|1/1]]&amp;lt;/span&amp;gt; ||
|| 1 || 190.196 || [[10_9|10/9]], 28/25 ||
|| 2 || 380.391 || &amp;lt;span style="color: #660000;"&amp;gt;[[5_4|5/4]]&amp;lt;/span&amp;gt; ||
|| 3 || 570.587 || [[7_5|7/5]] ||
|| 4 || 760.782 || &amp;lt;span style="color: #660000;"&amp;gt;[[14_9|14/9]]&amp;lt;/span&amp;gt; ||
|| 5 || 950.978 || 19/11? ||
|| 6 || 1141.173 || &amp;lt;span style="color: #660000;"&amp;gt;[[27_14|27/14]]&amp;lt;/span&amp;gt; ||
|| 7 || 1331.369 || 15/7 ([[15_14|15/14]] plus an octave) ||
|| 8 || 1521.564 || 12/5 (&amp;lt;span style="color: #660000;"&amp;gt;[[6_5|6/5]]&amp;lt;/span&amp;gt; plus an octave) ||
|| 9 || 1711.760 || 27/10 ||
|| 10 || 1901.955 || 3/1 ||


10edt, like [[5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 18px; line-height: 27px;"&amp;gt;&amp;lt;strong&amp;gt;10 Equal Divisions of the Tritave&amp;lt;/strong&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Degrees&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Approximate Ratios&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/1_1"&amp;gt;1/1&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;190.196&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/10_9"&amp;gt;10/9&amp;lt;/a&amp;gt;, 28/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;380.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;570.587&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;760.782&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/14_9"&amp;gt;14/9&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;950.978&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/11?&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1141.173&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/27_14"&amp;gt;27/14&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1331.369&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/7 (&amp;lt;a class="wiki_link" href="/15_14"&amp;gt;15/14&amp;lt;/a&amp;gt; plus an octave)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1521.564&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;12/5 (&amp;lt;span style="color: #660000;"&amp;gt;&amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; plus an octave)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1711.760&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1901.955&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10edt, like &amp;lt;a class="wiki_link" href="/5edt"&amp;gt;5edt&amp;lt;/a&amp;gt;, has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of &amp;amp;quot;meantone&amp;amp;quot; temperament.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>b6k9ipe1h0h1t4x2a0ppg0706p5vnfg</sha1>
    </revision>
    <revision>
      <id>33127</id>
      <parentid>40</parentid>
      <timestamp>2018-09-21T07:54:09Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <origin>33127</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1231" sha1="hooh59jocbibzkwae4rm1k2hn3zbufo" xml:space="preserve">&lt;span style="font-size: 18px; line-height: 27px;"&gt;'''10 Equal Divisions of the Tritave'''&lt;/span&gt;

{| class="wikitable"
|-
| | Degrees
| | Cents
| | Approximate Ratios
|-
| | 0
| | 0
| | &lt;span style="color: #660000;"&gt;[[1/1|1/1]]&lt;/span&gt;
|-
| | 1
| | 190.196
| | [[10/9|10/9]], [[28/25|28/25]]
|-
| | 2
| | 380.391
| | &lt;span style="color: #660000;"&gt;[[5/4|5/4]]&lt;/span&gt;
|-
| | 3
| | 570.587
| | [[7/5|7/5]]
|-
| | 4
| | 760.782
| | &lt;span style="color: #660000;"&gt;[[14/9|14/9]]&lt;/span&gt;
|-
| | 5
| | 950.978
| | [[19/11|19/11]]?
|-
| | 6
| | 1141.173
| | &lt;span style="color: #660000;"&gt;[[27/14|27/14]]&lt;/span&gt;
|-
| | 7
| | 1331.369
| | [[15/7|15/7]] ([[15/14|15/14]] plus an octave)
|-
| | 8
| | 1521.564
| | [[12/5|5/5]] (&lt;span style="color: #660000;"&gt;[[6/5|6/5]]&lt;/span&gt; plus an octave)
|-
| | 9
| | 1711.760
| | [[27/10|27/10]]
|-
| | 10
| | 1901.955
| | [[3/1|3/1]]
|}

10edt, like [[5edt|5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
[[Category:edt]]
[[category:macrotonal]]</text>
      <sha1>hooh59jocbibzkwae4rm1k2hn3zbufo</sha1>
    </revision>
    <revision>
      <id>33128</id>
      <parentid>33127</parentid>
      <timestamp>2018-09-21T07:54:53Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <comment>5/5?</comment>
      <origin>33128</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1227" sha1="c2z9slsmd5j81w424zpgfimfemfxwi2" xml:space="preserve">&lt;span style="font-size: 18px; line-height: 27px;"&gt;'''10 Equal Divisions of the Tritave'''&lt;/span&gt;

{| class="wikitable"
|-
| | Degrees
| | Cents
| | Approximate Ratios
|-
| | 0
| | 0
| | &lt;span style="color: #660000;"&gt;[[1/1|1/1]]&lt;/span&gt;
|-
| | 1
| | 190.196
| | [[10/9|10/9]], [[28/25|28/25]]
|-
| | 2
| | 380.391
| | &lt;span style="color: #660000;"&gt;[[5/4|5/4]]&lt;/span&gt;
|-
| | 3
| | 570.587
| | [[7/5|7/5]]
|-
| | 4
| | 760.782
| | &lt;span style="color: #660000;"&gt;[[14/9|14/9]]&lt;/span&gt;
|-
| | 5
| | 950.978
| | [[19/11|19/11]]?
|-
| | 6
| | 1141.173
| | &lt;span style="color: #660000;"&gt;[[27/14|27/14]]&lt;/span&gt;
|-
| | 7
| | 1331.369
| | [[15/7|15/7]] ([[15/14|15/14]] plus an octave)
|-
| | 8
| | 1521.564
| | [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5|6/5]]&lt;/span&gt; plus an octave)
|-
| | 9
| | 1711.760
| | [[27/10|27/10]]
|-
| | 10
| | 1901.955
| | [[3/1|3/1]]
|}

10edt, like [[5edt|5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
[[Category:edt]]
[[category:macrotonal]]</text>
      <sha1>c2z9slsmd5j81w424zpgfimfemfxwi2</sha1>
    </revision>
    <revision>
      <id>40112</id>
      <parentid>33128</parentid>
      <timestamp>2019-02-08T06:56:46Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>40112</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1343" sha1="37vdfz613stje94o84nk91wgqk5z71u" xml:space="preserve">'''10EDT''' is the [[Edt|equal division of the third harmonic (tritave)]] into ten parts of 190.1955 [[cent|cents]] each, corresponding to 6.3093 [[edo]]. It is related to the pocus temperament, which tempers out 169/168, 225/224, and 245/243 in the 2.3.5.7.13 subgroup.

{| class="wikitable"
|-
| | Degrees
| | Cents
| | Approximate Ratios
|-
| | 0
| | 0
| | &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| | 1
| | 190.196
| | [[10/9]], [[28/25]]
|-
| | 2
| | 380.391
| | &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| | 3
| | 570.587
| | [[7/5]]
|-
| | 4
| | 760.782
| | &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| | 5
| | 950.978
| | 45/26, [[26/15]]
|-
| | 6
| | 1141.173
| | &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| | 7
| | 1331.369
| | [[15/7]] ([[15/14]] plus an octave)
|-
| | 8
| | 1521.564
| | [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| | 9
| | 1711.760
| | [[27/20|27/10]]
|-
| | 10
| | 1901.955
| | [[3/1]]
|}

10edt, like [[5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
[[Category:edt]]
[[category:macrotonal]]</text>
      <sha1>37vdfz613stje94o84nk91wgqk5z71u</sha1>
    </revision>
    <revision>
      <id>41415</id>
      <parentid>40112</parentid>
      <timestamp>2019-04-10T17:09:47Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>41415</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1415" sha1="mq9b04kan6scskw3npgl28fgdsstfha" xml:space="preserve">'''10EDT''' is the [[Edt|equal division of the third harmonic (tritave)]] into ten parts of 190.1955 [[cent|cents]] each, corresponding to 6.3093 [[edo]]. It is related to the pocus temperament, which tempers out 169/168, 225/224, and 245/243 in the 2.3.5.7.13 subgroup.

{| class="wikitable"
|-
| | Degrees
| | Cents
|Hekts
| | Approximate Ratios
|-
| | 0
|  colspan="2"| 0
| | &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| | 1
| | 190.196
|130
| | [[10/9]], [[28/25]]
|-
| | 2
| | 380.391
|260
| | &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| | 3
| | 570.587
|390
| | [[7/5]]
|-
| | 4
| | 760.782
|520
| | &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| | 5
| | 950.978
|650
| | 45/26, [[26/15]]
|-
| | 6
| | 1141.173
|780
| | &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| | 7
| | 1331.369
|910
| | [[15/7]] ([[15/14]] plus an octave)
|-
| | 8
| | 1521.564
|1040
| | [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| | 9
| | 1711.760
|1170
| | [[27/20|27/10]]
|-
| | 10
| | 1901.955
|1300
| | [[3/1]]
|}

10edt, like [[5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
[[Category:edt]]
[[category:macrotonal]]</text>
      <sha1>mq9b04kan6scskw3npgl28fgdsstfha</sha1>
    </revision>
    <revision>
      <id>42096</id>
      <parentid>41415</parentid>
      <timestamp>2019-07-07T20:03:26Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>42096</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1409" sha1="s4k3aqu6xcvp6ubc3rm6e02btok7tu1" xml:space="preserve">'''10EDT''' is the [[Edt|equal division of the third harmonic (tritave)]] into ten parts of 190.1955 [[cent|cents]] each, corresponding to 6.3093 [[edo]]. It is related to the pocus temperament, which tempers out 169/168, 225/224, and 245/243 in the 2.3.5.7.13 subgroup.

{| class="wikitable"
|-
| | Degrees
| | Cents
|Hekts
| | Approximate Ratios
|-
|  colspan="3"| 0
| | &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| | 1
| | 190.196
|130
| | [[10/9]], [[28/25]]
|-
| | 2
| | 380.391
|260
| | &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| | 3
| | 570.587
|390
| | [[7/5]]
|-
| | 4
| | 760.782
|520
| | &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| | 5
| | 950.978
|650
| | 45/26, [[26/15]]
|-
| | 6
| | 1141.173
|780
| | &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| | 7
| | 1331.369
|910
| | [[15/7]] ([[15/14]] plus an octave)
|-
| | 8
| | 1521.564
|1040
| | [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| | 9
| | 1711.760
|1170
| | [[27/20|27/10]]
|-
| | 10
| | 1901.955
|1300
| | [[3/1]]
|}

10edt, like [[5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
[[Category:edt]]
[[category:macrotonal]]</text>
      <sha1>s4k3aqu6xcvp6ubc3rm6e02btok7tu1</sha1>
    </revision>
    <revision>
      <id>71093</id>
      <parentid>42096</parentid>
      <timestamp>2021-05-30T16:54:26Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>71093</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1409" sha1="ogld3lele66osmhahuxg3icu0ksz2u0" xml:space="preserve">'''10EDT''' is the [[Edt|equal division of the third harmonic (tritave)]] into ten parts of 190.1955 [[cent|cents]] each, corresponding to 6.3093 [[edo]]. It is related to the pocus temperament, which tempers out 169/168, 225/224, and 245/243 in the 2.3.5.7.13 subgroup.

{| class="wikitable"
|-
| | Degrees
| | Cents
!Hekts
| | Approximate Ratios
|-
! colspan="3" | 0
| | &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| | 1
| | 190.196
|130
| | [[10/9]], [[28/25]]
|-
| | 2
| | 380.391
|260
| | &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| | 3
| | 570.587
|390
| | [[7/5]]
|-
| | 4
| | 760.782
|520
| | &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| | 5
| | 950.978
|650
| | 45/26, [[26/15]]
|-
| | 6
| | 1141.173
|780
| | &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| | 7
| | 1331.369
|910
| | [[15/7]] ([[15/14]] plus an octave)
|-
| | 8
| | 1521.564
|1040
| | [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| | 9
| | 1711.760
|1170
| | [[27/20|27/10]]
|-
| | 10
| | 1901.955
|1300
| | [[3/1]]
|}

10edt, like [[5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
[[Category:edt]]
[[category:macrotonal]]</text>
      <sha1>ogld3lele66osmhahuxg3icu0ksz2u0</sha1>
    </revision>
    <revision>
      <id>96667</id>
      <parentid>71093</parentid>
      <timestamp>2022-10-05T19:44:16Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
      <origin>96667</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1424" sha1="ehctbgsfmgv5dxywo6v2w849kxftpic" xml:space="preserve">{{Infobox ET}}
'''10EDT''' is the [[Edt|equal division of the third harmonic (tritave)]] into ten parts of 190.1955 [[cent|cents]] each, corresponding to 6.3093 [[edo]]. It is related to the pocus temperament, which tempers out 169/168, 225/224, and 245/243 in the 2.3.5.7.13 subgroup.

{| class="wikitable"
|-
| | Degrees
| | Cents
!Hekts
| | Approximate Ratios
|-
! colspan="3" | 0
| | &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| | 1
| | 190.196
|130
| | [[10/9]], [[28/25]]
|-
| | 2
| | 380.391
|260
| | &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| | 3
| | 570.587
|390
| | [[7/5]]
|-
| | 4
| | 760.782
|520
| | &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| | 5
| | 950.978
|650
| | 45/26, [[26/15]]
|-
| | 6
| | 1141.173
|780
| | &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| | 7
| | 1331.369
|910
| | [[15/7]] ([[15/14]] plus an octave)
|-
| | 8
| | 1521.564
|1040
| | [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| | 9
| | 1711.760
|1170
| | [[27/20|27/10]]
|-
| | 10
| | 1901.955
|1300
| | [[3/1]]
|}

10edt, like [[5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
[[Category:edt]]
[[category:macrotonal]]</text>
      <sha1>ehctbgsfmgv5dxywo6v2w849kxftpic</sha1>
    </revision>
    <revision>
      <id>132034</id>
      <parentid>96667</parentid>
      <timestamp>2024-01-17T09:29:32Z</timestamp>
      <contributor>
        <username>Akselai</username>
        <id>437</id>
      </contributor>
      <comment>add harmonics in equal</comment>
      <origin>132034</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1477" sha1="hcyg00a8s0vhhyfndbktx12sd53r6uo" xml:space="preserve">{{Infobox ET}}
'''10EDT''' is the [[Edt|equal division of the third harmonic (tritave)]] into ten parts of 190.1955 [[cent|cents]] each, corresponding to 6.3093 [[edo]]. It is related to the pocus temperament, which tempers out 169/168, 225/224, and 245/243 in the 2.3.5.7.13 subgroup.

{| class="wikitable"
|-
| | Degrees
| | Cents
!Hekts
| | Approximate Ratios
|-
! colspan="3" | 0
| | &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| | 1
| | 190.196
|130
| | [[10/9]], [[28/25]]
|-
| | 2
| | 380.391
|260
| | &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| | 3
| | 570.587
|390
| | [[7/5]]
|-
| | 4
| | 760.782
|520
| | &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| | 5
| | 950.978
|650
| | 45/26, [[26/15]]
|-
| | 6
| | 1141.173
|780
| | &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| | 7
| | 1331.369
|910
| | [[15/7]] ([[15/14]] plus an octave)
|-
| | 8
| | 1521.564
|1040
| | [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| | 9
| | 1711.760
|1170
| | [[27/20|27/10]]
|-
| | 10
| | 1901.955
|1300
| | [[3/1]]
|}

== Prime harmonics ==
{{Harmonics in equal|10|3|1}}

10edt, like [[5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
[[Category:edt]]
[[category:macrotonal]]</text>
      <sha1>hcyg00a8s0vhhyfndbktx12sd53r6uo</sha1>
    </revision>
    <revision>
      <id>132040</id>
      <parentid>132034</parentid>
      <timestamp>2024-01-17T09:39:50Z</timestamp>
      <contributor>
        <username>Akselai</username>
        <id>437</id>
      </contributor>
      <comment>/* Prime harmonics */</comment>
      <origin>132040</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1493" sha1="e3ss60z1w7lofk2pqbnyrs8ibectneo" xml:space="preserve">{{Infobox ET}}
'''10EDT''' is the [[Edt|equal division of the third harmonic (tritave)]] into ten parts of 190.1955 [[cent|cents]] each, corresponding to 6.3093 [[edo]]. It is related to the pocus temperament, which tempers out 169/168, 225/224, and 245/243 in the 2.3.5.7.13 subgroup.

{| class="wikitable"
|-
| | Degrees
| | Cents
!Hekts
| | Approximate Ratios
|-
! colspan="3" | 0
| | &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| | 1
| | 190.196
|130
| | [[10/9]], [[28/25]]
|-
| | 2
| | 380.391
|260
| | &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| | 3
| | 570.587
|390
| | [[7/5]]
|-
| | 4
| | 760.782
|520
| | &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| | 5
| | 950.978
|650
| | 45/26, [[26/15]]
|-
| | 6
| | 1141.173
|780
| | &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| | 7
| | 1331.369
|910
| | [[15/7]] ([[15/14]] plus an octave)
|-
| | 8
| | 1521.564
|1040
| | [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| | 9
| | 1711.760
|1170
| | [[27/20|27/10]]
|-
| | 10
| | 1901.955
|1300
| | [[3/1]]
|}

== Prime harmonics ==
{{Harmonics in equal|10|3|1|intervals=prime}}

10edt, like [[5edt]], has very accurate 5-limit harmony for such a small number of steps per tritave. 10edt introduces some new harmonic properties though; notably the 571 cent tritone which can function as 7/5. It also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
[[Category:edt]]
[[category:macrotonal]]</text>
      <sha1>e3ss60z1w7lofk2pqbnyrs8ibectneo</sha1>
    </revision>
    <revision>
      <id>143721</id>
      <parentid>132040</parentid>
      <timestamp>2024-05-20T10:00:27Z</timestamp>
      <contributor>
        <username>Akselai</username>
        <id>437</id>
      </contributor>
      <comment>clean up</comment>
      <origin>143721</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1761" sha1="f69dfqrn113lt6c1xa6bvo085ntbs64" xml:space="preserve">{{Infobox ET}}
{{EDO intro}}

== Theory ==
10edt has very accurate 5-limit harmony for such a small number of steps per tritave, most notably the [[5/4]] inherited from 5edt. 10edt introduces some new harmonic properties though — such as the 571 cent tritone, which can function as [[7/5]]. We can use this to readily construct chords such as 4:5:7:12, although the [[7/4]], being 18 cents flat, does considerable damage to the consonance of this chord. 

10edt also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
 
One step of 10edt can serve as the generator for [[pocus]] temperament, a [[Temperament merging|merge]] of [[sensamagic]] and 2.3.5.7.13 [[catakleismic]], which tempers out [[169/168]], [[225/224]], and [[245/243]] in the 2.3.5.7.13 subgroup.

=== Harmonics ===
{{Harmonics in equal|10|3|1}}
{{Harmonics in equal|10|3|1|intervals=prime}}

=== Interval table ===
{| class="wikitable"
|-
| | Degrees
| | [[Cent]]s
| | [[Hekt]]s
| | Approximate Ratios
|-
! colspan="3" | 0
| | &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| | 1
| | 190.196
|130
| | [[10/9]], [[28/25]]
|-
| | 2
| | 380.391
|260
| | &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| | 3
| | 570.587
|390
| | [[7/5]]
|-
| | 4
| | 760.782
|520
| | &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| | 5
| | 950.978
|650
| | 45/26, [[26/15]]
|-
| | 6
| | 1141.173
|780
| | &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| | 7
| | 1331.369
|910
| | [[15/7]] ([[15/14]] plus an octave)
|-
| | 8
| | 1521.564
|1040
| | [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| | 9
| | 1711.760
|1170
| | [[27/20|27/10]]
|-
| | 10
| | 1901.955
|1300
| | [[3/1]]
|}

[[Category:Edt]]
[[Category:Macrotonal]]</text>
      <sha1>f69dfqrn113lt6c1xa6bvo085ntbs64</sha1>
    </revision>
    <revision>
      <id>181314</id>
      <parentid>143721</parentid>
      <timestamp>2025-02-18T16:49:57Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181314</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1700" sha1="sjy1fzfj9b2jt4r8zr8k70b5sxk32h0" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edt has very accurate 5-limit harmony for such a small number of steps per tritave, most notably the [[5/4]] inherited from 5edt. 10edt introduces some new harmonic properties though — such as the 571 cent tritone, which can function as [[7/5]]. We can use this to readily construct chords such as 4:5:7:12, although the [[7/4]], being 18 cents flat, does considerable damage to the consonance of this chord. 

10edt also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
 
One step of 10edt can serve as the generator for [[pocus]] temperament, a [[Temperament merging|merge]] of [[sensamagic]] and 2.3.5.7.13 [[catakleismic]], which tempers out [[169/168]], [[225/224]], and [[245/243]] in the 2.3.5.7.13 subgroup.

=== Harmonics ===
{{Harmonics in equal|10|3|1}}
{{Harmonics in equal|10|3|1|intervals=prime}}

=== Interval table ===
{| class="wikitable"
|-
| Degrees
| [[Cent]]s
| [[Hekt]]s
| Approximate Ratios
|-
! colspan="3" | 0
| &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| 1
| 190.196
| 130
| [[10/9]], [[28/25]]
|-
| 2
| 380.391
| 260
| &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| 3
| 570.587
| 390
| [[7/5]]
|-
| 4
| 760.782
| 520
| &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| 5
| 950.978
| 650
| 45/26, [[26/15]]
|-
| 6
| 1141.173
| 780
| &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| 7
| 1331.369
| 910
| [[15/7]] ([[15/14]] plus an octave)
|-
| 8
| 1521.564
| 1040
| [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| 9
| 1711.760
| 1170
| [[27/20|27/10]]
|-
| 10
| 1901.955
| 1300
| [[3/1]]
|}

[[Category:Edt]]
[[Category:Macrotonal]]</text>
      <sha1>sjy1fzfj9b2jt4r8zr8k70b5sxk32h0</sha1>
    </revision>
    <revision>
      <id>181315</id>
      <parentid>181314</parentid>
      <timestamp>2025-02-18T16:50:23Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181315</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1700" sha1="7j0e2jqjgicapbykw1z5fwlmmpucnj8" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edt has very accurate 5-limit harmony for such a small number of steps per tritave, most notably the [[5/4]] inherited from 5edt. 10edt introduces some new harmonic properties though — such as the 571 cent tritone, which can function as [[7/5]]. We can use this to readily construct chords such as 4:5:7:12, although the [[7/4]], being 18 cents flat, does considerable damage to the consonance of this chord. 

10edt also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
 
One step of 10edt can serve as the generator for [[pocus]] temperament, a [[Temperament merging|merge]] of [[sensamagic]] and 2.3.5.7.13 [[catakleismic]], which tempers out [[169/168]], [[225/224]], and [[245/243]] in the 2.3.5.7.13 subgroup.

=== Harmonics ===
{{Harmonics in equal|10|3|1}}
{{Harmonics in equal|10|3|1|intervals=prime}}

=== Interval table ===
{| class="wikitable"
|-
! Degrees
! [[Cent]]s
! [[Hekt]]s
! Approximate Ratios
|-
| colspan="3" | 0
| &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| 1
| 190.196
| 130
| [[10/9]], [[28/25]]
|-
| 2
| 380.391
| 260
| &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| 3
| 570.587
| 390
| [[7/5]]
|-
| 4
| 760.782
| 520
| &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| 5
| 950.978
| 650
| 45/26, [[26/15]]
|-
| 6
| 1141.173
| 780
| &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| 7
| 1331.369
| 910
| [[15/7]] ([[15/14]] plus an octave)
|-
| 8
| 1521.564
| 1040
| [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| 9
| 1711.760
| 1170
| [[27/20|27/10]]
|-
| 10
| 1901.955
| 1300
| [[3/1]]
|}

[[Category:Edt]]
[[Category:Macrotonal]]</text>
      <sha1>7j0e2jqjgicapbykw1z5fwlmmpucnj8</sha1>
    </revision>
    <revision>
      <id>205578</id>
      <parentid>181315</parentid>
      <timestamp>2025-07-31T15:31:25Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Edt]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>205578</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1683" sha1="kl13th8ibcovnu4j9ke5reuxm51pi40" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edt has very accurate 5-limit harmony for such a small number of steps per tritave, most notably the [[5/4]] inherited from 5edt. 10edt introduces some new harmonic properties though — such as the 571 cent tritone, which can function as [[7/5]]. We can use this to readily construct chords such as 4:5:7:12, although the [[7/4]], being 18 cents flat, does considerable damage to the consonance of this chord. 

10edt also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
 
One step of 10edt can serve as the generator for [[pocus]] temperament, a [[Temperament merging|merge]] of [[sensamagic]] and 2.3.5.7.13 [[catakleismic]], which tempers out [[169/168]], [[225/224]], and [[245/243]] in the 2.3.5.7.13 subgroup.

=== Harmonics ===
{{Harmonics in equal|10|3|1}}
{{Harmonics in equal|10|3|1|intervals=prime}}

=== Interval table ===
{| class="wikitable"
|-
! Degrees
! [[Cent]]s
! [[Hekt]]s
! Approximate Ratios
|-
| colspan="3" | 0
| &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| 1
| 190.196
| 130
| [[10/9]], [[28/25]]
|-
| 2
| 380.391
| 260
| &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| 3
| 570.587
| 390
| [[7/5]]
|-
| 4
| 760.782
| 520
| &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| 5
| 950.978
| 650
| 45/26, [[26/15]]
|-
| 6
| 1141.173
| 780
| &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| 7
| 1331.369
| 910
| [[15/7]] ([[15/14]] plus an octave)
|-
| 8
| 1521.564
| 1040
| [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| 9
| 1711.760
| 1170
| [[27/20|27/10]]
|-
| 10
| 1901.955
| 1300
| [[3/1]]
|}

[[Category:Macrotonal]]</text>
      <sha1>kl13th8ibcovnu4j9ke5reuxm51pi40</sha1>
    </revision>
    <revision>
      <id>231366</id>
      <parentid>205578</parentid>
      <timestamp>2026-05-31T02:48:13Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Harmonics */ integer and prime harmonics side-by-side is nonstandard and potentially confusing</comment>
      <origin>231366</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1676" sha1="jz7tc8tmme2rccaue7z4kcvezk376nj" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edt has very accurate 5-limit harmony for such a small number of steps per tritave, most notably the [[5/4]] inherited from 5edt. 10edt introduces some new harmonic properties though — such as the 571 cent tritone, which can function as [[7/5]]. We can use this to readily construct chords such as 4:5:7:12, although the [[7/4]], being 18 cents flat, does considerable damage to the consonance of this chord. 

10edt also splits the major third in half, categorizing this tuning as a fringe variety of "meantone" temperament.
 
One step of 10edt can serve as the generator for [[pocus]] temperament, a [[Temperament merging|merge]] of [[sensamagic]] and 2.3.5.7.13 [[catakleismic]], which tempers out [[169/168]], [[225/224]], and [[245/243]] in the 2.3.5.7.13 subgroup.

=== Harmonics ===
{{Harmonics in equal|10|3|1}}
{{Harmonics in equal|10|3|1|start=12}}

=== Interval table ===
{| class="wikitable"
|-
! Degrees
! [[Cent]]s
! [[Hekt]]s
! Approximate Ratios
|-
| colspan="3" | 0
| &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| 1
| 190.196
| 130
| [[10/9]], [[28/25]]
|-
| 2
| 380.391
| 260
| &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| 3
| 570.587
| 390
| [[7/5]]
|-
| 4
| 760.782
| 520
| &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| 5
| 950.978
| 650
| 45/26, [[26/15]]
|-
| 6
| 1141.173
| 780
| &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| 7
| 1331.369
| 910
| [[15/7]] ([[15/14]] plus an octave)
|-
| 8
| 1521.564
| 1040
| [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| 9
| 1711.760
| 1170
| [[27/20|27/10]]
|-
| 10
| 1901.955
| 1300
| [[3/1]]
|}

[[Category:Macrotonal]]</text>
      <sha1>jz7tc8tmme2rccaue7z4kcvezk376nj</sha1>
    </revision>
    <revision>
      <id>231376</id>
      <parentid>231366</parentid>
      <timestamp>2026-05-31T07:12:36Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>"Very accurate 5-limit harmony" what?</comment>
      <origin>231376</origin>
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      <text bytes="1646" sha1="qtugm94xkc4lq7l4f7ftj2oqaptzfey" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edt most notably inherits the [[5/4]] from 5edt, and introduces some new harmonic elements, such as the 571-cent tritone, which can function as [[7/5]]. We can use this to readily construct chords such as 4:5:7:12, although the [[7/4]], being 18 cents flat, does considerable damage to the consonance of this chord. 

10edt also splits the 5/4 in half, categorizing this tuning as a fringe variety of meantone.
 
10edt can serve as the generator chain for the [[pocus]] temperament, a [[temperament merging|merge]] of [[sensamagic]] and 2.3.5.7.13 [[catakleismic]], which tempers out [[169/168]], [[225/224]], and [[245/243]] in the 2.3.5.7.13 subgroup.

=== Harmonics ===
{{Harmonics in equal|10|3|1|columns=11}}
{{Harmonics in equal|10|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 10edt (continued)}}

== Interval table ==
{| class="wikitable"
|-
! Degrees
! [[Cent]]s
! [[Hekt]]s
! Approximate Ratios
|-
| colspan="3" | 0
| &lt;span style="color: #660000;"&gt;[[1/1]]&lt;/span&gt;
|-
| 1
| 190.196
| 130
| [[10/9]], [[28/25]]
|-
| 2
| 380.391
| 260
| &lt;span style="color: #660000;"&gt;[[5/4]]&lt;/span&gt;
|-
| 3
| 570.587
| 390
| [[7/5]]
|-
| 4
| 760.782
| 520
| &lt;span style="color: #660000;"&gt;[[14/9]]&lt;/span&gt;
|-
| 5
| 950.978
| 650
| 45/26, [[26/15]]
|-
| 6
| 1141.173
| 780
| &lt;span style="color: #660000;"&gt;[[27/14]]&lt;/span&gt;
|-
| 7
| 1331.369
| 910
| [[15/7]] ([[15/14]] plus an octave)
|-
| 8
| 1521.564
| 1040
| [[12/5]] (&lt;span style="color: #660000;"&gt;[[6/5]]&lt;/span&gt; plus an octave)
|-
| 9
| 1711.760
| 1170
| [[27/20|27/10]]
|-
| 10
| 1901.955
| 1300
| [[3/1]]
|}

[[Category:Macrotonal]]</text>
      <sha1>qtugm94xkc4lq7l4f7ftj2oqaptzfey</sha1>
    </revision>
    <revision>
      <id>231377</id>
      <parentid>231376</parentid>
      <timestamp>2026-05-31T07:14:29Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Interval table</comment>
      <origin>231377</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1381" sha1="hdbx6sj2stujkx40zuqcrhiep6edkdq" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
10edt most notably inherits the [[5/4]] from 5edt, and introduces some new harmonic elements, such as the 571-cent tritone, which can function as [[7/5]]. We can use this to readily construct chords such as 4:5:7:12, although the [[7/4]], being 18 cents flat, does considerable damage to the consonance of this chord. 

10edt also splits the 5/4 in half, categorizing this tuning as a fringe variety of meantone.
 
10edt can serve as the generator chain for the [[pocus]] temperament, a [[temperament merging|merge]] of [[sensamagic]] and 2.3.5.7.13 [[catakleismic]], which tempers out [[169/168]], [[225/224]], and [[245/243]] in the 2.3.5.7.13 subgroup.

=== Harmonics ===
{{Harmonics in equal|10|3|1|columns=11}}
{{Harmonics in equal|10|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 10edt (continued)}}

== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
! #
! [[Cent]]s
! [[Hekt]]s
! Approximate ratios
|-
| 0
| 0
| 0
| [[1/1]]
|-
| 1
| 190
| 130
| [[10/9]], [[28/25]]
|-
| 2
| 380
| 260
| [[5/4]]
|-
| 3
| 571
| 390
| [[7/5]]
|-
| 4
| 761
| 520
| [[14/9]]
|-
| 5
| 951
| 650
| [[26/15]], [[45/26]]
|-
| 6
| 1141
| 780
| [[27/14]]
|-
| 7
| 1331
| 910
| [[15/7]]
|-
| 8
| 1522
| 1040
| [[12/5]]
|-
| 9
| 1712
| 1170
| [[27/20|27/10]]
|-
| 10
| 1902
| 1300
| [[3/1]]
|}

[[Category:Macrotonal]]</text>
      <sha1>hdbx6sj2stujkx40zuqcrhiep6edkdq</sha1>
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  <page>
    <title>10highschool1</title>
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      <id>41</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
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      <text bytes="203" sha1="qc2mu1tin9q2epj7b9j0idkdhse4sa9" xml:space="preserve">! 10highschool1.scl

First 10-note Highschool scale

10

!

21/20 

9/8  

5/4 

4/3 

7/5 

3/2 

5/3 

7/4 

15/8 

2/1
[[Category:10-tone]]
[[Category:highschool]]
[[Category:just]]
[[Category:scale]]</text>
      <sha1>qc2mu1tin9q2epj7b9j0idkdhse4sa9</sha1>
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      <timestamp>2011-07-07T13:40:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
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      <comment>**Imported revision 240389919 - Original comment: **</comment>
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      <text bytes="1609" sha1="o2v6wkdr0jxnpluwqojb7esaxyrdrld" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-07-07 13:40:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;240389919&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 10highschool1.scl
First 10-note Highschool scale
10
!
21/20 
9/8  
5/4 
4/3 
7/5 
3/2 
5/3 
7/4 
15/8 
2/1
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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First 10-note Highschool scale&amp;lt;br /&amp;gt;
10&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
21/20 &amp;lt;br /&amp;gt;
9/8  &amp;lt;br /&amp;gt;
5/4 &amp;lt;br /&amp;gt;
4/3 &amp;lt;br /&amp;gt;
7/5 &amp;lt;br /&amp;gt;
3/2 &amp;lt;br /&amp;gt;
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7/4 &amp;lt;br /&amp;gt;
15/8 &amp;lt;br /&amp;gt;
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      <text bytes="203" sha1="j3gvl4fgqgffkgyq9fw0nzslrjq99l3" xml:space="preserve">! 10highschool1.scl

First 10-note Highschool scale

10

!

21/20 

9/8  

5/4 

4/3 

7/5 

3/2 

5/3 

7/4 

15/8 

2/1

[[Category:10-tone]]
[[Category:highschool]]
[[Category:Just intonation scales]]</text>
      <sha1>j3gvl4fgqgffkgyq9fw0nzslrjq99l3</sha1>
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    <revision>
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      <contributor>
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      <minor/>
      <comment>formatting, recat</comment>
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      <text bytes="246" sha1="155cclh4uwfijocx9zxhvwlraznrhs2" xml:space="preserve">&lt;pre&gt;
! 10highschool1.scl
First 10-note Highschool scale
10
!
21/20 
9/8  
5/4 
4/3 
7/5 
3/2 
5/3 
7/4 
15/8 
2/1
&lt;/pre&gt;

[[Category:10-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with Scala files]]</text>
      <sha1>155cclh4uwfijocx9zxhvwlraznrhs2</sha1>
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      <text bytes="457" sha1="nbdwqemka6uhux2aycu0cnojsd4w6oy" xml:space="preserve">A 10-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].&lt;pre&gt;
! 10highschool1.scl
First 10-note Highschool scale
10
!
21/20 
9/8  
5/4 
4/3 
7/5 
3/2 
5/3 
7/4 
15/8 
2/1
&lt;/pre&gt;

[[Category:10-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with Scala files]]</text>
      <sha1>nbdwqemka6uhux2aycu0cnojsd4w6oy</sha1>
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    <revision>
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      <contributor>
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        <id>480</id>
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      <minor/>
      <comment>Text replacement - "[[Category:Just intonation scales]]" to "[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>200911</origin>
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      <text bytes="506" sha1="0vaop4o468ash3iw25ipwuidkvnsrmb" xml:space="preserve">A 10-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].&lt;pre&gt;
! 10highschool1.scl
First 10-note Highschool scale
10
!
21/20 
9/8  
5/4 
4/3 
7/5 
3/2 
5/3 
7/4 
15/8 
2/1
&lt;/pre&gt;

[[Category:10-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Pages with Scala files]]</text>
      <sha1>0vaop4o468ash3iw25ipwuidkvnsrmb</sha1>
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  <page>
    <title>10highschool2</title>
    <ns>0</ns>
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    <revision>
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      <timestamp>2018-07-17T00:00:00Z</timestamp>
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      <text bytes="204" sha1="95e8tf96x6ihwnl7j9qk4dq3n17vwo2" xml:space="preserve">! 10highschool2.scl

Second 10-note Highschool scale

10

!

15/14 

9/8 

5/4 

4/3 

10/7 

3/2 

5/3 

7/4 

15/8 

2/1
[[Category:10-tone]]
[[Category:highschool]]
[[Category:just]]
[[Category:scale]]</text>
      <sha1>95e8tf96x6ihwnl7j9qk4dq3n17vwo2</sha1>
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      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 240390007 - Original comment: **</comment>
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      <text bytes="1611" sha1="otftdqjre3penh901rb1ps4kod7r809" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-07-07 13:41:12 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;240390007&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 10highschool2.scl
Second 10-note Highschool scale
10
!
15/14 
9/8 
5/4 
4/3 
10/7 
3/2 
5/3 
7/4 
15/8 
2/1
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10highschool2&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 10highschool2.scl&amp;lt;br /&amp;gt;
Second 10-note Highschool scale&amp;lt;br /&amp;gt;
10&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
15/14 &amp;lt;br /&amp;gt;
9/8 &amp;lt;br /&amp;gt;
5/4 &amp;lt;br /&amp;gt;
4/3 &amp;lt;br /&amp;gt;
10/7 &amp;lt;br /&amp;gt;
3/2 &amp;lt;br /&amp;gt;
5/3 &amp;lt;br /&amp;gt;
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15/8 &amp;lt;br /&amp;gt;
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    <revision>
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      <text bytes="204" sha1="70ubvpianodw586ve98ww8m2ieg1vnf" xml:space="preserve">! 10highschool2.scl

Second 10-note Highschool scale

10

!

15/14 

9/8 

5/4 

4/3 

10/7 

3/2 

5/3 

7/4 

15/8 

2/1

[[Category:10-tone]]
[[Category:highschool]]
[[Category:Just intonation scales]]</text>
      <sha1>70ubvpianodw586ve98ww8m2ieg1vnf</sha1>
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    <revision>
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      <contributor>
        <username>Xenwolf</username>
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      <minor/>
      <comment>formatting, recat</comment>
      <origin>62135</origin>
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      <format>text/x-wiki</format>
      <text bytes="247" sha1="qgwa2u1rlfjlnyogy6lil8kwsgckyo2" xml:space="preserve">&lt;pre&gt;
! 10highschool2.scl
Second 10-note Highschool scale
10
!
15/14 
9/8 
5/4 
4/3 
10/7 
3/2 
5/3 
7/4 
15/8 
2/1
&lt;/pre&gt;

[[Category:10-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with Scala files]]</text>
      <sha1>qgwa2u1rlfjlnyogy6lil8kwsgckyo2</sha1>
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      <text bytes="458" sha1="fi6hc1wd9n6n2obioyfnegwwwimubeh" xml:space="preserve">A 10-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].&lt;pre&gt;
! 10highschool2.scl
Second 10-note Highschool scale
10
!
15/14 
9/8 
5/4 
4/3 
10/7 
3/2 
5/3 
7/4 
15/8 
2/1
&lt;/pre&gt;

[[Category:10-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with Scala files]]</text>
      <sha1>fi6hc1wd9n6n2obioyfnegwwwimubeh</sha1>
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        <id>480</id>
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      <minor/>
      <comment>Text replacement - "[[Category:Just intonation scales]]" to "[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>200658</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="507" sha1="lkmoljl72b55scdahgrsfddu4mtkrlz" xml:space="preserve">A 10-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].&lt;pre&gt;
! 10highschool2.scl
Second 10-note Highschool scale
10
!
15/14 
9/8 
5/4 
4/3 
10/7 
3/2 
5/3 
7/4 
15/8 
2/1
&lt;/pre&gt;

[[Category:10-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Pages with Scala files]]</text>
      <sha1>lkmoljl72b55scdahgrsfddu4mtkrlz</sha1>
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  <page>
    <title>10L 11s</title>
    <ns>0</ns>
    <id>43</id>
    <revision>
      <id>43</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>43</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1338" sha1="r540grutg9080xk9ze1s1zps1lg0rjm" xml:space="preserve">This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114.286 cents).

{| class="wikitable"
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|-
| | 
| | 
| | 
| | 
| | 6/61
| | 118.033
|-
| | 
| | 
| | 
| | 5/51
| | 
| | 117.647
|-
| | 
| | 
| | 
| | 
| | 9/92
| | 117.391
|-
| | 
| | 
| | 
| | 
| | 
| | 117.171
|-
| | 
| | 
| | 4/41
| | 
| | 
| | 117.073
|-
| | 
| | 
| | 
| | 
| | 
| | 116.857
|-
| | 
| | 
| | 
| | 
| | 11/113
| | 116.814
|-
| | 
| | 
| | 
| | 
| | 
| | 116.7725
|-
| | 
| | 
| | 
| | 7/72
| | 
| | 116.667
|-
| | 
| | 
| | 
| | 
| | 10/103
| | 116.505
|-
| | 
| | 3/31
| | 
| | 
| | 
| | 116.192
|-
| | 
| | 
| | 
| | 
| | 11/114
| | 115.7895
|-
| | 
| | 
| | 
| | 
| | 
| | 115.763
|-
| | 
| | 
| | 
| | 8/83
| | 
| | 115.663
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|-
| | 
| | 
| | 
| | 
| | 13/135
| | 115.556
|-
| | 
| | 
| | 
| | 
| | 
| | 115.507
|-
| | 
| | 
| | 5/52
| | 
| | 
| | 115.385
|-
| | 
| | 
| | 
| | 
| | 12/125
| | 115.2
|-
| | 
| | 
| | 
| | 7/73
| | 
| | 115.0865
|-
| | 
| | 
| | 
| | 
| | 9/94
| | 114.894
|-
| | 2/21
| | 
| | 
| | 
| | 
| | 114.286
|}</text>
      <sha1>r540grutg9080xk9ze1s1zps1lg0rjm</sha1>
    </revision>
    <revision>
      <id>4179</id>
      <timestamp>2015-11-15T15:07:11Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 566524933 - Original comment: **</comment>
      <origin>4179</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9780" sha1="05b0fhmiauh1kzqv653yr4w98xqs959" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-15 15:07:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566524933&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114.286 cents).
|| 1/10 ||   ||   ||   ||   || 120 ||
||   ||   ||   ||   || 6/61 || 118.033 ||
||   ||   ||   || 5/51 ||   || 117.647 ||
||   ||   ||   ||   || 9/92 || 117.391 ||
||   ||   ||   ||   ||   || 117.171 ||
||   ||   || 4/41 ||   ||   || 117.073 ||
||   ||   ||   ||   ||   || 116.857 ||
||   ||   ||   ||   || 11/113 || 116.814 ||
||   ||   ||   ||   ||   || 116.7725 ||
||   ||   ||   || 7/72 ||   || 116.667 ||
||   ||   ||   ||   || 10/103 || 116.505 ||
||   || 3/31 ||   ||   ||   || 116.192 ||
||   ||   ||   ||   || 11/114 || 115.7895 ||
||   ||   ||   ||   ||   || 115.763 ||
||   ||   ||   || 8/83 ||   || 115.663 ||
||   ||   ||   ||   ||   || 115.585 ||
||   ||   ||   ||   || 13/135 || 115.556 ||
||   ||   ||   ||   ||   || 115.507 ||
||   ||   || 5/52 ||   ||   || 115.385 ||
||   ||   ||   ||   || 12/125 || 115.2 ||
||   ||   ||   || 7/73 ||   || 115.0865 ||
||   ||   ||   ||   || 9/94 || 114.894 ||
|| 2/21 ||   ||   ||   ||   || 114.286 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 11s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of &amp;amp;quot;practicality&amp;amp;quot; at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114.286 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;118.033&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;117.647&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/92&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;117.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;117.171&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;117.073&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.857&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.814&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.7725&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.667&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.192&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/114&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.7895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.763&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.663&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.585&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/135&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.556&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.507&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.385&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;12/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.0865&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;114.894&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;114.286&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>05b0fhmiauh1kzqv653yr4w98xqs959</sha1>
    </revision>
    <revision>
      <id>4180</id>
      <timestamp>2015-10-28T15:57:30Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 564248587 - Original comment: **</comment>
      <origin>4180</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9110" sha1="jhehe5uhjuavcopoaffd765r2y6utfn" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-10-28 15:57:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;564248587&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114.286 cents).
|| 1/10 ||   ||   ||   ||   || 120 ||
||   ||   ||   ||   || 6/61 || 118.033 ||
||   ||   ||   || 5/51 ||   || 117.647 ||
||   ||   ||   ||   || 9/92 || 117.391 ||
||   ||   ||   ||   ||   || 117.171 ||
||   ||   || 4/41 ||   ||   || 117.073 ||
||   ||   ||   ||   ||   || 116.857 ||
||   ||   ||   ||   || 11/113 || 116.814 ||
||   ||   ||   ||   ||   || 116.7725 ||
||   ||   ||   || 7/72 ||   || 116.667 ||
||   ||   ||   ||   || 10/103 || 116.505 ||
||   || 3/31 ||   ||   ||   || 116.192 ||
||   ||   ||   ||   || 11/114 || 115.7895 ||
||   ||   ||   || 8/83 ||   || 115.663 ||
||   ||   ||   ||   ||   || 115.585 ||
||   ||   ||   ||   || 13/135 || 115.556 ||
||   ||   || 5/52 ||   ||   || 115.385 ||
||   ||   ||   ||   || 12/125 || 115.2 ||
||   ||   ||   || 7/73 ||   || 115.0865 ||
||   ||   ||   ||   || 9/94 || 114.894 ||
|| 2/21 ||   ||   ||   ||   || 114.286 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 11s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of &amp;amp;quot;practicality&amp;amp;quot; at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114.286 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;118.033&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;5/51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;117.647&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/92&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;117.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;117.171&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;117.073&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.857&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.814&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.7725&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.667&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116.192&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/114&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.7895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.663&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.585&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;13/135&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.556&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;12/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.2&amp;lt;br /&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;7/73&amp;lt;br /&amp;gt;
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    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;114.894&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;114.286&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>jhehe5uhjuavcopoaffd765r2y6utfn</sha1>
    </revision>
    <revision>
      <id>4181</id>
      <timestamp>2015-05-24T14:07:32Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 552015824 - Original comment: **</comment>
      <origin>4181</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8486" sha1="0xvjep6r5hxvn36vhah2hbe4xi42ovb" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-24 14:07:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;552015824&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114 2/7 cents).
|| 1/10 ||   ||   ||   ||   || 120 ||
||   ||   ||   ||   || 6/61 || 118 2/61 ||
||   ||   ||   || 5/51 ||   || 117 11/17 ||
||   ||   ||   ||   || 9/92 || 117 9/23 ||
||   ||   ||   ||   ||   || 117.170847 ||
||   ||   || 4/41 ||   ||   || 117 3/41 ||
||   ||   ||   ||   ||   || 116.857225 ||
||   ||   ||   ||   || 11/113 || 116 92/113 ||
||   ||   ||   || 7/72 ||   || 116 2/3 ||
||   ||   ||   ||   || 10/103 || 116 52/103 ||
||   || 3/31 ||   ||   ||   || 116 4/31 ||
||   ||   ||   ||   || 11/114 || 115 15/19 ||
||   ||   ||   || 8/83 ||   || 115 55/83 ||
||   ||   ||   ||   || 13/135 || 115 5/9 ||
||   ||   || 5/52 ||   ||   || 115 5/13 ||
||   ||   ||   ||   || 12/125 || 115.2 ||
||   ||   ||   || 7/73 ||   || 115 5/73 ||
||   ||   ||   ||   || 9/94 || 114 42/47 ||
|| 2/21 ||   ||   ||   ||   || 114 2/7 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 11s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of &amp;amp;quot;practicality&amp;amp;quot; at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114 2/7 cents).&amp;lt;br /&amp;gt;


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        &amp;lt;td&amp;gt;117 3/41&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;116.857225&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;115.2&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;114 42/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/21&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;114 2/7&amp;lt;br /&amp;gt;
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&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>0xvjep6r5hxvn36vhah2hbe4xi42ovb</sha1>
    </revision>
    <revision>
      <id>4182</id>
      <timestamp>2015-02-13T15:03:23Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540924438 - Original comment: **</comment>
      <origin>4182</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8486" sha1="69xd9cm46i10i8qw3y0wr8pyt9h22dj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-13 15:03:23 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540924438&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114 2/7 cents).
|| 1/10 ||   ||   ||   ||   || 120 ||
||   ||   ||   ||   || 6/61 || 118 2/61 ||
||   ||   ||   || 5/51 ||   || 117 11/17 ||
||   ||   ||   ||   || 9/92 || 117 9/23 ||
||   ||   ||   ||   ||   || 117.092921 ||
||   ||   || 4/41 ||   ||   || 117 3/41 ||
||   ||   ||   ||   ||   || 117.053336 ||
||   ||   ||   ||   || 11/113 || 116 92/113 ||
||   ||   ||   || 7/72 ||   || 116 2/3 ||
||   ||   ||   ||   || 10/103 || 116 52/103 ||
||   || 3/31 ||   ||   ||   || 116 4/31 ||
||   ||   ||   ||   || 11/114 || 115 15/19 ||
||   ||   ||   || 8/83 ||   || 115 55/83 ||
||   ||   ||   ||   || 13/135 || 115 5/9 ||
||   ||   || 5/52 ||   ||   || 115 5/13 ||
||   ||   ||   ||   || 12/125 || 115.2 ||
||   ||   ||   || 7/73 ||   || 115 5/73 ||
||   ||   ||   ||   || 9/94 || 114 42/47 ||
|| 2/21 ||   ||   ||   ||   || 114 2/7 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 11s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of &amp;amp;quot;practicality&amp;amp;quot; at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114 2/7 cents).&amp;lt;br /&amp;gt;


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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>69xd9cm46i10i8qw3y0wr8pyt9h22dj</sha1>
    </revision>
    <revision>
      <id>4183</id>
      <timestamp>2014-11-05T11:02:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 529637272 - Original comment: **</comment>
      <origin>4183</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7804" sha1="ra4p8o3dj46fmkb257t6hesd3aul25f" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2014-11-05 11:02:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;529637272&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114 2/7 cents).
|| 1/10 ||   ||   ||   ||   || 120 ||
||   ||   ||   ||   || 6/61 || 118 2/61 ||
||   ||   ||   || 5/51 ||   || 117 11/17 ||
||   ||   ||   ||   || 9/92 || 117 9/23 ||
||   ||   || 4/41 ||   ||   || 117 3/41 ||
||   ||   ||   ||   || 11/113 || 116 92/113 ||
||   ||   ||   || 7/72 ||   || 116 2/3 ||
||   ||   ||   ||   || 10/103 || 116 52/103 ||
||   || 3/31 ||   ||   ||   || 116 4/31 ||
||   ||   ||   ||   || 11/114 || 115 15/19 ||
||   ||   ||   || 8/83 ||   || 115 55/83 ||
||   ||   ||   ||   || 13/135 || 115 5/9 ||
||   ||   || 5/52 ||   ||   || 115 5/13 ||
||   ||   ||   ||   || 12/125 || 115.2 ||
||   ||   ||   || 7/73 ||   || 115 5/73 ||
||   ||   ||   ||   || 9/94 || 114 42/47 ||
|| 2/21 ||   ||   ||   ||   || 114 2/7 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 11s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of &amp;amp;quot;practicality&amp;amp;quot; at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114 2/7 cents).&amp;lt;br /&amp;gt;


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      <sha1>ra4p8o3dj46fmkb257t6hesd3aul25f</sha1>
    </revision>
    <revision>
      <id>4184</id>
      <timestamp>2012-11-18T21:16:41Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 383843190 - Original comment: **</comment>
      <origin>4184</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7804" sha1="0ljnxu32bky5sqm7gu56g8malq9tu0u" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-11-18 21:16:41 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;383843190&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits its small steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114 2/7 cents).
|| 1/10 ||   ||   ||   ||   || 120 ||
||   ||   ||   ||   || 6/61 || 118 2/61 ||
||   ||   ||   || 5/51 ||   || 117 11/17 ||
||   ||   ||   ||   || 9/92 || 117 9/23 ||
||   ||   || 4/41 ||   ||   || 117 3/41 ||
||   ||   ||   ||   || 11/113 || 116 92/113 ||
||   ||   ||   || 7/72 ||   || 116 2/3 ||
||   ||   ||   ||   || 10/103 || 116 52/103 ||
||   || 3/31 ||   ||   ||   || 116 4/31 ||
||   ||   ||   ||   || 11/114 || 115 15/19 ||
||   ||   ||   || 8/83 ||   || 115 55/83 ||
||   ||   ||   ||   || 13/135 || 115 5/9 ||
||   ||   || 5/52 ||   ||   || 115 5/13 ||
||   ||   ||   ||   || 12/125 || 115.2 ||
||   ||   ||   || 7/73 ||   || 115 5/73 ||
||   ||   ||   ||   || 9/94 || 114 42/47 ||
|| 2/21 ||   ||   ||   ||   || 114 2/7 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 11s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits its small steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of &amp;amp;quot;practicality&amp;amp;quot; at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114 2/7 cents).&amp;lt;br /&amp;gt;


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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>0ljnxu32bky5sqm7gu56g8malq9tu0u</sha1>
    </revision>
    <revision>
      <id>4185</id>
      <timestamp>2012-11-18T21:15:56Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 383843024 - Original comment: **</comment>
      <origin>4185</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7762" sha1="6sp4g0a50ggs5u633qeulnuhgbniajn" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-11-18 21:15:56 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;383843024&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits its small steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114 2/7 cents).
|| 1/10 ||   ||   ||   ||   || 120 ||
||   ||   ||   ||   || 6/61 || 118 2/61 ||
||   ||   ||   || 5/51 ||   || 117 11/17 ||
||   ||   ||   ||   || 9/92 || 117 9/23 ||
||   ||   || 4/41 ||   ||   || 117 3/41 ||
||   ||   ||   ||   || 11/113 || 116 92/113 ||
||   ||   ||   || 7/72 ||   || 116 2/3 ||
||   ||   ||   ||   || 10/103 || 116 52/103 ||
||   || 3/31 ||   ||   ||   || 116 4/31 ||
||   ||   ||   ||   || 11/114 || 115 15/19 ||
||   ||   ||   || 8/83 ||   || 115 55/83 ||
||   ||   ||   ||   || 13/135 || 115 5/9 ||
||   ||   || 5/52 ||   ||   || 115 5/13 ||
||   ||   ||   ||   || 12/125 || 115.2 ||
||   ||   ||   || 7/73 ||   || 115 5/73 ||
||   ||   ||   ||   || 9/94 || 114 42/47 ||
|| 2/21 ||   ||   ||   ||   || 114 2/7 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 11s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits its small steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the boundary of &amp;amp;quot;practicality&amp;amp;quot; at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114 2/7 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;117 3/41&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;115 15/19&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;115 55/83&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;115 5/13&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;12/125&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;115.2&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;114 42/47&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;114 2/7&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>6sp4g0a50ggs5u633qeulnuhgbniajn</sha1>
    </revision>
    <revision>
      <id>4186</id>
      <timestamp>2012-11-13T20:48:16Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 382137502 - Original comment: **</comment>
      <origin>4186</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7762" sha1="iryf9mnlkvy8b6ko0parkrddpv9z11o" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-11-13 20:48:16 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;382137502&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits its small steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing 41edo at the boundray of "practicality" (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114 2/7 cents).
|| 1/10 ||   ||   ||   ||   || 120 ||
||   ||   ||   ||   || 6/61 || 118 2/61 ||
||   ||   ||   || 5/51 ||   || 117 11/17 ||
||   ||   ||   ||   || 9/92 || 117 9/23 ||
||   ||   || 4/41 ||   ||   || 117 3/41 ||
||   ||   ||   ||   || 11/113 || 116 92/113 ||
||   ||   ||   || 7/72 ||   || 116 2/3 ||
||   ||   ||   ||   || 10/103 || 116 52/103 ||
||   || 3/31 ||   ||   ||   || 116 4/31 ||
||   ||   ||   ||   || 11/114 || 115 15/19 ||
||   ||   ||   || 8/83 ||   || 115 55/83 ||
||   ||   ||   ||   || 13/135 || 115 5/9 ||
||   ||   || 5/52 ||   ||   || 115 5/13 ||
||   ||   ||   ||   || 12/125 || 115.2 ||
||   ||   ||   || 7/73 ||   || 115 5/73 ||
||   ||   ||   ||   || 9/94 || 114 42/47 ||
|| 2/21 ||   ||   ||   ||   || 114 2/7 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 11s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits its small steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing 41edo at the boundray of &amp;amp;quot;practicality&amp;amp;quot; (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114 2/7 cents).&amp;lt;br /&amp;gt;


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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116 92/113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116 52/103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116 4/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/114&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115 15/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115 55/83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/135&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115 5/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115 5/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;12/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115.2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115 5/73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;114 42/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;114 2/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>iryf9mnlkvy8b6ko0parkrddpv9z11o</sha1>
    </revision>
    <revision>
      <id>92225</id>
      <parentid>43</parentid>
      <timestamp>2022-05-29T14:34:36Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>added infobox</comment>
      <origin>92225</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1540" sha1="fpvi4hjlyasydqyt2se6h7vwr2k1yw6" xml:space="preserve">{{Infobox MOS
| Name = miracloid
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 11
| Equalized = 2
| Paucitonic = 1
| Pattern = sLsLsLsLsLsLsLsLsLsLs
}}
This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114.286 cents).
If this MOS had a name, it would be miracloid.

{| class="wikitable"
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|-
| | 
| | 
| | 
| | 
| | 6/61
| | 118.033
|-
| | 
| | 
| | 
| | 5/51
| | 
| | 117.647
|-
| | 
| | 
| | 
| | 
| | 9/92
| | 117.391
|-
| | 
| | 
| | 
| | 
| | 
| | 117.171
|-
| | 
| | 
| | 4/41
| | 
| | 
| | 117.073
|-
| | 
| | 
| | 
| | 
| | 
| | 116.857
|-
| | 
| | 
| | 
| | 
| | 11/113
| | 116.814
|-
| | 
| | 
| | 
| | 
| | 
| | 116.7725
|-
| | 
| | 
| | 
| | 7/72
| | 
| | 116.667
|-
| | 
| | 
| | 
| | 
| | 10/103
| | 116.505
|-
| | 
| | 3/31
| | 
| | 
| | 
| | 116.192
|-
| | 
| | 
| | 
| | 
| | 11/114
| | 115.7895
|-
| | 
| | 
| | 
| | 
| | 
| | 115.763
|-
| | 
| | 
| | 
| | 8/83
| | 
| | 115.663
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|-
| | 
| | 
| | 
| | 
| | 13/135
| | 115.556
|-
| | 
| | 
| | 
| | 
| | 
| | 115.507
|-
| | 
| | 
| | 5/52
| | 
| | 
| | 115.385
|-
| | 
| | 
| | 
| | 
| | 12/125
| | 115.2
|-
| | 
| | 
| | 
| | 7/73
| | 
| | 115.0865
|-
| | 
| | 
| | 
| | 
| | 9/94
| | 114.894
|-
| | 2/21
| | 
| | 
| | 
| | 
| | 114.286
|}</text>
      <sha1>fpvi4hjlyasydqyt2se6h7vwr2k1yw6</sha1>
    </revision>
    <revision>
      <id>92241</id>
      <parentid>92225</parentid>
      <timestamp>2022-05-29T18:27:02Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>The name isn't endorsed by TAMNAMS maintainers</comment>
      <origin>92241</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1557" sha1="lq7a6aipllmz7nzo9ybjdy2z9ulqvay" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 11
| Equalized = 2
| Paucitonic = 1
| Pattern = sLsLsLsLsLsLsLsLsLsLs
}}
This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114.286 cents). [[User:Eliora|Eliora]] proposes the name ''miracloid'' for this pattern.

{| class="wikitable"
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|-
| | 
| | 
| | 
| | 
| | 6/61
| | 118.033
|-
| | 
| | 
| | 
| | 5/51
| | 
| | 117.647
|-
| | 
| | 
| | 
| | 
| | 9/92
| | 117.391
|-
| | 
| | 
| | 
| | 
| | 
| | 117.171
|-
| | 
| | 
| | 4/41
| | 
| | 
| | 117.073
|-
| | 
| | 
| | 
| | 
| | 
| | 116.857
|-
| | 
| | 
| | 
| | 
| | 11/113
| | 116.814
|-
| | 
| | 
| | 
| | 
| | 
| | 116.7725
|-
| | 
| | 
| | 
| | 7/72
| | 
| | 116.667
|-
| | 
| | 
| | 
| | 
| | 10/103
| | 116.505
|-
| | 
| | 3/31
| | 
| | 
| | 
| | 116.192
|-
| | 
| | 
| | 
| | 
| | 11/114
| | 115.7895
|-
| | 
| | 
| | 
| | 
| | 
| | 115.763
|-
| | 
| | 
| | 
| | 8/83
| | 
| | 115.663
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|-
| | 
| | 
| | 
| | 
| | 13/135
| | 115.556
|-
| | 
| | 
| | 
| | 
| | 
| | 115.507
|-
| | 
| | 
| | 5/52
| | 
| | 
| | 115.385
|-
| | 
| | 
| | 
| | 
| | 12/125
| | 115.2
|-
| | 
| | 
| | 
| | 7/73
| | 
| | 115.0865
|-
| | 
| | 
| | 
| | 
| | 9/94
| | 114.894
|-
| | 2/21
| | 
| | 
| | 
| | 
| | 114.286
|}</text>
      <sha1>lq7a6aipllmz7nzo9ybjdy2z9ulqvay</sha1>
    </revision>
    <revision>
      <id>93348</id>
      <parentid>92241</parentid>
      <timestamp>2022-07-11T21:05:40Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>93348</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1704" sha1="sv1d0yryylefgjmvujycp7j9d6ycetx" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 11
| Equalized = 2
| Paucitonic = 1
| Pattern = sLsLsLsLsLsLsLsLsLsLs
}}
This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114.286 cents). [[User:Eliora|Eliora]] proposes the name ''miracloid'' for this pattern.

{| class="wikitable"
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|
|-
| | 
| | 
| | 
| | 
| | 6/61
| | 118.033
|
|-
| | 
| | 
| | 
| | 5/51
| | 
| | 117.647
|
|-
| | 
| | 
| | 
| | 
| | 9/92
| | 117.391
|
|-
| | 
| | 
| | 
| | 
| | 
| | 117.171
|
|-
| | 
| | 
| | 4/41
| | 
| | 
| | 117.073
|
|-
| | 
| | 
| | 
| | 
| | 
| | 116.857
|
|-
| | 
| | 
| | 
| | 
| | 11/113
| | 116.814
|
|-
| | 
| | 
| | 
| | 
| | 
| | 116.7725
|
|-
| | 
| | 
| | 
| | 7/72
| | 
| | 116.667
|
|-
| | 
| | 
| | 
| | 
| | 10/103
| | 116.505
|
|-
| | 
| | 3/31
| | 
| | 
| | 
| | 116.192
|46/43-based miracloid is around here.
|-
| | 
| | 
| | 
| | 
| | 11/114
| | 115.7895
|
|-
| | 
| | 
| | 
| | 
| | 
| | 115.763
|
|-
| | 
| | 
| | 
| | 8/83
| | 
| | 115.663
|
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|
|-
| | 
| | 
| | 
| | 
| | 13/135
| | 115.556
|
|-
| | 
| | 
| | 
| | 
| | 
| | 115.507
|
|-
| | 
| | 
| | 5/52
| | 
| | 
| | 115.385
|Tricesimoprimal miracloid (using 31/29 as a gen) is around here.
|-
| | 
| | 
| | 
| | 
| | 12/125
| | 115.2
|
|-
| | 
| | 
| | 
| | 7/73
| | 
| | 115.0865
|
|-
| | 
| | 
| | 
| | 
| | 9/94
| | 114.894
|
|-
| | 2/21
| | 
| | 
| | 
| | 
| | 114.286
|
|}</text>
      <sha1>sv1d0yryylefgjmvujycp7j9d6ycetx</sha1>
    </revision>
    <revision>
      <id>94498</id>
      <parentid>93348</parentid>
      <timestamp>2022-08-17T09:15:52Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94498</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1703" sha1="ql1lsefgvpah0oom2kuqbe6f6o9d59l" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 11
| Equalized = 2
| Collapsed = 1
| Pattern = sLsLsLsLsLsLsLsLsLsLs
}}
This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114.286 cents). [[User:Eliora|Eliora]] proposes the name ''miracloid'' for this pattern.

{| class="wikitable"
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|
|-
| | 
| | 
| | 
| | 
| | 6/61
| | 118.033
|
|-
| | 
| | 
| | 
| | 5/51
| | 
| | 117.647
|
|-
| | 
| | 
| | 
| | 
| | 9/92
| | 117.391
|
|-
| | 
| | 
| | 
| | 
| | 
| | 117.171
|
|-
| | 
| | 
| | 4/41
| | 
| | 
| | 117.073
|
|-
| | 
| | 
| | 
| | 
| | 
| | 116.857
|
|-
| | 
| | 
| | 
| | 
| | 11/113
| | 116.814
|
|-
| | 
| | 
| | 
| | 
| | 
| | 116.7725
|
|-
| | 
| | 
| | 
| | 7/72
| | 
| | 116.667
|
|-
| | 
| | 
| | 
| | 
| | 10/103
| | 116.505
|
|-
| | 
| | 3/31
| | 
| | 
| | 
| | 116.192
|46/43-based miracloid is around here.
|-
| | 
| | 
| | 
| | 
| | 11/114
| | 115.7895
|
|-
| | 
| | 
| | 
| | 
| | 
| | 115.763
|
|-
| | 
| | 
| | 
| | 8/83
| | 
| | 115.663
|
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|
|-
| | 
| | 
| | 
| | 
| | 13/135
| | 115.556
|
|-
| | 
| | 
| | 
| | 
| | 
| | 115.507
|
|-
| | 
| | 
| | 5/52
| | 
| | 
| | 115.385
|Tricesimoprimal miracloid (using 31/29 as a gen) is around here.
|-
| | 
| | 
| | 
| | 
| | 12/125
| | 115.2
|
|-
| | 
| | 
| | 
| | 7/73
| | 
| | 115.0865
|
|-
| | 
| | 
| | 
| | 
| | 9/94
| | 114.894
|
|-
| | 2/21
| | 
| | 
| | 
| | 
| | 114.286
|
|}</text>
      <sha1>ql1lsefgvpah0oom2kuqbe6f6o9d59l</sha1>
    </revision>
    <revision>
      <id>136607</id>
      <parentid>94498</parentid>
      <timestamp>2024-02-22T05:36:32Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Mos intro</comment>
      <origin>136607</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1743" sha1="9uecw8jznm1fewozxwglbkp4lmui59c" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 11
| Equalized = 2
| Collapsed = 1
| Pattern = sLsLsLsLsLsLsLsLsLsLs
}}
{{MOS intro|Other Names=miracloid}}
This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114.286 cents). [[User:Eliora|Eliora]] has proposed the name ''miracloid'' for this pattern.

{| class="wikitable"
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|
|-
| | 
| | 
| | 
| | 
| | 6/61
| | 118.033
|
|-
| | 
| | 
| | 
| | 5/51
| | 
| | 117.647
|
|-
| | 
| | 
| | 
| | 
| | 9/92
| | 117.391
|
|-
| | 
| | 
| | 
| | 
| | 
| | 117.171
|
|-
| | 
| | 
| | 4/41
| | 
| | 
| | 117.073
|
|-
| | 
| | 
| | 
| | 
| | 
| | 116.857
|
|-
| | 
| | 
| | 
| | 
| | 11/113
| | 116.814
|
|-
| | 
| | 
| | 
| | 
| | 
| | 116.7725
|
|-
| | 
| | 
| | 
| | 7/72
| | 
| | 116.667
|
|-
| | 
| | 
| | 
| | 
| | 10/103
| | 116.505
|
|-
| | 
| | 3/31
| | 
| | 
| | 
| | 116.192
|46/43-based miracloid is around here.
|-
| | 
| | 
| | 
| | 
| | 11/114
| | 115.7895
|
|-
| | 
| | 
| | 
| | 
| | 
| | 115.763
|
|-
| | 
| | 
| | 
| | 8/83
| | 
| | 115.663
|
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|
|-
| | 
| | 
| | 
| | 
| | 13/135
| | 115.556
|
|-
| | 
| | 
| | 
| | 
| | 
| | 115.507
|
|-
| | 
| | 
| | 5/52
| | 
| | 
| | 115.385
|Tricesimoprimal miracloid (using 31/29 as a gen) is around here.
|-
| | 
| | 
| | 
| | 
| | 12/125
| | 115.2
|
|-
| | 
| | 
| | 
| | 7/73
| | 
| | 115.0865
|
|-
| | 
| | 
| | 
| | 
| | 9/94
| | 114.894
|
|-
| | 2/21
| | 
| | 
| | 
| | 
| | 114.286
|
|}</text>
      <sha1>9uecw8jznm1fewozxwglbkp4lmui59c</sha1>
    </revision>
    <revision>
      <id>137596</id>
      <parentid>136607</parentid>
      <timestamp>2024-02-29T23:15:09Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Added todo-cleanup</comment>
      <origin>137596</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1810" sha1="ayig8e8doxbgr95quf52m04nhf82a1g" xml:space="preserve">{{Todo|cleanup|inline=1|text=Replace scale tree, transfer entries}}{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 11
| Equalized = 2
| Collapsed = 1
| Pattern = sLsLsLsLsLsLsLsLsLsLs
}}
{{MOS intro|Other Names=miracloid}}
This is the MOS which splits its large steps 1-1-1-1-1-1-1-1-1-2 between its small steps. It is the simplest MOS which is really useful for Miracle temperament, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1). Its diatonic semitone generator is no smaller than 2/21edo (114.286 cents). [[User:Eliora|Eliora]] has proposed the name ''miracloid'' for this pattern.

{| class="wikitable"
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|
|-
| | 
| | 
| | 
| | 
| | 6/61
| | 118.033
|
|-
| | 
| | 
| | 
| | 5/51
| | 
| | 117.647
|
|-
| | 
| | 
| | 
| | 
| | 9/92
| | 117.391
|
|-
| | 
| | 
| | 
| | 
| | 
| | 117.171
|
|-
| | 
| | 
| | 4/41
| | 
| | 
| | 117.073
|
|-
| | 
| | 
| | 
| | 
| | 
| | 116.857
|
|-
| | 
| | 
| | 
| | 
| | 11/113
| | 116.814
|
|-
| | 
| | 
| | 
| | 
| | 
| | 116.7725
|
|-
| | 
| | 
| | 
| | 7/72
| | 
| | 116.667
|
|-
| | 
| | 
| | 
| | 
| | 10/103
| | 116.505
|
|-
| | 
| | 3/31
| | 
| | 
| | 
| | 116.192
|46/43-based miracloid is around here.
|-
| | 
| | 
| | 
| | 
| | 11/114
| | 115.7895
|
|-
| | 
| | 
| | 
| | 
| | 
| | 115.763
|
|-
| | 
| | 
| | 
| | 8/83
| | 
| | 115.663
|
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|
|-
| | 
| | 
| | 
| | 
| | 13/135
| | 115.556
|
|-
| | 
| | 
| | 
| | 
| | 
| | 115.507
|
|-
| | 
| | 
| | 5/52
| | 
| | 
| | 115.385
|Tricesimoprimal miracloid (using 31/29 as a gen) is around here.
|-
| | 
| | 
| | 
| | 
| | 12/125
| | 115.2
|
|-
| | 
| | 
| | 
| | 7/73
| | 
| | 115.0865
|
|-
| | 
| | 
| | 
| | 
| | 9/94
| | 114.894
|
|-
| | 2/21
| | 
| | 
| | 
| | 
| | 114.286
|
|}</text>
      <sha1>ayig8e8doxbgr95quf52m04nhf82a1g</sha1>
    </revision>
    <revision>
      <id>137599</id>
      <parentid>137596</parentid>
      <timestamp>2024-03-01T00:15:22Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Rewrite intro, transferred scale tree info, added clarify to intro text</comment>
      <origin>137599</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="565" sha1="tfd6ctn59g0b2i56ba4b7nnf9zw1jt9" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 11
| Equalized = 2
| Collapsed = 1
| Pattern = sLsLsLsLsLsLsLsLsLsLs
}}
{{MOS intro|Other Names=miracloid}}
This is the simplest MOS for which [[miracle]] temperament can be used{{Clarify}}, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1).

[[User:Eliora|Eliora]] has proposed the name ''miracloid'' for this pattern.

{{Scale tree|Comments=3/2: Approximate range for using [[31/29]] as a generator;
2/1: Approximate range for using [[46/43]] as a generator}}</text>
      <sha1>tfd6ctn59g0b2i56ba4b7nnf9zw1jt9</sha1>
    </revision>
    <revision>
      <id>137600</id>
      <parentid>137599</parentid>
      <timestamp>2024-03-01T00:15:35Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <minor/>
      <comment>a space</comment>
      <origin>137600</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="563" sha1="fwcp5tpajuv2126wgz4mfrogqrlt75y" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 11
| Equalized = 2
| Collapsed = 1
| Pattern = sLsLsLsLsLsLsLsLsLsLs
}}
{{MOS intro|Other Names=miracloid}}
This is the simplest MOS for which [[miracle]] temperament can be used{{Clarify}}, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1).
[[User:Eliora|Eliora]] has proposed the name ''miracloid'' for this pattern.
{{Scale tree|Comments=3/2: Approximate range for using [[31/29]] as a generator;
2/1: Approximate range for using [[46/43]] as a generator}}</text>
      <sha1>fwcp5tpajuv2126wgz4mfrogqrlt75y</sha1>
    </revision>
    <revision>
      <id>170546</id>
      <parentid>137600</parentid>
      <timestamp>2024-12-15T06:41:05Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Modes intervals headings</comment>
      <origin>170546</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="644" sha1="peo2vjhoantp8mpgbikw88jatqlhqpo" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 11
| Equalized = 2
| Collapsed = 1
| Pattern = sLsLsLsLsLsLsLsLsLsLs
}}
{{MOS intro|Other Names=miracloid}}

This is the simplest MOS for which [[miracle]] temperament can be used{{Clarify}}, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1).
[[User:Eliora|Eliora]] has proposed the name ''miracloid'' for this pattern.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree|Comments=3/2: Approximate range for using [[31/29]] as a generator;
2/1: Approximate range for using [[46/43]] as a generator}}</text>
      <sha1>peo2vjhoantp8mpgbikw88jatqlhqpo</sha1>
    </revision>
    <revision>
      <id>170547</id>
      <parentid>170546</parentid>
      <timestamp>2024-12-15T06:41:24Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>-modes</comment>
      <origin>170547</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="617" sha1="3tzo3xtojo7rvx9c8dvhewcwaarudv6" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 11
| Equalized = 2
| Collapsed = 1
| Pattern = sLsLsLsLsLsLsLsLsLsLs
}}
{{MOS intro|Other Names=miracloid}}

This is the simplest MOS for which [[miracle]] temperament can be used{{Clarify}}, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1).
[[User:Eliora|Eliora]] has proposed the name ''miracloid'' for this pattern.

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree|Comments=3/2: Approximate range for using [[31/29]] as a generator;
2/1: Approximate range for using [[46/43]] as a generator}}</text>
      <sha1>3tzo3xtojo7rvx9c8dvhewcwaarudv6</sha1>
    </revision>
    <revision>
      <id>183722</id>
      <parentid>170547</parentid>
      <timestamp>2025-02-28T16:32:48Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183722</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="623" sha1="ory0n0fcfmg8dsf2hy07yu6rqqgivkz" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 11
| Equalized = 2
| Collapsed = 1
| Pattern = sLsLsLsLsLsLsLsLsLsLs
}}
{{MOS intro|Other Names=miracloid}}

This is the simplest MOS for which [[miracle]] temperament can be used{{Clarify}}, placing the low estimate for the boundary of "practicality" at 41edo (L:s = 3:1).
[[User:Eliora|Eliora]] has proposed the name ''miracloid'' for this pattern.

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 3/2 = Approximate range for using [[31/29]] as a generator
| 2/1 = Approximate range for using [[46/43]] as a generator
}}</text>
      <sha1>ory0n0fcfmg8dsf2hy07yu6rqqgivkz</sha1>
    </revision>
  </page>
  <page>
    <title>10L 1s</title>
    <ns>0</ns>
    <id>44</id>
    <revision>
      <id>44</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>44</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1201" sha1="dogpjjw84i95yi6m8sqnrvcs12pwhoy" xml:space="preserve">This MOS, having a generator of a diatonic semitone between 1/11edo (109.09 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.

{| class="wikitable"
|-
| | 1/11
| | 
| | 
| | 
| | 
| | 109.091
|-
| | 
| | 
| | 
| | 
| | 5/54
| | 111.111
|-
| | 
| | 
| | 
| | 4/43
| | 
| | 111.628
|-
| | 
| | 
| | 
| | 
| | 7/75
| | 112
|-
| | 
| | 
| | 3/32
| | 
| | 
| | 112.5
|-
| | 
| | 
| | 
| | 
| | 
| | 112.818
|-
| | 
| | 
| | 
| | 
| | 8/85
| | 112.941
|-
| | 
| | 
| | 
| | 
| | 
| | 113.15
|-
| | 
| | 
| | 
| | 5/53
| | 
| | 113.2075
|-
| | 
| | 
| | 
| | 
| | 
| | 113.45
|-
| | 
| | 
| | 
| | 
| | 7/74
| | 113.5135
|-
| | 
| | 2/21
| | 
| | 
| | 
| | 114.286
|-
| | 
| | 
| | 
| | 
| | 7/73
| | 115.0685
|-
| | 
| | 
| | 
| | 5/52
| | 
| | 115.308
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|-
| | 
| | 
| | 
| | 
| | 8/83
| | 115.663
|-
| | 
| | 
| | 
| | 
| | 
| | 115.742
|-
| | 
| | 
| | 3/31
| | 
| | 
| | 116.129
|-
| | 
| | 
| | 
| | 
| | 
| | 116.298
|-
| | 
| | 
| | 
| | 
| | 7/72
| | 116.667
|-
| | 
| | 
| | 
| | 4/41
| | 
| | 117.073
|-
| | 
| | 
| | 
| | 
| | 5/51
| | 117.647
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}</text>
      <sha1>dogpjjw84i95yi6m8sqnrvcs12pwhoy</sha1>
    </revision>
    <revision>
      <id>4187</id>
      <timestamp>2015-11-09T15:10:01Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 565776209 - Original comment: **</comment>
      <origin>4187</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9487" sha1="8bfp990tz3wkeqduz9hvy2teq3gr9bz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-09 15:10:01 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;565776209&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, having a generator of a diatonic semitone between 1/11edo (109.09 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.
|| 1/11 ||   ||   ||   ||   || 109.091 ||
||   ||   ||   ||   || 5/54 || 111.111 ||
||   ||   ||   || 4/43 ||   || 111.628 ||
||   ||   ||   ||   || 7/75 || 112 ||
||   ||   || 3/32 ||   ||   || 112.5 ||
||   ||   ||   ||   ||   || 112.818 ||
||   ||   ||   ||   || 8/85 || 112.941 ||
||   ||   ||   ||   ||   || 113.15 ||
||   ||   ||   || 5/53 ||   || 113.2075 ||
||   ||   ||   ||   ||   || 113.45 ||
||   ||   ||   ||   || 7/74 || 113.5135 ||
||   || 2/21 ||   ||   ||   || 114.286 ||
||   ||   ||   ||   || 7/73 || 115.0685 ||
||   ||   ||   || 5/52 ||   || 115.308 ||
||   ||   ||   ||   ||   || 115.585 ||
||   ||   ||   ||   || 8/83 || 115.663 ||
||   ||   ||   ||   ||   || 115.742 ||
||   ||   || 3/31 ||   ||   || 116.129 ||
||   ||   ||   ||   ||   || 116.298 ||
||   ||   ||   ||   || 7/72 || 116.667 ||
||   ||   ||   || 4/41 ||   || 117.073 ||
||   ||   ||   ||   || 5/51 || 117.647 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 1s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, having a generator of a diatonic semitone between 1/11edo (109.09 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;111.111&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;111.628&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;112&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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      <sha1>8bfp990tz3wkeqduz9hvy2teq3gr9bz</sha1>
    </revision>
    <revision>
      <id>4188</id>
      <timestamp>2015-05-23T17:12:26Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 551976552 - Original comment: **</comment>
      <origin>4188</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8215" sha1="7n46htgybfloo32rrsrqjwck7ceqike" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-23 17:12:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;551976552&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, having a generator of a diatonic semitone between 1/11edo (109 1/11 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.
|| 1/11 ||   ||   ||   ||   || 109 1/11 ||
||   ||   ||   ||   || 5/54 || 111 1/9 ||
||   ||   ||   || 4/43 ||   || 111.627907 ||
||   ||   ||   ||   || 7/75 || 112 ||
||   ||   || 3/32 ||   ||   || 112.5 ||
||   ||   ||   ||   || 8/85 || 112.9411765 ||
||   ||   ||   || 5/53 ||   || 113.207547 ||
||   ||   ||   ||   || 7/74 || 113 19/37 ||
||   || 2/21 ||   ||   ||   || 114 2/7 ||
||   ||   ||   ||   || 7/73 || 115.068493 ||
||   ||   ||   || 5/52 ||   || 115 4/13 ||
||   ||   ||   ||   || 8/83 || 115.662651 ||
||   ||   ||   ||   ||   || 115.742087 ||
||   ||   || 3/31 ||   ||   || 116.129032 ||
||   ||   ||   ||   ||   || 116.298116 ||
||   ||   ||   ||   || 7/72 || 116 2/3 ||
||   ||   ||   || 4/41 ||   || 117 3/41 ||
||   ||   ||   ||   || 5/51 || 117.647059 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 1s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, having a generator of a diatonic semitone between 1/11edo (109 1/11 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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      <sha1>7n46htgybfloo32rrsrqjwck7ceqike</sha1>
    </revision>
    <revision>
      <id>4189</id>
      <timestamp>2015-02-11T20:13:23Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540685400 - Original comment: **</comment>
      <origin>4189</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8215" sha1="9j226v2p1mvlao3mouvd6rrhapdtbze" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-11 20:13:23 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540685400&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, having a generator of a diatonic semitone between 1/11edo (109 1/11 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.
|| 1/11 ||   ||   ||   ||   || 109 1/11 ||
||   ||   ||   ||   || 5/54 || 111 1/9 ||
||   ||   ||   || 4/43 ||   || 111.627907 ||
||   ||   ||   ||   || 7/75 || 112 ||
||   ||   || 3/32 ||   ||   || 112.5 ||
||   ||   ||   ||   || 8/85 || 112.9411765 ||
||   ||   ||   || 5/53 ||   || 113.207547 ||
||   ||   ||   ||   || 7/74 || 113 19/37 ||
||   || 2/21 ||   ||   ||   || 114 2/7 ||
||   ||   ||   ||   || 7/73 || 115.068493 ||
||   ||   ||   || 5/52 ||   || 115 4/13 ||
||   ||   ||   ||   || 8/83 || 115.662651 ||
||   ||   ||   ||   ||   || 116.094261 ||
||   ||   || 3/31 ||   ||   || 116.129032 ||
||   ||   ||   ||   ||   || 116.163504 ||
||   ||   ||   ||   || 7/72 || 116 2/3 ||
||   ||   ||   || 4/41 ||   || 117 3/41 ||
||   ||   ||   ||   || 5/51 || 117.647059 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 1s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, having a generator of a diatonic semitone between 1/11edo (109 1/11 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109 1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;112.5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;113.207547&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/74&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;113 19/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115 4/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;7/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;117 3/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;117.647059&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;1/10&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9j226v2p1mvlao3mouvd6rrhapdtbze</sha1>
    </revision>
    <revision>
      <id>4190</id>
      <timestamp>2012-10-28T23:32:19Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 377208692 - Original comment: **</comment>
      <origin>4190</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7533" sha1="93lulde4nyoqqi6kmy9odoiuol1krtp" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-10-28 23:32:19 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;377208692&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, having a generator of a diatonic semitone between 1/11edo (109 1/11 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.
|| 1/11 ||   ||   ||   ||   || 109 1/11 ||
||   ||   ||   ||   || 5/54 || 111 1/9 ||
||   ||   ||   || 4/43 ||   || 111.627907 ||
||   ||   ||   ||   || 7/75 || 112 ||
||   ||   || 3/32 ||   ||   || 112.5 ||
||   ||   ||   ||   || 8/85 || 112.9411765 ||
||   ||   ||   || 5/53 ||   || 113.207547 ||
||   ||   ||   ||   || 7/74 || 113 19/37 ||
||   || 2/21 ||   ||   ||   || 114 2/7 ||
||   ||   ||   ||   || 7/73 || 115.068493 ||
||   ||   ||   || 5/52 ||   || 115 4/13 ||
||   ||   ||   ||   || 8/83 || 115.662651 ||
||   ||   || 3/31 ||   ||   || 116.129032 ||
||   ||   ||   ||   || 7/72 || 116 2/3 ||
||   ||   ||   || 4/41 ||   || 117 3/41 ||
||   ||   ||   ||   || 5/51 || 117.647059 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 1s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, having a generator of a diatonic semitone between 1/11edo (109 1/11 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/54&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/32&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/74&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;113 19/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;115 4/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/31&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;117 3/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>93lulde4nyoqqi6kmy9odoiuol1krtp</sha1>
    </revision>
    <revision>
      <id>64709</id>
      <parentid>44</parentid>
      <timestamp>2021-03-27T19:47:54Z</timestamp>
      <contributor>
        <username>Flirora</username>
        <id>236</id>
      </contributor>
      <comment>+infobox</comment>
      <origin>64709</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1337" sha1="m4569gvdtu4p8ug02lvatomdeqhvczo" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Paucitonic = 1
| Pattern = LLLLLLLLLLs
}}

This MOS, having a generator of a diatonic semitone between 1/11edo (109.09 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.

{| class="wikitable"
|-
| | 1/11
| | 
| | 
| | 
| | 
| | 109.091
|-
| | 
| | 
| | 
| | 
| | 5/54
| | 111.111
|-
| | 
| | 
| | 
| | 4/43
| | 
| | 111.628
|-
| | 
| | 
| | 
| | 
| | 7/75
| | 112
|-
| | 
| | 
| | 3/32
| | 
| | 
| | 112.5
|-
| | 
| | 
| | 
| | 
| | 
| | 112.818
|-
| | 
| | 
| | 
| | 
| | 8/85
| | 112.941
|-
| | 
| | 
| | 
| | 
| | 
| | 113.15
|-
| | 
| | 
| | 
| | 5/53
| | 
| | 113.2075
|-
| | 
| | 
| | 
| | 
| | 
| | 113.45
|-
| | 
| | 
| | 
| | 
| | 7/74
| | 113.5135
|-
| | 
| | 2/21
| | 
| | 
| | 
| | 114.286
|-
| | 
| | 
| | 
| | 
| | 7/73
| | 115.0685
|-
| | 
| | 
| | 
| | 5/52
| | 
| | 115.308
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|-
| | 
| | 
| | 
| | 
| | 8/83
| | 115.663
|-
| | 
| | 
| | 
| | 
| | 
| | 115.742
|-
| | 
| | 
| | 3/31
| | 
| | 
| | 116.129
|-
| | 
| | 
| | 
| | 
| | 
| | 116.298
|-
| | 
| | 
| | 
| | 
| | 7/72
| | 116.667
|-
| | 
| | 
| | 
| | 4/41
| | 
| | 117.073
|-
| | 
| | 
| | 
| | 
| | 5/51
| | 117.647
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}</text>
      <sha1>m4569gvdtu4p8ug02lvatomdeqhvczo</sha1>
    </revision>
    <revision>
      <id>68613</id>
      <parentid>64709</parentid>
      <timestamp>2021-04-24T19:59:11Z</timestamp>
      <contributor>
        <username>Keenan Pepper</username>
        <id>25</id>
      </contributor>
      <minor/>
      <comment>[[Category:Abstract MOS patterns]]</comment>
      <origin>68613</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1373" sha1="ejn57rvwooa5aqkldyez2fccdua7bai" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Paucitonic = 1
| Pattern = LLLLLLLLLLs
}}

This MOS, having a generator of a diatonic semitone between 1/11edo (109.09 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.

{| class="wikitable"
|-
| | 1/11
| | 
| | 
| | 
| | 
| | 109.091
|-
| | 
| | 
| | 
| | 
| | 5/54
| | 111.111
|-
| | 
| | 
| | 
| | 4/43
| | 
| | 111.628
|-
| | 
| | 
| | 
| | 
| | 7/75
| | 112
|-
| | 
| | 
| | 3/32
| | 
| | 
| | 112.5
|-
| | 
| | 
| | 
| | 
| | 
| | 112.818
|-
| | 
| | 
| | 
| | 
| | 8/85
| | 112.941
|-
| | 
| | 
| | 
| | 
| | 
| | 113.15
|-
| | 
| | 
| | 
| | 5/53
| | 
| | 113.2075
|-
| | 
| | 
| | 
| | 
| | 
| | 113.45
|-
| | 
| | 
| | 
| | 
| | 7/74
| | 113.5135
|-
| | 
| | 2/21
| | 
| | 
| | 
| | 114.286
|-
| | 
| | 
| | 
| | 
| | 7/73
| | 115.0685
|-
| | 
| | 
| | 
| | 5/52
| | 
| | 115.308
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|-
| | 
| | 
| | 
| | 
| | 8/83
| | 115.663
|-
| | 
| | 
| | 
| | 
| | 
| | 115.742
|-
| | 
| | 
| | 3/31
| | 
| | 
| | 116.129
|-
| | 
| | 
| | 
| | 
| | 
| | 116.298
|-
| | 
| | 
| | 
| | 
| | 7/72
| | 116.667
|-
| | 
| | 
| | 
| | 4/41
| | 
| | 117.073
|-
| | 
| | 
| | 
| | 
| | 5/51
| | 117.647
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}

[[Category:Abstract MOS patterns]]</text>
      <sha1>ejn57rvwooa5aqkldyez2fccdua7bai</sha1>
    </revision>
    <revision>
      <id>93945</id>
      <parentid>68613</parentid>
      <timestamp>2022-08-05T03:16:43Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Abstract MOS patterns]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>93945</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1337" sha1="m4569gvdtu4p8ug02lvatomdeqhvczo" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Paucitonic = 1
| Pattern = LLLLLLLLLLs
}}

This MOS, having a generator of a diatonic semitone between 1/11edo (109.09 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.

{| class="wikitable"
|-
| | 1/11
| | 
| | 
| | 
| | 
| | 109.091
|-
| | 
| | 
| | 
| | 
| | 5/54
| | 111.111
|-
| | 
| | 
| | 
| | 4/43
| | 
| | 111.628
|-
| | 
| | 
| | 
| | 
| | 7/75
| | 112
|-
| | 
| | 
| | 3/32
| | 
| | 
| | 112.5
|-
| | 
| | 
| | 
| | 
| | 
| | 112.818
|-
| | 
| | 
| | 
| | 
| | 8/85
| | 112.941
|-
| | 
| | 
| | 
| | 
| | 
| | 113.15
|-
| | 
| | 
| | 
| | 5/53
| | 
| | 113.2075
|-
| | 
| | 
| | 
| | 
| | 
| | 113.45
|-
| | 
| | 
| | 
| | 
| | 7/74
| | 113.5135
|-
| | 
| | 2/21
| | 
| | 
| | 
| | 114.286
|-
| | 
| | 
| | 
| | 
| | 7/73
| | 115.0685
|-
| | 
| | 
| | 
| | 5/52
| | 
| | 115.308
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|-
| | 
| | 
| | 
| | 
| | 8/83
| | 115.663
|-
| | 
| | 
| | 
| | 
| | 
| | 115.742
|-
| | 
| | 
| | 3/31
| | 
| | 
| | 116.129
|-
| | 
| | 
| | 
| | 
| | 
| | 116.298
|-
| | 
| | 
| | 
| | 
| | 7/72
| | 116.667
|-
| | 
| | 
| | 
| | 4/41
| | 
| | 117.073
|-
| | 
| | 
| | 
| | 
| | 5/51
| | 117.647
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}</text>
      <sha1>m4569gvdtu4p8ug02lvatomdeqhvczo</sha1>
    </revision>
    <revision>
      <id>94157</id>
      <parentid>93945</parentid>
      <timestamp>2022-08-13T08:42:38Z</timestamp>
      <contributor>
        <username>Frostburn</username>
        <id>326</id>
      </contributor>
      <comment>Add name.</comment>
      <origin>94157</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1347" sha1="20nnblbo1oiuiyjddt3wq4idjeehu55" xml:space="preserve">{{Infobox MOS
| Name = miraculoid
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Paucitonic = 1
| Pattern = LLLLLLLLLLs
}}

This MOS, having a generator of a diatonic semitone between 1/11edo (109.09 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.

{| class="wikitable"
|-
| | 1/11
| | 
| | 
| | 
| | 
| | 109.091
|-
| | 
| | 
| | 
| | 
| | 5/54
| | 111.111
|-
| | 
| | 
| | 
| | 4/43
| | 
| | 111.628
|-
| | 
| | 
| | 
| | 
| | 7/75
| | 112
|-
| | 
| | 
| | 3/32
| | 
| | 
| | 112.5
|-
| | 
| | 
| | 
| | 
| | 
| | 112.818
|-
| | 
| | 
| | 
| | 
| | 8/85
| | 112.941
|-
| | 
| | 
| | 
| | 
| | 
| | 113.15
|-
| | 
| | 
| | 
| | 5/53
| | 
| | 113.2075
|-
| | 
| | 
| | 
| | 
| | 
| | 113.45
|-
| | 
| | 
| | 
| | 
| | 7/74
| | 113.5135
|-
| | 
| | 2/21
| | 
| | 
| | 
| | 114.286
|-
| | 
| | 
| | 
| | 
| | 7/73
| | 115.0685
|-
| | 
| | 
| | 
| | 5/52
| | 
| | 115.308
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|-
| | 
| | 
| | 
| | 
| | 8/83
| | 115.663
|-
| | 
| | 
| | 
| | 
| | 
| | 115.742
|-
| | 
| | 
| | 3/31
| | 
| | 
| | 116.129
|-
| | 
| | 
| | 
| | 
| | 
| | 116.298
|-
| | 
| | 
| | 
| | 
| | 7/72
| | 116.667
|-
| | 
| | 
| | 
| | 4/41
| | 
| | 117.073
|-
| | 
| | 
| | 
| | 
| | 5/51
| | 117.647
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}</text>
      <sha1>20nnblbo1oiuiyjddt3wq4idjeehu55</sha1>
    </revision>
    <revision>
      <id>94499</id>
      <parentid>94157</parentid>
      <timestamp>2022-08-17T09:16:10Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94499</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1346" sha1="rjiw5hymugmg73k44y0h8hlzqyv9asq" xml:space="preserve">{{Infobox MOS
| Name = miraculoid
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLs
}}

This MOS, having a generator of a diatonic semitone between 1/11edo (109.09 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.

{| class="wikitable"
|-
| | 1/11
| | 
| | 
| | 
| | 
| | 109.091
|-
| | 
| | 
| | 
| | 
| | 5/54
| | 111.111
|-
| | 
| | 
| | 
| | 4/43
| | 
| | 111.628
|-
| | 
| | 
| | 
| | 
| | 7/75
| | 112
|-
| | 
| | 
| | 3/32
| | 
| | 
| | 112.5
|-
| | 
| | 
| | 
| | 
| | 
| | 112.818
|-
| | 
| | 
| | 
| | 
| | 8/85
| | 112.941
|-
| | 
| | 
| | 
| | 
| | 
| | 113.15
|-
| | 
| | 
| | 
| | 5/53
| | 
| | 113.2075
|-
| | 
| | 
| | 
| | 
| | 
| | 113.45
|-
| | 
| | 
| | 
| | 
| | 7/74
| | 113.5135
|-
| | 
| | 2/21
| | 
| | 
| | 
| | 114.286
|-
| | 
| | 
| | 
| | 
| | 7/73
| | 115.0685
|-
| | 
| | 
| | 
| | 5/52
| | 
| | 115.308
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|-
| | 
| | 
| | 
| | 
| | 8/83
| | 115.663
|-
| | 
| | 
| | 
| | 
| | 
| | 115.742
|-
| | 
| | 
| | 3/31
| | 
| | 
| | 116.129
|-
| | 
| | 
| | 
| | 
| | 
| | 116.298
|-
| | 
| | 
| | 
| | 
| | 7/72
| | 116.667
|-
| | 
| | 
| | 
| | 4/41
| | 
| | 117.073
|-
| | 
| | 
| | 
| | 
| | 5/51
| | 117.647
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}</text>
      <sha1>rjiw5hymugmg73k44y0h8hlzqyv9asq</sha1>
    </revision>
    <revision>
      <id>94500</id>
      <parentid>94499</parentid>
      <timestamp>2022-08-17T09:16:24Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <comment>not named in tamnams</comment>
      <origin>94500</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1336" sha1="3zg852533u30u4o88rmnm5k6fh1sbtq" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLs
}}

This MOS, having a generator of a diatonic semitone between 1/11edo (109.09 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.

{| class="wikitable"
|-
| | 1/11
| | 
| | 
| | 
| | 
| | 109.091
|-
| | 
| | 
| | 
| | 
| | 5/54
| | 111.111
|-
| | 
| | 
| | 
| | 4/43
| | 
| | 111.628
|-
| | 
| | 
| | 
| | 
| | 7/75
| | 112
|-
| | 
| | 
| | 3/32
| | 
| | 
| | 112.5
|-
| | 
| | 
| | 
| | 
| | 
| | 112.818
|-
| | 
| | 
| | 
| | 
| | 8/85
| | 112.941
|-
| | 
| | 
| | 
| | 
| | 
| | 113.15
|-
| | 
| | 
| | 
| | 5/53
| | 
| | 113.2075
|-
| | 
| | 
| | 
| | 
| | 
| | 113.45
|-
| | 
| | 
| | 
| | 
| | 7/74
| | 113.5135
|-
| | 
| | 2/21
| | 
| | 
| | 
| | 114.286
|-
| | 
| | 
| | 
| | 
| | 7/73
| | 115.0685
|-
| | 
| | 
| | 
| | 5/52
| | 
| | 115.308
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|-
| | 
| | 
| | 
| | 
| | 8/83
| | 115.663
|-
| | 
| | 
| | 
| | 
| | 
| | 115.742
|-
| | 
| | 
| | 3/31
| | 
| | 
| | 116.129
|-
| | 
| | 
| | 
| | 
| | 
| | 116.298
|-
| | 
| | 
| | 
| | 
| | 7/72
| | 116.667
|-
| | 
| | 
| | 
| | 4/41
| | 
| | 117.073
|-
| | 
| | 
| | 
| | 
| | 5/51
| | 117.647
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}</text>
      <sha1>3zg852533u30u4o88rmnm5k6fh1sbtq</sha1>
    </revision>
    <revision>
      <id>101010</id>
      <parentid>94500</parentid>
      <timestamp>2023-01-06T16:12:39Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Copying from [[Category:Abstract MOS patterns]] to [[Category:11-tone scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>101010</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1365" sha1="kkzi1iql8tzg6s2fl633q3jmif8yhjo" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLs
}}

This MOS, having a generator of a diatonic semitone between 1/11edo (109.09 cents) and 1/10edo (120), functions as the superdiatonic scale of the Miracle temperment and its unnamed close neighbors.

{| class="wikitable"
|-
| | 1/11
| | 
| | 
| | 
| | 
| | 109.091
|-
| | 
| | 
| | 
| | 
| | 5/54
| | 111.111
|-
| | 
| | 
| | 
| | 4/43
| | 
| | 111.628
|-
| | 
| | 
| | 
| | 
| | 7/75
| | 112
|-
| | 
| | 
| | 3/32
| | 
| | 
| | 112.5
|-
| | 
| | 
| | 
| | 
| | 
| | 112.818
|-
| | 
| | 
| | 
| | 
| | 8/85
| | 112.941
|-
| | 
| | 
| | 
| | 
| | 
| | 113.15
|-
| | 
| | 
| | 
| | 5/53
| | 
| | 113.2075
|-
| | 
| | 
| | 
| | 
| | 
| | 113.45
|-
| | 
| | 
| | 
| | 
| | 7/74
| | 113.5135
|-
| | 
| | 2/21
| | 
| | 
| | 
| | 114.286
|-
| | 
| | 
| | 
| | 
| | 7/73
| | 115.0685
|-
| | 
| | 
| | 
| | 5/52
| | 
| | 115.308
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|-
| | 
| | 
| | 
| | 
| | 8/83
| | 115.663
|-
| | 
| | 
| | 
| | 
| | 
| | 115.742
|-
| | 
| | 
| | 3/31
| | 
| | 
| | 116.129
|-
| | 
| | 
| | 
| | 
| | 
| | 116.298
|-
| | 
| | 
| | 
| | 
| | 7/72
| | 116.667
|-
| | 
| | 
| | 
| | 4/41
| | 
| | 117.073
|-
| | 
| | 
| | 
| | 
| | 5/51
| | 117.647
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}

[[Category:11-tone scales]]</text>
      <sha1>kkzi1iql8tzg6s2fl633q3jmif8yhjo</sha1>
    </revision>
    <revision>
      <id>135319</id>
      <parentid>101010</parentid>
      <timestamp>2024-02-12T10:34:18Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Intro &amp; modes</comment>
      <origin>135319</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1329" sha1="g113h051n8genyoxazkhdhl7og63klr" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLs
}}
{{MOS intro}}
This MOS functions as the superdiatonic scale of the [[Miracle]] temperment and its unnamed close neighbors.

==Modes==
{{MOS modes}}

==Scale tree==
{| class="wikitable"
|-
| | 1/11
| | 
| | 
| | 
| | 
| | 109.091
|-
| | 
| | 
| | 
| | 
| | 5/54
| | 111.111
|-
| | 
| | 
| | 
| | 4/43
| | 
| | 111.628
|-
| | 
| | 
| | 
| | 
| | 7/75
| | 112
|-
| | 
| | 
| | 3/32
| | 
| | 
| | 112.5
|-
| | 
| | 
| | 
| | 
| | 
| | 112.818
|-
| | 
| | 
| | 
| | 
| | 8/85
| | 112.941
|-
| | 
| | 
| | 
| | 
| | 
| | 113.15
|-
| | 
| | 
| | 
| | 5/53
| | 
| | 113.2075
|-
| | 
| | 
| | 
| | 
| | 
| | 113.45
|-
| | 
| | 
| | 
| | 
| | 7/74
| | 113.5135
|-
| | 
| | 2/21
| | 
| | 
| | 
| | 114.286
|-
| | 
| | 
| | 
| | 
| | 7/73
| | 115.0685
|-
| | 
| | 
| | 
| | 5/52
| | 
| | 115.308
|-
| | 
| | 
| | 
| | 
| | 
| | 115.585
|-
| | 
| | 
| | 
| | 
| | 8/83
| | 115.663
|-
| | 
| | 
| | 
| | 
| | 
| | 115.742
|-
| | 
| | 
| | 3/31
| | 
| | 
| | 116.129
|-
| | 
| | 
| | 
| | 
| | 
| | 116.298
|-
| | 
| | 
| | 
| | 
| | 7/72
| | 116.667
|-
| | 
| | 
| | 
| | 4/41
| | 
| | 117.073
|-
| | 
| | 
| | 
| | 
| | 5/51
| | 117.647
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}

[[Category:11-tone scales]]</text>
      <sha1>g113h051n8genyoxazkhdhl7og63klr</sha1>
    </revision>
    <revision>
      <id>136315</id>
      <parentid>135319</parentid>
      <timestamp>2024-02-19T19:56:08Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>/* Scale tree */ Replaced scale tree</comment>
      <origin>136315</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="340" sha1="kkfoo4g0n1ziod58jeo97q9c0tv7g8l" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLs
}}
{{MOS intro}}
This MOS functions as the superdiatonic scale of the [[Miracle]] temperment and its unnamed close neighbors.

==Modes==
{{MOS modes}}

==Scale tree==
{{Scale tree}}
[[Category:11-tone scales]]</text>
      <sha1>kkfoo4g0n1ziod58jeo97q9c0tv7g8l</sha1>
    </revision>
    <revision>
      <id>137815</id>
      <parentid>136315</parentid>
      <timestamp>2024-03-01T19:20:12Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Added clarify tag to intro; added miracle to scale tree</comment>
      <origin>137815</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="375" sha1="dgew4mez55zmd3nngnkgjnulgq4jgjy" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLs
}}
{{MOS intro}}
This MOS functions as the superdiatonic scale{{Clarify}} of [[Miracle]] temperment.

==Modes==
{{MOS modes}}

==Scale tree==
{{Scale tree|Comments=7/2: [[Miracle]];
2/1: Simplest tuning for miracle}}
[[Category:11-tone scales]]</text>
      <sha1>dgew4mez55zmd3nngnkgjnulgq4jgjy</sha1>
    </revision>
    <revision>
      <id>170550</id>
      <parentid>137815</parentid>
      <timestamp>2024-12-15T06:43:13Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>== Intervals == {{MOS intervals}}</comment>
      <origin>170550</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="410" sha1="spbclzh6oy8tuevg6q04179r80hpyi7" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLs
}}
{{MOS intro}}
This MOS functions as the superdiatonic scale{{Clarify}} of [[Miracle]] temperment.

==Modes==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

==Scale tree==
{{Scale tree|Comments=7/2: [[Miracle]];
2/1: Simplest tuning for miracle}}
[[Category:11-tone scales]]</text>
      <sha1>spbclzh6oy8tuevg6q04179r80hpyi7</sha1>
    </revision>
    <revision>
      <id>183723</id>
      <parentid>170550</parentid>
      <timestamp>2025-02-28T16:35:50Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183723</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="421" sha1="50a20hr94e56pa4btz63u1p42eiea0f" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLs
}}
{{MOS intro}}
This MOS functions as the superdiatonic scale{{Clarify}} of [[Miracle]] temperment.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 7/2 = [[Miracle]]
| 2/1 = Simplest tuning for miracle
}}

[[Category:11-tone scales]]</text>
      <sha1>50a20hr94e56pa4btz63u1p42eiea0f</sha1>
    </revision>
    <revision>
      <id>185255</id>
      <parentid>183723</parentid>
      <timestamp>2025-03-10T11:12:51Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <comment>added name "secoric"</comment>
      <origin>185255</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="637" sha1="n35q8u95oqg1cl35qnbqs7vpgbaj5sd" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLs
}}
{{MOS intro}}
This MOS functions as the superdiatonic scale{{Clarify}} of [[Miracle]] temperament. Its generator range spans the interval range of [[secor]]s, and therefore the name '''secoric''' has been proposed for this MOS pattern independently by [[Praveen Venkataramana]] and [[User:Lériendil|Lériendil]].

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 7/2 = [[Miracle]]
| 2/1 = Simplest tuning for miracle
}}

[[Category:11-tone scales]]</text>
      <sha1>n35q8u95oqg1cl35qnbqs7vpgbaj5sd</sha1>
    </revision>
  </page>
  <page>
    <title>10L 2s</title>
    <ns>0</ns>
    <id>45</id>
    <revision>
      <id>45</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>45</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1810" sha1="dl118hegwzw3ushvny15rx66ukwnqwx" xml:space="preserve">The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]][12], which is simply [[2L_8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg_propriety|proper]], because there is only one small step per period.

{| class="wikitable"
|-
! colspan="4" | Generator
! | 
! | 
! | Cents
! | Comments
|-
| | 1\12
| | 
| | 
| | 
| | 
| | 
| | 100
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\70
| | 102.86
| | 
|-
| | 
| | 
| | 
| | 
| | 5\58
| | 
| | 103.45
| | 
|-
| | 
| | 
| | 
| | 4\46
| | 
| | 
| | 104.35
| style="text-align:center;" | 
|-
| | 
| | 
| | 3\34
| | 
| | 
| | 
| | 105.88
| style="text-align:center;" | Srutal/pajara is around here
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.45
| | 
|-
| | 
| | 
| | 
| | 
| | 8\90
| | 
| | 106.67
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.8
| | 
|-
| | 
| | 
| | 
| | 5\56
| | 
| | 
| | 107.14
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 107.58
| | 
|-
| | 
| | 2\22
| | 
| | 
| | 
| | 
| | 109.09
| style="text-align:center;" | Boundary of propriety for near-MOS

Optimal rank range (L/s=2/1) for MOS
|-
| | 
| | 
| | 
| | 5\54
| | 
| | 
| | 111.11
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.48
| | 
|-
| | 
| | 
| | 
| | 
| | 8\86
| | 
| | 111.63
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.78
| | 
|-
| | 
| | 
| | 3\32
| | 
| | 
| | 
| | 112.5
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 112.82
| | 
|-
| | 
| | 
| | 
| | 4\42
| | 
| | 
| | 114.29
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 5\52
| | 
| | 115.385
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\62
| | 116.13
| | 
|-
| | 1\10
| | 
| | 
| | 
| | 
| | 
| | 120
| style="text-align:center;" | 
|}</text>
      <sha1>dl118hegwzw3ushvny15rx66ukwnqwx</sha1>
    </revision>
    <revision>
      <id>4191</id>
      <timestamp>2015-11-10T18:26:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 565973661 - Original comment: **</comment>
      <origin>4191</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11957" sha1="903uaa236i38rmrzzzsut603vdswsb7" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-10 18:26:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;565973661&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic family|srutal/pajara]][12], which is simply [[2L 8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg propriety|proper]], because there is only one small step per period.

||||||||~ Generator ||~   ||~   ||~ Cents ||~ Comments ||
|| 1\12 ||   ||   ||   ||   ||   || 100 ||=   ||
||   ||   ||   ||   ||   || 6\70 || 102.86 ||   ||
||   ||   ||   ||   || 5\58 ||   || 103.45 ||   ||
||   ||   ||   || 4\46 ||   ||   || 104.35 ||=   ||
||   ||   || 3\34 ||   ||   ||   || 105.88 ||= Srutal/pajara is around here ||
||   ||   ||   ||   ||   ||   || 106.45 ||   ||
||   ||   ||   ||   || 8\90 ||   || 106.67 ||   ||
||   ||   ||   ||   ||   ||   || 106.8 ||   ||
||   ||   ||   || 5\56 ||   ||   || 107.14 ||=   ||
||   ||   ||   ||   ||   ||   || 107.58 ||   ||
||   || 2\22 ||   ||   ||   ||   || 109.09 ||= Boundary of propriety for near-MOS
Optimal rank range (L/s=2/1) for MOS ||
||   ||   ||   || 5\54 ||   ||   || 111.11 ||=   ||
||   ||   ||   ||   ||   ||   || 111.48 ||   ||
||   ||   ||   ||   || 8\86 ||   || 111.63 ||   ||
||   ||   ||   ||   ||   ||   || 111.78 ||   ||
||   ||   || 3\32 ||   ||   ||   || 112.5 ||=   ||
||   ||   ||   ||   ||   ||   || 112.82 ||   ||
||   ||   ||   || 4\42 ||   ||   || 114.29 ||=   ||
||   ||   ||   ||   || 5\52 ||   || 115.385 ||   ||
||   ||   ||   ||   ||   || 6\62 || 116.13 ||   ||
|| 1\10 ||   ||   ||   ||   ||   || 120 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The only significant harmonic entropy minimum with this MOS pattern is &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;srutal/pajara&amp;lt;/a&amp;gt;[12], which is simply &amp;lt;a class="wiki_link" href="/2L%208s"&amp;gt;decatonic&amp;lt;/a&amp;gt; with two more notes added in.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
All scales with this MOS pattern are &amp;lt;a class="wiki_link" href="/Rothenberg%20propriety"&amp;gt;proper&amp;lt;/a&amp;gt;, because there is only one small step per period.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th colspan="4"&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Comments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6\70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Srutal/pajara is around here&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8\90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106.67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106.8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety for near-MOS&amp;lt;br /&amp;gt;
Optimal rank range (L/s=2/1) for MOS&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;115.385&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;6\62&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>903uaa236i38rmrzzzsut603vdswsb7</sha1>
    </revision>
    <revision>
      <id>4192</id>
      <timestamp>2015-05-23T17:41:58Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 551977726 - Original comment: **</comment>
      <origin>4192</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9371" sha1="81q2eb7nuu8v269b9wqf29s06yk9has" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-23 17:41:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;551977726&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic family|srutal/pajara]][12], which is simply [[2L 8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg propriety|proper]], because there is only one small step per period.

||||||||~ Generator ||~   ||~   ||~ Cents ||~ Comments ||
|| 1\12 ||   ||   ||   ||   ||   || 100 ||=   ||
||   ||   ||   ||   ||   || 6\70 || 102.86 ||   ||
||   ||   ||   ||   || 5\58 ||   || 103.45 ||   ||
||   ||   ||   || 4\46 ||   ||   || 104.35 ||=   ||
||   ||   || 3\34 ||   ||   ||   || 105.88 ||= Srutal/pajara is around here ||
||   ||   ||   || 5\56 ||   ||   || 107.14 ||=   ||
||   || 2\22 ||   ||   ||   ||   || 109.09 ||= Boundary of propriety for near-MOS
Optimal rank range (L/s=2/1) for MOS ||
||   ||   ||   || 5\54 ||   ||   || 111.11 ||=   ||
||   ||   ||   ||   ||   ||   || 111.78 ||   ||
||   ||   || 3\32 ||   ||   ||   || 112.5 ||=   ||
||   ||   ||   ||   ||   ||   || 112.82 ||   ||
||   ||   ||   || 4\42 ||   ||   || 114.29 ||=   ||
||   ||   ||   ||   || 5\52 ||   || 115.385 ||   ||
||   ||   ||   ||   ||   || 6\62 || 116.13 ||   ||
|| 1\10 ||   ||   ||   ||   ||   || 120 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The only significant harmonic entropy minimum with this MOS pattern is &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;srutal/pajara&amp;lt;/a&amp;gt;[12], which is simply &amp;lt;a class="wiki_link" href="/2L%208s"&amp;gt;decatonic&amp;lt;/a&amp;gt; with two more notes added in.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
All scales with this MOS pattern are &amp;lt;a class="wiki_link" href="/Rothenberg%20propriety"&amp;gt;proper&amp;lt;/a&amp;gt;, because there is only one small step per period.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;th colspan="4"&amp;gt;Generator&amp;lt;br /&amp;gt;
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&amp;lt;/th&amp;gt;
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&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Comments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;6\70&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;102.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;103.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.35&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Srutal/pajara is around here&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety for near-MOS&amp;lt;br /&amp;gt;
Optimal rank range (L/s=2/1) for MOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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      <sha1>81q2eb7nuu8v269b9wqf29s06yk9has</sha1>
    </revision>
    <revision>
      <id>4193</id>
      <timestamp>2015-02-11T21:16:55Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540690628 - Original comment: **</comment>
      <origin>4193</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9375" sha1="7i1c9hrq7k82wg68g1j79jz3krny1ip" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-11 21:16:55 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540690628&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic family|srutal/pajara]][12], which is simply [[2L 8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg propriety|proper]], because there is only one small step per period.

||||||||~ Generator ||~   ||~   ||~ Cents ||~ Comments ||
|| 1\12 ||   ||   ||   ||   ||   || 100 ||=   ||
||   ||   ||   ||   ||   || 6\70 || 102.86 ||   ||
||   ||   ||   ||   || 5\58 ||   || 103.45 ||   ||
||   ||   ||   || 4\46 ||   ||   || 104.35 ||=   ||
||   ||   || 3\34 ||   ||   ||   || 105.88 ||= Srutal/pajara is around here ||
||   ||   ||   || 5\56 ||   ||   || 107.14 ||=   ||
||   || 2\22 ||   ||   ||   ||   || 109.09 ||= Boundary of propriety for near-MOS
Optimal rank range (L/s=2/1) for MOS ||
||   ||   ||   || 5\54 ||   ||   || 111.11 ||=   ||
||   ||   ||   ||   ||   ||   || 112.435 ||   ||
||   ||   || 3\32 ||   ||   ||   || 112.5 ||=   ||
||   ||   ||   ||   ||   ||   || 112.565 ||   ||
||   ||   ||   || 4\42 ||   ||   || 114.29 ||=   ||
||   ||   ||   ||   || 5\52 ||   || 115.385 ||   ||
||   ||   ||   ||   ||   || 6\62 || 116.13 ||   ||
|| 1\10 ||   ||   ||   ||   ||   || 120 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The only significant harmonic entropy minimum with this MOS pattern is &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;srutal/pajara&amp;lt;/a&amp;gt;[12], which is simply &amp;lt;a class="wiki_link" href="/2L%208s"&amp;gt;decatonic&amp;lt;/a&amp;gt; with two more notes added in.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
All scales with this MOS pattern are &amp;lt;a class="wiki_link" href="/Rothenberg%20propriety"&amp;gt;proper&amp;lt;/a&amp;gt;, because there is only one small step per period.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;th colspan="4"&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
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&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Comments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
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        &amp;lt;td&amp;gt;103.45&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;3\34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Srutal/pajara is around here&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;107.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety for near-MOS&amp;lt;br /&amp;gt;
Optimal rank range (L/s=2/1) for MOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;112.435&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;112.5&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>7i1c9hrq7k82wg68g1j79jz3krny1ip</sha1>
    </revision>
    <revision>
      <id>4194</id>
      <timestamp>2014-11-06T22:50:02Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 529922256 - Original comment: **</comment>
      <origin>4194</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8513" sha1="bedt9yl8muexrbu33p0c5575jz2l5r2" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2014-11-06 22:50:02 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;529922256&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic family|srutal/pajara]][12], which is simply [[2L 8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg propriety|proper]], because there is only one small step per period.

||||||||~ Generator ||~   ||~   ||~ Cents ||~ Comments ||
|| 1\12 ||   ||   ||   ||   ||   || 100 ||=   ||
||   ||   ||   ||   ||   || 6\70 || 102.86 ||   ||
||   ||   ||   ||   || 5\58 ||   || 103.45 ||   ||
||   ||   ||   || 4\46 ||   ||   || 104.35 ||=   ||
||   ||   || 3\34 ||   ||   ||   || 105.88 ||= Srutal/pajara is around here ||
||   ||   ||   || 5\56 ||   ||   || 107.14 ||=   ||
||   || 2\22 ||   ||   ||   ||   || 109.09 ||= Boundary of propriety for near-MOS
Optimal rank range (L/s=2/1) for MOS ||
||   ||   ||   || 5\54 ||   ||   || 111.11 ||=   ||
||   ||   || 3\32 ||   ||   ||   || 112.5 ||=   ||
||   ||   ||   || 4\42 ||   ||   || 114.29 ||=   ||
||   ||   ||   ||   || 5\52 ||   || 115.385 ||   ||
||   ||   ||   ||   ||   || 6\62 || 116.13 ||   ||
|| 1\10 ||   ||   ||   ||   ||   || 120 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The only significant harmonic entropy minimum with this MOS pattern is &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;srutal/pajara&amp;lt;/a&amp;gt;[12], which is simply &amp;lt;a class="wiki_link" href="/2L%208s"&amp;gt;decatonic&amp;lt;/a&amp;gt; with two more notes added in.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
All scales with this MOS pattern are &amp;lt;a class="wiki_link" href="/Rothenberg%20propriety"&amp;gt;proper&amp;lt;/a&amp;gt;, because there is only one small step per period.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;105.88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Srutal/pajara is around here&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety for near-MOS&amp;lt;br /&amp;gt;
Optimal rank range (L/s=2/1) for MOS&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>bedt9yl8muexrbu33p0c5575jz2l5r2</sha1>
    </revision>
    <revision>
      <id>4195</id>
      <timestamp>2012-10-05T01:30:07Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;keenanpepper</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 370445872 - Original comment: **</comment>
      <origin>4195</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8517" sha1="eboogwzv6dh0g2r2kgwo7d1h9qnl5zp" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on &lt;tt&gt;2012-10-05 01:30:07 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;370445872&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic family|srutal/pajara]][12], which is simply [[2L 8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg propriety|proper]], because there is only one small step per period.

||||||||~ Generator ||~   ||~   ||~ Cents ||~ Comments ||
|| 1\12 ||   ||   ||   ||   ||   || 100 ||=   ||
||   ||   ||   ||   ||   || 6\70 || 102.86 ||   ||
||   ||   ||   ||   || 5\58 ||   || 103.45 ||   ||
||   ||   ||   || 4\46 ||   ||   || 104.35 ||=   ||
||   ||   || 3\34 ||   ||   ||   || 105.88 ||= Srutal/pajara is around here ||
||   ||   ||   || 5\56 ||   ||   || 107.14 ||=   ||
||   || 2\22 ||   ||   ||   ||   || 109.09 ||= Boundary of propriety for near-MOS
Optimal rank range (L/s=1/2) for MOS ||
||   ||   ||   || 5\54 ||   ||   || 111.11 ||=   ||
||   ||   || 3\32 ||   ||   ||   || 112.5 ||=   ||
||   ||   ||   || 4\42 ||   ||   || 114.29 ||=   ||
||   ||   ||   ||   || 5\52 ||   || 115.385 ||   ||
||   ||   ||   ||   ||   || 6\62 || 116.13 ||   ||
|| 1\10 ||   ||   ||   ||   ||   || 120 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The only significant harmonic entropy minimum with this MOS pattern is &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;srutal/pajara&amp;lt;/a&amp;gt;[12], which is simply &amp;lt;a class="wiki_link" href="/2L%208s"&amp;gt;decatonic&amp;lt;/a&amp;gt; with two more notes added in.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
All scales with this MOS pattern are &amp;lt;a class="wiki_link" href="/Rothenberg%20propriety"&amp;gt;proper&amp;lt;/a&amp;gt;, because there is only one small step per period.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


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        &amp;lt;th colspan="4"&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
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&amp;lt;/th&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;6\70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.86&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;105.88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Srutal/pajara is around here&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety for near-MOS&amp;lt;br /&amp;gt;
Optimal rank range (L/s=1/2) for MOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;111.11&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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      <sha1>eboogwzv6dh0g2r2kgwo7d1h9qnl5zp</sha1>
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    <revision>
      <id>4196</id>
      <timestamp>2012-06-26T03:42:22Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 348036912 - Original comment: **</comment>
      <origin>4196</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17334" sha1="d1s86xx4h76ips7iqzhggb6lx3rubn0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-06-26 03:42:22 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;348036912&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic family|srutal/pajara]][12], which is simply [[2L 8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg propriety|proper]], because there is only one small step per period.

||||||||~ Generator ||~   ||~   ||~ Cents ||~ Comments ||
|| 1\12 ||   ||   ||   ||   ||   || 100 ||=   ||
||   ||   ||   ||   ||   || 6\70 || 102.86 ||   ||
||   ||   ||   ||   || 5\58 ||   || 103.45 ||   ||
||   ||   ||   ||   ||   || 9\104 || 103.84 ||   ||
||   ||   ||   || 4\46 ||   ||   || 104.35 ||=   ||
||   ||   ||   ||   ||   || 11\126 || 104.76 ||   ||
||   ||   ||   ||   || 7\80 ||   || 105 ||   ||
||   ||   ||   ||   ||   || 10\114 || 105.26 ||   ||
||   ||   || 3\34 ||   ||   ||   || 105.88 ||= Srutal/pajara is around here ||
||   ||   ||   ||   ||   || 11\124 || 106.45 ||   ||
||   ||   ||   ||   || 8\90 ||   || 106.__6__ ||   ||
||   ||   ||   ||   ||   || 13\146 || 106.85 ||   ||
||   ||   ||   || 5\56 ||   ||   || 107.14 ||=   ||
||   ||   ||   ||   ||   || 12\134 || 107.46 ||   ||
||   ||   ||   ||   || 7\78 ||   || 107.69 ||   ||
||   ||   ||   ||   ||   || 9\100 || 108 ||   ||
||   || 2\22 ||   ||   ||   ||   || 109.__09__ ||= Boundary of propriety for near-MOS
Optimal rank range (L/s=1/2) for MOS ||
||   ||   ||   ||   ||   || 9\98 || 110.2 ||   ||
||   ||   ||   ||   || 7\76 ||   || 110.53 ||   ||
||   ||   ||   ||   ||   || 12\130 || 110.77 ||   ||
||   ||   ||   || 5\54 ||   ||   || 111.__1__ ||=   ||
||   ||   ||   ||   ||   || 13\140 || 111.43 ||   ||
||   ||   ||   ||   || 8\86 ||   || 111.63 ||   ||
||   ||   ||   ||   ||   || 11\118 || 111.86 ||   ||
||   ||   || 3\32 ||   ||   ||   || 112.5 ||=   ||
||   ||   ||   ||   ||   || 10\106 || 113.21 ||   ||
||   ||   ||   ||   || 7\74 ||   || 113.51 ||   ||
||   ||   ||   ||   ||   || 11\116 || 113.79 ||   ||
||   ||   ||   || 4\42 ||   ||   || 114.29 ||=   ||
||   ||   ||   ||   ||   || 9\94 || 114.89 ||   ||
||   ||   ||   ||   || 5\52 ||   || 115.385 ||   ||
||   ||   ||   ||   ||   || 6\62 || 116.13 ||   ||
|| 1\10 ||   ||   ||   ||   ||   || 120 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The only significant harmonic entropy minimum with this MOS pattern is &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;srutal/pajara&amp;lt;/a&amp;gt;[12], which is simply &amp;lt;a class="wiki_link" href="/2L%208s"&amp;gt;decatonic&amp;lt;/a&amp;gt; with two more notes added in.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
All scales with this MOS pattern are &amp;lt;a class="wiki_link" href="/Rothenberg%20propriety"&amp;gt;proper&amp;lt;/a&amp;gt;, because there is only one small step per period.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th colspan="4"&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
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&amp;lt;/th&amp;gt;
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        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Comments&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;103.45&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;104.35&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;104.76&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;7\80&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;10\114&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;105.26&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;105.88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Srutal/pajara is around here&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;106.&amp;lt;u&amp;gt;6&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;107.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;107.69&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;109.&amp;lt;u&amp;gt;09&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety for near-MOS&amp;lt;br /&amp;gt;
Optimal rank range (L/s=1/2) for MOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;9\98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;110.2&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;110.53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;12\130&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;110.77&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;111.&amp;lt;u&amp;gt;1&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;111.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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      <sha1>d1s86xx4h76ips7iqzhggb6lx3rubn0</sha1>
    </revision>
    <revision>
      <id>4197</id>
      <timestamp>2012-06-26T03:41:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 348036802 - Original comment: **</comment>
      <origin>4197</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17334" sha1="b2u4jaiz1gjib5cbjh7oyb5f4k21i43" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-06-26 03:41:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;348036802&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic family|srutal/pajara]][12], which is simply [[2L 8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg propriety|proper]], because there is only one small step per period.

||||||||~ Generator ||~   ||~   ||~ Cents ||~ Comments ||
|| 1\12 ||   ||   ||   ||   ||   || 100 ||=   ||
||   ||   ||   ||   ||   || 6\70 || 102.86 ||   ||
||   ||   ||   ||   || 5\58 ||   || 103.45 ||   ||
||   ||   ||   ||   ||   || 9\104 || 103.84 ||   ||
||   ||   ||   || 4\46 ||   ||   || 104.35 ||=   ||
||   ||   ||   ||   ||   || 11\126 || 104.76 ||   ||
||   ||   ||   ||   || 7/80 ||   || 105 ||   ||
||   ||   ||   ||   ||   || 10\114 || 105.26 ||   ||
||   ||   || 3\34 ||   ||   ||   || 105.88 ||= Srutal/pajara is around here ||
||   ||   ||   ||   ||   || 11\124 || 106.45 ||   ||
||   ||   ||   ||   || 8\90 ||   || 106.__6__ ||   ||
||   ||   ||   ||   ||   || 13\146 || 106.85 ||   ||
||   ||   ||   || 5\56 ||   ||   || 107.14 ||=   ||
||   ||   ||   ||   ||   || 12\134 || 107.46 ||   ||
||   ||   ||   ||   || 7\78 ||   || 107.69 ||   ||
||   ||   ||   ||   ||   || 9\100 || 108 ||   ||
||   || 2\22 ||   ||   ||   ||   || 109.__09__ ||= Boundary of propriety for near-MOS
Optimal rank range (L/s=1/2) for MOS ||
||   ||   ||   ||   ||   || 9\98 || 110.2 ||   ||
||   ||   ||   ||   || 7\76 ||   || 110.53 ||   ||
||   ||   ||   ||   ||   || 12\130 || 110.77 ||   ||
||   ||   ||   || 5\54 ||   ||   || 111.__1__ ||=   ||
||   ||   ||   ||   ||   || 13\140 || 111.43 ||   ||
||   ||   ||   ||   || 8\86 ||   || 111.63 ||   ||
||   ||   ||   ||   ||   || 11\118 || 111.86 ||   ||
||   ||   || 3\32 ||   ||   ||   || 112.5 ||=   ||
||   ||   ||   ||   ||   || 10\106 || 113.21 ||   ||
||   ||   ||   ||   || 7\74 ||   || 113.51 ||   ||
||   ||   ||   ||   ||   || 11\116 || 113.79 ||   ||
||   ||   ||   || 4\42 ||   ||   || 114.29 ||=   ||
||   ||   ||   ||   ||   || 9\94 || 114.89 ||   ||
||   ||   ||   ||   || 5\52 ||   || 115.385 ||   ||
||   ||   ||   ||   ||   || 6\62 || 116.13 ||   ||
|| 1\10 ||   ||   ||   ||   ||   || 120 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The only significant harmonic entropy minimum with this MOS pattern is &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;srutal/pajara&amp;lt;/a&amp;gt;[12], which is simply &amp;lt;a class="wiki_link" href="/2L%208s"&amp;gt;decatonic&amp;lt;/a&amp;gt; with two more notes added in.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
All scales with this MOS pattern are &amp;lt;a class="wiki_link" href="/Rothenberg%20propriety"&amp;gt;proper&amp;lt;/a&amp;gt;, because there is only one small step per period.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


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        &amp;lt;td&amp;gt;105.88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Srutal/pajara is around here&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;106.&amp;lt;u&amp;gt;6&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;107.69&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;109.&amp;lt;u&amp;gt;09&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety for near-MOS&amp;lt;br /&amp;gt;
Optimal rank range (L/s=1/2) for MOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;110.2&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;110.53&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;12\130&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;110.77&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;111.&amp;lt;u&amp;gt;1&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;111.63&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;112.5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;113.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;11\116&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;114.29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;115.385&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>b2u4jaiz1gjib5cbjh7oyb5f4k21i43</sha1>
    </revision>
    <revision>
      <id>4198</id>
      <timestamp>2012-06-25T22:10:30Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 347988884 - Original comment: **</comment>
      <origin>4198</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17328" sha1="gqymld7aqg12tr4vq78k1sqmlpmhfz1" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-06-25 22:10:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;347988884&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic family|srutal/pajara]][12], which is simply [[2L 8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg propriety|proper]], because there is only one small step per period.

||||||||~ Generator ||~   ||~   ||~ Cents ||~ Comments ||
|| 1\12 ||   ||   ||   ||   ||   || 100 ||=   ||
||   ||   ||   ||   ||   || 6/70 || 102.86 ||   ||
||   ||   ||   ||   || 5\58 ||   || 103.45 ||   ||
||   ||   ||   ||   ||   || 9/104 || 103.84 ||   ||
||   ||   ||   || 4\46 ||   ||   || 104.35 ||=   ||
||   ||   ||   ||   ||   || 11/126 || 104.76 ||   ||
||   ||   ||   ||   || 7/80 ||   || 105 ||   ||
||   ||   ||   ||   ||   || 10\114 || 105.26 ||   ||
||   ||   || 3\34 ||   ||   ||   || 105.88 ||= Srutal/pajara is around here ||
||   ||   ||   ||   ||   || 11\124 || 106.45 ||   ||
||   ||   ||   ||   || 8\90 ||   || 106.__6__ ||   ||
||   ||   ||   ||   ||   || 13\146 || 106.85 ||   ||
||   ||   ||   || 5\56 ||   ||   || 107.14 ||=   ||
||   ||   ||   ||   ||   || 12/134 || 107.46 ||   ||
||   ||   ||   ||   || 7\78 ||   || 107.69 ||   ||
||   ||   ||   ||   ||   || 9/100 || 108 ||   ||
||   || 2\22 ||   ||   ||   ||   || 109.__09__ ||= Boundary of propriety for near-MOS
Optimal rank range (L/s=1/2) for MOS ||
||   ||   ||   ||   ||   || 9\98 || 110.2 ||   ||
||   ||   ||   ||   || 7/76 ||   || 110.53 ||   ||
||   ||   ||   ||   ||   || 12\130 || 110.77 ||   ||
||   ||   ||   || 5/54 ||   ||   || 111.__1__ ||=   ||
||   ||   ||   ||   ||   || 13\140 || 111.43 ||   ||
||   ||   ||   ||   || 8/86 ||   || 111.63 ||   ||
||   ||   ||   ||   ||   || 11/118 || 111.86 ||   ||
||   ||   || 3\32 ||   ||   ||   || 112.5 ||=   ||
||   ||   ||   ||   ||   || 10\106 || 113.21 ||   ||
||   ||   ||   ||   || 7/74 ||   || 113.51 ||   ||
||   ||   ||   ||   ||   || 11\116 || 113.79 ||   ||
||   ||   ||   || 4/42 ||   ||   || 114.29 ||=   ||
||   ||   ||   ||   ||   || 9/94 || 114.89 ||   ||
||   ||   ||   ||   || 5/52 ||   || 115.385 ||   ||
||   ||   ||   ||   ||   || 6\62 || 116.13 ||   ||
|| 1\10 ||   ||   ||   ||   ||   || 120 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The only significant harmonic entropy minimum with this MOS pattern is &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;srutal/pajara&amp;lt;/a&amp;gt;[12], which is simply &amp;lt;a class="wiki_link" href="/2L%208s"&amp;gt;decatonic&amp;lt;/a&amp;gt; with two more notes added in.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
All scales with this MOS pattern are &amp;lt;a class="wiki_link" href="/Rothenberg%20propriety"&amp;gt;proper&amp;lt;/a&amp;gt;, because there is only one small step per period.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;th colspan="4"&amp;gt;Generator&amp;lt;br /&amp;gt;
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        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
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        &amp;lt;th&amp;gt;Comments&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;102.86&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;103.45&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;Srutal/pajara is around here&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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Optimal rank range (L/s=1/2) for MOS&amp;lt;br /&amp;gt;
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      <sha1>gqymld7aqg12tr4vq78k1sqmlpmhfz1</sha1>
    </revision>
    <revision>
      <id>4199</id>
      <timestamp>2012-04-25T13:42:41Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 325149726 - Original comment: **</comment>
      <origin>4199</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5679" sha1="hw8ghmv42cvf1ja2wu38qrkgubuhw48" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2012-04-25 13:42:41 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;325149726&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic family|srutal/pajara]][12], which is simply [[2L 8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg propriety|proper]], because there is only one small step per period.

||||||||~ Generator ||~ Cents ||~ Comments ||
|| 1\12 ||   ||   ||   || 100 ||=   ||
||   ||   ||   || 4\46 || 104.35 ||=   ||
||   ||   || 3\34 ||   || 105.88 ||= Srutal/pajara is around here ||
||   ||   ||   || 5\56 || 107.14 ||   ||
||   || 2\22 ||   ||   || 109.09 ||= Boundary of propriety for near-MOS
Optimal rank range (L/s=1/2) for MOS ||
||   ||   ||   || 5/54 || 111.11 ||   ||
||   ||   || 3\32 ||   || 112.5 ||   ||
||   ||   ||   || 4/42 || 114.29 ||   ||
|| 1\10 ||   ||   ||   || 120 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The only significant harmonic entropy minimum with this MOS pattern is &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;srutal/pajara&amp;lt;/a&amp;gt;[12], which is simply &amp;lt;a class="wiki_link" href="/2L%208s"&amp;gt;decatonic&amp;lt;/a&amp;gt; with two more notes added in.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
All scales with this MOS pattern are &amp;lt;a class="wiki_link" href="/Rothenberg%20propriety"&amp;gt;proper&amp;lt;/a&amp;gt;, because there is only one small step per period.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Srutal/pajara is around here&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety for near-MOS&amp;lt;br /&amp;gt;
Optimal rank range (L/s=1/2) for MOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;111.11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\32&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;112.5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;114.29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>hw8ghmv42cvf1ja2wu38qrkgubuhw48</sha1>
    </revision>
    <revision>
      <id>4200</id>
      <timestamp>2011-08-29T04:31:46Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 249050815 - Original comment: **</comment>
      <origin>4200</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4999" sha1="4iy4169stpiwbjkz4w9h9b7k9hsj5eh" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2011-08-29 04:31:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;249050815&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic family|srutal/pajara]][12], which is simply [[2L 8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg propriety|proper]], because there is only one small step per period.

||||||||~ Generator ||~ Cents ||~ Comments ||
|| 1\12 ||   ||   ||   || 100 ||=   ||
||   ||   ||   || 4\46 || 104.35 ||=   ||
||   ||   || 3\34 ||   || 105.88 ||= Srutal/pajara is around here ||
||   ||   ||   || 5\56 || 107.14 ||   ||
||   || 2\22 ||   ||   || 109.09 ||= Boundary of propriety for near-MOS
Optimal rank range (L/s=1/2) for MOS ||
||   ||   || 3\32 ||   || 112.5 ||   ||
|| 1\10 ||   ||   ||   || 120 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The only significant harmonic entropy minimum with this MOS pattern is &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;srutal/pajara&amp;lt;/a&amp;gt;[12], which is simply &amp;lt;a class="wiki_link" href="/2L%208s"&amp;gt;decatonic&amp;lt;/a&amp;gt; with two more notes added in.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
All scales with this MOS pattern are &amp;lt;a class="wiki_link" href="/Rothenberg%20propriety"&amp;gt;proper&amp;lt;/a&amp;gt;, because there is only one small step per period.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th colspan="4"&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Comments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
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        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Srutal/pajara is around here&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107.14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety for near-MOS&amp;lt;br /&amp;gt;
Optimal rank range (L/s=1/2) for MOS&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;112.5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>4iy4169stpiwbjkz4w9h9b7k9hsj5eh</sha1>
    </revision>
    <revision>
      <id>4201</id>
      <timestamp>2011-05-13T03:59:58Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;keenanpepper</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 228122066 - Original comment: **</comment>
      <origin>4201</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4690" sha1="213rn2bib36ob4tqc4hw5zq16m55dgn" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on &lt;tt&gt;2011-05-13 03:59:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;228122066&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic family|srutal/pajara]][12], which is simply [[2L 8s|decatonic]] with two more notes added in.

||||||||~ Generator ||~ Cents ||~ Comments ||
|| 1\12 ||   ||   ||   || 100 ||=   ||
||   ||   ||   || 4\46 || 104.35 ||=   ||
||   ||   || 3\34 ||   || 105.88 ||= Srutal/pajara is around here ||
||   ||   ||   || 5\56 || 107.14 ||   ||
||   || 2\22 ||   ||   || 109.09 ||= Boundary of propriety (generators
smaller than this are proper) ||
||   ||   || 3\32 ||   || 112.5 ||   ||
|| 1\10 ||   ||   ||   || 120 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The only significant harmonic entropy minimum with this MOS pattern is &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;srutal/pajara&amp;lt;/a&amp;gt;[12], which is simply &amp;lt;a class="wiki_link" href="/2L%208s"&amp;gt;decatonic&amp;lt;/a&amp;gt; with two more notes added in.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th colspan="4"&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1\10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>213rn2bib36ob4tqc4hw5zq16m55dgn</sha1>
    </revision>
    <revision>
      <id>68621</id>
      <parentid>45</parentid>
      <timestamp>2021-04-24T20:00:13Z</timestamp>
      <contributor>
        <username>Keenan Pepper</username>
        <id>25</id>
      </contributor>
      <minor/>
      <comment>[[Category:Abstract MOS patterns]]</comment>
      <origin>68621</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1846" sha1="pqtlgvto272gyu9683xwb9qxx33n0oy" xml:space="preserve">The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]][12], which is simply [[2L_8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg_propriety|proper]], because there is only one small step per period.

{| class="wikitable"
|-
! colspan="4" | Generator
! | 
! | 
! | Cents
! | Comments
|-
| | 1\12
| | 
| | 
| | 
| | 
| | 
| | 100
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\70
| | 102.86
| | 
|-
| | 
| | 
| | 
| | 
| | 5\58
| | 
| | 103.45
| | 
|-
| | 
| | 
| | 
| | 4\46
| | 
| | 
| | 104.35
| style="text-align:center;" | 
|-
| | 
| | 
| | 3\34
| | 
| | 
| | 
| | 105.88
| style="text-align:center;" | Srutal/pajara is around here
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.45
| | 
|-
| | 
| | 
| | 
| | 
| | 8\90
| | 
| | 106.67
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.8
| | 
|-
| | 
| | 
| | 
| | 5\56
| | 
| | 
| | 107.14
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 107.58
| | 
|-
| | 
| | 2\22
| | 
| | 
| | 
| | 
| | 109.09
| style="text-align:center;" | Boundary of propriety for near-MOS

Optimal rank range (L/s=2/1) for MOS
|-
| | 
| | 
| | 
| | 5\54
| | 
| | 
| | 111.11
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.48
| | 
|-
| | 
| | 
| | 
| | 
| | 8\86
| | 
| | 111.63
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.78
| | 
|-
| | 
| | 
| | 3\32
| | 
| | 
| | 
| | 112.5
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 112.82
| | 
|-
| | 
| | 
| | 
| | 4\42
| | 
| | 
| | 114.29
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 5\52
| | 
| | 115.385
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\62
| | 116.13
| | 
|-
| | 1\10
| | 
| | 
| | 
| | 
| | 
| | 120
| style="text-align:center;" | 
|}

[[Category:Abstract MOS patterns]]</text>
      <sha1>pqtlgvto272gyu9683xwb9qxx33n0oy</sha1>
    </revision>
    <revision>
      <id>76475</id>
      <parentid>68621</parentid>
      <timestamp>2021-08-17T19:40:14Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>76475</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1836" sha1="dk3yheghwa49600o32dtpkc7myjlaj5" xml:space="preserve">The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]][12], which is simply [[2L_8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg_propriety|proper]], because there is only one small step per period.

{| class="wikitable"
|-
! colspan="6" | Generator
! | Cents
! | Comments
|-
| | 1\12
| | 
| | 
| | 
| | 
| | 
| | 100
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\70
| | 102.86
| | 
|-
| | 
| | 
| | 
| | 
| | 5\58
| | 
| | 103.45
| | 
|-
| | 
| | 
| | 
| | 4\46
| | 
| | 
| | 104.35
| style="text-align:center;" | 
|-
| | 
| | 
| | 3\34
| | 
| | 
| | 
| | 105.88
| style="text-align:center;" | Srutal/pajara is around here
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.45
| | 
|-
| | 
| | 
| | 
| | 
| | 8\90
| | 
| | 106.67
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.8
| | 
|-
| | 
| | 
| | 
| | 5\56
| | 
| | 
| | 107.14
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 107.58
| | 
|-
| | 
| | 2\22
| | 
| | 
| | 
| | 
| | 109.09
| style="text-align:center;" | Boundary of propriety for near-MOS

Optimal rank range (L/s=2/1) for MOS
|-
| | 
| | 
| | 
| | 5\54
| | 
| | 
| | 111.11
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.48
| | 
|-
| | 
| | 
| | 
| | 
| | 8\86
| | 
| | 111.63
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.78
| | 
|-
| | 
| | 
| | 3\32
| | 
| | 
| | 
| | 112.5
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 112.82
| | 
|-
| | 
| | 
| | 
| | 4\42
| | 
| | 
| | 114.29
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 5\52
| | 
| | 115.385
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\62
| | 116.13
| | 
|-
| | 1\10
| | 
| | 
| | 
| | 
| | 
| | 120
| style="text-align:center;" | 
|}

[[Category:Abstract MOS patterns]]</text>
      <sha1>dk3yheghwa49600o32dtpkc7myjlaj5</sha1>
    </revision>
    <revision>
      <id>93943</id>
      <parentid>76475</parentid>
      <timestamp>2022-08-05T03:16:43Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Abstract MOS patterns]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>93943</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1800" sha1="jm8gfytg3pjicreh9pal3zv985m3zu8" xml:space="preserve">The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]][12], which is simply [[2L_8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg_propriety|proper]], because there is only one small step per period.

{| class="wikitable"
|-
! colspan="6" | Generator
! | Cents
! | Comments
|-
| | 1\12
| | 
| | 
| | 
| | 
| | 
| | 100
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\70
| | 102.86
| | 
|-
| | 
| | 
| | 
| | 
| | 5\58
| | 
| | 103.45
| | 
|-
| | 
| | 
| | 
| | 4\46
| | 
| | 
| | 104.35
| style="text-align:center;" | 
|-
| | 
| | 
| | 3\34
| | 
| | 
| | 
| | 105.88
| style="text-align:center;" | Srutal/pajara is around here
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.45
| | 
|-
| | 
| | 
| | 
| | 
| | 8\90
| | 
| | 106.67
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.8
| | 
|-
| | 
| | 
| | 
| | 5\56
| | 
| | 
| | 107.14
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 107.58
| | 
|-
| | 
| | 2\22
| | 
| | 
| | 
| | 
| | 109.09
| style="text-align:center;" | Boundary of propriety for near-MOS

Optimal rank range (L/s=2/1) for MOS
|-
| | 
| | 
| | 
| | 5\54
| | 
| | 
| | 111.11
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.48
| | 
|-
| | 
| | 
| | 
| | 
| | 8\86
| | 
| | 111.63
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.78
| | 
|-
| | 
| | 
| | 3\32
| | 
| | 
| | 
| | 112.5
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 112.82
| | 
|-
| | 
| | 
| | 
| | 4\42
| | 
| | 
| | 114.29
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 5\52
| | 
| | 115.385
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\62
| | 116.13
| | 
|-
| | 1\10
| | 
| | 
| | 
| | 
| | 
| | 120
| style="text-align:center;" | 
|}</text>
      <sha1>jm8gfytg3pjicreh9pal3zv985m3zu8</sha1>
    </revision>
    <revision>
      <id>94031</id>
      <parentid>93943</parentid>
      <timestamp>2022-08-05T03:47:10Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox, name still missing, misc. edits</comment>
      <origin>94031</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2051" sha1="1y7ltegdo441ie7jmqadkhq0slyirvv" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 2
| nLargeSteps = 10
| nSmallSteps = 2
| Equalized = 1
| Paucitonic = 1
| Pattern = LLLLLsLLLLLs
}}
The '''10L 2s''' (or '''5L 1s&lt;1\2&gt;''') [[MOS scale]] contains two [[5L 1s]] patterns per octave.

The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]][12], which is simply [[2L_8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg_propriety|proper]], because there is only one small step per period.

== Scale tree ==
{| class="wikitable"
|-
! colspan="6" | Generator
! | Cents
! | Comments
|-
| | 1\12
| | 
| | 
| | 
| | 
| | 
| | 100
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\70
| | 102.86
| | 
|-
| | 
| | 
| | 
| | 
| | 5\58
| | 
| | 103.45
| | 
|-
| | 
| | 
| | 
| | 4\46
| | 
| | 
| | 104.35
| style="text-align:center;" | 
|-
| | 
| | 
| | 3\34
| | 
| | 
| | 
| | 105.88
| style="text-align:center;" | Srutal/pajara is around here
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.45
| | 
|-
| | 
| | 
| | 
| | 
| | 8\90
| | 
| | 106.67
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.8
| | 
|-
| | 
| | 
| | 
| | 5\56
| | 
| | 
| | 107.14
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 107.58
| | 
|-
| | 
| | 2\22
| | 
| | 
| | 
| | 
| | 109.09
| style="text-align:center;" | Boundary of propriety for near-MOS

Optimal rank range (L/s=2/1) for MOS
|-
| | 
| | 
| | 
| | 5\54
| | 
| | 
| | 111.11
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.48
| | 
|-
| | 
| | 
| | 
| | 
| | 8\86
| | 
| | 111.63
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.78
| | 
|-
| | 
| | 
| | 3\32
| | 
| | 
| | 
| | 112.5
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 112.82
| | 
|-
| | 
| | 
| | 
| | 4\42
| | 
| | 
| | 114.29
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 5\52
| | 
| | 115.385
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\62
| | 116.13
| | 
|-
| | 1\10
| | 
| | 
| | 
| | 
| | 
| | 120
| style="text-align:center;" | 
|}</text>
      <sha1>1y7ltegdo441ie7jmqadkhq0slyirvv</sha1>
    </revision>
    <revision>
      <id>94197</id>
      <parentid>94031</parentid>
      <timestamp>2022-08-14T06:10:24Z</timestamp>
      <contributor>
        <username>Frostburn</username>
        <id>326</id>
      </contributor>
      <comment>Add name.</comment>
      <origin>94197</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2059" sha1="bis6i8h7hvwr9jidq4v54z832au5wjf" xml:space="preserve">{{Infobox MOS
| Name = pajaroid
| Periods = 2
| nLargeSteps = 10
| nSmallSteps = 2
| Equalized = 1
| Paucitonic = 1
| Pattern = LLLLLsLLLLLs
}}
The '''10L 2s''' (or '''5L 1s&lt;1\2&gt;''') [[MOS scale]] contains two [[5L 1s]] patterns per octave.

The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]][12], which is simply [[2L_8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg_propriety|proper]], because there is only one small step per period.

== Scale tree ==
{| class="wikitable"
|-
! colspan="6" | Generator
! | Cents
! | Comments
|-
| | 1\12
| | 
| | 
| | 
| | 
| | 
| | 100
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\70
| | 102.86
| | 
|-
| | 
| | 
| | 
| | 
| | 5\58
| | 
| | 103.45
| | 
|-
| | 
| | 
| | 
| | 4\46
| | 
| | 
| | 104.35
| style="text-align:center;" | 
|-
| | 
| | 
| | 3\34
| | 
| | 
| | 
| | 105.88
| style="text-align:center;" | Srutal/pajara is around here
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.45
| | 
|-
| | 
| | 
| | 
| | 
| | 8\90
| | 
| | 106.67
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.8
| | 
|-
| | 
| | 
| | 
| | 5\56
| | 
| | 
| | 107.14
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 107.58
| | 
|-
| | 
| | 2\22
| | 
| | 
| | 
| | 
| | 109.09
| style="text-align:center;" | Boundary of propriety for near-MOS

Optimal rank range (L/s=2/1) for MOS
|-
| | 
| | 
| | 
| | 5\54
| | 
| | 
| | 111.11
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.48
| | 
|-
| | 
| | 
| | 
| | 
| | 8\86
| | 
| | 111.63
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.78
| | 
|-
| | 
| | 
| | 3\32
| | 
| | 
| | 
| | 112.5
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 112.82
| | 
|-
| | 
| | 
| | 
| | 4\42
| | 
| | 
| | 114.29
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 5\52
| | 
| | 115.385
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\62
| | 116.13
| | 
|-
| | 1\10
| | 
| | 
| | 
| | 
| | 
| | 120
| style="text-align:center;" | 
|}</text>
      <sha1>bis6i8h7hvwr9jidq4v54z832au5wjf</sha1>
    </revision>
    <revision>
      <id>94501</id>
      <parentid>94197</parentid>
      <timestamp>2022-08-17T09:16:42Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94501</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2050" sha1="8ax2mmofnvwtnfkwcbwa4qfjeyki1i7" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 2
| nLargeSteps = 10
| nSmallSteps = 2
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLsLLLLLs
}}
The '''10L 2s''' (or '''5L 1s&lt;1\2&gt;''') [[MOS scale]] contains two [[5L 1s]] patterns per octave.

The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]][12], which is simply [[2L_8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg_propriety|proper]], because there is only one small step per period.

== Scale tree ==
{| class="wikitable"
|-
! colspan="6" | Generator
! | Cents
! | Comments
|-
| | 1\12
| | 
| | 
| | 
| | 
| | 
| | 100
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\70
| | 102.86
| | 
|-
| | 
| | 
| | 
| | 
| | 5\58
| | 
| | 103.45
| | 
|-
| | 
| | 
| | 
| | 4\46
| | 
| | 
| | 104.35
| style="text-align:center;" | 
|-
| | 
| | 
| | 3\34
| | 
| | 
| | 
| | 105.88
| style="text-align:center;" | Srutal/pajara is around here
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.45
| | 
|-
| | 
| | 
| | 
| | 
| | 8\90
| | 
| | 106.67
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.8
| | 
|-
| | 
| | 
| | 
| | 5\56
| | 
| | 
| | 107.14
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 107.58
| | 
|-
| | 
| | 2\22
| | 
| | 
| | 
| | 
| | 109.09
| style="text-align:center;" | Boundary of propriety for near-MOS

Optimal rank range (L/s=2/1) for MOS
|-
| | 
| | 
| | 
| | 5\54
| | 
| | 
| | 111.11
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.48
| | 
|-
| | 
| | 
| | 
| | 
| | 8\86
| | 
| | 111.63
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.78
| | 
|-
| | 
| | 
| | 3\32
| | 
| | 
| | 
| | 112.5
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 112.82
| | 
|-
| | 
| | 
| | 
| | 4\42
| | 
| | 
| | 114.29
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 5\52
| | 
| | 115.385
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\62
| | 116.13
| | 
|-
| | 1\10
| | 
| | 
| | 
| | 
| | 
| | 120
| style="text-align:center;" | 
|}</text>
      <sha1>8ax2mmofnvwtnfkwcbwa4qfjeyki1i7</sha1>
    </revision>
    <revision>
      <id>101019</id>
      <parentid>94501</parentid>
      <timestamp>2023-01-06T16:13:28Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Copying from [[Category:Abstract MOS patterns]] to [[Category:12-tone scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>101019</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2079" sha1="4gebmbtu9s9r9mcjk9k4gfsl26cmhd1" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 2
| nLargeSteps = 10
| nSmallSteps = 2
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLsLLLLLs
}}
The '''10L 2s''' (or '''5L 1s&lt;1\2&gt;''') [[MOS scale]] contains two [[5L 1s]] patterns per octave.

The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]][12], which is simply [[2L_8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg_propriety|proper]], because there is only one small step per period.

== Scale tree ==
{| class="wikitable"
|-
! colspan="6" | Generator
! | Cents
! | Comments
|-
| | 1\12
| | 
| | 
| | 
| | 
| | 
| | 100
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\70
| | 102.86
| | 
|-
| | 
| | 
| | 
| | 
| | 5\58
| | 
| | 103.45
| | 
|-
| | 
| | 
| | 
| | 4\46
| | 
| | 
| | 104.35
| style="text-align:center;" | 
|-
| | 
| | 
| | 3\34
| | 
| | 
| | 
| | 105.88
| style="text-align:center;" | Srutal/pajara is around here
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.45
| | 
|-
| | 
| | 
| | 
| | 
| | 8\90
| | 
| | 106.67
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.8
| | 
|-
| | 
| | 
| | 
| | 5\56
| | 
| | 
| | 107.14
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 107.58
| | 
|-
| | 
| | 2\22
| | 
| | 
| | 
| | 
| | 109.09
| style="text-align:center;" | Boundary of propriety for near-MOS

Optimal rank range (L/s=2/1) for MOS
|-
| | 
| | 
| | 
| | 5\54
| | 
| | 
| | 111.11
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.48
| | 
|-
| | 
| | 
| | 
| | 
| | 8\86
| | 
| | 111.63
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.78
| | 
|-
| | 
| | 
| | 3\32
| | 
| | 
| | 
| | 112.5
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 112.82
| | 
|-
| | 
| | 
| | 
| | 4\42
| | 
| | 
| | 114.29
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 5\52
| | 
| | 115.385
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\62
| | 116.13
| | 
|-
| | 1\10
| | 
| | 
| | 
| | 
| | 
| | 120
| style="text-align:center;" | 
|}

[[Category:12-tone scales]]</text>
      <sha1>4gebmbtu9s9r9mcjk9k4gfsl26cmhd1</sha1>
    </revision>
    <revision>
      <id>106775</id>
      <parentid>101019</parentid>
      <timestamp>2023-04-02T01:44:09Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>106775</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1960" sha1="ssijjyfanoj3pzvsvwigvcyhn0iowud" xml:space="preserve">{{Infobox MOS}}
The '''10L 2s''' (or '''5L 1s&lt;1\2&gt;''') [[MOS scale]] contains two [[5L 1s]] patterns per octave.

The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]][12], which is simply [[2L_8s|decatonic]] with two more notes added in.

All scales with this MOS pattern are [[Rothenberg_propriety|proper]], because there is only one small step per period.

== Scale tree ==
{| class="wikitable"
|-
! colspan="6" | Generator
! | Cents
! | Comments
|-
| | 1\12
| | 
| | 
| | 
| | 
| | 
| | 100
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\70
| | 102.86
| | 
|-
| | 
| | 
| | 
| | 
| | 5\58
| | 
| | 103.45
| | 
|-
| | 
| | 
| | 
| | 4\46
| | 
| | 
| | 104.35
| style="text-align:center;" | 
|-
| | 
| | 
| | 3\34
| | 
| | 
| | 
| | 105.88
| style="text-align:center;" | Srutal/pajara is around here
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.45
| | 
|-
| | 
| | 
| | 
| | 
| | 8\90
| | 
| | 106.67
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.8
| | 
|-
| | 
| | 
| | 
| | 5\56
| | 
| | 
| | 107.14
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 107.58
| | 
|-
| | 
| | 2\22
| | 
| | 
| | 
| | 
| | 109.09
| style="text-align:center;" | Boundary of propriety for near-MOS

Optimal rank range (L/s=2/1) for MOS
|-
| | 
| | 
| | 
| | 5\54
| | 
| | 
| | 111.11
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.48
| | 
|-
| | 
| | 
| | 
| | 
| | 8\86
| | 
| | 111.63
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.78
| | 
|-
| | 
| | 
| | 3\32
| | 
| | 
| | 
| | 112.5
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 112.82
| | 
|-
| | 
| | 
| | 
| | 4\42
| | 
| | 
| | 114.29
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 5\52
| | 
| | 115.385
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\62
| | 116.13
| | 
|-
| | 1\10
| | 
| | 
| | 
| | 
| | 
| | 120
| style="text-align:center;" | 
|}

[[Category:12-tone scales]]</text>
      <sha1>ssijjyfanoj3pzvsvwigvcyhn0iowud</sha1>
    </revision>
    <revision>
      <id>134205</id>
      <parentid>106775</parentid>
      <timestamp>2024-02-03T11:53:39Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Mos intro, rothenprop</comment>
      <origin>134205</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1755" sha1="bhlp6p6vv5kzrgd91vzn8lw6yvhxkuj" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]][12], which is simply [[2L_8s|decatonic]] with two more notes added in.
== Scale tree ==
{| class="wikitable"
|-
! colspan="6" | Generator
! | Cents
! | Comments
|-
| | 1\12
| | 
| | 
| | 
| | 
| | 
| | 100
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\70
| | 102.86
| | 
|-
| | 
| | 
| | 
| | 
| | 5\58
| | 
| | 103.45
| | 
|-
| | 
| | 
| | 
| | 4\46
| | 
| | 
| | 104.35
| style="text-align:center;" | 
|-
| | 
| | 
| | 3\34
| | 
| | 
| | 
| | 105.88
| style="text-align:center;" | Srutal/pajara is around here
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.45
| | 
|-
| | 
| | 
| | 
| | 
| | 8\90
| | 
| | 106.67
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 106.8
| | 
|-
| | 
| | 
| | 
| | 5\56
| | 
| | 
| | 107.14
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 107.58
| | 
|-
| | 
| | 2\22
| | 
| | 
| | 
| | 
| | 109.09
| style="text-align:center;" | Boundary of propriety for near-MOS

Optimal rank range (L/s=2/1) for MOS
|-
| | 
| | 
| | 
| | 5\54
| | 
| | 
| | 111.11
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.48
| | 
|-
| | 
| | 
| | 
| | 
| | 8\86
| | 
| | 111.63
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 111.78
| | 
|-
| | 
| | 
| | 3\32
| | 
| | 
| | 
| | 112.5
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 112.82
| | 
|-
| | 
| | 
| | 
| | 4\42
| | 
| | 
| | 114.29
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 5\52
| | 
| | 115.385
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 6\62
| | 116.13
| | 
|-
| | 1\10
| | 
| | 
| | 
| | 
| | 
| | 120
| style="text-align:center;" | 
|}

[[Category:12-tone scales]]</text>
      <sha1>bhlp6p6vv5kzrgd91vzn8lw6yvhxkuj</sha1>
    </revision>
    <revision>
      <id>136095</id>
      <parentid>134205</parentid>
      <timestamp>2024-02-18T23:43:29Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>/* Scale tree */ Replaced scale tree</comment>
      <origin>136095</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="313" sha1="7zt6a9wxz1wno7yx2wu4g72zcg0lwih" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]][12], which is simply [[2L_8s|decatonic]] with two more notes added in.
== Scale tree ==
{{Scale tree|
Comments=
3/2: ↕ [[Srutal]] / [[pajara]]
}}
[[Category:12-tone scales]]</text>
      <sha1>7zt6a9wxz1wno7yx2wu4g72zcg0lwih</sha1>
    </revision>
    <revision>
      <id>136221</id>
      <parentid>136095</parentid>
      <timestamp>2024-02-19T06:48:03Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <minor/>
      <comment>decatonic meant 2L 8s, so reworded intro</comment>
      <origin>136221</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="298" sha1="2nue2bto0l2fvw3h28k6onee5ja3cv4" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]], which appears as a 12-note scale of srutal/pajara[12].
== Scale tree ==
{{Scale tree|
Comments=
3/2: ↕ [[Srutal]] / [[pajara]]
}}
[[Category:12-tone scales]]</text>
      <sha1>2nue2bto0l2fvw3h28k6onee5ja3cv4</sha1>
    </revision>
    <revision>
      <id>170551</id>
      <parentid>136221</parentid>
      <timestamp>2024-12-15T06:44:13Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Add modes intervals</comment>
      <origin>170551</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="361" sha1="643hcok6c225kqohshtr6sdtwybo2gl" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]], which appears as a 12-note scale of srutal/pajara[12].

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree|
Comments=
3/2: ↕ [[Srutal]] / [[pajara]]
}}
[[Category:12-tone scales]]</text>
      <sha1>643hcok6c225kqohshtr6sdtwybo2gl</sha1>
    </revision>
    <revision>
      <id>183724</id>
      <parentid>170551</parentid>
      <timestamp>2025-02-28T16:36:30Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183724</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="363" sha1="6xs9fg3141auulkwdfl8yrsgdbxpck4" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]], which appears as a 12-note scale of srutal/pajara[12].

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 3/2 = ↕ [[Srutal]] / [[pajara]]
}}

[[Category:12-tone scales]]</text>
      <sha1>6xs9fg3141auulkwdfl8yrsgdbxpck4</sha1>
    </revision>
    <revision>
      <id>185261</id>
      <parentid>183724</parentid>
      <timestamp>2025-03-10T11:40:16Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <comment>added name "telluric"</comment>
      <origin>185261</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="536" sha1="0e1iiu17byuatftcg6uc34yij0mtnlo" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]], which appears as a 12-note scale of srutal/pajara[12]. The name '''telluric''' has been proposed for this scale by [[User:Lériendil|Lériendil]], after the personal use of "tellurian" for the LLLsLLLLLLsL [[MODMOS]] of 10L 2s.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 3/2 = ↕ [[Srutal]] / [[pajara]]
}}

[[Category:12-tone scales]]</text>
      <sha1>0e1iiu17byuatftcg6uc34yij0mtnlo</sha1>
    </revision>
    <revision>
      <id>196167</id>
      <parentid>185261</parentid>
      <timestamp>2025-05-14T12:53:43Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Scale tree */</comment>
      <origin>196167</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="541" sha1="hmi6617dvkjfhfoxnxkbyuv2gzrlzqc" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
The only significant harmonic entropy minimum with this MOS pattern is [[Diaschismic_family|srutal/pajara]], which appears as a 12-note scale of srutal/pajara[12]. The name '''telluric''' has been proposed for this scale by [[User:Lériendil|Lériendil]], after the personal use of "tellurian" for the LLLsLLLLLLsL [[MODMOS]] of 10L 2s.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 3/2 = ↕&amp;nbsp;[[Srutal]] / [[pajara]]
}}

[[Category:12-tone scales]]</text>
      <sha1>hmi6617dvkjfhfoxnxkbyuv2gzrlzqc</sha1>
    </revision>
  </page>
  <page>
    <title>10L 3s</title>
    <ns>0</ns>
    <id>46</id>
    <revision>
      <id>46</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>46</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5760" sha1="2a8pmcl1s70b6r9shnoagwgiqh2j2t9" xml:space="preserve">This is the chromatic scale of the Sephiroth temperament explicated on the page for [[3L_7s|3L 7s]]. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.

{| class="wikitable"
|-
! | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;(Circle of fifths)&lt;/span&gt;
! | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;(ascending scale)&lt;/span&gt;
! | 
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&lt;/span&gt;
| | L L L L s L L L s L L L s
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;Mode Tishrei&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;x-I&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L L s L L L s L L L s L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Cheshvan&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L s L L L s L L L s L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Kislev&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;ii-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L s L L L s L L L s L L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Tevet&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;s L L L s L L L s L L L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Shvat&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;vii-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L L s L L L s L L L L s&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Adar minor&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;iii-vii&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L s L L L s L L L L s L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Adar major&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;xii-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L s L L L s L L L L s L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Nisan&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;viii-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;s L L L s L L L L s L L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Iyar&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;iv-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L L s L L L L s L L L s&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Sivan&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L s L L L L s L L L s L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Tammuz&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L s L L L L s L L L s L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Av&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;v-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;s L L L L s L L L s L L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Elul&lt;/span&gt;
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}</text>
      <sha1>2a8pmcl1s70b6r9shnoagwgiqh2j2t9</sha1>
    </revision>
    <revision>
      <id>4202</id>
      <timestamp>2015-11-13T15:32:12Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 566387165 - Original comment: **</comment>
      <origin>4202</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18595" sha1="q1evjhjx8tl53b66uay7t3zlplhdfgs" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-13 15:32:12 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566387165&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for [[3L 7s]]. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.
||~ &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(Circle of fifths)&amp;lt;/span&amp;gt; ||~ &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(ascending scale)&amp;lt;/span&amp;gt; ||~   ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&amp;lt;/span&amp;gt; || L L L L s L L L s L L L s || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;Mode Tishrei&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;x-I&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Cheshvan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Kislev&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;ii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tevet&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L s L L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Shvat&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;vii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L L s&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar minor&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iii-vii&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar major&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;xii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Nisan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;viii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Iyar&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iv-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L L s L L L s&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Sivan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L L s L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tammuz&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L L s L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Av&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;v-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L L s L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Elul&amp;lt;/span&amp;gt; ||

|| 3/10 ||   || 360 ||
|| 16/53 ||   || 362.264 ||
|| 13/43 ||   || 362.791 ||
||   ||   || 363.487 ||
|| 10/33 ||   || 363.818 ||
||   ||   || 363.976 ||
||   || 27/89 || 364.045 ||
||   ||   || 364.112 ||
||   || 17/56 || 364.286 ||
|| 7/23 ||   || 365.217 ||
||   ||   || 365.904 ||
||   || 18/59 || 366.102 ||
||   ||   || 366.256 ||
||   || 29/95 || 366.316 ||
||   ||   || 366.414 ||
|| 11/36 ||   || 366.667 ||
|| 15/49 ||   || 367.347 ||
|| 19/62 ||   || 367.742 ||
|| 4/13 ||   || 369.231 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for &amp;lt;a class="wiki_link" href="/3L%207s"&amp;gt;3L 7s&amp;lt;/a&amp;gt;. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these &amp;amp;quot;Luach&amp;amp;quot; modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(Circle of fifths)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(ascending scale)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;L L L L s L L L s L L L s&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;Mode Tishrei&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;x-I&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Cheshvan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Kislev&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;ii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tevet&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L s L L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Shvat&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;vii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L L s&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar minor&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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      <sha1>q1evjhjx8tl53b66uay7t3zlplhdfgs</sha1>
    </revision>
    <revision>
      <id>4203</id>
      <timestamp>2015-11-13T15:21:02Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 566385845 - Original comment: **</comment>
      <origin>4203</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18395" sha1="62dphnu5afwbf4l64v7w5r0f9bm5cba" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-13 15:21:02 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566385845&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for [[3L 7s]]. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.
||~ &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(Circle of fifths)&amp;lt;/span&amp;gt; ||~ &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(ascending scale)&amp;lt;/span&amp;gt; ||~   ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&amp;lt;/span&amp;gt; || L L L L s L L L s L L L s || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;Mode Tishrei&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;x-I&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Cheshvan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Kislev&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;ii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tevet&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L s L L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Shvat&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;vii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L L s&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar minor&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iii-vii&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar major&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;xii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Nisan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;viii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Iyar&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iv-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L L s L L L s&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Sivan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L L s L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tammuz&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L L s L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Av&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;v-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L L s L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Elul&amp;lt;/span&amp;gt; ||

|| 3/10 ||   || 360 ||
|| 16/53 ||   || 362.264 ||
|| 13/43 ||   || 362.791 ||
||   ||   || 363.487 ||
|| 10/33 ||   || 363.818 ||
||   ||   || 363.976 ||
||   || 27/89 || 364.045 ||
||   ||   || 364.112 ||
||   || 17/56 || 364.286 ||
|| 7/23 ||   || 365.217 ||
||   ||   || 365.904 ||
||   || 18/59 || 366.102 ||
||   ||   || 366.256 ||
||   || 29/95 || 366.316 ||
||   ||   || 366.414 ||
|| 11/36 ||   || 366.667 ||
|| 15/49 ||   || 367.347 ||
|| 4/13 ||   || 369.231 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for &amp;lt;a class="wiki_link" href="/3L%207s"&amp;gt;3L 7s&amp;lt;/a&amp;gt;. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these &amp;amp;quot;Luach&amp;amp;quot; modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(Circle of fifths)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(ascending scale)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;L L L L s L L L s L L L s&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;Mode Tishrei&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;x-I&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Cheshvan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Kislev&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;ii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tevet&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L s L L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Shvat&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;vii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L L s&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar minor&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iii-vii&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar major&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;xii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Nisan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;viii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Iyar&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iv-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L L s L L L s&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Sivan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L L s L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tammuz&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L L s L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Av&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;v-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L L s L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Elul&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362.264&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362.791&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;363.487&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;363.818&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;363.976&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27/89&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;364.045&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;364.112&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;364.286&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;365.217&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;365.904&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;366.102&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;366.256&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29/95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;366.316&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;366.414&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;366.667&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;367.347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;369.231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>62dphnu5afwbf4l64v7w5r0f9bm5cba</sha1>
    </revision>
    <revision>
      <id>4204</id>
      <timestamp>2015-05-23T22:38:56Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 551986252 - Original comment: **</comment>
      <origin>4204</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17265" sha1="fre5fhhocttq5op179fqxki0e2gmd0g" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-23 22:38:56 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;551986252&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for [[3L 7s]]. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.
||~ &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(Circle of fifths)&amp;lt;/span&amp;gt; ||~ &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(ascending scale)&amp;lt;/span&amp;gt; ||~   ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&amp;lt;/span&amp;gt; || L L L L s L L L s L L L s || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;Mode Tishrei&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;x-I&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Cheshvan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Kislev&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;ii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tevet&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L s L L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Shvat&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;vii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L L s&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar minor&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iii-vii&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar major&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;xii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Nisan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;viii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Iyar&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iv-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L L s L L L s&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Sivan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L L s L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tammuz&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L L s L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Av&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;v-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L L s L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Elul&amp;lt;/span&amp;gt; ||

|| 3/10 ||   || 360 ||
|| 16/53 ||   || 362 14/53 ||
|| 13/43 ||   || 362 34/43 ||
||   ||   || 363.486758 ||
|| 10/33 ||   || 363 9/11 ||
||   ||   || 363.975772 ||
||   || 17/56 || 364 2/7 ||
|| 7/23 ||   || 365 5/23 ||
|| 11/36 ||   || 366 2/3 ||
|| 15/49 ||   || 367 17/49 ||
|| 19/62 ||   || 367 18/31 ||
|| 4/13 ||   || 369 3/13 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for &amp;lt;a class="wiki_link" href="/3L%207s"&amp;gt;3L 7s&amp;lt;/a&amp;gt;. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these &amp;amp;quot;Luach&amp;amp;quot; modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(Circle of fifths)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(ascending scale)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;L L L L s L L L s L L L s&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;Mode Tishrei&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;x-I&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Cheshvan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Kislev&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;ii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tevet&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L s L L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Shvat&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;vii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L L s&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar minor&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iii-vii&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar major&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;xii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Nisan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;viii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Iyar&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Sivan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tammuz&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L L s L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Av&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L L s L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Elul&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;3/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;16/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362 14/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362 34/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;366 2/3&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;4/13&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;369 3/13&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>fre5fhhocttq5op179fqxki0e2gmd0g</sha1>
    </revision>
    <revision>
      <id>4205</id>
      <timestamp>2015-02-12T13:49:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540788472 - Original comment: **</comment>
      <origin>4205</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17265" sha1="awaccn8bognm35yb4qx0ct0hvyv9mwh" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-12 13:49:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540788472&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for [[3L 7s]]. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.
||~ &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(Circle of fifths)&amp;lt;/span&amp;gt; ||~ &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(ascending scale)&amp;lt;/span&amp;gt; ||~   ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&amp;lt;/span&amp;gt; || L L L L s L L L s L L L s || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;Mode Tishrei&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;x-I&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Cheshvan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Kislev&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;ii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tevet&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L s L L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Shvat&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;vii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L L s&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar minor&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iii-vii&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar major&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;xii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Nisan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;viii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Iyar&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iv-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L L s L L L s&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Sivan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L L s L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tammuz&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L L s L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Av&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;v-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L L s L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Elul&amp;lt;/span&amp;gt; ||

|| 3/10 ||   || 360 ||
|| 16/53 ||   || 362 14/53 ||
|| 13/43 ||   || 362 34/43 ||
||   ||   || 363.605927 ||
|| 10/33 ||   || 363 9/11 ||
||   ||   || 363.667031 ||
||   || 17/56 || 364 2/7 ||
|| 7/23 ||   || 365 5/23 ||
|| 11/36 ||   || 366 2/3 ||
|| 15/49 ||   || 367 17/49 ||
|| 19/62 ||   || 367 18/31 ||
|| 4/13 ||   || 369 3/13 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for &amp;lt;a class="wiki_link" href="/3L%207s"&amp;gt;3L 7s&amp;lt;/a&amp;gt;. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these &amp;amp;quot;Luach&amp;amp;quot; modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(Circle of fifths)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
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&amp;lt;/th&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Cheshvan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Kislev&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;ii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tevet&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L s L L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Shvat&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;vii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L L s&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar minor&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iii-vii&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar major&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;xii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Nisan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;viii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Iyar&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iv-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L L s L L L s&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Sivan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L L s L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tammuz&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L L s L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Av&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;v-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L L s L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Elul&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362 14/53&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;13/43&amp;lt;br /&amp;gt;
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      <sha1>awaccn8bognm35yb4qx0ct0hvyv9mwh</sha1>
    </revision>
    <revision>
      <id>4206</id>
      <timestamp>2014-11-27T19:50:37Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 532965840 - Original comment: **</comment>
      <origin>4206</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16871" sha1="gndb2uyzzd5p14hcjyqkbnjb4862q3u" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2014-11-27 19:50:37 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;532965840&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for [[3L 7s]]. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.
||~ &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(Circle of fifths)&amp;lt;/span&amp;gt; ||~ &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(ascending scale)&amp;lt;/span&amp;gt; ||~   ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&amp;lt;/span&amp;gt; || L L L L s L L L s L L L s || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;Mode Tishrei&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;x-I&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Cheshvan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Kislev&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;ii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tevet&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L s L L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Shvat&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;vii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L L s&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar minor&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iii-vii&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar major&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;xii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Nisan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;viii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Iyar&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iv-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L L s L L L s&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Sivan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L L s L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tammuz&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L L s L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Av&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;v-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L L s L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Elul&amp;lt;/span&amp;gt; ||

|| 3/10 ||   || 360 ||
|| 16/53 ||   || 362 14/53 ||
|| 13/43 ||   || 362 34/43 ||
|| 10/33 ||   || 363 9/11 ||
||   || 17/56 || 364 2/7 ||
|| 7/23 ||   || 365 5/23 ||
|| 11/36 ||   || 366 2/3 ||
|| 15/49 ||   || 367 17/49 ||
|| 19/62 ||   || 367 18/31 ||
|| 4/13 ||   || 369 3/13 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for &amp;lt;a class="wiki_link" href="/3L%207s"&amp;gt;3L 7s&amp;lt;/a&amp;gt;. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these &amp;amp;quot;Luach&amp;amp;quot; modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(Circle of fifths)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(ascending scale)&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;L L L L s L L L s L L L s&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;Mode Tishrei&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;x-I&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Cheshvan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Kislev&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;ii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tevet&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L s L L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Shvat&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;vii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L L s&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar minor&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iii-vii&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar major&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;xii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Nisan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;viii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Iyar&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iv-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L L s L L L s&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Sivan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L L s L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tammuz&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L L s L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Av&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;v-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L L s L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Elul&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362 14/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362 34/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;363 9/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;364 2/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;365 5/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;366 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;367 17/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19/62&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;367 18/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;369 3/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>gndb2uyzzd5p14hcjyqkbnjb4862q3u</sha1>
    </revision>
    <revision>
      <id>4207</id>
      <timestamp>2014-11-27T18:42:54Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 532960614 - Original comment: **</comment>
      <origin>4207</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16320" sha1="9dzjjamqa0kq4snna7enexpnoxd2qz0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2014-11-27 18:42:54 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;532960614&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for [[3L 7s]]. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.
(Circle of fifths) (ascending scale)
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&amp;lt;/span&amp;gt; || L L L L s L L L s L L L s || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;Mode Tishrei&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;x-I&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Cheshvan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Kislev&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;ii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tevet&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L s L L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Shvat&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;vii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L L s&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar minor&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iii-vii&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar major&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;xii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Nisan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;viii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Iyar&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iv-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L L s L L L s&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Sivan&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L L s L L L s L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tammuz&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L L s L L L s L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Av&amp;lt;/span&amp;gt; ||
|| &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;v-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L L s L L L s L L L&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Elul&amp;lt;/span&amp;gt; ||

|| 3/10 ||   || 360 ||
|| 16/53 ||   || 362 14/53 ||
|| 13/43 ||   || 362 34/43 ||
|| 10/33 ||   || 363 9/11 ||
||   || 17/56 || 364 2/7 ||
|| 7/23 ||   || 365 5/23 ||
|| 11/36 ||   || 366 2/3 ||
|| 15/49 ||   || 367 17/49 ||
|| 19/62 ||   || 367 18/31 ||
|| 4/13 ||   || 369 3/13 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for &amp;lt;a class="wiki_link" href="/3L%207s"&amp;gt;3L 7s&amp;lt;/a&amp;gt;. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these &amp;amp;quot;Luach&amp;amp;quot; modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.&amp;lt;br /&amp;gt;
(Circle of fifths) (ascending scale)&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;L L L L s L L L s L L L s&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;Mode Tishrei&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;x-I&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Cheshvan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Kislev&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;ii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tevet&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L s L L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Shvat&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;vii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L s L L L L s&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar minor&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iii-vii&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L s L L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Adar major&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;xii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L s L L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Nisan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;viii-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L s L L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Iyar&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;iv-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L L s L L L L s L L L s&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Sivan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L L s L L L L s L L L s L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Tammuz&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;L s L L L L s L L L s L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Av&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;v-&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;s L L L L s L L L s L L L&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&amp;gt;Elul&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9dzjjamqa0kq4snna7enexpnoxd2qz0</sha1>
    </revision>
    <revision>
      <id>4208</id>
      <timestamp>2014-11-26T21:36:24Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 532861392 - Original comment: **</comment>
      <origin>4208</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7460" sha1="829od41i1y2jtgtw6e6af5tqcrj65ib" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2014-11-26 21:36:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;532861392&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for [[3L 7s]]. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.
(Circle of fifths)
I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X: Mode Tishrei
x-I&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI: Cheshvan&amp;lt;/span&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II: Kislev&amp;lt;/span&amp;gt;
ii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI: Tevet&amp;lt;/span&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII: Shvat&amp;lt;/span&amp;gt;
vii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III: Adar minor&amp;lt;/span&amp;gt;
iii-vii&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII: Adar major&amp;lt;/span&amp;gt;
xii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII: Nisan&amp;lt;/span&amp;gt;
viii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV: Iyar&amp;lt;/span&amp;gt;
iv-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII: Sivan&amp;lt;/span&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX: Tammuz&amp;lt;/span&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V: Av&amp;lt;/span&amp;gt;
v-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I: Elul&amp;lt;/span&amp;gt;
|| 3/10 ||   || 360 ||
|| 16/53 ||   || 362 14/53 ||
|| 13/43 ||   || 362 34/43 ||
|| 10/33 ||   || 363 9/11 ||
||   || 17/56 || 364 2/7 ||
|| 7/23 ||   || 365 5/23 ||
|| 11/36 ||   || 366 2/3 ||
|| 15/49 ||   || 367 17/49 ||
|| 19/62 ||   || 367 18/31 ||
|| 4/13 ||   || 369 3/13 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for &amp;lt;a class="wiki_link" href="/3L%207s"&amp;gt;3L 7s&amp;lt;/a&amp;gt;. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these &amp;amp;quot;Luach&amp;amp;quot; modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.&amp;lt;br /&amp;gt;
(Circle of fifths)&amp;lt;br /&amp;gt;
I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X: Mode Tishrei&amp;lt;br /&amp;gt;
x-I&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI: Cheshvan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II: Kislev&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
ii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI: Tevet&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII: Shvat&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
vii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III: Adar minor&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
iii-vii&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII: Adar major&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
xii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII: Nisan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
viii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV: Iyar&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
iv-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII: Sivan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX: Tammuz&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V: Av&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
v-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I: Elul&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;3/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;16/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362 14/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362 34/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;15/49&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;4/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;369 3/13&amp;lt;br /&amp;gt;
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&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>829od41i1y2jtgtw6e6af5tqcrj65ib</sha1>
    </revision>
    <revision>
      <id>4209</id>
      <timestamp>2014-11-26T21:23:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 532860372 - Original comment: **</comment>
      <origin>4209</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6828" sha1="lx9y0snl4q5ewc6gdnjamdurz39exdk" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2014-11-26 21:23:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;532860372&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for [[3L 7s]]. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.
(Circle of fifths)
I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X: Mode Tishrei
x-I&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI: Cheshvan&amp;lt;/span&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II: Kislev&amp;lt;/span&amp;gt;
ii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI: Tevet&amp;lt;/span&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII: Shvat&amp;lt;/span&amp;gt;
vii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III: Adar minor&amp;lt;/span&amp;gt;
iii-vii&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII: Adar major&amp;lt;/span&amp;gt;
xii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII: Nisan&amp;lt;/span&amp;gt;
viii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV: Iyar&amp;lt;/span&amp;gt;
iv-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII: Sivan&amp;lt;/span&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX: Tammuz&amp;lt;/span&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V: Av&amp;lt;/span&amp;gt;
v-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I: Elul&amp;lt;/span&amp;gt;
|| 3/10 || 360 ||
|| 16/53 || 362 14/53 ||
|| 13/43 || 362 34/43 ||
|| 10/33 || 363 9/11 ||
|| 7/23 || 365 5/23 ||
|| 11/36 || 366 2/3 ||
|| 15/49 || 367 17/49 ||
|| 19/62 || 367 18/31 ||
|| 4/13 || 369 3/13 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for &amp;lt;a class="wiki_link" href="/3L%207s"&amp;gt;3L 7s&amp;lt;/a&amp;gt;. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these &amp;amp;quot;Luach&amp;amp;quot; modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.&amp;lt;br /&amp;gt;
(Circle of fifths)&amp;lt;br /&amp;gt;
I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X: Mode Tishrei&amp;lt;br /&amp;gt;
x-I&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI: Cheshvan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II: Kislev&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
ii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI: Tevet&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII: Shvat&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
vii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III: Adar minor&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
iii-vii&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII: Adar major&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
xii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII: Nisan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
viii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV: Iyar&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
iv-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII: Sivan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX: Tammuz&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V: Av&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
v-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I: Elul&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362 14/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362 34/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;363 9/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;365 5/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;366 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;367 17/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19/62&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;367 18/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;369 3/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>lx9y0snl4q5ewc6gdnjamdurz39exdk</sha1>
    </revision>
    <revision>
      <id>4210</id>
      <timestamp>2014-11-26T21:22:26Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 532860328 - Original comment: **</comment>
      <origin>4210</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6884" sha1="ipdpxmujlzmq4espsf575105mqjrj5h" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2014-11-26 21:22:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;532860328&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for [[3L 7s]]. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.
(Circle of fifths)
I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X: Mode Tishrei
x-I&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI: Cheshvan&amp;lt;/span&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II: Kislev&amp;lt;/span&amp;gt;
ii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI: Tevet&amp;lt;/span&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII: Shvat&amp;lt;/span&amp;gt;
vii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III: Adar minor&amp;lt;/span&amp;gt;


iii-vii&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII: Adar major&amp;lt;/span&amp;gt;
xii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII: Nisan&amp;lt;/span&amp;gt;
viii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV: Iyar&amp;lt;/span&amp;gt;
iv-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII: Sivan&amp;lt;/span&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX: Tammuz&amp;lt;/span&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V: Av&amp;lt;/span&amp;gt;


v-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I: Elul&amp;lt;/span&amp;gt;
|| 3/10 || 360 ||
|| 16/53 || 362 14/53 ||
|| 13/43 || 362 34/43 ||
|| 10/33 || 363 9/11 ||
|| 7/23 || 365 5/23 ||
|| 11/36 || 366 2/3 ||
|| 15/49 || 367 17/49 ||
|| 19/62 || 367 18/31 ||
|| 4/13 || 369 3/13 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the chromatic scale of the Sephiroth temperament explicated on the page for &amp;lt;a class="wiki_link" href="/3L%207s"&amp;gt;3L 7s&amp;lt;/a&amp;gt;. However, a tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these &amp;amp;quot;Luach&amp;amp;quot; modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.&amp;lt;br /&amp;gt;
(Circle of fifths)&amp;lt;br /&amp;gt;
I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X: Mode Tishrei&amp;lt;br /&amp;gt;
x-I&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI: Cheshvan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II: Kislev&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
ii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI: Tevet&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII: Shvat&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
vii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III: Adar minor&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
iii-vii&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII: Adar major&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
xii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII: Nisan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
viii-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV: Iyar&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
iv-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII: Sivan&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX: Tammuz&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V: Av&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
v-&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I: Elul&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362 14/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;362 34/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;363 9/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;365 5/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;366 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;367 17/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;19/62&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;367 18/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;369 3/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ipdpxmujlzmq4espsf575105mqjrj5h</sha1>
    </revision>
    <revision>
      <id>66221</id>
      <parentid>46</parentid>
      <timestamp>2021-04-11T18:43:07Z</timestamp>
      <contributor>
        <username>Ayceman</username>
        <id>284</id>
      </contributor>
      <origin>66221</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6001" sha1="dvuu5zfs4br9qsunyh37179e3vx9nf5" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{Infobox MOS
| Name = luachoid
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' or '''luachoid''' is scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.

{| class="wikitable"
|-
! | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;Circle of fifths&lt;/span&gt;
! | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;Ascending scale&lt;/span&gt;
! | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;Mode&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&lt;/span&gt;
| | L L L L s L L L s L L L s
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;Tishrei&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;x-I&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L L s L L L s L L L s L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Cheshvan&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L s L L L s L L L s L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Kislev&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;ii-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L s L L L s L L L s L L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Tevet&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;s L L L s L L L s L L L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Shvat&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;vii-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L L s L L L s L L L L s&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Adar minor&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;iii-vii&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L s L L L s L L L L s L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Adar major&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;xii-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L s L L L s L L L L s L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Nisan&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;viii-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;s L L L s L L L L s L L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Iyar&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;iv-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L L s L L L L s L L L s&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Sivan&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L s L L L L s L L L s L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Tammuz&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L s L L L L s L L L s L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Av&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;v-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;s L L L L s L L L s L L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Elul&lt;/span&gt;
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}</text>
      <sha1>dvuu5zfs4br9qsunyh37179e3vx9nf5</sha1>
    </revision>
    <revision>
      <id>66223</id>
      <parentid>66221</parentid>
      <timestamp>2021-04-11T18:44:47Z</timestamp>
      <contributor>
        <username>Ayceman</username>
        <id>284</id>
      </contributor>
      <minor/>
      <comment>Cats</comment>
      <origin>66223</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6081" sha1="9uhe15d7up675ayoqowrovd2x1yd2ck" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{Infobox MOS
| Name = luachoid
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' or '''luachoid''' is scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.

{| class="wikitable"
|-
! | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;Circle of fifths&lt;/span&gt;
! | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;Ascending scale&lt;/span&gt;
! | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;Mode&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X&lt;/span&gt;
| | L L L L s L L L s L L L s
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;Tishrei&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;x-I&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L L s L L L s L L L s L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Cheshvan&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L s L L L s L L L s L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Kislev&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;ii-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L s L L L s L L L s L L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Tevet&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;s L L L s L L L s L L L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Shvat&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;vii-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L L s L L L s L L L L s&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Adar minor&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;iii-vii&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L s L L L s L L L L s L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Adar major&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;xii-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L s L L L s L L L L s L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Nisan&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;viii-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;s L L L s L L L L s L L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Iyar&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;iv-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L L s L L L L s L L L s&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Sivan&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L L s L L L L s L L L s L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Tammuz&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;L s L L L L s L L L s L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Av&lt;/span&gt;
|-
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;v-&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px;"&gt;ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px;"&gt;s L L L L s L L L s L L L&lt;/span&gt;
| | &lt;span style="font-size: 12.8000001907349px; line-height: 15.3600006103516px;"&gt;Elul&lt;/span&gt;
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]</text>
      <sha1>9uhe15d7up675ayoqowrovd2x1yd2ck</sha1>
    </revision>
    <revision>
      <id>66229</id>
      <parentid>66223</parentid>
      <timestamp>2021-04-11T20:29:09Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>66229</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2576" sha1="3b02udhyi3hg4ij3m8xaavtlqrveydn" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{Infobox MOS
| Name = luachoid
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' or '''luachoid''' is scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]</text>
      <sha1>3b02udhyi3hg4ij3m8xaavtlqrveydn</sha1>
    </revision>
    <revision>
      <id>66230</id>
      <parentid>66229</parentid>
      <timestamp>2021-04-11T20:30:13Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>66230</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2658" sha1="lid8i5bbardzah313kmympp1rhyhz38" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{Infobox MOS
| Name = luachoid
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' or '''luachoid''' (since its modes are named after months in the Hebrew calendar, or the ''Luach'') is scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]</text>
      <sha1>lid8i5bbardzah313kmympp1rhyhz38</sha1>
    </revision>
    <revision>
      <id>66232</id>
      <parentid>66230</parentid>
      <timestamp>2021-04-11T21:31:23Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>66232</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2660" sha1="q53x74eqbxvyd3pm38bnb5of2kxo8dq" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{Infobox MOS
| Name = luachoid
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' or '''luachoid''' (since its modes are named after months in the Hebrew calendar, or the ''Luach'') is a scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]</text>
      <sha1>q53x74eqbxvyd3pm38bnb5of2kxo8dq</sha1>
    </revision>
    <revision>
      <id>66983</id>
      <parentid>66232</parentid>
      <timestamp>2021-04-14T02:42:59Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>66983</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2622" sha1="6qvj4a81rp4l8mq6u838rrsz8kthk1f" xml:space="preserve">{{Infobox MOS
| Name = luachoid
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' or '''luachoid''' (since its modes are named after months in the Hebrew calendar, or the ''Luach'') is a scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered is descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]</text>
      <sha1>6qvj4a81rp4l8mq6u838rrsz8kthk1f</sha1>
    </revision>
    <revision>
      <id>67608</id>
      <parentid>66983</parentid>
      <timestamp>2021-04-17T00:07:36Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>TAMNAMS consistency correction</comment>
      <origin>67608</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2504" sha1="feu501dhap86nmzd36lq0tlinnbyuvm" xml:space="preserve">{{Infobox MOS
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' is a scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]</text>
      <sha1>feu501dhap86nmzd36lq0tlinnbyuvm</sha1>
    </revision>
    <revision>
      <id>68556</id>
      <parentid>67608</parentid>
      <timestamp>2021-04-24T19:44:10Z</timestamp>
      <contributor>
        <username>Keenan Pepper</username>
        <id>25</id>
      </contributor>
      <minor/>
      <comment>Clean up categories</comment>
      <origin>68556</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2480" sha1="0w1bb8ks0yoii2sobwoyrnq58dl3rot" xml:space="preserve">{{Infobox MOS
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' is a scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:Scales]]
[[Category:Abstract MOS patterns]]</text>
      <sha1>0w1bb8ks0yoii2sobwoyrnq58dl3rot</sha1>
    </revision>
    <revision>
      <id>87063</id>
      <parentid>68556</parentid>
      <timestamp>2022-02-25T14:54:41Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87063</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2460" sha1="igjp1tg1kvaqsz40jn54b8yratcs2hj" xml:space="preserve">{{Infobox MOS
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' is a scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking.

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:Abstract MOS patterns]]</text>
      <sha1>igjp1tg1kvaqsz40jn54b8yratcs2hj</sha1>
    </revision>
    <revision>
      <id>90869</id>
      <parentid>87063</parentid>
      <timestamp>2022-04-06T19:44:31Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>90869</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2563" sha1="o1lbazb8pm7u40g0sf7roq05mdhnjhq" xml:space="preserve">{{Infobox MOS
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' is a scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 10L 3s does not have a [[TAMNAMS]] name, but other proposed names include "luachoid"

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:Abstract MOS patterns]]</text>
      <sha1>o1lbazb8pm7u40g0sf7roq05mdhnjhq</sha1>
    </revision>
    <revision>
      <id>90870</id>
      <parentid>90869</parentid>
      <timestamp>2022-04-06T19:44:45Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>90870</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2566" sha1="n9u5heb1ufcdn5p236lc41uocmdhne0" xml:space="preserve">{{Infobox MOS
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' is a scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

10L 3s does not have a [[TAMNAMS]] name, but other proposed names include "luachoid".

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:Abstract MOS patterns]]</text>
      <sha1>n9u5heb1ufcdn5p236lc41uocmdhne0</sha1>
    </revision>
    <revision>
      <id>93949</id>
      <parentid>90870</parentid>
      <timestamp>2022-08-05T03:16:43Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Abstract MOS patterns]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>93949</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2530" sha1="e6b0sahbtssjsbltayyuhyw5w29kk7d" xml:space="preserve">{{Infobox MOS
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' is a scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

10L 3s does not have a [[TAMNAMS]] name, but other proposed names include "luachoid".

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}</text>
      <sha1>e6b0sahbtssjsbltayyuhyw5w29kk7d</sha1>
    </revision>
    <revision>
      <id>94502</id>
      <parentid>93949</parentid>
      <timestamp>2022-08-17T09:16:53Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94502</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2529" sha1="kjg5eunbitgqmp617aiz1xji7rcz4ib" xml:space="preserve">{{Infobox MOS
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Collapsed = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' is a scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

10L 3s does not have a [[TAMNAMS]] name, but other proposed names include "luachoid".

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}</text>
      <sha1>kjg5eunbitgqmp617aiz1xji7rcz4ib</sha1>
    </revision>
    <revision>
      <id>101026</id>
      <parentid>94502</parentid>
      <timestamp>2023-01-06T16:14:29Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Copying from [[Category:Abstract MOS patterns]] to [[Category:13-tone scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>101026</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2558" sha1="cx9h7vhb16ero96ebdb6vy239r442wd" xml:space="preserve">{{Infobox MOS
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 3
| Equalized = 4
| Collapsed = 3
| Pattern = LLLLsLLLsLLLs
}}

'''10L 3s''' is a scale pattern that expands on sephiroid ([[3L 7s]]). A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

10L 3s does not have a [[TAMNAMS]] name, but other proposed names include "luachoid".

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:13-tone scales]]</text>
      <sha1>cx9h7vhb16ero96ebdb6vy239r442wd</sha1>
    </revision>
    <revision>
      <id>134227</id>
      <parentid>101026</parentid>
      <timestamp>2024-02-03T21:48:32Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Mos intro</comment>
      <origin>134227</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2354" sha1="95js56muh4m27cmvacy1a29fvy96pt9" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

'''Luachoid''' is a proposed name for this scale.

{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:13-tone scales]]</text>
      <sha1>95js56muh4m27cmvacy1a29fvy96pt9</sha1>
    </revision>
    <revision>
      <id>135332</id>
      <parentid>134227</parentid>
      <timestamp>2024-02-12T11:14:10Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>135332</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2396" sha1="rpmvbb7poqiodmp3tgkxh5h5wsxiyx9" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

'''Luachoid''' is a proposed name for this scale.

== Modes ==
{{MOS modes}}

==Scale tree==
{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:13-tone scales]]</text>
      <sha1>rpmvbb7poqiodmp3tgkxh5h5wsxiyx9</sha1>
    </revision>
    <revision>
      <id>135333</id>
      <parentid>135332</parentid>
      <timestamp>2024-02-12T11:15:53Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>135333</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2383" sha1="g5g4xshljo20y5a7jz4uvoft6yggg6e" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

'''Luachoid''' is a proposed name for this scale.

== Modes ==
{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

== Scale tree ==
{| class="wikitable"
|-
| | 3/10
| | 
| | 360
|-
| | 16/53
| | 
| | 362.264
|-
| | 13/43
| | 
| | 362.791
|-
| | 
| | 
| | 363.487
|-
| | 10/33
| | 
| | 363.818
|-
| | 
| | 
| | 363.976
|-
| | 
| | 27/89
| | 364.045
|-
| | 
| | 
| | 364.112
|-
| | 
| | 17/56
| | 364.286
|-
| | 7/23
| | 
| | 365.217
|-
| | 
| | 
| | 365.904
|-
| | 
| | 18/59
| | 366.102
|-
| | 
| | 
| | 366.256
|-
| | 
| | 29/95
| | 366.316
|-
| | 
| | 
| | 366.414
|-
| | 11/36
| | 
| | 366.667
|-
| | 15/49
| | 
| | 367.347
|-
| | 19/62
| | 
| | 367.742
|-
| | 4/13
| | 
| | 369.231
|}

[[Category:13-tone scales]]</text>
      <sha1>g5g4xshljo20y5a7jz4uvoft6yggg6e</sha1>
    </revision>
    <revision>
      <id>136137</id>
      <parentid>135333</parentid>
      <timestamp>2024-02-19T01:22:38Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>/* Scale tree */ Replaced scale tree</comment>
      <origin>136137</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1840" sha1="77t62f4pvyaguvo4gcxekzv6sdhv01o" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

'''Luachoid''' is a proposed name for this scale.

== Modes ==
{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

== Scale tree ==
{{Scale tree}}
[[Category:13-tone scales]]</text>
      <sha1>77t62f4pvyaguvo4gcxekzv6sdhv01o</sha1>
    </revision>
    <revision>
      <id>148726</id>
      <parentid>136137</parentid>
      <timestamp>2024-07-25T23:39:16Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>/* Modes */ Proposed names</comment>
      <origin>148726</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1864" sha1="kmgntgu4ysk0eli5b5l8waizb2bxooo" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

'''Luachoid''' is a proposed name for this scale.

== Modes ==

=== Proposed names ===
{| class="wikitable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

== Scale tree ==
{{Scale tree}}
[[Category:13-tone scales]]</text>
      <sha1>kmgntgu4ysk0eli5b5l8waizb2bxooo</sha1>
    </revision>
    <revision>
      <id>149013</id>
      <parentid>148726</parentid>
      <timestamp>2024-07-29T10:44:22Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>/* Modes */ Adopt modes template; to move proposed names...</comment>
      <origin>149013</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1887" sha1="4wjdbsvkacj7g1aqlm3e48z9uyovglx" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

'''Luachoid''' is a proposed name for this scale.

== Modes ==

=== Proposed names ===
{{MOS modes}}
{| class="wikitable sortable"
|-
! | Circle of fifths
! | Ascending scale
! | Mode
|-
| | I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI-X
| | L L L L s L L L s L L L s
| | Tishrei
|-
| | x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II-VI
| | L L L s L L L s L L L s L
| | Cheshvan
|-
| | vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI-II
| | L L s L L L s L L L s L L
| | Kislev
|-
| | ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII-XI
| | L s L L L s L L L s L L L
| | Tevet
|-
| | xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III-VII
| | s L L L s L L L s L L L L
| | Shvat
|-
| | vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII-III
| | L L L s L L L s L L L L s
| | Adar minor
|-
| | iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII-XII
| | L L s L L L s L L L L s L
| | Adar major
|-
| | xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV-VIII
| | L s L L L s L L L L s L L
| | Nisan
|-
| | viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII-IV
| | s L L L s L L L L s L L L
| | Iyar
|-
| | iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX-XIII
| | L L L s L L L L s L L L s
| | Sivan
|-
| | xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V-IX
| | L L s L L L L s L L L s L
| | Tammuz
|-
| | ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I-V
| | L s L L L L s L L L s L L
| | Av
|-
| | v-ix-xiii-iv-viii-xii-iii-vii-xi-ii-vi-x-I
| | s L L L L s L L L s L L L
| | Elul
|}

== Scale tree ==
{{Scale tree}}
[[Category:13-tone scales]]</text>
      <sha1>4wjdbsvkacj7g1aqlm3e48z9uyovglx</sha1>
    </revision>
    <revision>
      <id>149014</id>
      <parentid>149013</parentid>
      <timestamp>2024-07-29T10:48:28Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>/* Modes */ Adopt templates; sections</comment>
      <origin>149014</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="853" sha1="nxgxd9253jj6oowdz0e9maxfafnc7u1" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}} 

A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

'''Luachoid''' is a proposed name for this scale.

== Modes ==
{{MOS mode degrees}}

=== Proposed names ===
Names based on the [[wikipedia:Hebrew_calendar|Hebrew calendar]] have been proposed as mode names.
{{MOS modes|Mode Names=Tishrei; Sivan; Adar minor; Cheshvan; Tammuz; Adar major; Kislev; Av; Nisan; Tevet; Elul; Iyar; Shvat}}
== Scale tree ==
{{Scale tree}}
[[Category:13-tone scales]]</text>
      <sha1>nxgxd9253jj6oowdz0e9maxfafnc7u1</sha1>
    </revision>
    <revision>
      <id>170552</id>
      <parentid>149014</parentid>
      <timestamp>2024-12-15T06:46:05Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Intervals, reorder for better formatting</comment>
      <origin>170552</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="906" sha1="fstqwn5wuyl4nv1p8d85p5ihzw6c8cy" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}} 

A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

'''Luachoid''' is a proposed name for this scale.

== Modes ==
=== Proposed names ===
Names based on the [[wikipedia:Hebrew_calendar|Hebrew calendar]] have been proposed as mode names.
{{MOS modes|Mode Names=Tishrei; Sivan; Adar minor; Cheshvan; Tammuz; Adar major; Kislev; Av; Nisan; Tevet; Elul; Iyar; Shvat}}

=== Degrees ===
{{MOS mode degrees}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree}}

[[Category:13-tone scales]]</text>
      <sha1>fstqwn5wuyl4nv1p8d85p5ihzw6c8cy</sha1>
    </revision>
    <revision>
      <id>183725</id>
      <parentid>170552</parentid>
      <timestamp>2025-02-28T16:36:41Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183725</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="915" sha1="a565lzwnb40j3b77asws5ecs1w3wyxu" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}} 

A tempered-flat chain of the 13th harmonic really comes into its own as a distinct scale when extended to 13 tones. In these "Luach" (Hebrew calendar) modes the degrees are to be taken as numbered in descending order so that the the tempered 13th harmonic is fifth rather than tenth in the scale. This inverts the meaning of major and minor so that the Ionian-like mode reaches its degrees almost exclusively by backtracking. 

'''Luachoid''' is a proposed name for this scale.

== Modes ==
=== Proposed names ===
Names based on the [[wikipedia:Hebrew_calendar|Hebrew calendar]] have been proposed as mode names.
{{MOS modes|Mode Names=Tishrei; Sivan; Adar minor; Cheshvan; Tammuz; Adar major; Kislev; Av; Nisan; Tevet; Elul; Iyar; Shvat}}

=== Degrees ===
{{MOS mode degrees}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum}}

[[Category:13-tone scales]]</text>
      <sha1>a565lzwnb40j3b77asws5ecs1w3wyxu</sha1>
    </revision>
  </page>
  <page>
    <title>10L 5s</title>
    <ns>0</ns>
    <id>47</id>
    <revision>
      <id>47</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>47</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1452" sha1="flw3uu8ba6gqaaq7gxerlaggn9u2gpb" xml:space="preserve">This MOS has five periods of 2L 1s and its generator is a neutral or diatonic semitone of 1/15edo (80 cents) to 1/10edo (120). Because it is a Blacksmith scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].

{| class="wikitable"
|-
| | 1/15
| | 
| | 
| | 80
|-
| | 5/70
| | 
| | 
| | 85.714
|-
| | 
| | 9/125
| | 
| | 86.4
|-
| | 4/55
| | 
| | 
| | 87.273
|-
| | 
| | 7/95
| | 
| | 88.421
|-
| | 3/40
| | 
| | 
| | 90
|-
| | 
| | 
| | 
| | 91.026
|-
| | 
| | 8/105
| | 
| | 91.429
|-
| | 
| | 
| | 
| | 91.672
|-
| | 
| | 5/65
| | 
| | 92.308
|-
| | 
| | 
| | 
| | 93.119
|-
| | 
| | 7/90
| | 
| | 93.333
|-
| | 
| | 9/115
| | 
| | 93.913
|-
| | 
| | 11/140
| | 
| | 94.286
|-
| | 
| | 13/165
| | 
| | 94,5455
|-
| | 
| | 15/190
| | 
| | 94.737
|-
| | 
| | 17/215
| | 
| | 94.884
|-
| | 
| | 19/240
| | 
| | 95
|-
| | 2/25
| | 
| | 
| | 96
|-
| | 
| | 11/135
| | 
| | 97.778
|-
| | 
| | 9/110
| | 
| | 98.182
|-
| | 
| | 7/85
| | 
| | 98.8235
|-
| | 
| | 
| | 12/145
| | 99.31
|-
| | 
| | 5/60
| | 
| | 100
|-
| | 
| | 
| | 
| | 100.757
|-
| | 
| | 8/95
| | 
| | 101.053
|-
| | 
| | 
| | 
| | 101.3565
|-
| | 3/35
| | 
| | 
| | 102.857
|-
| | 
| | 
| | 
| | 103.524
|-
| | 
| | 10/115
| | 
| | 104.348
|-
| | 
| | 7/80
| | 
| | 105
|-
| | 4/45
| | 
| | 
| | 106.667
|-
| | 
| | 13/145
| | 
| | 107.586
|-
| | 
| | 9/100
| | 
| | 108
|-
| | 5/55
| | 
| | 
| | 109.111
|-
| | 1/10
| | 
| | 
| | 120
|}</text>
      <sha1>flw3uu8ba6gqaaq7gxerlaggn9u2gpb</sha1>
    </revision>
    <revision>
      <id>4211</id>
      <timestamp>2015-11-14T15:38:31Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 566466985 - Original comment: **</comment>
      <origin>4211</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10555" sha1="ho6f7srdomh77l8ju7egursbh2ls53a" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-14 15:38:31 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566466985&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS has five periods of 2L 1s and its generator is a neutral or diatonic semitone of 1/15edo (80 cents) to 1/10edo (120). Because it is a Blacksmith scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].
|| 1/15 ||   ||   || 80 ||
|| 5/70 ||   ||   || 85.714 ||
||   || 9/125 ||   || 86.4 ||
|| 4/55 ||   ||   || 87.273 ||
||   || 7/95 ||   || 88.421 ||
|| 3/40 ||   ||   || 90 ||
||   ||   ||   || 91.026 ||
||   || 8/105 ||   || 91.429 ||
||   ||   ||   || 91.672 ||
||   || 5/65 ||   || 92.308 ||
||   ||   ||   || 93.119 ||
||   || 7/90 ||   || 93.333 ||
||   || 9/115 ||   || 93.913 ||
||   || 11/140 ||   || 94.286 ||
||   || 13/165 ||   || 94,5455 ||
||   || 15/190 ||   || 94.737 ||
||   || 17/215 ||   || 94.884 ||
||   || 19/240 ||   || 95 ||
|| 2/25 ||   ||   || 96 ||
||   || 11/135 ||   || 97.778 ||
||   || 9/110 ||   || 98.182 ||
||   || 7/85 ||   || 98.8235 ||
||   ||   || 12/145 || 99.31 ||
||   || 5/60 ||   || 100 ||
||   ||   ||   || 100.757 ||
||   || 8/95 ||   || 101.053 ||
||   ||   ||   || 101.3565 ||
|| 3/35 ||   ||   || 102.857 ||
||   ||   ||   || 103.524 ||
||   || 10/115 ||   || 104.348 ||
||   || 7/80 ||   || 105 ||
|| 4/45 ||   ||   || 106.667 ||
||   || 13/145 ||   || 107.586 ||
||   || 9/100 ||   || 108 ||
|| 5/55 ||   ||   || 109.111 ||
|| 1/10 ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 5s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS has five periods of 2L 1s and its generator is a neutral or diatonic semitone of 1/15edo (80 cents) to 1/10edo (120). Because it is a Blacksmith scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;86.4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;87.273&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.421&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.026&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.429&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.672&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/65&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92.308&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.119&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/115&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.913&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/140&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.286&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/165&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94,5455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/190&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.737&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/215&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.884&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/240&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ho6f7srdomh77l8ju7egursbh2ls53a</sha1>
    </revision>
    <revision>
      <id>4212</id>
      <timestamp>2015-10-26T14:15:26Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 563954933 - Original comment: **</comment>
      <origin>4212</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10318" sha1="i1mnrr4kp3lagvkluy28dmy5uflcg36" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-10-26 14:15:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;563954933&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS has five periods of 2L 1s and its generator is a neutral or diatonic semitone of 1/15edo (80 cents) to 1/10edo (120). Because it is a Blacksmith scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].
|| 1/15 ||   ||   || 80 ||
||   || 5/70 ||   || 85.714 ||
||   ||   || 9/125 || 86.4 ||
||   || 4/55 ||   || 87.273 ||
||   ||   || 7/95 || 88.421 ||
||   || 3/40 ||   || 90 ||
||   ||   ||   || 91.026 ||
||   ||   || 8/105 || 91.429 ||
||   ||   ||   || 91.672 ||
||   || 5/65 ||   || 92.308 ||
||   || 7/90 ||   || 93.333 ||
||   || 9/115 ||   || 93.913 ||
||   || 11/140 ||   || 94.286 ||
||   || 13/165 ||   || 94,5455 ||
||   || 15/190 ||   || 94.737 ||
||   || 17/215 ||   || 94.884 ||
||   || 19/240 ||   || 95 ||
|| 2/25 ||   ||   || 96 ||
||   || 11/135 ||   || 97.778 ||
||   || 9/110 ||   || 98.182 ||
||   || 7/85 ||   || 98.8235 ||
||   ||   || 12/145 || 99.31 ||
||   || 5/60 ||   || 100 ||
||   ||   ||   || 100.757 ||
||   ||   || 8/95 || 101.053 ||
||   ||   ||   || 101.3565 ||
||   || 3/35 ||   || 102.857 ||
||   ||   ||   || 103.524 ||
||   ||   || 10/115 || 104.348 ||
||   ||   || 7/80 || 105 ||
||   || 4/45 ||   || 106.667 ||
||   ||   || 13/145 || 107.586 ||
||   ||   || 9/100 || 108 ||
||   || 5/55 ||   || 109.111 ||
|| 1/10 ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 5s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS has five periods of 2L 1s and its generator is a neutral or diatonic semitone of 1/15edo (80 cents) to 1/10edo (120). Because it is a Blacksmith scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.913&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.737&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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      <sha1>i1mnrr4kp3lagvkluy28dmy5uflcg36</sha1>
    </revision>
    <revision>
      <id>4213</id>
      <timestamp>2015-05-24T10:40:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 552006424 - Original comment: **</comment>
      <origin>4213</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8255" sha1="s21242a7ioa7tgnrenpomqkvl3ubgal" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-24 10:40:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;552006424&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS has five periods of 2L 1s and its generator is a neutral or diatonic semitone of 1/15edo (80 cents) to 1/10edo (120). Because it is a Blacksmith scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].
|| 1/15 ||   ||   ||   ||   || 80 ||
||   ||   ||   ||   || 5/70 || 85 5/7 ||
||   ||   ||   || 4/55 ||   || 87 3/11 ||
||   ||   ||   ||   || 7/95 || 88.421053 ||
||   ||   || 3/40 ||   ||   || 90 ||
||   ||   ||   ||   || 8/105 || 91 3/7 ||
||   ||   ||   || 5/65 ||   || 92 4/13 ||
||   ||   ||   ||   || 7/90 || 93 1/3 ||
||   || 2/25 ||   ||   ||   || 96 ||
||   ||   ||   ||   || 7/85 || 98.82353 ||
||   ||   ||   || 5/60 ||   || 100 ||
||   ||   ||   ||   || 8/95 || 101.052632 ||
||   ||   ||   ||   ||   || 101.356512 ||
||   ||   || 3/35 ||   ||   || 102 6/7 ||
||   ||   ||   ||   ||   || 103.523693 ||
||   ||   ||   ||   || 7/80 || 105 ||
||   ||   ||   || 4/45 ||   || 106 2/3 ||
||   ||   ||   ||   || 5/55 || 109 1/9 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 5s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS has five periods of 2L 1s and its generator is a neutral or diatonic semitone of 1/15edo (80 cents) to 1/10edo (120). Because it is a Blacksmith scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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      <sha1>s21242a7ioa7tgnrenpomqkvl3ubgal</sha1>
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    <revision>
      <id>4214</id>
      <timestamp>2015-02-13T12:06:46Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540902768 - Original comment: **</comment>
      <origin>4214</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8255" sha1="40gn8zngfye3d0esmga3ofpcinudw2t" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-13 12:06:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540902768&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS has five periods of 2L 1s and its generator is a neutral or diatonic semitone of 1/15edo (80 cents) to 1/10edo (120). Because it is a Blacksmith scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].
|| 1/15 ||   ||   ||   ||   || 80 ||
||   ||   ||   ||   || 5/70 || 85 5/7 ||
||   ||   ||   || 4/55 ||   || 87 3/11 ||
||   ||   ||   ||   || 7/95 || 88.421053 ||
||   ||   || 3/40 ||   ||   || 90 ||
||   ||   ||   ||   || 8/105 || 91 3/7 ||
||   ||   ||   || 5/65 ||   || 92 4/13 ||
||   ||   ||   ||   || 7/90 || 93 1/3 ||
||   || 2/25 ||   ||   ||   || 96 ||
||   ||   ||   ||   || 7/85 || 98.82353 ||
||   ||   ||   || 5/60 ||   || 100 ||
||   ||   ||   ||   || 8/95 || 101.052632 ||
||   ||   ||   ||   ||   || 102.720894 ||
||   ||   || 3/35 ||   ||   || 102 6/7 ||
||   ||   ||   ||   ||   || 102.992495 ||
||   ||   ||   ||   || 7/80 || 105 ||
||   ||   ||   || 4/45 ||   || 106 2/3 ||
||   ||   ||   ||   || 5/55 || 109 1/9 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 5s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS has five periods of 2L 1s and its generator is a neutral or diatonic semitone of 1/15edo (80 cents) to 1/10edo (120). Because it is a Blacksmith scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].&amp;lt;br /&amp;gt;


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      <sha1>40gn8zngfye3d0esmga3ofpcinudw2t</sha1>
    </revision>
    <revision>
      <id>4215</id>
      <timestamp>2014-03-01T12:06:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 492886100 - Original comment: **</comment>
      <origin>4215</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7579" sha1="jfxosdngpfb9k3jj645jfr77811oark" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2014-03-01 12:06:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;492886100&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS has five periods of 2L 1s and its generator is a neutral or diatonic semitone of 1/15edo (80 cents) to 1/10edo (120). Because it is a Blacksmith scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].
|| 1/15 ||   ||   ||   ||   || 80 ||
||   ||   ||   ||   || 5/70 || 85 5/7 ||
||   ||   ||   || 4/55 ||   || 87 3/11 ||
||   ||   ||   ||   || 7/95 || 88.421053 ||
||   ||   || 3/40 ||   ||   || 90 ||
||   ||   ||   ||   || 8/105 || 91 3/7 ||
||   ||   ||   || 5/65 ||   || 92 4/13 ||
||   ||   ||   ||   || 7/90 || 93 1/3 ||
||   || 2/25 ||   ||   ||   || 96 ||
||   ||   ||   ||   || 7/85 || 98.82353 ||
||   ||   ||   || 5/60 ||   || 100 ||
||   ||   ||   ||   || 8/95 || 101.052632 ||
||   ||   || 3/35 ||   ||   || 102 6/7 ||
||   ||   ||   ||   || 7/80 || 105 ||
||   ||   ||   || 4/45 ||   || 106 2/3 ||
||   ||   ||   ||   || 5/55 || 109 1/9 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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    <revision>
      <id>4216</id>
      <timestamp>2012-11-01T23:41:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 378536550 - Original comment: **</comment>
      <origin>4216</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7559" sha1="lof9y5mjigq0en18ewxu73fh0zoro4g" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-11-01 23:41:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;378536550&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS has five periods of 2L 1s and its generator is a neutral or diatonic semitone of 1/15edo (80 cents) to 1/10edo (120). Because it is an Augene scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Augene[15].
|| 1/15 ||   ||   ||   ||   || 80 ||
||   ||   ||   ||   || 5/70 || 85 5/7 ||
||   ||   ||   || 4/55 ||   || 87 3/11 ||
||   ||   ||   ||   || 7/95 || 88.421053 ||
||   ||   || 3/40 ||   ||   || 90 ||
||   ||   ||   ||   || 8/105 || 91 3/7 ||
||   ||   ||   || 5/65 ||   || 92 4/13 ||
||   ||   ||   ||   || 7/90 || 93 1/3 ||
||   || 2/25 ||   ||   ||   || 96 ||
||   ||   ||   ||   || 7/85 || 98.82353 ||
||   ||   ||   || 5/60 ||   || 100 ||
||   ||   ||   ||   || 8/95 || 101.052632 ||
||   ||   || 3/35 ||   ||   || 102 6/7 ||
||   ||   ||   ||   || 7/80 || 105 ||
||   ||   ||   || 4/45 ||   || 106 2/3 ||
||   ||   ||   ||   || 5/55 || 109 1/9 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91 3/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/65&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92 4/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93 1/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/85&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.82353&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.052632&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 6/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109 1/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>lof9y5mjigq0en18ewxu73fh0zoro4g</sha1>
    </revision>
    <revision>
      <id>94330</id>
      <parentid>47</parentid>
      <timestamp>2022-08-15T16:01:17Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox, name still missing, misc. edits</comment>
      <origin>94330</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1679" sha1="na3jkz9xlvrd4kq8ej2ckprpw803v2l" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Paucitonic = 1
| Pattern = LLsLLsLLsLLsLLs
}}
The '''10L 5s''' (or '''2L 1s&lt;1\5&gt;''') [[MOS scale]] contains five [[2L 1s]] patterns per octave.

Its generator is a neutral or diatonic semitone of 1\[[15edo|15]] (80{{cent}}) to 1\[[10edo|10]] (120{{cent}}). Because it is a [[Blacksmith]] scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].

{| class="wikitable"
|-
| | 1/15
| | 
| | 
| | 80
|-
| | 5/70
| | 
| | 
| | 85.714
|-
| | 
| | 9/125
| | 
| | 86.4
|-
| | 4/55
| | 
| | 
| | 87.273
|-
| | 
| | 7/95
| | 
| | 88.421
|-
| | 3/40
| | 
| | 
| | 90
|-
| | 
| | 
| | 
| | 91.026
|-
| | 
| | 8/105
| | 
| | 91.429
|-
| | 
| | 
| | 
| | 91.672
|-
| | 
| | 5/65
| | 
| | 92.308
|-
| | 
| | 
| | 
| | 93.119
|-
| | 
| | 7/90
| | 
| | 93.333
|-
| | 
| | 9/115
| | 
| | 93.913
|-
| | 
| | 11/140
| | 
| | 94.286
|-
| | 
| | 13/165
| | 
| | 94,5455
|-
| | 
| | 15/190
| | 
| | 94.737
|-
| | 
| | 17/215
| | 
| | 94.884
|-
| | 
| | 19/240
| | 
| | 95
|-
| | 2/25
| | 
| | 
| | 96
|-
| | 
| | 11/135
| | 
| | 97.778
|-
| | 
| | 9/110
| | 
| | 98.182
|-
| | 
| | 7/85
| | 
| | 98.8235
|-
| | 
| | 
| | 12/145
| | 99.31
|-
| | 
| | 5/60
| | 
| | 100
|-
| | 
| | 
| | 
| | 100.757
|-
| | 
| | 8/95
| | 
| | 101.053
|-
| | 
| | 
| | 
| | 101.3565
|-
| | 3/35
| | 
| | 
| | 102.857
|-
| | 
| | 
| | 
| | 103.524
|-
| | 
| | 10/115
| | 
| | 104.348
|-
| | 
| | 7/80
| | 
| | 105
|-
| | 4/45
| | 
| | 
| | 106.667
|-
| | 
| | 13/145
| | 
| | 107.586
|-
| | 
| | 9/100
| | 
| | 108
|-
| | 5/55
| | 
| | 
| | 109.111
|-
| | 1/10
| | 
| | 
| | 120
|}</text>
      <sha1>na3jkz9xlvrd4kq8ej2ckprpw803v2l</sha1>
    </revision>
    <revision>
      <id>94342</id>
      <parentid>94330</parentid>
      <timestamp>2022-08-15T16:49:11Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Headers</comment>
      <origin>94342</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1696" sha1="luczo10a41vwsh05l630uvpx9k95blf" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Paucitonic = 1
| Pattern = LLsLLsLLsLLsLLs
}}
The '''10L 5s''' (or '''2L 1s&lt;1\5&gt;''') [[MOS scale]] contains five [[2L 1s]] patterns per octave.

Its generator is a neutral or diatonic semitone of 1\[[15edo|15]] (80{{cent}}) to 1\[[10edo|10]] (120{{cent}}). Because it is a [[Blacksmith]] scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].

== Scale tree ==
{| class="wikitable"
|-
| | 1/15
| | 
| | 
| | 80
|-
| | 5/70
| | 
| | 
| | 85.714
|-
| | 
| | 9/125
| | 
| | 86.4
|-
| | 4/55
| | 
| | 
| | 87.273
|-
| | 
| | 7/95
| | 
| | 88.421
|-
| | 3/40
| | 
| | 
| | 90
|-
| | 
| | 
| | 
| | 91.026
|-
| | 
| | 8/105
| | 
| | 91.429
|-
| | 
| | 
| | 
| | 91.672
|-
| | 
| | 5/65
| | 
| | 92.308
|-
| | 
| | 
| | 
| | 93.119
|-
| | 
| | 7/90
| | 
| | 93.333
|-
| | 
| | 9/115
| | 
| | 93.913
|-
| | 
| | 11/140
| | 
| | 94.286
|-
| | 
| | 13/165
| | 
| | 94,5455
|-
| | 
| | 15/190
| | 
| | 94.737
|-
| | 
| | 17/215
| | 
| | 94.884
|-
| | 
| | 19/240
| | 
| | 95
|-
| | 2/25
| | 
| | 
| | 96
|-
| | 
| | 11/135
| | 
| | 97.778
|-
| | 
| | 9/110
| | 
| | 98.182
|-
| | 
| | 7/85
| | 
| | 98.8235
|-
| | 
| | 
| | 12/145
| | 99.31
|-
| | 
| | 5/60
| | 
| | 100
|-
| | 
| | 
| | 
| | 100.757
|-
| | 
| | 8/95
| | 
| | 101.053
|-
| | 
| | 
| | 
| | 101.3565
|-
| | 3/35
| | 
| | 
| | 102.857
|-
| | 
| | 
| | 
| | 103.524
|-
| | 
| | 10/115
| | 
| | 104.348
|-
| | 
| | 7/80
| | 
| | 105
|-
| | 4/45
| | 
| | 
| | 106.667
|-
| | 
| | 13/145
| | 
| | 107.586
|-
| | 
| | 9/100
| | 
| | 108
|-
| | 5/55
| | 
| | 
| | 109.111
|-
| | 1/10
| | 
| | 
| | 120
|}</text>
      <sha1>luczo10a41vwsh05l630uvpx9k95blf</sha1>
    </revision>
    <revision>
      <id>94503</id>
      <parentid>94342</parentid>
      <timestamp>2022-08-17T09:17:07Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94503</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1695" sha1="qanbpd8bqvugdqq0by4hkr90bz3yikb" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
The '''10L 5s''' (or '''2L 1s&lt;1\5&gt;''') [[MOS scale]] contains five [[2L 1s]] patterns per octave.

Its generator is a neutral or diatonic semitone of 1\[[15edo|15]] (80{{cent}}) to 1\[[10edo|10]] (120{{cent}}). Because it is a [[Blacksmith]] scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].

== Scale tree ==
{| class="wikitable"
|-
| | 1/15
| | 
| | 
| | 80
|-
| | 5/70
| | 
| | 
| | 85.714
|-
| | 
| | 9/125
| | 
| | 86.4
|-
| | 4/55
| | 
| | 
| | 87.273
|-
| | 
| | 7/95
| | 
| | 88.421
|-
| | 3/40
| | 
| | 
| | 90
|-
| | 
| | 
| | 
| | 91.026
|-
| | 
| | 8/105
| | 
| | 91.429
|-
| | 
| | 
| | 
| | 91.672
|-
| | 
| | 5/65
| | 
| | 92.308
|-
| | 
| | 
| | 
| | 93.119
|-
| | 
| | 7/90
| | 
| | 93.333
|-
| | 
| | 9/115
| | 
| | 93.913
|-
| | 
| | 11/140
| | 
| | 94.286
|-
| | 
| | 13/165
| | 
| | 94,5455
|-
| | 
| | 15/190
| | 
| | 94.737
|-
| | 
| | 17/215
| | 
| | 94.884
|-
| | 
| | 19/240
| | 
| | 95
|-
| | 2/25
| | 
| | 
| | 96
|-
| | 
| | 11/135
| | 
| | 97.778
|-
| | 
| | 9/110
| | 
| | 98.182
|-
| | 
| | 7/85
| | 
| | 98.8235
|-
| | 
| | 
| | 12/145
| | 99.31
|-
| | 
| | 5/60
| | 
| | 100
|-
| | 
| | 
| | 
| | 100.757
|-
| | 
| | 8/95
| | 
| | 101.053
|-
| | 
| | 
| | 
| | 101.3565
|-
| | 3/35
| | 
| | 
| | 102.857
|-
| | 
| | 
| | 
| | 103.524
|-
| | 
| | 10/115
| | 
| | 104.348
|-
| | 
| | 7/80
| | 
| | 105
|-
| | 4/45
| | 
| | 
| | 106.667
|-
| | 
| | 13/145
| | 
| | 107.586
|-
| | 
| | 9/100
| | 
| | 108
|-
| | 5/55
| | 
| | 
| | 109.111
|-
| | 1/10
| | 
| | 
| | 120
|}</text>
      <sha1>qanbpd8bqvugdqq0by4hkr90bz3yikb</sha1>
    </revision>
    <revision>
      <id>101036</id>
      <parentid>94503</parentid>
      <timestamp>2023-01-06T16:16:17Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Copying from [[Category:Abstract MOS patterns]] to [[Category:15-tone scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>101036</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1724" sha1="bwkfbijjpkgslka4c8vh0x5aamyesof" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
The '''10L 5s''' (or '''2L 1s&lt;1\5&gt;''') [[MOS scale]] contains five [[2L 1s]] patterns per octave.

Its generator is a neutral or diatonic semitone of 1\[[15edo|15]] (80{{cent}}) to 1\[[10edo|10]] (120{{cent}}). Because it is a [[Blacksmith]] scale which appears in 5n-edos, it may be referred to more clearly as the Quinary mode of Blacksmith[15].

== Scale tree ==
{| class="wikitable"
|-
| | 1/15
| | 
| | 
| | 80
|-
| | 5/70
| | 
| | 
| | 85.714
|-
| | 
| | 9/125
| | 
| | 86.4
|-
| | 4/55
| | 
| | 
| | 87.273
|-
| | 
| | 7/95
| | 
| | 88.421
|-
| | 3/40
| | 
| | 
| | 90
|-
| | 
| | 
| | 
| | 91.026
|-
| | 
| | 8/105
| | 
| | 91.429
|-
| | 
| | 
| | 
| | 91.672
|-
| | 
| | 5/65
| | 
| | 92.308
|-
| | 
| | 
| | 
| | 93.119
|-
| | 
| | 7/90
| | 
| | 93.333
|-
| | 
| | 9/115
| | 
| | 93.913
|-
| | 
| | 11/140
| | 
| | 94.286
|-
| | 
| | 13/165
| | 
| | 94,5455
|-
| | 
| | 15/190
| | 
| | 94.737
|-
| | 
| | 17/215
| | 
| | 94.884
|-
| | 
| | 19/240
| | 
| | 95
|-
| | 2/25
| | 
| | 
| | 96
|-
| | 
| | 11/135
| | 
| | 97.778
|-
| | 
| | 9/110
| | 
| | 98.182
|-
| | 
| | 7/85
| | 
| | 98.8235
|-
| | 
| | 
| | 12/145
| | 99.31
|-
| | 
| | 5/60
| | 
| | 100
|-
| | 
| | 
| | 
| | 100.757
|-
| | 
| | 8/95
| | 
| | 101.053
|-
| | 
| | 
| | 
| | 101.3565
|-
| | 3/35
| | 
| | 
| | 102.857
|-
| | 
| | 
| | 
| | 103.524
|-
| | 
| | 10/115
| | 
| | 104.348
|-
| | 
| | 7/80
| | 
| | 105
|-
| | 4/45
| | 
| | 
| | 106.667
|-
| | 
| | 13/145
| | 
| | 107.586
|-
| | 
| | 9/100
| | 
| | 108
|-
| | 5/55
| | 
| | 
| | 109.111
|-
| | 1/10
| | 
| | 
| | 120
|}

[[Category:15-tone scales]]</text>
      <sha1>bwkfbijjpkgslka4c8vh0x5aamyesof</sha1>
    </revision>
    <revision>
      <id>113379</id>
      <parentid>101036</parentid>
      <timestamp>2023-05-15T22:13:51Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>113379</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1586" sha1="d3aziot3bsks7d3rofil3ly4zjt7185" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
The '''10L 5s''' (or '''2L 1s&lt;1\5&gt;''') [[MOS scale]] contains five [[2L 1s]] patterns per octave. Its generator is a neutral or diatonic semitone of 1\[[15edo|15]] (80{{cent}}) to 1\[[10edo|10]] (120{{cent}}).

== Scale tree ==
{| class="wikitable"
|-
| | 1/15
| | 
| | 
| | 80
|-
| | 5/70
| | 
| | 
| | 85.714
|-
| | 
| | 9/125
| | 
| | 86.4
|-
| | 4/55
| | 
| | 
| | 87.273
|-
| | 
| | 7/95
| | 
| | 88.421
|-
| | 3/40
| | 
| | 
| | 90
|-
| | 
| | 
| | 
| | 91.026
|-
| | 
| | 8/105
| | 
| | 91.429
|-
| | 
| | 
| | 
| | 91.672
|-
| | 
| | 5/65
| | 
| | 92.308
|-
| | 
| | 
| | 
| | 93.119
|-
| | 
| | 7/90
| | 
| | 93.333
|-
| | 
| | 9/115
| | 
| | 93.913
|-
| | 
| | 11/140
| | 
| | 94.286
|-
| | 
| | 13/165
| | 
| | 94,5455
|-
| | 
| | 15/190
| | 
| | 94.737
|-
| | 
| | 17/215
| | 
| | 94.884
|-
| | 
| | 19/240
| | 
| | 95
|-
| | 2/25
| | 
| | 
| | 96
|-
| | 
| | 11/135
| | 
| | 97.778
|-
| | 
| | 9/110
| | 
| | 98.182
|-
| | 
| | 7/85
| | 
| | 98.8235
|-
| | 
| | 
| | 12/145
| | 99.31
|-
| | 
| | 5/60
| | 
| | 100
|-
| | 
| | 
| | 
| | 100.757
|-
| | 
| | 8/95
| | 
| | 101.053
|-
| | 
| | 
| | 
| | 101.3565
|-
| | 3/35
| | 
| | 
| | 102.857
|-
| | 
| | 
| | 
| | 103.524
|-
| | 
| | 10/115
| | 
| | 104.348
|-
| | 
| | 7/80
| | 
| | 105
|-
| | 4/45
| | 
| | 
| | 106.667
|-
| | 
| | 13/145
| | 
| | 107.586
|-
| | 
| | 9/100
| | 
| | 108
|-
| | 5/55
| | 
| | 
| | 109.111
|-
| | 1/10
| | 
| | 
| | 120
|}

[[Category:15-tone scales]]</text>
      <sha1>d3aziot3bsks7d3rofil3ly4zjt7185</sha1>
    </revision>
    <revision>
      <id>113457</id>
      <parentid>113379</parentid>
      <timestamp>2023-05-16T21:38:07Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>113457</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1632" sha1="kott4e8t74b79twndqd9a2bpd2uuzsw" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
The '''10L 5s''' (or '''2L 1s&lt;1\5&gt;''') [[MOS scale]] contains five [[2L 1s]] patterns per octave. Its generator is a neutral or diatonic semitone of 1\[[15edo|15]] (80{{cent}}) to 1\[[10edo|10]] (120{{cent}}).

== Scale tree ==
{| class="wikitable"
|-
| | 1/15
| | 
| | 
| | 80
|-
| | 5/70
| | 
| | 
| | 85.714
|-
| | 
| | 9/125
| | 
| | 86.4
|-
| | 4/55
| | 
| | 
| | 87.273
|-
| | 
| | 7/95
| | 
| | 88.421
|-
|
|10/135
|
|88.889
|-
|
|13/175
|
|89.143
|-
| | 3/40
| | 
| | 
| | 90
|-
| | 
| | 
| | 
| | 91.026
|-
| | 
| | 8/105
| | 
| | 91.429
|-
| | 
| | 
| | 
| | 91.672
|-
| | 
| | 5/65
| | 
| | 92.308
|-
| | 
| | 
| | 
| | 93.119
|-
| | 
| | 7/90
| | 
| | 93.333
|-
| | 
| | 9/115
| | 
| | 93.913
|-
| | 
| | 11/140
| | 
| | 94.286
|-
| | 
| | 13/165
| | 
| | 94,5455
|-
| | 
| | 15/190
| | 
| | 94.737
|-
| | 
| | 17/215
| | 
| | 94.884
|-
| | 
| | 19/240
| | 
| | 95
|-
| | 2/25
| | 
| | 
| | 96
|-
| | 
| | 11/135
| | 
| | 97.778
|-
| | 
| | 9/110
| | 
| | 98.182
|-
| | 
| | 7/85
| | 
| | 98.8235
|-
| | 
| | 
| | 12/145
| | 99.31
|-
| | 
| | 5/60
| | 
| | 100
|-
| | 
| | 
| | 
| | 100.757
|-
| | 
| | 8/95
| | 
| | 101.053
|-
| | 
| | 
| | 
| | 101.3565
|-
| | 3/35
| | 
| | 
| | 102.857
|-
| | 
| | 
| | 
| | 103.524
|-
| | 
| | 10/115
| | 
| | 104.348
|-
| | 
| | 7/80
| | 
| | 105
|-
| | 4/45
| | 
| | 
| | 106.667
|-
| | 
| | 13/145
| | 
| | 107.586
|-
| | 
| | 9/100
| | 
| | 108
|-
| | 5/55
| | 
| | 
| | 109.091
|-
| | 1/10
| | 
| | 
| | 120
|}

[[Category:15-tone scales]]</text>
      <sha1>kott4e8t74b79twndqd9a2bpd2uuzsw</sha1>
    </revision>
    <revision>
      <id>133691</id>
      <parentid>113457</parentid>
      <timestamp>2024-01-30T08:13:30Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>+modes</comment>
      <origin>133691</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1658" sha1="nos6gr6ou76zu5piw8khcvu4aazqrpa" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
The '''10L 5s''' (or '''2L 1s&lt;1\5&gt;''') [[MOS scale]] contains five [[2L 1s]] patterns per octave. Its generator is a neutral or diatonic semitone of 1\[[15edo|15]] (80{{cent}}) to 1\[[10edo|10]] (120{{cent}}).
== Modes ===
{{MOS modes}}
== Scale tree ==
{| class="wikitable"
|-
| | 1/15
| | 
| | 
| | 80
|-
| | 5/70
| | 
| | 
| | 85.714
|-
| | 
| | 9/125
| | 
| | 86.4
|-
| | 4/55
| | 
| | 
| | 87.273
|-
| | 
| | 7/95
| | 
| | 88.421
|-
|
|10/135
|
|88.889
|-
|
|13/175
|
|89.143
|-
| | 3/40
| | 
| | 
| | 90
|-
| | 
| | 
| | 
| | 91.026
|-
| | 
| | 8/105
| | 
| | 91.429
|-
| | 
| | 
| | 
| | 91.672
|-
| | 
| | 5/65
| | 
| | 92.308
|-
| | 
| | 
| | 
| | 93.119
|-
| | 
| | 7/90
| | 
| | 93.333
|-
| | 
| | 9/115
| | 
| | 93.913
|-
| | 
| | 11/140
| | 
| | 94.286
|-
| | 
| | 13/165
| | 
| | 94,5455
|-
| | 
| | 15/190
| | 
| | 94.737
|-
| | 
| | 17/215
| | 
| | 94.884
|-
| | 
| | 19/240
| | 
| | 95
|-
| | 2/25
| | 
| | 
| | 96
|-
| | 
| | 11/135
| | 
| | 97.778
|-
| | 
| | 9/110
| | 
| | 98.182
|-
| | 
| | 7/85
| | 
| | 98.8235
|-
| | 
| | 
| | 12/145
| | 99.31
|-
| | 
| | 5/60
| | 
| | 100
|-
| | 
| | 
| | 
| | 100.757
|-
| | 
| | 8/95
| | 
| | 101.053
|-
| | 
| | 
| | 
| | 101.3565
|-
| | 3/35
| | 
| | 
| | 102.857
|-
| | 
| | 
| | 
| | 103.524
|-
| | 
| | 10/115
| | 
| | 104.348
|-
| | 
| | 7/80
| | 
| | 105
|-
| | 4/45
| | 
| | 
| | 106.667
|-
| | 
| | 13/145
| | 
| | 107.586
|-
| | 
| | 9/100
| | 
| | 108
|-
| | 5/55
| | 
| | 
| | 109.091
|-
| | 1/10
| | 
| | 
| | 120
|}

[[Category:15-tone scales]]</text>
      <sha1>nos6gr6ou76zu5piw8khcvu4aazqrpa</sha1>
    </revision>
    <revision>
      <id>133692</id>
      <parentid>133691</parentid>
      <timestamp>2024-01-30T08:14:33Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>133692</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1657" sha1="m1pbl93t3y9lud1k3tlkn9g12upkg3w" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
The '''10L 5s''' (or '''2L 1s&lt;1\5&gt;''') [[MOS scale]] contains five [[2L 1s]] patterns per octave. Its generator is a neutral or diatonic semitone of 1\[[15edo|15]] (80{{cent}}) to 1\[[10edo|10]] (120{{cent}}).
== Modes ==
{{MOS modes}}
== Scale tree ==
{| class="wikitable"
|-
| | 1/15
| | 
| | 
| | 80
|-
| | 5/70
| | 
| | 
| | 85.714
|-
| | 
| | 9/125
| | 
| | 86.4
|-
| | 4/55
| | 
| | 
| | 87.273
|-
| | 
| | 7/95
| | 
| | 88.421
|-
|
|10/135
|
|88.889
|-
|
|13/175
|
|89.143
|-
| | 3/40
| | 
| | 
| | 90
|-
| | 
| | 
| | 
| | 91.026
|-
| | 
| | 8/105
| | 
| | 91.429
|-
| | 
| | 
| | 
| | 91.672
|-
| | 
| | 5/65
| | 
| | 92.308
|-
| | 
| | 
| | 
| | 93.119
|-
| | 
| | 7/90
| | 
| | 93.333
|-
| | 
| | 9/115
| | 
| | 93.913
|-
| | 
| | 11/140
| | 
| | 94.286
|-
| | 
| | 13/165
| | 
| | 94,5455
|-
| | 
| | 15/190
| | 
| | 94.737
|-
| | 
| | 17/215
| | 
| | 94.884
|-
| | 
| | 19/240
| | 
| | 95
|-
| | 2/25
| | 
| | 
| | 96
|-
| | 
| | 11/135
| | 
| | 97.778
|-
| | 
| | 9/110
| | 
| | 98.182
|-
| | 
| | 7/85
| | 
| | 98.8235
|-
| | 
| | 
| | 12/145
| | 99.31
|-
| | 
| | 5/60
| | 
| | 100
|-
| | 
| | 
| | 
| | 100.757
|-
| | 
| | 8/95
| | 
| | 101.053
|-
| | 
| | 
| | 
| | 101.3565
|-
| | 3/35
| | 
| | 
| | 102.857
|-
| | 
| | 
| | 
| | 103.524
|-
| | 
| | 10/115
| | 
| | 104.348
|-
| | 
| | 7/80
| | 
| | 105
|-
| | 4/45
| | 
| | 
| | 106.667
|-
| | 
| | 13/145
| | 
| | 107.586
|-
| | 
| | 9/100
| | 
| | 108
|-
| | 5/55
| | 
| | 
| | 109.091
|-
| | 1/10
| | 
| | 
| | 120
|}

[[Category:15-tone scales]]</text>
      <sha1>m1pbl93t3y9lud1k3tlkn9g12upkg3w</sha1>
    </revision>
    <revision>
      <id>136596</id>
      <parentid>133692</parentid>
      <timestamp>2024-02-22T05:23:20Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Mos intro</comment>
      <origin>136596</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1539" sha1="s50mfwukja0lbv0qna29p3r1942p3xr" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
{{MOS intro}}

Its generator is a neutral or diatonic semitone (80{{cent}} to 120{{cent}}).
== Modes ==
{{MOS modes}}
== Scale tree ==
{| class="wikitable"
|-
| | 1/15
| | 
| | 
| | 80
|-
| | 5/70
| | 
| | 
| | 85.714
|-
| | 
| | 9/125
| | 
| | 86.4
|-
| | 4/55
| | 
| | 
| | 87.273
|-
| | 
| | 7/95
| | 
| | 88.421
|-
|
|10/135
|
|88.889
|-
|
|13/175
|
|89.143
|-
| | 3/40
| | 
| | 
| | 90
|-
| | 
| | 
| | 
| | 91.026
|-
| | 
| | 8/105
| | 
| | 91.429
|-
| | 
| | 
| | 
| | 91.672
|-
| | 
| | 5/65
| | 
| | 92.308
|-
| | 
| | 
| | 
| | 93.119
|-
| | 
| | 7/90
| | 
| | 93.333
|-
| | 
| | 9/115
| | 
| | 93.913
|-
| | 
| | 11/140
| | 
| | 94.286
|-
| | 
| | 13/165
| | 
| | 94,5455
|-
| | 
| | 15/190
| | 
| | 94.737
|-
| | 
| | 17/215
| | 
| | 94.884
|-
| | 
| | 19/240
| | 
| | 95
|-
| | 2/25
| | 
| | 
| | 96
|-
| | 
| | 11/135
| | 
| | 97.778
|-
| | 
| | 9/110
| | 
| | 98.182
|-
| | 
| | 7/85
| | 
| | 98.8235
|-
| | 
| | 
| | 12/145
| | 99.31
|-
| | 
| | 5/60
| | 
| | 100
|-
| | 
| | 
| | 
| | 100.757
|-
| | 
| | 8/95
| | 
| | 101.053
|-
| | 
| | 
| | 
| | 101.3565
|-
| | 3/35
| | 
| | 
| | 102.857
|-
| | 
| | 
| | 
| | 103.524
|-
| | 
| | 10/115
| | 
| | 104.348
|-
| | 
| | 7/80
| | 
| | 105
|-
| | 4/45
| | 
| | 
| | 106.667
|-
| | 
| | 13/145
| | 
| | 107.586
|-
| | 
| | 9/100
| | 
| | 108
|-
| | 5/55
| | 
| | 
| | 109.091
|-
| | 1/10
| | 
| | 
| | 120
|}

[[Category:15-tone scales]]</text>
      <sha1>s50mfwukja0lbv0qna29p3r1942p3xr</sha1>
    </revision>
    <revision>
      <id>136619</id>
      <parentid>136596</parentid>
      <timestamp>2024-02-22T06:09:55Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <minor/>
      <comment>a space</comment>
      <origin>136619</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1538" sha1="k2zl81i6emi3lquvtic8ozc8lv5c3ms" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
{{MOS intro}}
Its generator is a neutral or diatonic semitone (80{{cent}} to 120{{cent}}).
== Modes ==
{{MOS modes}}
== Scale tree ==
{| class="wikitable"
|-
| | 1/15
| | 
| | 
| | 80
|-
| | 5/70
| | 
| | 
| | 85.714
|-
| | 
| | 9/125
| | 
| | 86.4
|-
| | 4/55
| | 
| | 
| | 87.273
|-
| | 
| | 7/95
| | 
| | 88.421
|-
|
|10/135
|
|88.889
|-
|
|13/175
|
|89.143
|-
| | 3/40
| | 
| | 
| | 90
|-
| | 
| | 
| | 
| | 91.026
|-
| | 
| | 8/105
| | 
| | 91.429
|-
| | 
| | 
| | 
| | 91.672
|-
| | 
| | 5/65
| | 
| | 92.308
|-
| | 
| | 
| | 
| | 93.119
|-
| | 
| | 7/90
| | 
| | 93.333
|-
| | 
| | 9/115
| | 
| | 93.913
|-
| | 
| | 11/140
| | 
| | 94.286
|-
| | 
| | 13/165
| | 
| | 94,5455
|-
| | 
| | 15/190
| | 
| | 94.737
|-
| | 
| | 17/215
| | 
| | 94.884
|-
| | 
| | 19/240
| | 
| | 95
|-
| | 2/25
| | 
| | 
| | 96
|-
| | 
| | 11/135
| | 
| | 97.778
|-
| | 
| | 9/110
| | 
| | 98.182
|-
| | 
| | 7/85
| | 
| | 98.8235
|-
| | 
| | 
| | 12/145
| | 99.31
|-
| | 
| | 5/60
| | 
| | 100
|-
| | 
| | 
| | 
| | 100.757
|-
| | 
| | 8/95
| | 
| | 101.053
|-
| | 
| | 
| | 
| | 101.3565
|-
| | 3/35
| | 
| | 
| | 102.857
|-
| | 
| | 
| | 
| | 103.524
|-
| | 
| | 10/115
| | 
| | 104.348
|-
| | 
| | 7/80
| | 
| | 105
|-
| | 4/45
| | 
| | 
| | 106.667
|-
| | 
| | 13/145
| | 
| | 107.586
|-
| | 
| | 9/100
| | 
| | 108
|-
| | 5/55
| | 
| | 
| | 109.091
|-
| | 1/10
| | 
| | 
| | 120
|}

[[Category:15-tone scales]]</text>
      <sha1>k2zl81i6emi3lquvtic8ozc8lv5c3ms</sha1>
    </revision>
    <revision>
      <id>137506</id>
      <parentid>136619</parentid>
      <timestamp>2024-02-29T05:24:53Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Removed redundant mention of generator range, replaced scale tree</comment>
      <origin>137506</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="238" sha1="hevwhnju79tr4mdnn9vfmuwbp9zgt5p" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
{{MOS intro}}

== Modes ==
{{MOS modes}}
== Scale tree ==
{{Scale tree}}
[[Category:15-tone scales]]</text>
      <sha1>hevwhnju79tr4mdnn9vfmuwbp9zgt5p</sha1>
    </revision>
    <revision>
      <id>137507</id>
      <parentid>137506</parentid>
      <timestamp>2024-02-29T05:25:02Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <minor/>
      <comment>a space</comment>
      <origin>137507</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="237" sha1="ewqe5n8upr322c2jnqw7pz4k031peyj" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
{{MOS intro}}
== Modes ==
{{MOS modes}}
== Scale tree ==
{{Scale tree}}
[[Category:15-tone scales]]</text>
      <sha1>ewqe5n8upr322c2jnqw7pz4k031peyj</sha1>
    </revision>
    <revision>
      <id>138069</id>
      <parentid>137507</parentid>
      <timestamp>2024-03-03T23:02:59Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>stub</comment>
      <origin>138069</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="246" sha1="fge7xbha3yjmk3r3iosq374ubiyt4y6" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
{{MOS intro}}
== Modes ==
{{MOS modes}}
== Scale tree ==
{{Scale tree}} {{stub}}
[[Category:15-tone scales]]</text>
      <sha1>fge7xbha3yjmk3r3iosq374ubiyt4y6</sha1>
    </revision>
    <revision>
      <id>170555</id>
      <parentid>138069</parentid>
      <timestamp>2024-12-15T06:48:08Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Add intervals</comment>
      <origin>170555</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="282" sha1="bfzksz4k6ah5e90i0xj6ji5u6pimv1m" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
{{MOS intro}}
== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree}} {{stub}}
[[Category:15-tone scales]]</text>
      <sha1>bfzksz4k6ah5e90i0xj6ji5u6pimv1m</sha1>
    </revision>
    <revision>
      <id>170565</id>
      <parentid>170555</parentid>
      <timestamp>2024-12-15T06:53:24Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Stub</comment>
      <origin>170565</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="285" sha1="rcharf0rmzvkg9fknf55ooblg6ioxhz" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
{{MOS intro}}
== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree}} 


{{stub}}
[[Category:15-tone scales]]</text>
      <sha1>rcharf0rmzvkg9fknf55ooblg6ioxhz</sha1>
    </revision>
    <revision>
      <id>183726</id>
      <parentid>170565</parentid>
      <timestamp>2025-02-28T16:36:54Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183726</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="295" sha1="06m67uu1ri1xqztiluv4837fpkpyvei" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
{{MOS intro}}

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum}} 

{{stub}}

[[Category:15-tone scales]]</text>
      <sha1>06m67uu1ri1xqztiluv4837fpkpyvei</sha1>
    </revision>
    <revision>
      <id>196025</id>
      <parentid>183726</parentid>
      <timestamp>2025-05-12T13:29:12Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>196025</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="354" sha1="fp29gcdgmr48geen0i853ouphupt9ee" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 5
| nLargeSteps = 10
| nSmallSteps = 5
| Equalized = 1
| Collapsed = 1
| Pattern = LLsLLsLLsLLsLLs
}}
{{MOS intro}}

== Theory ==
=== Intervals ===
{{MOS intervals}}

=== Generator chain ===
{{MOS genchain}}

=== Modes ===
{{MOS modes}}

== Scale tree ==
{{MOS tuning spectrum}} 

{{stub}}

[[Category:15-tone scales]]</text>
      <sha1>fp29gcdgmr48geen0i853ouphupt9ee</sha1>
    </revision>
  </page>
  <page>
    <title>10L 7s</title>
    <ns>0</ns>
    <id>48</id>
    <revision>
      <id>48</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>48</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1309" sha1="ibszi4905rx3jflzisq0voyookdsf5n" xml:space="preserve">This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352.941) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).

{| class="wikitable"
|-
| | 3/10
| | 
| | 
| | 
| | 
| | 360
|-
| | 
| | 
| | 
| | 
| | 17/57
| | 357.895
|-
| | 
| | 
| | 
| | 14/47
| | 
| | 357.447
|-
| | 
| | 
| | 
| | 
| | 25/84
| | 357.143
|-
| | 
| | 
| | 
| | 
| | 
| | 356.876
|-
| | 
| | 
| | 11/37
| | 
| | 
| | 356.757
|-
| | 
| | 
| | 
| | 
| | 
| | 356.4895
|-
| | 
| | 
| | 
| | 
| | 30/101
| | 356.436
|-
| | 
| | 
| | 
| | 
| | 
| | 356.383
|-
| | 
| | 
| | 
| | 19/64
| | 
| | 356.25
|-
| | 
| | 
| | 
| | 
| | 27/91
| | 356.044
|-
| | 
| | 8/27
| | 
| | 
| | 
| | 355.556
|-
| | 
| | 
| | 
| | 
| | 29/98
| | 355.102
|-
| | 
| | 
| | 
| | 
| | 
| | 355.066
|-
| | 
| | 
| | 
| | 21/71
| | 
| | 354.93
|-
| | 
| | 
| | 
| | 
| | 
| | 354.823
|-
| | 
| | 
| | 
| | 
| | 34/115
| | 354.783
|-
| | 
| | 
| | 
| | 
| | 
| | 354.7155
|-
| | 
| | 
| | 13/44
| | 
| | 
| | 354.5455
|-
| | 
| | 
| | 
| | 
| | 31/105
| | 354.286
|-
| | 
| | 
| | 
| | 18/61
| | 
| | 354.098
|-
| | 
| | 
| | 
| | 
| | 23/78
| | 353.846
|-
| | 5/17
| | 
| | 
| | 
| | 
| | 352.941
|}</text>
      <sha1>ibszi4905rx3jflzisq0voyookdsf5n</sha1>
    </revision>
    <revision>
      <id>4217</id>
      <timestamp>2015-11-14T16:22:55Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 566469231 - Original comment: **</comment>
      <origin>4217</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9703" sha1="jgihe5hpap2bv70iyhi7riaonjnk3km" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-14 16:22:55 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566469231&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352.941) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).
|| 3/10 ||   ||   ||   ||   || 360 ||
||   ||   ||   ||   || 17/57 || 357.895 ||
||   ||   ||   || 14/47 ||   || 357.447 ||
||   ||   ||   ||   || 25/84 || 357.143 ||
||   ||   ||   ||   ||   || 356.876 ||
||   ||   || 11/37 ||   ||   || 356.757 ||
||   ||   ||   ||   ||   || 356.4895 ||
||   ||   ||   ||   || 30/101 || 356.436 ||
||   ||   ||   ||   ||   || 356.383 ||
||   ||   ||   || 19/64 ||   || 356.25 ||
||   ||   ||   ||   || 27/91 || 356.044 ||
||   || 8/27 ||   ||   ||   || 355.556 ||
||   ||   ||   ||   || 29/98 || 355.102 ||
||   ||   ||   ||   ||   || 355.066 ||
||   ||   ||   || 21/71 ||   || 354.93 ||
||   ||   ||   ||   ||   || 354.823 ||
||   ||   ||   ||   || 34/115 || 354.783 ||
||   ||   ||   ||   ||   || 354.7155 ||
||   ||   || 13/44 ||   ||   || 354.5455 ||
||   ||   ||   ||   || 31/105 || 354.286 ||
||   ||   ||   || 18/61 ||   || 354.098 ||
||   ||   ||   ||   || 23/78 || 353.846 ||
|| 5/17 ||   ||   ||   ||   || 352.941 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 7s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352.941) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357.895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357.447&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25/84&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357.143&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.876&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.757&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.4895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.383&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.044&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;355.556&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;355.102&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;355.066&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21/71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354.823&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34/115&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354.783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354.7155&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/44&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354.5455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;31/105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354.286&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18/61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354.098&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23/78&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;353.846&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;352.941&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>jgihe5hpap2bv70iyhi7riaonjnk3km</sha1>
    </revision>
    <revision>
      <id>4218</id>
      <timestamp>2015-10-26T21:01:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 564010301 - Original comment: **</comment>
      <origin>4218</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9368" sha1="dsr9t4ghu8f03cc3l9q0zlto3kvrg4s" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-10-26 21:01:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;564010301&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352.941) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).
|| 3/10 ||   ||   ||   ||   || 360 ||
||   ||   ||   ||   || 17/57 || 357.895 ||
||   ||   ||   || 14/47 ||   || 357.447 ||
||   ||   ||   ||   || 25/84 || 357.143 ||
||   ||   ||   ||   ||   || 356.876 ||
||   ||   || 11/37 ||   ||   || 356.757 ||
||   ||   ||   ||   ||   || 356.4895 ||
||   ||   ||   ||   || 30/101 || 356.436 ||
||   ||   ||   ||   ||   || 356.383 ||
||   ||   ||   || 19/64 ||   || 356.25 ||
||   ||   ||   ||   || 27/91 || 356.044 ||
||   || 8/27 ||   ||   ||   || 355.556 ||
||   ||   ||   ||   || 29/98 || 355.102 ||
||   ||   ||   || 21/71 ||   || 354.93 ||
||   ||   ||   ||   ||   || 354.823 ||
||   ||   ||   ||   || 34/115 || 354.783 ||
||   ||   ||   ||   ||   || 354.7155 ||
||   ||   || 13/44 ||   ||   || 354.5455 ||
||   ||   ||   ||   || 31/105 || 354.286 ||
||   ||   ||   || 18/61 ||   || 354.098 ||
||   ||   ||   ||   || 23/78 || 353.846 ||
|| 5/17 ||   ||   ||   ||   || 352.941 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 7s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352.941) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357.895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357.447&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25/84&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357.143&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.876&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.757&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.4895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.383&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.044&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;355.556&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;355.102&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21/71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354.823&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34/115&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354.783&amp;lt;br /&amp;gt;
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      <sha1>dsr9t4ghu8f03cc3l9q0zlto3kvrg4s</sha1>
    </revision>
    <revision>
      <id>4219</id>
      <timestamp>2015-06-11T23:01:35Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 553737652 - Original comment: **</comment>
      <origin>4219</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8427" sha1="ck7dtx6iqc8ma96riuw5hf1puh27x8d" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-06-11 23:01:35 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;553737652&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is like the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352 16/17) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).
|| 3/10 ||   ||   ||   ||   || 360 ||
||   ||   ||   ||   || 17/57 || 357 17/19 ||
||   ||   ||   || 14/47 ||   || 357 21/47 ||
||   ||   ||   ||   || 25/84 || 357 1/7 ||
||   ||   ||   ||   ||   || 356.876296 ||
||   ||   || 11/37 ||   ||   || 356 28/37 ||
||   ||   ||   ||   ||   || 356.489464 ||
||   ||   ||   ||   || 30/101 || 356.435644 ||
||   ||   ||   || 19/64 ||   || 356.25 ||
||   ||   ||   ||   || 27/91 || 356 4/91 ||
||   || 8/27 ||   ||   ||   || 355 5/9 ||
||   ||   ||   ||   || 29/98 || 355 5/49 ||
||   ||   ||   || 21/71 ||   || 354 66/71 ||
||   ||   ||   ||   || 34/115 || 354 18/23 ||
||   ||   || 13/44 ||   ||   || 354 6/11 ||
||   ||   ||   ||   || 31/105 || 354 2/7 ||
||   ||   ||   || 18/61 ||   || 354 6/61 ||
||   ||   ||   ||   || 23/78 || 353 11/13 ||
|| 5/17 ||   ||   ||   ||   || 352 16/17 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 7s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is like the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352 16/17) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;25/84&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;357 1/7&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;356.876296&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;11/37&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;356.489464&amp;lt;br /&amp;gt;
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&amp;lt;/table&amp;gt;

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      <sha1>ck7dtx6iqc8ma96riuw5hf1puh27x8d</sha1>
    </revision>
    <revision>
      <id>4220</id>
      <timestamp>2015-05-24T11:41:46Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 552009120 - Original comment: **</comment>
      <origin>4220</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8427" sha1="lamqzut5t43iu1zui7pt0p76r1fw1tg" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-24 11:41:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;552009120&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is like the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352 16/17) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).
|| 3/10 ||   ||   ||   ||   || 360 ||
||   ||   ||   ||   || 17/57 || 357 17/19 ||
||   ||   ||   || 14/47 ||   || 357 21/47 ||
||   ||   ||   ||   || 25/84 || 357 1/7 ||
||   ||   ||   ||   ||   || 356.876296 ||
||   ||   || 11/37 ||   ||   || 356 28/37 ||
||   ||   ||   ||   ||   || 356.489464 ||
||   ||   ||   ||   || 30/101 || 356 44/101 ||
||   ||   ||   || 19/64 ||   || 356.25 ||
||   ||   ||   ||   || 27/91 || 356 4/91 ||
||   || 8/27 ||   ||   ||   || 355 5/9 ||
||   ||   ||   ||   || 29/98 || 355 5/49 ||
||   ||   ||   || 21/71 ||   || 354 66/71 ||
||   ||   ||   ||   || 34/115 || 354 18/23 ||
||   ||   || 13/44 ||   ||   || 354 6/11 ||
||   ||   ||   ||   || 31/105 || 354 2/7 ||
||   ||   ||   || 18/61 ||   || 354 6/61 ||
||   ||   ||   ||   || 23/78 || 353 11/13 ||
|| 5/17 ||   ||   ||   ||   || 352 16/17 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 7s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is like the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352 16/17) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/10&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357 21/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25/84&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357 1/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.876296&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356 28/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.489464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356 44/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356 4/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;355 5/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;355 5/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21/71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 66/71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34/115&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 18/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/44&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 6/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;31/105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 2/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18/61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 6/61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23/78&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;353 11/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;352 16/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>lamqzut5t43iu1zui7pt0p76r1fw1tg</sha1>
    </revision>
    <revision>
      <id>4221</id>
      <timestamp>2015-02-13T13:48:03Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540915052 - Original comment: **</comment>
      <origin>4221</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8425" sha1="4inevd15c0l2x34uk7cav27lf93mzy9" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-13 13:48:03 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540915052&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is like the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352 16/17) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).
|| 3/10 ||   ||   ||   ||   || 360 ||
||   ||   ||   ||   || 17/57 || 357 17/19 ||
||   ||   ||   || 14/47 ||   || 357 21/47 ||
||   ||   ||   ||   || 25/84 || 357 1/7 ||
||   ||   ||   ||   ||   || 356.78099 ||
||   ||   || 11/37 ||   ||   || 356 28/37 ||
||   ||   ||   ||   ||   || 356.732384 ||
||   ||   ||   ||   || 30/101 || 356 44/101 ||
||   ||   ||   || 19/64 ||   || 356.25 ||
||   ||   ||   ||   || 27/91 || 356 4/91 ||
||   || 8/27 ||   ||   ||   || 355 5/9 ||
||   ||   ||   ||   || 29/98 || 355 5/49 ||
||   ||   ||   || 21/71 ||   || 354 66/71 ||
||   ||   ||   ||   || 34/115 || 354 18/23 ||
||   ||   || 13/44 ||   ||   || 354 6/11 ||
||   ||   ||   ||   || 31/105 || 354 2/7 ||
||   ||   ||   || 18/61 ||   || 354 6/61 ||
||   ||   ||   ||   || 23/78 || 353 11/13 ||
|| 5/17 ||   ||   ||   ||   || 352 16/17 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 7s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is like the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352 16/17) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357 17/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;357 21/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;25/84&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357 1/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.78099&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356 28/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.732384&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356 44/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356 4/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;355 5/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;355 5/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21/71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 66/71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34/115&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 18/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/44&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 6/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;31/105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 2/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18/61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 6/61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23/78&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;353 11/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;352 16/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>4inevd15c0l2x34uk7cav27lf93mzy9</sha1>
    </revision>
    <revision>
      <id>4222</id>
      <timestamp>2014-11-08T09:33:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 530093828 - Original comment: **</comment>
      <origin>4222</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7745" sha1="pp9qsyv25j0g7a554d1k7so1dnrv193" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2014-11-08 09:33:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;530093828&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is like the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352 16/17) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).
|| 3/10 ||   ||   ||   ||   || 360 ||
||   ||   ||   ||   || 17/57 || 357 17/19 ||
||   ||   ||   || 14/47 ||   || 357 21/47 ||
||   ||   ||   ||   || 25/84 || 357 1/7 ||
||   ||   || 11/37 ||   ||   || 356 28/37 ||
||   ||   ||   ||   || 30/101 || 356 44/101 ||
||   ||   ||   || 19/64 ||   || 356.25 ||
||   ||   ||   ||   || 27/91 || 356 4/91 ||
||   || 8/27 ||   ||   ||   || 355 5/9 ||
||   ||   ||   ||   || 29/98 || 355 5/49 ||
||   ||   ||   || 21/71 ||   || 354 66/71 ||
||   ||   ||   ||   || 34/115 || 354 18/23 ||
||   ||   || 13/44 ||   ||   || 354 6/11 ||
||   ||   ||   ||   || 31/105 || 354 2/7 ||
||   ||   ||   || 18/61 ||   || 354 6/61 ||
||   ||   ||   ||   || 23/78 || 353 11/13 ||
|| 5/17 ||   ||   ||   ||   || 352 16/17 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 7s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is like the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352 16/17) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357 17/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357 21/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25/84&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357 1/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356 28/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356 44/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356 4/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;354 2/7&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;352 16/17&amp;lt;br /&amp;gt;
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      <sha1>pp9qsyv25j0g7a554d1k7so1dnrv193</sha1>
    </revision>
    <revision>
      <id>4223</id>
      <timestamp>2012-11-05T21:29:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 379546512 - Original comment: **</comment>
      <origin>4223</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7753" sha1="crikesz757qrb4b37bp52nqx1tpuv0v" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-11-05 21:29:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;379546512&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is the like the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352 16/17) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).
|| 3/10 ||   ||   ||   ||   || 360 ||
||   ||   ||   ||   || 17/57 || 357 17/19 ||
||   ||   ||   || 14/47 ||   || 357 21/47 ||
||   ||   ||   ||   || 25/84 || 357 1/7 ||
||   ||   || 11/37 ||   ||   || 356 28/37 ||
||   ||   ||   ||   || 30/101 || 356 44/101 ||
||   ||   ||   || 19/64 ||   || 356.25 ||
||   ||   ||   ||   || 27/91 || 356 4/91 ||
||   || 8/27 ||   ||   ||   || 355 5/9 ||
||   ||   ||   ||   || 29/98 || 355 5/49 ||
||   ||   ||   || 21/71 ||   || 354 66/71 ||
||   ||   ||   ||   || 34/115 || 354 18/23 ||
||   ||   || 13/44 ||   ||   || 354 6/11 ||
||   ||   ||   ||   || 31/105 || 354 2/7 ||
||   ||   ||   || 18/61 ||   || 354 6/61 ||
||   ||   ||   ||   || 23/78 || 353 11/13 ||
|| 5/17 ||   ||   ||   ||   || 352 16/17 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 7s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, with intervals of 2L-L-2L-L-2L-L-L between its small steps, is the like the neutral thirds version of Schismic temperament. It has a generator between 5/17edo (352 16/17) and 3/10edo (360 cents), making it an improper MOS of Beatles temperament (generator = c. 19/64 ~ 356.25 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;360&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;17/57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;357 17/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;357 21/47&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;25/84&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;357 1/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;11/37&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356 28/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;30/101&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;356 44/101&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;356 4/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/27&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21/71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;354 66/71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 18/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;354 6/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;31/105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 2/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;354 6/61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;353 11/13&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;352 16/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>crikesz757qrb4b37bp52nqx1tpuv0v</sha1>
    </revision>
    <revision>
      <id>94347</id>
      <parentid>48</parentid>
      <timestamp>2022-08-15T16:53:09Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox, name still missing, misc. edits</comment>
      <origin>94347</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1524" sha1="3y0tewhdrmh3t0jx9324snowe4eit40" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 7
| Equalized = 5
| Paucitonic = 3
| Pattern = LLsLsLLsLsLLsLsLs
}}
The '''10L 7s''' [[MOS scale]], with intervals of 2L-L-2L-L-2L-L-L between its small steps, is the neutral thirds version of [[Schismic]] temperament.

It has a generator between 5\[[17edo|17]] (352.941{{cent}}) and 3\[[10edo|10]] (360{{cent}}), making it an improper MOS of [[Beatles]] temperament (generator = c. 19\64 ~ 356.25{{cent}}).

== Scale tree ==
{| class="wikitable"
|-
| | 3/10
| | 
| | 
| | 
| | 
| | 360
|-
| | 
| | 
| | 
| | 
| | 17/57
| | 357.895
|-
| | 
| | 
| | 
| | 14/47
| | 
| | 357.447
|-
| | 
| | 
| | 
| | 
| | 25/84
| | 357.143
|-
| | 
| | 
| | 
| | 
| | 
| | 356.876
|-
| | 
| | 
| | 11/37
| | 
| | 
| | 356.757
|-
| | 
| | 
| | 
| | 
| | 
| | 356.4895
|-
| | 
| | 
| | 
| | 
| | 30/101
| | 356.436
|-
| | 
| | 
| | 
| | 
| | 
| | 356.383
|-
| | 
| | 
| | 
| | 19/64
| | 
| | 356.25
|-
| | 
| | 
| | 
| | 
| | 27/91
| | 356.044
|-
| | 
| | 8/27
| | 
| | 
| | 
| | 355.556
|-
| | 
| | 
| | 
| | 
| | 29/98
| | 355.102
|-
| | 
| | 
| | 
| | 
| | 
| | 355.066
|-
| | 
| | 
| | 
| | 21/71
| | 
| | 354.93
|-
| | 
| | 
| | 
| | 
| | 
| | 354.823
|-
| | 
| | 
| | 
| | 
| | 34/115
| | 354.783
|-
| | 
| | 
| | 
| | 
| | 
| | 354.7155
|-
| | 
| | 
| | 13/44
| | 
| | 
| | 354.5455
|-
| | 
| | 
| | 
| | 
| | 31/105
| | 354.286
|-
| | 
| | 
| | 
| | 18/61
| | 
| | 354.098
|-
| | 
| | 
| | 
| | 
| | 23/78
| | 353.846
|-
| | 5/17
| | 
| | 
| | 
| | 
| | 352.941
|}</text>
      <sha1>3y0tewhdrmh3t0jx9324snowe4eit40</sha1>
    </revision>
    <revision>
      <id>94504</id>
      <parentid>94347</parentid>
      <timestamp>2022-08-17T09:18:34Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94504</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1523" sha1="m566jfoxgzjsjvy4w8q17ezf5moxdiv" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 7
| Equalized = 5
| Collapsed = 3
| Pattern = LLsLsLLsLsLLsLsLs
}}
The '''10L 7s''' [[MOS scale]], with intervals of 2L-L-2L-L-2L-L-L between its small steps, is the neutral thirds version of [[Schismic]] temperament.

It has a generator between 5\[[17edo|17]] (352.941{{cent}}) and 3\[[10edo|10]] (360{{cent}}), making it an improper MOS of [[Beatles]] temperament (generator = c. 19\64 ~ 356.25{{cent}}).

== Scale tree ==
{| class="wikitable"
|-
| | 3/10
| | 
| | 
| | 
| | 
| | 360
|-
| | 
| | 
| | 
| | 
| | 17/57
| | 357.895
|-
| | 
| | 
| | 
| | 14/47
| | 
| | 357.447
|-
| | 
| | 
| | 
| | 
| | 25/84
| | 357.143
|-
| | 
| | 
| | 
| | 
| | 
| | 356.876
|-
| | 
| | 
| | 11/37
| | 
| | 
| | 356.757
|-
| | 
| | 
| | 
| | 
| | 
| | 356.4895
|-
| | 
| | 
| | 
| | 
| | 30/101
| | 356.436
|-
| | 
| | 
| | 
| | 
| | 
| | 356.383
|-
| | 
| | 
| | 
| | 19/64
| | 
| | 356.25
|-
| | 
| | 
| | 
| | 
| | 27/91
| | 356.044
|-
| | 
| | 8/27
| | 
| | 
| | 
| | 355.556
|-
| | 
| | 
| | 
| | 
| | 29/98
| | 355.102
|-
| | 
| | 
| | 
| | 
| | 
| | 355.066
|-
| | 
| | 
| | 
| | 21/71
| | 
| | 354.93
|-
| | 
| | 
| | 
| | 
| | 
| | 354.823
|-
| | 
| | 
| | 
| | 
| | 34/115
| | 354.783
|-
| | 
| | 
| | 
| | 
| | 
| | 354.7155
|-
| | 
| | 
| | 13/44
| | 
| | 
| | 354.5455
|-
| | 
| | 
| | 
| | 
| | 31/105
| | 354.286
|-
| | 
| | 
| | 
| | 18/61
| | 
| | 354.098
|-
| | 
| | 
| | 
| | 
| | 23/78
| | 353.846
|-
| | 5/17
| | 
| | 
| | 
| | 
| | 352.941
|}</text>
      <sha1>m566jfoxgzjsjvy4w8q17ezf5moxdiv</sha1>
    </revision>
    <revision>
      <id>136566</id>
      <parentid>94504</parentid>
      <timestamp>2024-02-22T03:14:25Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Mos intro</comment>
      <origin>136566</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1403" sha1="io15qekk0gme9oa3eue6ycc2gjx8kxg" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 7
| Equalized = 5
| Collapsed = 3
| Pattern = LLsLsLLsLsLLsLsLs
}}
{{MOS intro}}
This scale is the neutral thirds version of [[Schismic]] temperament. The range of the bright generator also makes it an [[improper]] MOS of [[Beatles]] temperament (generator = c. 19\64 ~ 356.25{{cent}}).

== Scale tree ==
{| class="wikitable"
|-
| | 3/10
| | 
| | 
| | 
| | 
| | 360
|-
| | 
| | 
| | 
| | 
| | 17/57
| | 357.895
|-
| | 
| | 
| | 
| | 14/47
| | 
| | 357.447
|-
| | 
| | 
| | 
| | 
| | 25/84
| | 357.143
|-
| | 
| | 
| | 
| | 
| | 
| | 356.876
|-
| | 
| | 
| | 11/37
| | 
| | 
| | 356.757
|-
| | 
| | 
| | 
| | 
| | 
| | 356.4895
|-
| | 
| | 
| | 
| | 
| | 30/101
| | 356.436
|-
| | 
| | 
| | 
| | 
| | 
| | 356.383
|-
| | 
| | 
| | 
| | 19/64
| | 
| | 356.25
|-
| | 
| | 
| | 
| | 
| | 27/91
| | 356.044
|-
| | 
| | 8/27
| | 
| | 
| | 
| | 355.556
|-
| | 
| | 
| | 
| | 
| | 29/98
| | 355.102
|-
| | 
| | 
| | 
| | 
| | 
| | 355.066
|-
| | 
| | 
| | 
| | 21/71
| | 
| | 354.93
|-
| | 
| | 
| | 
| | 
| | 
| | 354.823
|-
| | 
| | 
| | 
| | 
| | 34/115
| | 354.783
|-
| | 
| | 
| | 
| | 
| | 
| | 354.7155
|-
| | 
| | 
| | 13/44
| | 
| | 
| | 354.5455
|-
| | 
| | 
| | 
| | 
| | 31/105
| | 354.286
|-
| | 
| | 
| | 
| | 18/61
| | 
| | 354.098
|-
| | 
| | 
| | 
| | 
| | 23/78
| | 353.846
|-
| | 5/17
| | 
| | 
| | 
| | 
| | 352.941
|}</text>
      <sha1>io15qekk0gme9oa3eue6ycc2gjx8kxg</sha1>
    </revision>
    <revision>
      <id>137550</id>
      <parentid>136566</parentid>
      <timestamp>2024-02-29T19:18:27Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Replaced scale tree</comment>
      <origin>137550</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="392" sha1="fdg5eonp37d9x02893jpdgfagzyn7hj" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 7
| Equalized = 5
| Collapsed = 3
| Pattern = LLsLsLLsLsLLsLsLs
}}
{{MOS intro}}
This scale is the neutral thirds version of [[Schismic]] temperament. The range of the bright generator also makes it an [[improper]] MOS of [[Beatles]] temperament (generator = c. 19\64 ~ 356.25{{cent}}).

== Scale tree ==
{{Scale tree}}</text>
      <sha1>fdg5eonp37d9x02893jpdgfagzyn7hj</sha1>
    </revision>
    <revision>
      <id>170558</id>
      <parentid>137550</parentid>
      <timestamp>2024-12-15T06:50:14Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Modes intervals</comment>
      <origin>170558</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="454" sha1="l5a8pghfb25j9p2rnj7rv7m27wln4xa" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 7
| Equalized = 5
| Collapsed = 3
| Pattern = LLsLsLLsLsLLsLsLs
}}
{{MOS intro}}
This scale is the neutral thirds version of [[Schismic]] temperament. The range of the bright generator also makes it an [[improper]] MOS of [[Beatles]] temperament (generator = c. 19\64 ~ 356.25{{cent}}).

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree}}</text>
      <sha1>l5a8pghfb25j9p2rnj7rv7m27wln4xa</sha1>
    </revision>
    <revision>
      <id>183727</id>
      <parentid>170558</parentid>
      <timestamp>2025-02-28T16:37:08Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183727</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="463" sha1="cq8enuzwvjobjvlbgriuuqs63zdtstl" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 7
| Equalized = 5
| Collapsed = 3
| Pattern = LLsLsLLsLsLLsLsLs
}}
{{MOS intro}}
This scale is the neutral thirds version of [[Schismic]] temperament. The range of the bright generator also makes it an [[improper]] MOS of [[Beatles]] temperament (generator = c. 19\64 ~ 356.25{{cent}}).

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum}}</text>
      <sha1>cq8enuzwvjobjvlbgriuuqs63zdtstl</sha1>
    </revision>
    <revision>
      <id>199892</id>
      <parentid>183727</parentid>
      <timestamp>2025-06-18T03:32:18Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Removed outright wrong info</comment>
      <origin>199892</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="388" sha1="7q00et65zumqtwm226hec6qve7y7v1d" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 7
| Equalized = 5
| Collapsed = 3
| Pattern = LLsLsLLsLsLLsLsLs
}}
{{MOS intro}} The range of the bright generator makes it an [[improper]] MOS of [[Beatles]] temperament (generator = c. 19\64 ~ 356.25{{cent}}).

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum}}</text>
      <sha1>7q00et65zumqtwm226hec6qve7y7v1d</sha1>
    </revision>
  </page>
  <page>
    <title>10L 9s</title>
    <ns>0</ns>
    <id>49</id>
    <revision>
      <id>49</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>49</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1194" sha1="nenatj8gs69q02exbtnc23o866rngxa" xml:space="preserve">This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).

{| class="wikitable"
|-
| | 2/19
| | 
| | 
| | 
| | 
| | 126.316
|-
| | 
| | 
| | 
| | 
| | 9/86
| | 125.581
|-
| | 
| | 
| | 
| | 7/67
| | 
| | 125.373
|-
| | 
| | 
| | 
| | 
| | 12/115
| | 125.217
|-
| | 
| | 
| | 5/48
| | 
| | 
| | 125
|-
| | 
| | 
| | 
| | 
| | 
| | 124.857
|-
| | 
| | 
| | 
| | 
| | 13/125
| | 124.8
|-
| | 
| | 
| | 
| | 
| | 
| | 124.766
|-
| | 
| | 
| | 
| | 8/77
| | 
| | 124.675
|-
| | 
| | 
| | 
| | 
| | 
| | 124.559
|-
| | 
| | 
| | 
| | 
| | 11/106
| | 124.528
|-
| | 
| | 3/29
| | 
| | 
| | 
| | 124.138
|-
| | 
| | 
| | 
| | 
| | 10/97
| | 123.711
|-
| | 
| | 
| | 
| | 7/68
| | 
| | 123.529
|-
| | 
| | 
| | 
| | 
| | 
| | 123.411
|-
| | 
| | 
| | 
| | 
| | 11/107
| | 123.3645
|-
| | 
| | 
| | 
| | 
| | 
| | 123.3165
|-
| | 
| | 
| | 4/39
| | 
| | 
| | 123.077
|-
| | 
| | 
| | 
| | 
| | 
| | 122.969
|-
| | 
| | 
| | 
| | 
| | 9/88
| | 122.727
|-
| | 
| | 
| | 
| | 5/49
| | 
| | 122.449
|-
| | 
| | 
| | 
| | 
| | 6/59
| | 122.034
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}</text>
      <sha1>nenatj8gs69q02exbtnc23o866rngxa</sha1>
    </revision>
    <revision>
      <id>4224</id>
      <timestamp>2016-12-29T10:30:10Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602892898 - Original comment: removed tel-links**</comment>
      <origin>4224</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9484" sha1="8lhvbzot2w46qxe0pnf05o8zsxrwikh" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2016-12-29 10:30:10 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602892898&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;removed tel-links&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).
|| 2/19 ||   ||   ||   ||   || 126.316 ||
||   ||   ||   ||   || 9/86 || 125.581 ||
||   ||   ||   || 7/67 ||   || 125.373 ||
||   ||   ||   ||   || 12/115 || 125.217 ||
||   ||   || 5/48 ||   ||   || 125 ||
||   ||   ||   ||   ||   || 124.857 ||
||   ||   ||   ||   || 13/125 || 124.8 ||
||   ||   ||   ||   ||   || 124.766 ||
||   ||   ||   || 8/77 ||   || 124.675 ||
||   ||   ||   ||   ||   || 124.559 ||
||   ||   ||   ||   || 11/106 || 124.528 ||
||   || 3/29 ||   ||   ||   || 124.138 ||
||   ||   ||   ||   || 10/97 || 123.711 ||
||   ||   ||   || 7/68 ||   || 123.529 ||
||   ||   ||   ||   ||   || 123.411 ||
||   ||   ||   ||   || 11/107 || 123.3645 ||
||   ||   ||   ||   ||   || 123.3165 ||
||   ||   || 4/39 ||   ||   || 123.077 ||
||   ||   ||   ||   ||   || 122.969 ||
||   ||   ||   ||   || 9/88 || 122.727 ||
||   ||   ||   || 5/49 ||   || 122.449 ||
||   ||   ||   ||   || 6/59 || 122.034 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 9s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;126.316&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;125.581&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;123.3645&amp;lt;br /&amp;gt;
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&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>8lhvbzot2w46qxe0pnf05o8zsxrwikh</sha1>
    </revision>
    <revision>
      <id>4225</id>
      <timestamp>2016-10-25T10:19:57Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 596972712 - Original comment: **</comment>
      <origin>4225</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9665" sha1="h83uyf15j8hqsm6a7k5azv3kj9argsv" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-25 10:19:57 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;596972712&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).
|| 2/19 ||   ||   ||   ||   || 126.316 ||
||   ||   ||   ||   || 9/86 || 125.581 ||
||   ||   ||   || 7/67 ||   || 125.373 ||
||   ||   ||   ||   || 12/115 || 125.217 ||
||   ||   || 5/48 ||   ||   || 125 ||
||   ||   ||   ||   ||   || 124.857 ||
||   ||   ||   ||   || 13/125 || 124.8 ||
||   ||   ||   ||   ||   || 124.766 ||
||   ||   ||   || 8/77 ||   || 124.675 ||
||   ||   ||   ||   ||   || 124.559 ||
||   ||   ||   ||   || 11/106 || 124.528 ||
||   || 3/29 ||   ||   ||   || 124.138 ||
||   ||   ||   ||   || 10/97 || 123.711 ||
||   ||   ||   || 7/68 ||   || 123.529 ||
||   ||   ||   ||   ||   || 123.411 ||
||   ||   ||   ||   || 11/107 || [[tel:123.3645|123.3645]] ||
||   ||   ||   ||   ||   || [[tel:123.3165|123.3165]] ||
||   ||   || 4/39 ||   ||   || 123.077 ||
||   ||   ||   ||   ||   || 122.969 ||
||   ||   ||   ||   || 9/88 || 122.727 ||
||   ||   ||   || 5/49 ||   || 122.449 ||
||   ||   ||   ||   || 6/59 || 122.034 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 9s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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      <sha1>h83uyf15j8hqsm6a7k5azv3kj9argsv</sha1>
    </revision>
    <revision>
      <id>4226</id>
      <timestamp>2015-11-14T16:50:21Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 566470629 - Original comment: **</comment>
      <origin>4226</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9473" sha1="67uy1kmhheimxw29s7g5rr72aj2xu27" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-14 16:50:21 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566470629&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).
|| 2/19 ||   ||   ||   ||   || 126.316 ||
||   ||   ||   ||   || 9/86 || 125.581 ||
||   ||   ||   || 7/67 ||   || 125.373 ||
||   ||   ||   ||   || 12/115 || 125.217 ||
||   ||   || 5/48 ||   ||   || 125 ||
||   ||   ||   ||   ||   || 124.857 ||
||   ||   ||   ||   || 13/125 || 124.8 ||
||   ||   ||   ||   ||   || 124.766 ||
||   ||   ||   || 8/77 ||   || 124.675 ||
||   ||   ||   ||   ||   || 124.559 ||
||   ||   ||   ||   || 11/106 || 124.528 ||
||   || 3/29 ||   ||   ||   || 124.138 ||
||   ||   ||   ||   || 10/97 || 123.711 ||
||   ||   ||   || 7/68 ||   || 123.529 ||
||   ||   ||   ||   ||   || 123.411 ||
||   ||   ||   ||   || 11/107 || 123.3645 ||
||   ||   ||   ||   ||   || 123.3165 ||
||   ||   || 4/39 ||   ||   || 123.077 ||
||   ||   ||   ||   ||   || 122.969 ||
||   ||   ||   ||   || 9/88 || 122.727 ||
||   ||   ||   || 5/49 ||   || 122.449 ||
||   ||   ||   ||   || 6/59 || 122/034 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 9s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>67uy1kmhheimxw29s7g5rr72aj2xu27</sha1>
    </revision>
    <revision>
      <id>4227</id>
      <timestamp>2015-10-27T19:49:31Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 564145781 - Original comment: **</comment>
      <origin>4227</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9138" sha1="edl4ne4b8jnn6x5lsmwwwrbzbyuzxnp" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-10-27 19:49:31 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;564145781&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).
|| 2/19 ||   ||   ||   ||   || 126.316 ||
||   ||   ||   ||   || 9/86 || 125.581 ||
||   ||   ||   || 7/67 ||   || 125.373 ||
||   ||   ||   ||   || 12/115 || 125.217 ||
||   ||   || 5/48 ||   ||   || 125 ||
||   ||   ||   ||   ||   || 124.857 ||
||   ||   ||   ||   || 13/125 || 124.8 ||
||   ||   ||   ||   ||   || 124.766 ||
||   ||   ||   || 8/77 ||   || 124.675 ||
||   ||   ||   ||   || 11/106 || 124.528 ||
||   || 3/29 ||   ||   ||   || 124.138 ||
||   ||   ||   ||   || 10/97 || 123.711 ||
||   ||   ||   || 7/68 ||   || 123.529 ||
||   ||   ||   ||   ||   || 123.411 ||
||   ||   ||   ||   || 11/107 || 123.3645 ||
||   ||   ||   ||   ||   || 123.3165 ||
||   ||   || 4/39 ||   ||   || 123.077 ||
||   ||   ||   ||   ||   || 122.969 ||
||   ||   ||   ||   || 9/88 || 122.727 ||
||   ||   ||   || 5/49 ||   || 122.449 ||
||   ||   ||   ||   || 6/59 || 122/034 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 9s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;2/19&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;124.8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;8/77&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;124.675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>edl4ne4b8jnn6x5lsmwwwrbzbyuzxnp</sha1>
    </revision>
    <revision>
      <id>4228</id>
      <timestamp>2015-06-11T23:10:33Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 553737984 - Original comment: **</comment>
      <origin>4228</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8189" sha1="15kl6diivuc45ne9bhrp3el7jofrgd3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-06-11 23:10:33 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;553737984&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126 6/19 cents).
|| 2/19 ||   ||   ||   ||   || 126 6/19 ||
||   ||   ||   ||   || 9/86 || 125 25/43 ||
||   ||   ||   || 7/67 ||   || 125 25/67 ||
||   ||   ||   ||   || 12/115 || 125 5/23 ||
||   ||   || 5/48 ||   ||   || 125 ||
||   ||   ||   ||   || 13/125 || 124.8 ||
||   ||   ||   || 8/77 ||   || 124 52/77 ||
||   ||   ||   ||   || 11/106 || 124 28/53 ||
||   || 3/29 ||   ||   ||   || 124 4/29 ||
||   ||   ||   ||   || 10/97 || 123 69/97 ||
||   ||   ||   || 7/68 ||   || 123 9/17 ||
||   ||   ||   ||   || 11/107 || 123.364486 ||
||   ||   ||   ||   ||   || 123.316491 ||
||   ||   || 4/39 ||   ||   || 123 1/13 ||
||   ||   ||   ||   ||   || 122.9691265 ||
||   ||   ||   ||   || 9/88 || 122 8/11 ||
||   ||   ||   || 5/49 ||   || 122 22/19 ||
||   ||   ||   ||   || 6/59 || 122 2/59 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 9s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126 6/19 cents).&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;126 6/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/106&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;124 28/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/97&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;123 1/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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      <sha1>15kl6diivuc45ne9bhrp3el7jofrgd3</sha1>
    </revision>
    <revision>
      <id>4229</id>
      <timestamp>2015-05-24T13:22:38Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 552013500 - Original comment: **</comment>
      <origin>4229</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8189" sha1="q8v477tdlfxzttwmtarje7geit5rsel" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-24 13:22:38 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;552013500&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126 6/19 cents).
|| 2/19 ||   ||   ||   ||   || 126 6/19 ||
||   ||   ||   ||   || 9/86 || 125 25/43 ||
||   ||   ||   || 7/67 ||   || 125 25/67 ||
||   ||   ||   ||   || 12/115 || 125 5/23 ||
||   ||   || 5/48 ||   ||   || 125 ||
||   ||   ||   ||   || 13/125 || 124.8 ||
||   ||   ||   || 8/77 ||   || 124 52/77 ||
||   ||   ||   ||   || 11/106 || 124 28/53 ||
||   || 3/29 ||   ||   ||   || 124 4/29 ||
||   ||   ||   ||   || 10/97 || 123 69/97 ||
||   ||   ||   || 7/68 ||   || 123 9/17 ||
||   ||   ||   ||   || 11/107 || 123 39/107 ||
||   ||   ||   ||   ||   || 123.316491 ||
||   ||   || 4/39 ||   ||   || 123 1/13 ||
||   ||   ||   ||   ||   || 122.9691265 ||
||   ||   ||   ||   || 9/88 || 122 8/11 ||
||   ||   ||   || 5/49 ||   || 122 22/19 ||
||   ||   ||   ||   || 6/59 || 122 2/59 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 9s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126 6/19 cents).&amp;lt;br /&amp;gt;


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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>q8v477tdlfxzttwmtarje7geit5rsel</sha1>
    </revision>
    <revision>
      <id>4230</id>
      <timestamp>2015-02-13T14:17:48Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540918788 - Original comment: **</comment>
      <origin>4230</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8187" sha1="smgzv0zkgqtcqrxsd08l98dlw422r52" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-13 14:17:48 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540918788&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126 6/19 cents).
|| 2/19 ||   ||   ||   ||   || 126 6/19 ||
||   ||   ||   ||   || 9/86 || 125 25/43 ||
||   ||   ||   || 7/67 ||   || 125 25/67 ||
||   ||   ||   ||   || 12/115 || 125 5/23 ||
||   ||   || 5/48 ||   ||   || 125 ||
||   ||   ||   ||   || 13/125 || 124.8 ||
||   ||   ||   || 8/77 ||   || 124 52/77 ||
||   ||   ||   ||   || 11/106 || 124 28/53 ||
||   || 3/29 ||   ||   ||   || 124 4/29 ||
||   ||   ||   ||   || 10/97 || 123 69/97 ||
||   ||   ||   || 7/68 ||   || 123 9/17 ||
||   ||   ||   ||   || 11/107 || 123 39/107 ||
||   ||   ||   ||   ||   || 123.098852 ||
||   ||   || 4/39 ||   ||   || 123 1/13 ||
||   ||   ||   ||   ||   || 123.055103 ||
||   ||   ||   ||   || 9/88 || 122 8/11 ||
||   ||   ||   || 5/49 ||   || 122 22/19 ||
||   ||   ||   ||   || 6/59 || 122 2/59 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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      <sha1>smgzv0zkgqtcqrxsd08l98dlw422r52</sha1>
    </revision>
    <revision>
      <id>4231</id>
      <timestamp>2012-11-08T00:04:24Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 380336200 - Original comment: **</comment>
      <origin>4231</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7505" sha1="sl4luvejpors7wgkz93hwilml4g0jt0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-11-08 00:04:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;380336200&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126 6/19 cents).
|| 2/19 ||   ||   ||   ||   || 126 6/19 ||
||   ||   ||   ||   || 9/86 || 125 25/43 ||
||   ||   ||   || 7/67 ||   || 125 25/67 ||
||   ||   ||   ||   || 12/115 || 125 5/23 ||
||   ||   || 5/48 ||   ||   || 125 ||
||   ||   ||   ||   || 13/125 || 124.8 ||
||   ||   ||   || 8/77 ||   || 124 52/77 ||
||   ||   ||   ||   || 11/106 || 124 28/53 ||
||   || 3/29 ||   ||   ||   || 124 4/29 ||
||   ||   ||   ||   || 10/97 || 123 69/97 ||
||   ||   ||   || 7/68 ||   || 123 9/17 ||
||   ||   ||   ||   || 11/107 || 123 39/107 ||
||   ||   || 4/39 ||   ||   || 123 1/13 ||
||   ||   ||   ||   || 9/88 || 122 8/11 ||
||   ||   ||   || 5/49 ||   || 122 22/19 ||
||   ||   ||   ||   || 6/59 || 122 2/59 ||
|| 1/10 ||   ||   ||   ||   || 120 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;10L 9s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS divides its large steps 1-1-1-1-1-1-1-1-2 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126 6/19 cents).&amp;lt;br /&amp;gt;


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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;12/115&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;125 5/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;124.8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;124 52/77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/106&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;124 28/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;124 4/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/97&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;123 69/97&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/68&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;123 9/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/107&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;123 39/107&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;123 1/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/88&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;122 8/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;122 22/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;122 2/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>sl4luvejpors7wgkz93hwilml4g0jt0</sha1>
    </revision>
    <revision>
      <id>94359</id>
      <parentid>49</parentid>
      <timestamp>2022-08-15T18:03:58Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox, name still missing, chroma-positive, misc. edits</comment>
      <origin>94359</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1443" sha1="hg7btjdgq4bjkvqa80lh2i8ter5lfsm" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 9
| Equalized = 17
| Paucitonic = 9
| Pattern = LLsLsLsLsLsLsLsLsLs
}}
The '''10L 9s''' [[MOS scale]] divides its large steps 2-1-1-1-1-1-1-1-1 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).

== Scale tree ==
''Note: the following table uses the chroma-negative generator.''
{| class="wikitable"
|-
| | 2/19
| | 
| | 
| | 
| | 
| | 126.316
|-
| | 
| | 
| | 
| | 
| | 9/86
| | 125.581
|-
| | 
| | 
| | 
| | 7/67
| | 
| | 125.373
|-
| | 
| | 
| | 
| | 
| | 12/115
| | 125.217
|-
| | 
| | 
| | 5/48
| | 
| | 
| | 125
|-
| | 
| | 
| | 
| | 
| | 
| | 124.857
|-
| | 
| | 
| | 
| | 
| | 13/125
| | 124.8
|-
| | 
| | 
| | 
| | 
| | 
| | 124.766
|-
| | 
| | 
| | 
| | 8/77
| | 
| | 124.675
|-
| | 
| | 
| | 
| | 
| | 
| | 124.559
|-
| | 
| | 
| | 
| | 
| | 11/106
| | 124.528
|-
| | 
| | 3/29
| | 
| | 
| | 
| | 124.138
|-
| | 
| | 
| | 
| | 
| | 10/97
| | 123.711
|-
| | 
| | 
| | 
| | 7/68
| | 
| | 123.529
|-
| | 
| | 
| | 
| | 
| | 
| | 123.411
|-
| | 
| | 
| | 
| | 
| | 11/107
| | 123.3645
|-
| | 
| | 
| | 
| | 
| | 
| | 123.3165
|-
| | 
| | 
| | 4/39
| | 
| | 
| | 123.077
|-
| | 
| | 
| | 
| | 
| | 
| | 122.969
|-
| | 
| | 
| | 
| | 
| | 9/88
| | 122.727
|-
| | 
| | 
| | 
| | 5/49
| | 
| | 122.449
|-
| | 
| | 
| | 
| | 
| | 6/59
| | 122.034
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}</text>
      <sha1>hg7btjdgq4bjkvqa80lh2i8ter5lfsm</sha1>
    </revision>
    <revision>
      <id>94505</id>
      <parentid>94359</parentid>
      <timestamp>2022-08-17T09:18:48Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94505</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1442" sha1="hfoq1z7eogs96d5bh5jm0q9p3c60bsv" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 9
| Equalized = 17
| Collapsed = 9
| Pattern = LLsLsLsLsLsLsLsLsLs
}}
The '''10L 9s''' [[MOS scale]] divides its large steps 2-1-1-1-1-1-1-1-1 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).

== Scale tree ==
''Note: the following table uses the chroma-negative generator.''
{| class="wikitable"
|-
| | 2/19
| | 
| | 
| | 
| | 
| | 126.316
|-
| | 
| | 
| | 
| | 
| | 9/86
| | 125.581
|-
| | 
| | 
| | 
| | 7/67
| | 
| | 125.373
|-
| | 
| | 
| | 
| | 
| | 12/115
| | 125.217
|-
| | 
| | 
| | 5/48
| | 
| | 
| | 125
|-
| | 
| | 
| | 
| | 
| | 
| | 124.857
|-
| | 
| | 
| | 
| | 
| | 13/125
| | 124.8
|-
| | 
| | 
| | 
| | 
| | 
| | 124.766
|-
| | 
| | 
| | 
| | 8/77
| | 
| | 124.675
|-
| | 
| | 
| | 
| | 
| | 
| | 124.559
|-
| | 
| | 
| | 
| | 
| | 11/106
| | 124.528
|-
| | 
| | 3/29
| | 
| | 
| | 
| | 124.138
|-
| | 
| | 
| | 
| | 
| | 10/97
| | 123.711
|-
| | 
| | 
| | 
| | 7/68
| | 
| | 123.529
|-
| | 
| | 
| | 
| | 
| | 
| | 123.411
|-
| | 
| | 
| | 
| | 
| | 11/107
| | 123.3645
|-
| | 
| | 
| | 
| | 
| | 
| | 123.3165
|-
| | 
| | 
| | 4/39
| | 
| | 
| | 123.077
|-
| | 
| | 
| | 
| | 
| | 
| | 122.969
|-
| | 
| | 
| | 
| | 
| | 9/88
| | 122.727
|-
| | 
| | 
| | 
| | 5/49
| | 
| | 122.449
|-
| | 
| | 
| | 
| | 
| | 6/59
| | 122.034
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}</text>
      <sha1>hfoq1z7eogs96d5bh5jm0q9p3c60bsv</sha1>
    </revision>
    <revision>
      <id>136601</id>
      <parentid>94505</parentid>
      <timestamp>2024-02-22T05:28:26Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Mos intro</comment>
      <origin>136601</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1457" sha1="7415an4dtsxjrfz3ddn9p03q3kn9kr4" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 9
| Equalized = 17
| Collapsed = 9
| Pattern = LLsLsLsLsLsLsLsLsLs
}}
{{MOS intro}}

The '''10L 9s''' [[MOS scale]] divides its large steps 2-1-1-1-1-1-1-1-1 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).

== Scale tree ==
''Note: the following table uses the chroma-negative generator.''
{| class="wikitable"
|-
| | 2/19
| | 
| | 
| | 
| | 
| | 126.316
|-
| | 
| | 
| | 
| | 
| | 9/86
| | 125.581
|-
| | 
| | 
| | 
| | 7/67
| | 
| | 125.373
|-
| | 
| | 
| | 
| | 
| | 12/115
| | 125.217
|-
| | 
| | 
| | 5/48
| | 
| | 
| | 125
|-
| | 
| | 
| | 
| | 
| | 
| | 124.857
|-
| | 
| | 
| | 
| | 
| | 13/125
| | 124.8
|-
| | 
| | 
| | 
| | 
| | 
| | 124.766
|-
| | 
| | 
| | 
| | 8/77
| | 
| | 124.675
|-
| | 
| | 
| | 
| | 
| | 
| | 124.559
|-
| | 
| | 
| | 
| | 
| | 11/106
| | 124.528
|-
| | 
| | 3/29
| | 
| | 
| | 
| | 124.138
|-
| | 
| | 
| | 
| | 
| | 10/97
| | 123.711
|-
| | 
| | 
| | 
| | 7/68
| | 
| | 123.529
|-
| | 
| | 
| | 
| | 
| | 
| | 123.411
|-
| | 
| | 
| | 
| | 
| | 11/107
| | 123.3645
|-
| | 
| | 
| | 
| | 
| | 
| | 123.3165
|-
| | 
| | 
| | 4/39
| | 
| | 
| | 123.077
|-
| | 
| | 
| | 
| | 
| | 
| | 122.969
|-
| | 
| | 
| | 
| | 
| | 9/88
| | 122.727
|-
| | 
| | 
| | 
| | 5/49
| | 
| | 122.449
|-
| | 
| | 
| | 
| | 
| | 6/59
| | 122.034
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}</text>
      <sha1>7415an4dtsxjrfz3ddn9p03q3kn9kr4</sha1>
    </revision>
    <revision>
      <id>136616</id>
      <parentid>136601</parentid>
      <timestamp>2024-02-22T05:57:18Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <minor/>
      <comment>a space</comment>
      <origin>136616</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1456" sha1="p9y3leinmuftwhge8iqw3rqb0j8nd1s" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 9
| Equalized = 17
| Collapsed = 9
| Pattern = LLsLsLsLsLsLsLsLsLs
}}
{{MOS intro}}
The '''10L 9s''' [[MOS scale]] divides its large steps 2-1-1-1-1-1-1-1-1 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).

== Scale tree ==
''Note: the following table uses the chroma-negative generator.''
{| class="wikitable"
|-
| | 2/19
| | 
| | 
| | 
| | 
| | 126.316
|-
| | 
| | 
| | 
| | 
| | 9/86
| | 125.581
|-
| | 
| | 
| | 
| | 7/67
| | 
| | 125.373
|-
| | 
| | 
| | 
| | 
| | 12/115
| | 125.217
|-
| | 
| | 
| | 5/48
| | 
| | 
| | 125
|-
| | 
| | 
| | 
| | 
| | 
| | 124.857
|-
| | 
| | 
| | 
| | 
| | 13/125
| | 124.8
|-
| | 
| | 
| | 
| | 
| | 
| | 124.766
|-
| | 
| | 
| | 
| | 8/77
| | 
| | 124.675
|-
| | 
| | 
| | 
| | 
| | 
| | 124.559
|-
| | 
| | 
| | 
| | 
| | 11/106
| | 124.528
|-
| | 
| | 3/29
| | 
| | 
| | 
| | 124.138
|-
| | 
| | 
| | 
| | 
| | 10/97
| | 123.711
|-
| | 
| | 
| | 
| | 7/68
| | 
| | 123.529
|-
| | 
| | 
| | 
| | 
| | 
| | 123.411
|-
| | 
| | 
| | 
| | 
| | 11/107
| | 123.3645
|-
| | 
| | 
| | 
| | 
| | 
| | 123.3165
|-
| | 
| | 
| | 4/39
| | 
| | 
| | 123.077
|-
| | 
| | 
| | 
| | 
| | 
| | 122.969
|-
| | 
| | 
| | 
| | 
| | 9/88
| | 122.727
|-
| | 
| | 
| | 
| | 5/49
| | 
| | 122.449
|-
| | 
| | 
| | 
| | 
| | 6/59
| | 122.034
|-
| | 1/10
| | 
| | 
| | 
| | 
| | 120
|}</text>
      <sha1>p9y3leinmuftwhge8iqw3rqb0j8nd1s</sha1>
    </revision>
    <revision>
      <id>137569</id>
      <parentid>136616</parentid>
      <timestamp>2024-02-29T19:59:12Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Replaced scale tree</comment>
      <origin>137569</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="392" sha1="8bk8ettwkuomtq24u9eky9qjpfytaur" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 9
| Equalized = 17
| Collapsed = 9
| Pattern = LLsLsLsLsLsLsLsLsLs
}}
{{MOS intro}}
The '''10L 9s''' [[MOS scale]] divides its large steps 2-1-1-1-1-1-1-1-1 like the 10edo nonatonic scale. Its 'trienthird' generator falls in the narrow range of 1/10edo (120) to 2/19edo (126.316 cents).

== Scale tree ==
{{Scale tree}}</text>
      <sha1>8bk8ettwkuomtq24u9eky9qjpfytaur</sha1>
    </revision>
    <revision>
      <id>138060</id>
      <parentid>137569</parentid>
      <timestamp>2024-03-03T22:44:01Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Removed redundant mention of generator range, stub</comment>
      <origin>138060</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="196" sha1="ka6zj5e7syk12rbkxe840lgj3mlv20h" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 9
| Equalized = 17
| Collapsed = 9
| Pattern = LLsLsLsLsLsLsLsLsLs
}}
{{MOS intro}}

==Scale tree==
{{Scale tree}}
{{Stub}}</text>
      <sha1>ka6zj5e7syk12rbkxe840lgj3mlv20h</sha1>
    </revision>
    <revision>
      <id>138061</id>
      <parentid>138060</parentid>
      <timestamp>2024-03-03T22:44:08Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <minor/>
      <comment>a space</comment>
      <origin>138061</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="195" sha1="oz7xkgurbnelg7tghq4kg38w4pz7aws" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 9
| Equalized = 17
| Collapsed = 9
| Pattern = LLsLsLsLsLsLsLsLsLs
}}
{{MOS intro}}
==Scale tree==
{{Scale tree}}
{{Stub}}</text>
      <sha1>oz7xkgurbnelg7tghq4kg38w4pz7aws</sha1>
    </revision>
    <revision>
      <id>170568</id>
      <parentid>138061</parentid>
      <timestamp>2024-12-15T06:55:07Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>== Intervals == {{MOS intervals}}</comment>
      <origin>170568</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="233" sha1="dmzu7x6l4lefe1ys3dghi36pzxqd83q" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 9
| Equalized = 17
| Collapsed = 9
| Pattern = LLsLsLsLsLsLsLsLsLs
}}
{{MOS intro}}

== Intervals ==
{{MOS intervals}}

==Scale tree==
{{Scale tree}}


{{Stub}}</text>
      <sha1>dmzu7x6l4lefe1ys3dghi36pzxqd83q</sha1>
    </revision>
    <revision>
      <id>183728</id>
      <parentid>170568</parentid>
      <timestamp>2025-02-28T16:37:21Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183728</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="241" sha1="pgxr030ilfi33h0x77oerdceuzo9uq2" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 10
| nSmallSteps = 9
| Equalized = 17
| Collapsed = 9
| Pattern = LLsLsLsLsLsLsLsLsLs
}}
{{MOS intro}}

== Intervals ==
{{MOS intervals}}

==Scale tree==
{{MOS tuning spectrum}}

{{Stub}}</text>
      <sha1>pgxr030ilfi33h0x77oerdceuzo9uq2</sha1>
    </revision>
  </page>
  <page>
    <title>11 limit</title>
    <ns>0</ns>
    <id>51</id>
    <redirect title="11-limit" />
    <revision>
      <id>51</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>51</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22" sha1="8uhre5bfjhusxi9cwo17czsk89t6w3f" xml:space="preserve">#REDIRECT [[11-limit]]</text>
      <sha1>8uhre5bfjhusxi9cwo17czsk89t6w3f</sha1>
    </revision>
  </page>
  <page>
    <title>User:AthiTrydhen/11-Hendrix diamond</title>
    <ns>2</ns>
    <id>52</id>
    <revision>
      <id>52</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>52</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="954" sha1="ry7up0tuv5imfmynl01u5zckq8llvca" xml:space="preserve">The '''11-''''''Hendrix diamond''' is the tonality diamond generated from the ratios 1/1, 5/4, 11/8, 3/2 and 11/6.

{| class="wikitable"
|-
! | 
! | 
! | 
! | 
! | 
! | 
|-
! | 
| | 1/1
| | 5/4
| | 11/8
| | 3/2
| | 11/6
|-
! | 
| | 8/5
| | 1/1
| | 11/10
| | 6/5
| | 22/15
|-
! | 
| | 16/11
| | 20/11
| | 1/1
| | 12/11
| | 4/3
|-
! | 
| | 4/3
| | 5/3
| | 11/6
| | 1/1
| | 11/9
|-
! | 
| | 12/11
| | 15/11
| | 3/2
| | 18/11
| | 1/1
|}
All notes in the Hendrix diamond are also found in the [[15-limit_tonality_diamond|15-limit tonality diamond]], because 1/1:5/4:11/8:3/2:11/6 is a 15-limit [[Anomalous_Saturated_Suspensions|anomalous saturated suspension]].

==Symmetry group==

The symmetry group of the 11-Hendrix diamond has 16 elements. Its generators are:

&lt;ul&gt;&lt;li&gt;3:2, 5:4, 11:8 → 11:6, 5:3, 11:9&lt;/li&gt;&lt;li&gt;3:2, 5:4, 11:8 → 3:2, 15:11, 18:11&lt;/li&gt;&lt;li&gt;3:2, 5:4, 11:8 → 4:3, 8:5, 16:11&lt;/li&gt;&lt;/ul&gt;

===Orbits and Invariant Subsets===
[to be included]</text>
      <sha1>ry7up0tuv5imfmynl01u5zckq8llvca</sha1>
    </revision>
    <revision>
      <id>4255</id>
      <timestamp>2016-08-04T01:29:42Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Praimhin</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 588772216 - Original comment: **</comment>
      <origin>4255</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5242" sha1="4gmzd0fmjvgx0xkfyr6j7mvu5djbej5" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Praimhin|Praimhin]] and made on &lt;tt&gt;2016-08-04 01:29:42 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;588772216&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The **11-****Hendrix diamond** is the tonality diamond generated from the ratios 1/1, 5/4, 11/8, 3/2 and 11/6.
||~   ||~   ||~   ||~   ||~   ||~   ||
||~   || 1/1 || 5/4 || 11/8 || 3/2 || 11/6 ||
||~   || 8/5 || 1/1 || 11/10 || 6/5 || 22/15 ||
||~   || 16/11 || 20/11 || 1/1 || 12/11 || 4/3 ||
||~   || 4/3 || 5/3 || 11/6 || 1/1 || 11/9 ||
||~   || 12/11 || 15/11 || 3/2 || 18/11 || 1/1 ||
All notes in the Hendrix diamond are also found in the [[15-limit tonality diamond]], because 1/1:5/4:11/8:3/2:11/6 is a 15-limit [[Anomalous Saturated Suspensions|anomalous saturated suspension]].

==Symmetry group== 

The symmetry group of the 11-Hendrix diamond has 16 elements. Its generators are:
* 3:2, 5:4, 11:8 → 11:6, 5:3, 11:9
* 3:2, 5:4, 11:8 → 3:2, 15:11, 18:11
* 3:2, 5:4, 11:8 → 4:3, 8:5, 16:11

===Orbits and Invariant Subsets=== 
[to be included]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-Hendrix diamond&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;strong&amp;gt;11-&amp;lt;/strong&amp;gt;&amp;lt;strong&amp;gt;Hendrix diamond&amp;lt;/strong&amp;gt; is the tonality diamond generated from the ratios 1/1, 5/4, 11/8, 3/2 and 11/6.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;16/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;12/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;12/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

All notes in the Hendrix diamond are also found in the &amp;lt;a class="wiki_link" href="/15-limit%20tonality%20diamond"&amp;gt;15-limit tonality diamond&amp;lt;/a&amp;gt;, because 1/1:5/4:11/8:3/2:11/6 is a 15-limit &amp;lt;a class="wiki_link" href="/Anomalous%20Saturated%20Suspensions"&amp;gt;anomalous saturated suspension&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Symmetry group"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Symmetry group&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
The symmetry group of the 11-Hendrix diamond has 16 elements. Its generators are:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;3:2, 5:4, 11:8 → 11:6, 5:3, 11:9&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;3:2, 5:4, 11:8 → 3:2, 15:11, 18:11&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;3:2, 5:4, 11:8 → 4:3, 8:5, 16:11&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x-Symmetry group-Orbits and Invariant Subsets"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Orbits and Invariant Subsets&amp;lt;/h3&amp;gt;
 [to be included]&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>4gmzd0fmjvgx0xkfyr6j7mvu5djbej5</sha1>
    </revision>
    <revision>
      <id>4256</id>
      <timestamp>2016-08-04T01:29:04Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Praimhin</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 588772204 - Original comment: **</comment>
      <origin>4256</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5236" sha1="giywxn1t51xxrulik862odffkcrp4j4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Praimhin|Praimhin]] and made on &lt;tt&gt;2016-08-04 01:29:04 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;588772204&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The **11-****Hendrix diamond** is the tonality diamond generated from the ratios 1/1, 5/4, 11/8, 3/2 and 11/6.
||~   ||~   ||~   ||~   ||~   ||~   ||
||~   || 1/1 || 5/4 || 11/8 || 3/2 || 11/6 ||
||~   || 8/5 || 1/1 || 11/10 || 6/5 || 22/15 ||
||~   || 16/11 || 20/11 || 1/1 || 12/11 || 4/3 ||
||~   || 4/3 || 5/3 || 11/6 || 1/1 || 11/9 ||
||~   || 12/11 || 15/11 || 3/2 || 18/11 || 1/1 ||
All notes in the Hendrix diamond are also found in the [[15-limit tonality diamond]], because 1/1:5/4:11/8:3/2:11/6 is a 15-limit [[Anomalous Saturated Suspensions|anomalous saturated suspension]].

==Symmetry group== 

The symmetry group of the Hendrix diamond has 16 elements. Its generators are:
* 3:2, 5:4, 11:8 → 11:6, 5:3, 11:9
* 3:2, 5:4, 11:8 → 3:2, 15:11, 18:11
* 3:2, 5:4, 11:8 → 4:3, 8:5, 16:11

===Orbits and Invariant Subsets=== 
[to be included]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-Hendrix diamond&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;strong&amp;gt;11-&amp;lt;/strong&amp;gt;&amp;lt;strong&amp;gt;Hendrix diamond&amp;lt;/strong&amp;gt; is the tonality diamond generated from the ratios 1/1, 5/4, 11/8, 3/2 and 11/6.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;16/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;12/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;12/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

All notes in the Hendrix diamond are also found in the &amp;lt;a class="wiki_link" href="/15-limit%20tonality%20diamond"&amp;gt;15-limit tonality diamond&amp;lt;/a&amp;gt;, because 1/1:5/4:11/8:3/2:11/6 is a 15-limit &amp;lt;a class="wiki_link" href="/Anomalous%20Saturated%20Suspensions"&amp;gt;anomalous saturated suspension&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Symmetry group"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Symmetry group&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
The symmetry group of the Hendrix diamond has 16 elements. Its generators are:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;3:2, 5:4, 11:8 → 11:6, 5:3, 11:9&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;3:2, 5:4, 11:8 → 3:2, 15:11, 18:11&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;3:2, 5:4, 11:8 → 4:3, 8:5, 16:11&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x-Symmetry group-Orbits and Invariant Subsets"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Orbits and Invariant Subsets&amp;lt;/h3&amp;gt;
 [to be included]&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>giywxn1t51xxrulik862odffkcrp4j4</sha1>
    </revision>
    <revision>
      <id>4257</id>
      <timestamp>2016-08-04T01:26:20Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Praimhin</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 588772154 - Original comment: **</comment>
      <origin>4257</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5157" sha1="hkmmnv7wx8zs1ftj6pbe2vaszdg9omi" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Praimhin|Praimhin]] and made on &lt;tt&gt;2016-08-04 01:26:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;588772154&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The **11-****Hendrix diamond** is the tonality diamond generated from the ratios 1/1, 5/4, 11/8, 3/2 and 11/6.
||~   ||~   ||~   ||~   ||~   ||~   ||
||~   || 1/1 || 5/4 || 11/8 || 3/2 || 11/6 ||
||~   || 8/5 || 1/1 || 11/10 || 6/5 || 22/15 ||
||~   || 16/11 || 20/11 || 1/1 || 12/11 || 4/3 ||
||~   || 4/3 || 5/3 || 11/6 || 1/1 || 11/9 ||
||~   || 12/11 || 15/11 || 3/2 || 18/11 || 1/1 ||
All notes in the Hendrix diamond are also found in the 15-limit tonality diamond, because 1/1:5/4:11/8:3/2:11/6 is a 15-limit [[Anomalous Saturated Suspensions|anomalous saturated suspension]].

==Symmetry group== 

The symmetry group of the Hendrix diamond has 16 elements. Its generators are:
* 3:2, 5:4, 11:8 → 11:6, 5:3, 11:9
* 3:2, 5:4, 11:8 → 3:2, 15:11, 18:11
* 3:2, 5:4, 11:8 → 4:3, 8:5, 16:11

===Orbits and Invariant Subsets=== 
[to be included]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-Hendrix diamond&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;strong&amp;gt;11-&amp;lt;/strong&amp;gt;&amp;lt;strong&amp;gt;Hendrix diamond&amp;lt;/strong&amp;gt; is the tonality diamond generated from the ratios 1/1, 5/4, 11/8, 3/2 and 11/6.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;16/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;12/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;12/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

All notes in the Hendrix diamond are also found in the 15-limit tonality diamond, because 1/1:5/4:11/8:3/2:11/6 is a 15-limit &amp;lt;a class="wiki_link" href="/Anomalous%20Saturated%20Suspensions"&amp;gt;anomalous saturated suspension&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Symmetry group"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Symmetry group&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
The symmetry group of the Hendrix diamond has 16 elements. Its generators are:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;3:2, 5:4, 11:8 → 11:6, 5:3, 11:9&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;3:2, 5:4, 11:8 → 3:2, 15:11, 18:11&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;3:2, 5:4, 11:8 → 4:3, 8:5, 16:11&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x-Symmetry group-Orbits and Invariant Subsets"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Orbits and Invariant Subsets&amp;lt;/h3&amp;gt;
 [to be included]&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>hkmmnv7wx8zs1ftj6pbe2vaszdg9omi</sha1>
    </revision>
    <revision>
      <id>55628</id>
      <parentid>52</parentid>
      <timestamp>2020-12-05T15:17:29Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>simplify links</comment>
      <origin>55628</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="896" sha1="9csvde2tvclq4kawndaw0kg57haizzk" xml:space="preserve">The '''11-''''''Hendrix diamond''' is the tonality diamond generated from the ratios 1/1, 5/4, 11/8, 3/2 and 11/6.

{| class="wikitable"
|-
! | 
! | 
! | 
! | 
! | 
! | 
|-
! | 
| | 1/1
| | 5/4
| | 11/8
| | 3/2
| | 11/6
|-
! | 
| | 8/5
| | 1/1
| | 11/10
| | 6/5
| | 22/15
|-
! | 
| | 16/11
| | 20/11
| | 1/1
| | 12/11
| | 4/3
|-
! | 
| | 4/3
| | 5/3
| | 11/6
| | 1/1
| | 11/9
|-
! | 
| | 12/11
| | 15/11
| | 3/2
| | 18/11
| | 1/1
|}
All notes in the Hendrix diamond are also found in the [[15-limit tonality diamond]], because 1/1:5/4:11/8:3/2:11/6 is a 15-limit [[anomalous saturated suspension]].

==Symmetry group==

The symmetry group of the 11-Hendrix diamond has 16 elements. Its generators are:

&lt;ul&gt;&lt;li&gt;3:2, 5:4, 11:8 → 11:6, 5:3, 11:9&lt;/li&gt;&lt;li&gt;3:2, 5:4, 11:8 → 3:2, 15:11, 18:11&lt;/li&gt;&lt;li&gt;3:2, 5:4, 11:8 → 4:3, 8:5, 16:11&lt;/li&gt;&lt;/ul&gt;

===Orbits and Invariant Subsets===
[to be included]</text>
      <sha1>9csvde2tvclq4kawndaw0kg57haizzk</sha1>
    </revision>
    <revision>
      <id>93921</id>
      <parentid>55628</parentid>
      <timestamp>2022-08-04T19:26:23Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Categories</comment>
      <origin>93921</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="918" sha1="7upw4ct3dchs0dyoe1geaog5ezjqzwu" xml:space="preserve">The '''11-''''''Hendrix diamond''' is the tonality diamond generated from the ratios 1/1, 5/4, 11/8, 3/2 and 11/6.

{| class="wikitable"
|-
! | 
! | 
! | 
! | 
! | 
! | 
|-
! | 
| | 1/1
| | 5/4
| | 11/8
| | 3/2
| | 11/6
|-
! | 
| | 8/5
| | 1/1
| | 11/10
| | 6/5
| | 22/15
|-
! | 
| | 16/11
| | 20/11
| | 1/1
| | 12/11
| | 4/3
|-
! | 
| | 4/3
| | 5/3
| | 11/6
| | 1/1
| | 11/9
|-
! | 
| | 12/11
| | 15/11
| | 3/2
| | 18/11
| | 1/1
|}
All notes in the Hendrix diamond are also found in the [[15-limit tonality diamond]], because 1/1:5/4:11/8:3/2:11/6 is a 15-limit [[anomalous saturated suspension]].

==Symmetry group==

The symmetry group of the 11-Hendrix diamond has 16 elements. Its generators are:

&lt;ul&gt;&lt;li&gt;3:2, 5:4, 11:8 → 11:6, 5:3, 11:9&lt;/li&gt;&lt;li&gt;3:2, 5:4, 11:8 → 3:2, 15:11, 18:11&lt;/li&gt;&lt;li&gt;3:2, 5:4, 11:8 → 4:3, 8:5, 16:11&lt;/li&gt;&lt;/ul&gt;

===Orbits and Invariant Subsets===
[to be included]

[[Category:Diamond]]</text>
      <sha1>7upw4ct3dchs0dyoe1geaog5ezjqzwu</sha1>
    </revision>
    <revision>
      <id>93935</id>
      <parentid>93921</parentid>
      <timestamp>2022-08-05T02:09:18Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Misc. edit, categories</comment>
      <origin>93935</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="933" sha1="qu5devi7zwx2nd7pbmowjqae9f9hn1s" xml:space="preserve">The '''11-Hendrix diamond''' is the tonality diamond generated from the ratios 1/1, 5/4, 11/8, 3/2 and 11/6.

{| class="wikitable"
|-
! | 
! | 
! | 
! | 
! | 
! | 
|-
! | 
| | 1/1
| | 5/4
| | 11/8
| | 3/2
| | 11/6
|-
! | 
| | 8/5
| | 1/1
| | 11/10
| | 6/5
| | 22/15
|-
! | 
| | 16/11
| | 20/11
| | 1/1
| | 12/11
| | 4/3
|-
! | 
| | 4/3
| | 5/3
| | 11/6
| | 1/1
| | 11/9
|-
! | 
| | 12/11
| | 15/11
| | 3/2
| | 18/11
| | 1/1
|}
All notes in the Hendrix diamond are also found in the [[15-limit tonality diamond]], because 1/1:5/4:11/8:3/2:11/6 is a 15-limit [[anomalous saturated suspension]].

==Symmetry group==

The symmetry group of the 11-Hendrix diamond has 16 elements. Its generators are:

&lt;ul&gt;&lt;li&gt;3:2, 5:4, 11:8 → 11:6, 5:3, 11:9&lt;/li&gt;&lt;li&gt;3:2, 5:4, 11:8 → 3:2, 15:11, 18:11&lt;/li&gt;&lt;li&gt;3:2, 5:4, 11:8 → 4:3, 8:5, 16:11&lt;/li&gt;&lt;/ul&gt;

===Orbits and Invariant Subsets===
[to be included]

[[Category:Diamond]]
[[Category:Hendrix]]</text>
      <sha1>qu5devi7zwx2nd7pbmowjqae9f9hn1s</sha1>
    </revision>
    <revision>
      <id>170208</id>
      <parentid>93935</parentid>
      <timestamp>2024-12-14T05:47:30Z</timestamp>
      <contributor>
        <username>AthiTrydhen</username>
        <id>314</id>
      </contributor>
      <minor/>
      <comment>AthiTrydhen moved page [[11-Hendrix diamond]] to [[User:AthiTrydhen/11-Hendrix diamond]]</comment>
      <origin>93935</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="933" sha1="qu5devi7zwx2nd7pbmowjqae9f9hn1s" xml:space="preserve">The '''11-Hendrix diamond''' is the tonality diamond generated from the ratios 1/1, 5/4, 11/8, 3/2 and 11/6.

{| class="wikitable"
|-
! | 
! | 
! | 
! | 
! | 
! | 
|-
! | 
| | 1/1
| | 5/4
| | 11/8
| | 3/2
| | 11/6
|-
! | 
| | 8/5
| | 1/1
| | 11/10
| | 6/5
| | 22/15
|-
! | 
| | 16/11
| | 20/11
| | 1/1
| | 12/11
| | 4/3
|-
! | 
| | 4/3
| | 5/3
| | 11/6
| | 1/1
| | 11/9
|-
! | 
| | 12/11
| | 15/11
| | 3/2
| | 18/11
| | 1/1
|}
All notes in the Hendrix diamond are also found in the [[15-limit tonality diamond]], because 1/1:5/4:11/8:3/2:11/6 is a 15-limit [[anomalous saturated suspension]].

==Symmetry group==

The symmetry group of the 11-Hendrix diamond has 16 elements. Its generators are:

&lt;ul&gt;&lt;li&gt;3:2, 5:4, 11:8 → 11:6, 5:3, 11:9&lt;/li&gt;&lt;li&gt;3:2, 5:4, 11:8 → 3:2, 15:11, 18:11&lt;/li&gt;&lt;li&gt;3:2, 5:4, 11:8 → 4:3, 8:5, 16:11&lt;/li&gt;&lt;/ul&gt;

===Orbits and Invariant Subsets===
[to be included]

[[Category:Diamond]]
[[Category:Hendrix]]</text>
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  </page>
  <page>
    <title>26th-octave temperaments</title>
    <ns>0</ns>
    <id>53</id>
    <revision>
      <id>53</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>53</origin>
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      <text bytes="952" sha1="tggx6zsrp50x9ck6ki0jn5ve4uvl9cj" xml:space="preserve">__FORCETOC__
=Bosonic=
Commas: 321489/320000, 589824/588245

POTE generator: ~3/2 = 701.250

Map: [&amp;lt;26 0 -22 73|, &amp;lt;0 1 2 0|]

Wedgie: &amp;lt;&amp;lt;26 52 0 22 -73 -146||

EDOs: 26, 130, 546, 676, 806c

Badness: 0.1558

==11-limit==
Commas: 441/440, 8019/8000, 65536/65219

POTE generator: ~3/2 = 701.559

Map: [&amp;lt;26 0 -22 73 90|, &amp;lt;0 1 2 0 0|]

EDOs: 26, 104c, 130

Badness: 0.0652

==13-limit==
Commas: 351/350, 364/363, 441/440, 15379/15360

POTE generator: ~3/2 = 701.546

Map: [&amp;lt;26 0 -22 73 90 55|, &amp;lt;0 1 2 0 0 1|]

EDOs: 26, 104c, 130

Badness: 0.0329

=Fermionic=
Commas: 540/539, 78408/78125, 177147/176000

POTE generator: ~3/2 = 701.077

Map: [&amp;lt;26 0 -22 73 -116|, &amp;lt;0 1 2 0 5|]

EDOs: 130, 286, 416, 546, 1508bcd

Badness: 0.0906

==13-limit==
Commas: 351/350, 540/539, 40656/40625, 142884/142805

POTE generator: ~3/2 = 701.038

Map: [&amp;lt;26 0 -22 73 -116 55|, &amp;lt;0 1 2 0 5 1|]

EDOs: 130, 286, 416, 546, 962bf

Badness: 0.0436</text>
      <sha1>tggx6zsrp50x9ck6ki0jn5ve4uvl9cj</sha1>
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    <revision>
      <id>4277</id>
      <timestamp>2012-02-09T19:41:35Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 300337262 - Original comment: **</comment>
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      <text bytes="5433" sha1="ns674ddec13y7dgf6r9au5vykkmwqmu" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-02-09 19:41:35 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;300337262&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[toc|flat]]

=Bosonic=
Commas: 321489/320000, 589824/588245

POTE generator: ~3/2 = 701.250

Map: [&amp;lt;26 0 -22 73|, &amp;lt;0 1 2 0|]
Wedgie: &amp;lt;&amp;lt;26 52 0 22 -73 -146||
EDOs: 26, 130, 546, 676, 806c
Badness: 0.1558

==11-limit==
Commas: 441/440, 8019/8000, 65536/65219

POTE generator: ~3/2 = 701.559

Map: [&amp;lt;26 0 -22 73 90|, &amp;lt;0 1 2 0 0|]
EDOs: 26, 104c, 130
Badness: 0.0652

==13-limit==
Commas: 351/350, 364/363, 441/440, 15379/15360

POTE generator: ~3/2 = 701.546

Map: [&amp;lt;26 0 -22 73 90 55|, &amp;lt;0 1 2 0 0 1|]
EDOs: 26, 104c, 130
Badness: 0.0329

=Fermionic=
Commas: 540/539, 78408/78125, 177147/176000

POTE generator: ~3/2 = 701.077

Map: [&amp;lt;26 0 -22 73 -116|, &amp;lt;0 1 2 0 5|]
EDOs: 130, 286, 416, 546, 1508bcd
Badness: 0.0906

==13-limit==
Commas: 351/350, 540/539, 40656/40625, 142884/142805

POTE generator: ~3/2 = 701.038

Map: [&amp;lt;26 0 -22 73 -116 55|, &amp;lt;0 1 2 0 5 1|]
EDOs: 130, 286, 416, 546, 962bf
Badness: 0.0436

 &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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Commas: 321489/320000, 589824/588245&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
POTE generator: ~3/2 = 701.250&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Map: [&amp;amp;lt;26 0 -22 73|, &amp;amp;lt;0 1 2 0|]&amp;lt;br /&amp;gt;
Wedgie: &amp;amp;lt;&amp;amp;lt;26 52 0 22 -73 -146||&amp;lt;br /&amp;gt;
EDOs: 26, 130, 546, 676, 806c&amp;lt;br /&amp;gt;
Badness: 0.1558&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="Bosonic-11-limit"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;11-limit&amp;lt;/h2&amp;gt;
Commas: 441/440, 8019/8000, 65536/65219&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
POTE generator: ~3/2 = 701.559&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Map: [&amp;amp;lt;26 0 -22 73 90|, &amp;amp;lt;0 1 2 0 0|]&amp;lt;br /&amp;gt;
EDOs: 26, 104c, 130&amp;lt;br /&amp;gt;
Badness: 0.0652&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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Commas: 351/350, 364/363, 441/440, 15379/15360&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
POTE generator: ~3/2 = 701.546&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Map: [&amp;amp;lt;26 0 -22 73 90 55|, &amp;amp;lt;0 1 2 0 0 1|]&amp;lt;br /&amp;gt;
EDOs: 26, 104c, 130&amp;lt;br /&amp;gt;
Badness: 0.0329&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Fermionic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Fermionic&amp;lt;/h1&amp;gt;
Commas: 540/539, 78408/78125, 177147/176000&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
POTE generator: ~3/2 = 701.077&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Map: [&amp;amp;lt;26 0 -22 73 -116|, &amp;amp;lt;0 1 2 0 5|]&amp;lt;br /&amp;gt;
EDOs: 130, 286, 416, 546, 1508bcd&amp;lt;br /&amp;gt;
Badness: 0.0906&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Fermionic-13-limit"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;13-limit&amp;lt;/h2&amp;gt;
Commas: 351/350, 540/539, 40656/40625, 142884/142805&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
POTE generator: ~3/2 = 701.038&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Map: [&amp;amp;lt;26 0 -22 73 -116 55|, &amp;amp;lt;0 1 2 0 5 1|]&amp;lt;br /&amp;gt;
EDOs: 130, 286, 416, 546, 962bf&amp;lt;br /&amp;gt;
Badness: 0.0436&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ns674ddec13y7dgf6r9au5vykkmwqmu</sha1>
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    <revision>
      <id>4278</id>
      <timestamp>2012-02-07T01:17:54Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 299181754 - Original comment: **</comment>
      <origin>4278</origin>
      <model>wikitext</model>
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      <text bytes="3888" sha1="4vswdnfgvl21n3cxo0404xnl5a478ot" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-02-07 01:17:54 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;299181754&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[toc|flat]]

=Bosonic=
Commas: 321489/320000, 589824/588245

POTE generator: ~3/2 = 701.250

Map: [&amp;lt;26 0 -22 73|, &amp;lt;0 1 2 0|]
Wedgie: &amp;lt;&amp;lt;26 52 0 22 -73 -146||
EDOs: 26, 130, 546, 676, 806c
Badness: 0.1558

==11-limit==
Commas: 441/440, 8019/8000, 65536/65219

POTE generator: ~3/2 = 701.559

Map: [&amp;lt;26 0 -22 73 90|, &amp;lt;0 1 2 0 0|]
EDOs: 26, 104c, 130
Badness: 0.0652

==13-limit==
Commas: 351/350, 364/363, 441/440, 15379/15360

POTE generator: ~3/2 = 701.546

Map: [&amp;lt;26 0 -22 73 90 55|, &amp;lt;0 1 2 0 0 1|]
EDOs: 26, 104c, 130
Badness: 0.0329




 &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;!-- ws:end:WikiTextTocRule:10 --&amp;gt;&amp;lt;br /&amp;gt;
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Commas: 321489/320000, 589824/588245&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
POTE generator: ~3/2 = 701.250&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Map: [&amp;amp;lt;26 0 -22 73|, &amp;amp;lt;0 1 2 0|]&amp;lt;br /&amp;gt;
Wedgie: &amp;amp;lt;&amp;amp;lt;26 52 0 22 -73 -146||&amp;lt;br /&amp;gt;
EDOs: 26, 130, 546, 676, 806c&amp;lt;br /&amp;gt;
Badness: 0.1558&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="Bosonic-11-limit"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;11-limit&amp;lt;/h2&amp;gt;
Commas: 441/440, 8019/8000, 65536/65219&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
POTE generator: ~3/2 = 701.559&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Map: [&amp;amp;lt;26 0 -22 73 90|, &amp;amp;lt;0 1 2 0 0|]&amp;lt;br /&amp;gt;
EDOs: 26, 104c, 130&amp;lt;br /&amp;gt;
Badness: 0.0652&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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Commas: 351/350, 364/363, 441/440, 15379/15360&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
POTE generator: ~3/2 = 701.546&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Map: [&amp;amp;lt;26 0 -22 73 90 55|, &amp;amp;lt;0 1 2 0 0 1|]&amp;lt;br /&amp;gt;
EDOs: 26, 104c, 130&amp;lt;br /&amp;gt;
Badness: 0.0329&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>4vswdnfgvl21n3cxo0404xnl5a478ot</sha1>
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    <revision>
      <id>4279</id>
      <timestamp>2012-02-07T01:17:25Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 299181712 - Original comment: **</comment>
      <origin>4279</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3206" sha1="9y4gr4bp1wopd5kop91gx4dl1rwbvd1" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-02-07 01:17:25 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;299181712&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;

=Bosonic=
Commas: 321489/320000, 589824/588245

POTE generator: ~3/2 = 701.250

Map: [&amp;lt;26 0 -22 73|, &amp;lt;0 1 2 0|]
Wedgie: &amp;lt;&amp;lt;26 52 0 22 -73 -146||
EDOs: 26, 130, 546, 676, 806c
Badness: 0.1558

==11-limit==
Commas: 441/440, 8019/8000, 65536/65219

POTE generator: ~3/2 = 701.559

Map: [&amp;lt;26 0 -22 73 90|, &amp;lt;0 1 2 0 0|]
EDOs: 26, 104c, 130
Badness: 0.0652

==13-limit==
Commas: 351/350, 364/363, 441/440, 15379/15360

POTE generator: ~3/2 = 701.546

Map: [&amp;lt;26 0 -22 73 90 55|, &amp;lt;0 1 2 0 0 1|]
EDOs: 26, 104c, 130
Badness: 0.0329




 &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Bosonic"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Bosonic&amp;lt;/h1&amp;gt;
Commas: 321489/320000, 589824/588245&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
POTE generator: ~3/2 = 701.250&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Map: [&amp;amp;lt;26 0 -22 73|, &amp;amp;lt;0 1 2 0|]&amp;lt;br /&amp;gt;
Wedgie: &amp;amp;lt;&amp;amp;lt;26 52 0 22 -73 -146||&amp;lt;br /&amp;gt;
EDOs: 26, 130, 546, 676, 806c&amp;lt;br /&amp;gt;
Badness: 0.1558&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="Bosonic-11-limit"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;11-limit&amp;lt;/h2&amp;gt;
Commas: 441/440, 8019/8000, 65536/65219&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
POTE generator: ~3/2 = 701.559&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Map: [&amp;amp;lt;26 0 -22 73 90|, &amp;amp;lt;0 1 2 0 0|]&amp;lt;br /&amp;gt;
EDOs: 26, 104c, 130&amp;lt;br /&amp;gt;
Badness: 0.0652&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="Bosonic-13-limit"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;13-limit&amp;lt;/h2&amp;gt;
Commas: 351/350, 364/363, 441/440, 15379/15360&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
POTE generator: ~3/2 = 701.546&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Map: [&amp;amp;lt;26 0 -22 73 90 55|, &amp;amp;lt;0 1 2 0 0 1|]&amp;lt;br /&amp;gt;
EDOs: 26, 104c, 130&amp;lt;br /&amp;gt;
Badness: 0.0329&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9y4gr4bp1wopd5kop91gx4dl1rwbvd1</sha1>
    </revision>
    <revision>
      <id>44019</id>
      <parentid>53</parentid>
      <timestamp>2020-04-26T02:34:45Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <comment>TallKite moved page [[11-limit comma temperaments]] to [[26th-octave temperaments]]: Not all temperaments on this page are 11-limit, and most 11-limit temperaments are not on this page</comment>
      <origin>44019</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="952" sha1="tggx6zsrp50x9ck6ki0jn5ve4uvl9cj" xml:space="preserve">__FORCETOC__
=Bosonic=
Commas: 321489/320000, 589824/588245

POTE generator: ~3/2 = 701.250

Map: [&amp;lt;26 0 -22 73|, &amp;lt;0 1 2 0|]

Wedgie: &amp;lt;&amp;lt;26 52 0 22 -73 -146||

EDOs: 26, 130, 546, 676, 806c

Badness: 0.1558

==11-limit==
Commas: 441/440, 8019/8000, 65536/65219

POTE generator: ~3/2 = 701.559

Map: [&amp;lt;26 0 -22 73 90|, &amp;lt;0 1 2 0 0|]

EDOs: 26, 104c, 130

Badness: 0.0652

==13-limit==
Commas: 351/350, 364/363, 441/440, 15379/15360

POTE generator: ~3/2 = 701.546

Map: [&amp;lt;26 0 -22 73 90 55|, &amp;lt;0 1 2 0 0 1|]

EDOs: 26, 104c, 130

Badness: 0.0329

=Fermionic=
Commas: 540/539, 78408/78125, 177147/176000

POTE generator: ~3/2 = 701.077

Map: [&amp;lt;26 0 -22 73 -116|, &amp;lt;0 1 2 0 5|]

EDOs: 130, 286, 416, 546, 1508bcd

Badness: 0.0906

==13-limit==
Commas: 351/350, 540/539, 40656/40625, 142884/142805

POTE generator: ~3/2 = 701.038

Map: [&amp;lt;26 0 -22 73 -116 55|, &amp;lt;0 1 2 0 5 1|]

EDOs: 130, 286, 416, 546, 962bf

Badness: 0.0436</text>
      <sha1>tggx6zsrp50x9ck6ki0jn5ve4uvl9cj</sha1>
    </revision>
    <revision>
      <id>44021</id>
      <parentid>44019</parentid>
      <timestamp>2020-04-26T02:38:07Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>added a 2-sentence overview</comment>
      <origin>44021</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1129" sha1="0utucc69ynmfcqtg9muy3mmv7fk35v7" xml:space="preserve">__FORCETOC__

All temperaments on this page have a period that is 1/26th of an octave, i.e. their [[pergen]] is (P8/26, P5). However, the [-41 26&gt; comma is not part of these temperaments. 

=Bosonic=
Commas: 321489/320000, 589824/588245

POTE generator: ~3/2 = 701.250

Map: [&amp;lt;26 0 -22 73|, &amp;lt;0 1 2 0|]

Wedgie: &amp;lt;&amp;lt;26 52 0 22 -73 -146||

EDOs: 26, 130, 546, 676, 806c

Badness: 0.1558

==11-limit==
Commas: 441/440, 8019/8000, 65536/65219

POTE generator: ~3/2 = 701.559

Map: [&amp;lt;26 0 -22 73 90|, &amp;lt;0 1 2 0 0|]

EDOs: 26, 104c, 130

Badness: 0.0652

==13-limit==
Commas: 351/350, 364/363, 441/440, 15379/15360

POTE generator: ~3/2 = 701.546

Map: [&amp;lt;26 0 -22 73 90 55|, &amp;lt;0 1 2 0 0 1|]

EDOs: 26, 104c, 130

Badness: 0.0329

=Fermionic=
Commas: 540/539, 78408/78125, 177147/176000

POTE generator: ~3/2 = 701.077

Map: [&amp;lt;26 0 -22 73 -116|, &amp;lt;0 1 2 0 5|]

EDOs: 130, 286, 416, 546, 1508bcd

Badness: 0.0906

==13-limit==
Commas: 351/350, 540/539, 40656/40625, 142884/142805

POTE generator: ~3/2 = 701.038

Map: [&amp;lt;26 0 -22 73 -116 55|, &amp;lt;0 1 2 0 5 1|]

EDOs: 130, 286, 416, 546, 962bf

Badness: 0.0436</text>
      <sha1>0utucc69ynmfcqtg9muy3mmv7fk35v7</sha1>
    </revision>
    <revision>
      <id>44022</id>
      <parentid>44021</parentid>
      <timestamp>2020-04-26T04:07:03Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <origin>44022</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1206" sha1="qaa1icblpwm602dyefsbblemvi3km22" xml:space="preserve">__FORCETOC__

All temperaments on this page have a period that is 1/26th of an octave, i.e. their [[pergen]] is (P8/26, P5). However, the [-41 26&gt; comma is not tempered out. Thus the 3/2 is not 26-edo's 3/2. However, 7/4 is, as is 11/8 in the Bosonic temperaments. 

=Bosonic=
Commas: 321489/320000, 589824/588245

POTE generator: ~3/2 = 701.250

Map: [&amp;lt;26 0 -22 73|, &amp;lt;0 1 2 0|]

Wedgie: &amp;lt;&amp;lt;26 52 0 22 -73 -146||

EDOs: 26, 130, 546, 676, 806c

Badness: 0.1558

==11-limit==
Commas: 441/440, 8019/8000, 65536/65219

POTE generator: ~3/2 = 701.559

Map: [&amp;lt;26 0 -22 73 90|, &amp;lt;0 1 2 0 0|]

EDOs: 26, 104c, 130

Badness: 0.0652

==13-limit==
Commas: 351/350, 364/363, 441/440, 15379/15360

POTE generator: ~3/2 = 701.546

Map: [&amp;lt;26 0 -22 73 90 55|, &amp;lt;0 1 2 0 0 1|]

EDOs: 26, 104c, 130

Badness: 0.0329

=Fermionic=
Commas: 540/539, 78408/78125, 177147/176000

POTE generator: ~3/2 = 701.077

Map: [&amp;lt;26 0 -22 73 -116|, &amp;lt;0 1 2 0 5|]

EDOs: 130, 286, 416, 546, 1508bcd

Badness: 0.0906

==13-limit==
Commas: 351/350, 540/539, 40656/40625, 142884/142805

POTE generator: ~3/2 = 701.038

Map: [&amp;lt;26 0 -22 73 -116 55|, &amp;lt;0 1 2 0 5 1|]

EDOs: 130, 286, 416, 546, 962bf

Badness: 0.0436</text>
      <sha1>qaa1icblpwm602dyefsbblemvi3km22</sha1>
    </revision>
    <revision>
      <id>69801</id>
      <parentid>44022</parentid>
      <timestamp>2021-05-13T22:41:45Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>69801</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1450" sha1="8f48a7dcsljxhkjfhgyzlbdv1nokapt" xml:space="preserve">All temperaments on this page have a period that is 1/26th of an octave, i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo|-41 26}} comma is not tempered out. Thus the 3/2 is not 26-edo's 3/2. However, 7/4 is, as is 11/8 in the Bosonic temperaments. 

= Bosonic =
[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val|26 0 -22 73}}, {{val|0 1 2 0}}]

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Val list|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

== 11-limit ==
Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val|26 0 -22 73 90}}, {{val|0 1 2 0 0}}]

POTE generator: ~3/2 = 701.559

Vals: {{Val list| 26, 104c, 130 }}

Badness: 0.065219

== 13-limit ==
Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val|26 0 -22 73 90 55}}, {{val|0 1 2 0 0 1}}]

POTE generator: ~3/2 = 701.546

Vals: {{Val list| 26, 104c, 130 }}

Badness: 0.032946

= Fermionic =
Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val|26 0 -22 73 -116}}, {{val|0 1 2 0 5}}]

POTE generator: ~3/2 = 701.077

Vals: {{Val list| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

== 13-limit ==
Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val|26 0 -22 73 -116 55}}, {{val|0 1 2 0 5 1}}]

POTE generator: ~3/2 = 701.038

Vals: {{Val list| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

[[Category:Temperament]]
[[Category:Temperament collection]]</text>
      <sha1>8f48a7dcsljxhkjfhgyzlbdv1nokapt</sha1>
    </revision>
    <revision>
      <id>72424</id>
      <parentid>69801</parentid>
      <timestamp>2021-06-11T22:31:07Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <minor/>
      <origin>72424</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1595" sha1="2r6qp0xoi47n7u225lyb7288npi4bzm" xml:space="preserve">All temperaments on this page have a period that is 1/26th of an octave, i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo|-41 26}} comma is not tempered out. Thus the 3/2 is not [[26EDO]]'s 3/2. However, 7/4 is, as is 11/8 in the Bosonic temperaments. 

== Bosonic ==
Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val|26 0 -22 73}}, {{val|0 1 2 0}}]

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Val list|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val|26 0 -22 73 90}}, {{val|0 1 2 0 0}}]

POTE generator: ~3/2 = 701.559

Vals: {{Val list| 26, 104c, 130 }}

Badness: 0.065219

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val|26 0 -22 73 90 55}}, {{val|0 1 2 0 0 1}}]

POTE generator: ~3/2 = 701.546

Vals: {{Val list| 26, 104c, 130 }}

Badness: 0.032946

== Fermionic ==
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val|26 0 -22 73 -116}}, {{val|0 1 2 0 5}}]

POTE generator: ~3/2 = 701.077

Vals: {{Val list| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val|26 0 -22 73 -116 55}}, {{val|0 1 2 0 5 1}}]

POTE generator: ~3/2 = 701.038

Vals: {{Val list| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

[[Category:26edo]]
[[Category:Temperament]]
[[Category:Temperament collection]]</text>
      <sha1>2r6qp0xoi47n7u225lyb7288npi4bzm</sha1>
    </revision>
    <revision>
      <id>72425</id>
      <parentid>72424</parentid>
      <timestamp>2021-06-11T22:35:57Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <minor/>
      <origin>72425</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1633" sha1="3638k4c85tbtifj5y9pj9vj1wk5cjdk" xml:space="preserve">All temperaments on this page have a period that is 1/26th of an octave, i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo|-41 26}} comma is not tempered out. Thus the 3/2 is not [[26edo|26EDO]]'s 3/2. However, 7/4 is, as is 11/8 in the Bosonic temperaments. 

== Bosonic ==
Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val|26 0 -22 73}}, {{val|0 1 2 0}}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Val list|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val|26 0 -22 73 90}}, {{val|0 1 2 0 0}}]

POTE generator: ~3/2 = 701.559

Vals: {{Val list| 26, 104c, 130 }}

Badness: 0.065219

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val|26 0 -22 73 90 55}}, {{val|0 1 2 0 0 1}}]

POTE generator: ~3/2 = 701.546

Vals: {{Val list| 26, 104c, 130 }}

Badness: 0.032946

== Fermionic ==
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val|26 0 -22 73 -116}}, {{val|0 1 2 0 5}}]

POTE generator: ~3/2 = 701.077

Vals: {{Val list| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val|26 0 -22 73 -116 55}}, {{val|0 1 2 0 5 1}}]

POTE generator: ~3/2 = 701.038

Vals: {{Val list| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

[[Category:26edo]]
[[Category:Temperament]]
[[Category:Temperament collection]]</text>
      <sha1>3638k4c85tbtifj5y9pj9vj1wk5cjdk</sha1>
    </revision>
    <revision>
      <id>78592</id>
      <parentid>72425</parentid>
      <timestamp>2021-09-26T15:27:45Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Improve categories</comment>
      <origin>78592</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1668" sha1="q30pezympf4nqmofg8ha9ddn4j6tevn" xml:space="preserve">All temperaments on this page have a period that is 1/26th of an octave, i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo|-41 26}} comma is not tempered out. Thus the 3/2 is not [[26edo|26EDO]]'s 3/2. However, 7/4 is, as is 11/8 in the Bosonic temperaments. 

== Bosonic ==
Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val|26 0 -22 73}}, {{val|0 1 2 0}}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Val list|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val|26 0 -22 73 90}}, {{val|0 1 2 0 0}}]

POTE generator: ~3/2 = 701.559

Vals: {{Val list| 26, 104c, 130 }}

Badness: 0.065219

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val|26 0 -22 73 90 55}}, {{val|0 1 2 0 0 1}}]

POTE generator: ~3/2 = 701.546

Vals: {{Val list| 26, 104c, 130 }}

Badness: 0.032946

== Fermionic ==
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val|26 0 -22 73 -116}}, {{val|0 1 2 0 5}}]

POTE generator: ~3/2 = 701.077

Vals: {{Val list| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val|26 0 -22 73 -116 55}}, {{val|0 1 2 0 5 1}}]

POTE generator: ~3/2 = 701.038

Vals: {{Val list| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

[[Category:26edo]]
[[Category:Regular temperament theory]]
[[Category:Temperament collection]]
[[Category:Rank 2]]</text>
      <sha1>q30pezympf4nqmofg8ha9ddn4j6tevn</sha1>
    </revision>
    <revision>
      <id>81290</id>
      <parentid>78592</parentid>
      <timestamp>2021-11-27T00:37:35Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <minor/>
      <origin>81290</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1759" sha1="fuimqkslrx8rmf4tlw46muqf32pucbk" xml:space="preserve">All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]], i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo|-41 26}} comma is not tempered out. Thus the 3/2 is not [[26edo|26EDO]]'s 3/2. However, 7/4 is, as is 11/8 in the Bosonic temperaments. 

== Bosonic ==
{{see also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val|26 0 -22 73}}, {{val|0 1 2 0}}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Val list|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val|26 0 -22 73 90}}, {{val|0 1 2 0 0}}]

POTE generator: ~3/2 = 701.559

Vals: {{Val list| 26, 104c, 130 }}

Badness: 0.065219

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val|26 0 -22 73 90 55}}, {{val|0 1 2 0 0 1}}]

POTE generator: ~3/2 = 701.546

Vals: {{Val list| 26, 104c, 130 }}

Badness: 0.032946

== Fermionic ==
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val|26 0 -22 73 -116}}, {{val|0 1 2 0 5}}]

POTE generator: ~3/2 = 701.077

Vals: {{Val list| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val|26 0 -22 73 -116 55}}, {{val|0 1 2 0 5 1}}]

POTE generator: ~3/2 = 701.038

Vals: {{Val list| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

[[Category:26edo]]
[[Category:Regular temperament theory]]
[[Category:Temperament collection]]
[[Category:Rank 2]]</text>
      <sha1>fuimqkslrx8rmf4tlw46muqf32pucbk</sha1>
    </revision>
    <revision>
      <id>83446</id>
      <parentid>81290</parentid>
      <timestamp>2022-01-08T16:05:06Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Fix header levels</comment>
      <origin>83446</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1781" sha1="qhhsag30hr6gmnsp27k2reywkpn3jl1" xml:space="preserve">All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]], i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not [[26edo]]'s 3/2. However, 7/4 is, as is 11/8 in the Bosonic temperaments. 

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Val list|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

Vals: {{Val list| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

Vals: {{Val list| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

Vals: {{Val list| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

Vals: {{Val list| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

[[Category:26edo]]
[[Category:Regular temperament theory]]
[[Category:Temperament collection]]
[[Category:Rank 2]]</text>
      <sha1>qhhsag30hr6gmnsp27k2reywkpn3jl1</sha1>
    </revision>
    <revision>
      <id>86220</id>
      <parentid>83446</parentid>
      <timestamp>2022-02-22T04:41:36Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>86220</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2128" sha1="eek2ir7rdvwk33en4r6s6u8tyj5hfsp" xml:space="preserve">All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]], i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the 26-7-comma ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Val list|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

Optimal GPV sequence: {{Val list| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

Optimal GPV sequence: {{Val list| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

[[Category:26edo]]
[[Category:Regular temperament theory]]
[[Category:Temperament collection]]
[[Category:Rank 2]]</text>
      <sha1>eek2ir7rdvwk33en4r6s6u8tyj5hfsp</sha1>
    </revision>
    <revision>
      <id>87744</id>
      <parentid>86220</parentid>
      <timestamp>2022-02-28T17:08:50Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Temperament collection]] to [[Category:Temperament collections]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87744</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2129" sha1="4ro0jk6h04azts2zambnfs6e40202pv" xml:space="preserve">All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]], i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the 26-7-comma ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Val list|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

Optimal GPV sequence: {{Val list| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

Optimal GPV sequence: {{Val list| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

[[Category:26edo]]
[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Rank 2]]</text>
      <sha1>4ro0jk6h04azts2zambnfs6e40202pv</sha1>
    </revision>
    <revision>
      <id>88757</id>
      <parentid>87744</parentid>
      <timestamp>2022-03-02T21:40:02Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Regular temperament theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>88757</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2089" sha1="f45jamgupjx4k69pps1o11n0o6ymd5b" xml:space="preserve">All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]], i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the 26-7-comma ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Val list|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

Optimal GPV sequence: {{Val list| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

Optimal GPV sequence: {{Val list| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

[[Category:26edo]]
[[Category:Temperament collections]]
[[Category:Rank 2]]</text>
      <sha1>f45jamgupjx4k69pps1o11n0o6ymd5b</sha1>
    </revision>
    <revision>
      <id>101677</id>
      <parentid>88757</parentid>
      <timestamp>2023-01-19T19:20:34Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>iron</comment>
      <origin>101677</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2891" sha1="izmn1vg5jlp6ozdat1x1fzzjqch13oj" xml:space="preserve">All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]], i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the 26-7-comma ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Val list|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

Optimal GPV sequence: {{Val list| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

Optimal GPV sequence: {{Val list| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

Subgroup: 2.3.5.7

Comma list: 2460375/2458624, 2147483648/2144153025

Mapping: 26 26 106 73, 0 1 -3 0

Mapping generators: ~17280/16807 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

Vals: 130, 364, 494, 624, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: 26 26 106 73 166, 0 1 -3 0 -5

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

Vals: 130, 364, 494, 624, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: 26 26 106 73 166 81, 0 1 -3 0 -5 1

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

Vals: 130, 364, 494, 624,...
[[Category:26edo]]
[[Category:Temperament collections]]
[[Category:Rank 2]]</text>
      <sha1>izmn1vg5jlp6ozdat1x1fzzjqch13oj</sha1>
    </revision>
    <revision>
      <id>101678</id>
      <parentid>101677</parentid>
      <timestamp>2023-01-19T19:22:11Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Iron */</comment>
      <origin>101678</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2983" sha1="6z8sg0c6j4l5gkhjk8ih99vd8wmlr55" xml:space="preserve">All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]], i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the 26-7-comma ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Val list|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

Optimal GPV sequence: {{Val list| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

Optimal GPV sequence: {{Val list| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

Optimal GPV sequence: {{Val list| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

Vals: {{EDOs|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

Vals: 130, 364, 494, 624, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

Vals: 130, 364, 494, 624,...
[[Category:26edo]]
[[Category:Temperament collections]]
[[Category:Rank 2]]</text>
      <sha1>6z8sg0c6j4l5gkhjk8ih99vd8wmlr55</sha1>
    </revision>
    <revision>
      <id>112497</id>
      <parentid>101678</parentid>
      <timestamp>2023-05-07T16:19:31Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence</comment>
      <origin>112497</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2986" sha1="jethjyhl66u305pk6vv54ws6n43c4q4" xml:space="preserve">All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]], i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the 26-7-comma ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

Vals: {{EDOs|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

Vals: 130, 364, 494, 624, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

Vals: 130, 364, 494, 624,...
[[Category:26edo]]
[[Category:Temperament collections]]
[[Category:Rank 2]]</text>
      <sha1>jethjyhl66u305pk6vv54ws6n43c4q4</sha1>
    </revision>
    <revision>
      <id>112747</id>
      <parentid>112497</parentid>
      <timestamp>2023-05-07T19:37:48Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence</comment>
      <origin>112747</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3041" sha1="k7ultw4sjjmita4tge5fzxjvy4dvtox" xml:space="preserve">All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]], i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the 26-7-comma ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|130, 364, 494, 624}}, ...
[[Category:26edo]]
[[Category:Temperament collections]]
[[Category:Rank 2]]</text>
      <sha1>k7ultw4sjjmita4tge5fzxjvy4dvtox</sha1>
    </revision>
    <revision>
      <id>112749</id>
      <parentid>112747</parentid>
      <timestamp>2023-05-07T19:39:15Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence</comment>
      <origin>112749</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3059" sha1="enl4dce101w2pqjb8drepw62eyc4lhi" xml:space="preserve">All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]], i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the 26-7-comma ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...
[[Category:26edo]]
[[Category:Temperament collections]]
[[Category:Rank 2]]</text>
      <sha1>enl4dce101w2pqjb8drepw62eyc4lhi</sha1>
    </revision>
    <revision>
      <id>119025</id>
      <parentid>112749</parentid>
      <timestamp>2023-07-07T15:11:27Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>119025</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3095" sha1="9s4wrne3qw6p9xhmi9ssoexmz5vsprw" xml:space="preserve">{{Fractional-octave navigation|26}}
All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]], i.e. their [[pergen]] is (P8/26, P5). However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the 26-7-comma ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...
[[Category:26edo]]
[[Category:Temperament collections]]
[[Category:Rank 2]]</text>
      <sha1>9s4wrne3qw6p9xhmi9ssoexmz5vsprw</sha1>
    </revision>
    <revision>
      <id>119053</id>
      <parentid>119025</parentid>
      <timestamp>2023-07-07T23:34:09Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>generator isn't necessarily P5 or 3/2, important to note (although both existing examples only use that generator)</comment>
      <origin>119053</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3057" sha1="bzrprkx8c0nc564xe1106wjxlfeo2i9" xml:space="preserve">{{Fractional-octave navigation|26}}
All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]]. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the 26-7-comma ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...
[[Category:26edo]]
[[Category:Temperament collections]]
[[Category:Rank 2]]</text>
      <sha1>bzrprkx8c0nc564xe1106wjxlfeo2i9</sha1>
    </revision>
    <revision>
      <id>124353</id>
      <parentid>119053</parentid>
      <timestamp>2023-09-17T12:05:01Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>124353</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3037" sha1="lhfemsdxxz70uuou29z62bi84nd5cnt" xml:space="preserve">{{Fractional-octave navigation|26}}
All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]]. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the 26-7-comma ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...
[[Category:26edo]]
[[Category:Temperament collections]]</text>
      <sha1>lhfemsdxxz70uuou29z62bi84nd5cnt</sha1>
    </revision>
    <revision>
      <id>165252</id>
      <parentid>124353</parentid>
      <timestamp>2024-11-13T23:43:50Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>165252</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3041" sha1="6r3rt765uj5uy9r4z7duzr4j3e2o85b" xml:space="preserve">{{Fractional-octave navigation|26}}
All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]]. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...
[[Category:26edo]]
[[Category:Temperament collections]]</text>
      <sha1>6r3rt765uj5uy9r4z7duzr4j3e2o85b</sha1>
    </revision>
    <revision>
      <id>169041</id>
      <parentid>165252</parentid>
      <timestamp>2024-12-05T07:24:23Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>169041</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3035" sha1="ng24y6n3e96kiu7lh4n2whzerzks36e" xml:space="preserve">{{Fractional-octave navigation|26}}
All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]]. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

{{Navbox fractional-octave}}

[[Category:26edo]]</text>
      <sha1>ng24y6n3e96kiu7lh4n2whzerzks36e</sha1>
    </revision>
    <revision>
      <id>180323</id>
      <parentid>169041</parentid>
      <timestamp>2025-02-11T19:22:19Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>180323</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3031" sha1="cjqru04ku305nahdpzd07ngm2pz5i7x" xml:space="preserve">{{navbox fractional-octave|26}}
All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]]. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

{{Navbox fractional-octave}}

[[Category:26edo]]</text>
      <sha1>cjqru04ku305nahdpzd07ngm2pz5i7x</sha1>
    </revision>
    <revision>
      <id>180325</id>
      <parentid>180323</parentid>
      <timestamp>2025-02-11T19:23:07Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Undid revision 180323 by [[Special:Contributions/ArrowHead294|ArrowHead294]] ([[User talk:ArrowHead294|talk]])</comment>
      <origin>180325</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3035" sha1="ng24y6n3e96kiu7lh4n2whzerzks36e" xml:space="preserve">{{Fractional-octave navigation|26}}
All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]]. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

{{Navbox fractional-octave}}

[[Category:26edo]]</text>
      <sha1>ng24y6n3e96kiu7lh4n2whzerzks36e</sha1>
    </revision>
    <revision>
      <id>180629</id>
      <parentid>180325</parentid>
      <timestamp>2025-02-14T03:39:24Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>180629</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3032" sha1="cyjv888h5submne65k2ubpi4ta736vb" xml:space="preserve">{{Infobox fractional-octave|26}}
All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]]. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

{{Navbox fractional-octave}}

[[Category:26edo]]</text>
      <sha1>cyjv888h5submne65k2ubpi4ta736vb</sha1>
    </revision>
    <revision>
      <id>191615</id>
      <parentid>180629</parentid>
      <timestamp>2025-04-13T10:16:02Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>{{Technical data page}}&lt;br&gt;&lt;br&gt;</comment>
      <origin>191615</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3064" sha1="814xxsy6o2uvc5y00ix87l7k7pswyrq" xml:space="preserve">{{Technical data page}}&lt;br&gt;&lt;br&gt;
{{Infobox fractional-octave|26}}
All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]]. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

{{Navbox fractional-octave}}

[[Category:26edo]]</text>
      <sha1>814xxsy6o2uvc5y00ix87l7k7pswyrq</sha1>
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    <revision>
      <id>191812</id>
      <parentid>191615</parentid>
      <timestamp>2025-04-13T13:45:58Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
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      <text bytes="3056" sha1="k6z90pb5z4c392dqg212zbj32f8h0eb" xml:space="preserve">{{Technical data page}}
{{Infobox fractional-octave|26}}
All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]]. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

{{Multival|legend=1|26 52 0 22 -73 -146}}

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

{{Navbox fractional-octave}}

[[Category:26edo]]</text>
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      <text bytes="3013" sha1="cpuhgvyquleieyynv5iia6rrrpfieaq" xml:space="preserve">{{Technical data page}}
{{Infobox fractional-octave|26}}
All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]]. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| High badness temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

{{Navbox fractional-octave}}

[[Category:26edo]]</text>
      <sha1>cpuhgvyquleieyynv5iia6rrrpfieaq</sha1>
    </revision>
    <revision>
      <id>231491</id>
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      <timestamp>2026-06-03T01:45:56Z</timestamp>
      <contributor>
        <username>Hotcrystal0</username>
        <id>614</id>
      </contributor>
      <comment>fix link</comment>
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      <text bytes="3022" sha1="f0cn8wxt03ookt97d8d5f82usdh83ug" xml:space="preserve">{{Technical data page}}
{{Infobox fractional-octave|26}}
All temperaments on this page have a period that is [[Fractional-octave temperaments|1/26th of an octave]]. However, the {{monzo| -41 26 }} comma is not tempered out. Thus the 3/2 is not that of [[26edo]]. However, 7/4 is, as is 11/8 in the Bosonic temperaments.

26edo is very accurate for 7th harmonic, the [[26-7-comma]] ({{monzo|73 0 0 -26}}, the amount by which 26 septimal whole tones ([[8/7]]) exceed 5 octaves) is tempered out by 26-fold multiple EDOs up to 1456 (such as [[26edo|26]], [[130edo|130]], [[286edo|286]] or [[546edo|546]] EDO).

== Bosonic ==
{{See also| Miscellaneous 5-limit temperaments #Tridecatonic }}

Subgroup: 2.3.5.7

[[Comma list]]: 321489/320000, 589824/588245

[[Mapping]]: [{{val| 26 0 -22 73 }}, {{val| 0 1 2 0 }}]

Mapping generators: ~36/35, ~3

[[POTE generator]]: ~3/2 = 701.250

{{Optimal ET sequence|legend=1| 26, 130, 546, 676, 806c }}

[[Badness]]: 0.155827

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 8019/8000, 65536/65219

Mapping: [{{val| 26 0 -22 73 90 }}, {{val| 0 1 2 0 0 }}]

POTE generator: ~3/2 = 701.559

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.065219

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 15379/15360

Mapping: [{{val| 26 0 -22 73 90 55 }}, {{val| 0 1 2 0 0 1 }}]

POTE generator: ~3/2 = 701.546

{{Optimal ET sequence|legend=1| 26, 104c, 130 }}

Badness: 0.032946

=== Fermionic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 78408/78125, 177147/176000

Mapping: [{{val| 26 0 -22 73 -116 }}, {{val| 0 1 2 0 5 }}]

POTE generator: ~3/2 = 701.077

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 1508bcd }}

Badness: 0.090642

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 40656/40625, 142884/142805

Mapping: [{{val| 26 0 -22 73 -116 55 }}, {{val| 0 1 2 0 5 1 }}]

POTE generator: ~3/2 = 701.038

{{Optimal ET sequence|legend=1| 130, 286, 416, 546, 962bf }}

Badness: 0.043581

== Iron ==
''Iron'' is named after the 26th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2460375/2458624, 2147483648/2144153025

[[Mapping]]: [{{val|26 26 106 73}}, {{val|0 1 -3 0}}]

[[Mapping]] [[generators]]: ~17280/16807 = 1\26, ~3/2 = 702.017

[[Optimal tuning]] ([[CTE]]):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 131072/130977, 759375/758912

Mapping: [{{val|26 26 106 73 166}}, {{val|0 1 -3 0 -5}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.017

Optimal tuning (CTE):  ~3/2 = 702.017

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 4096/4095, 91125/91091

Mapping: [{{val|26 26 106 73 166 81}}], {{val|0 1 -3 0 -5 1}}]

Mapping generators: ~77/75 = 1\26, ~3/2 = 702.018

Optimal tuning (CTE):  ~3/2 = 702.018

{{Optimal ET sequence|legend=1|130, 364, 494, 624}}, ...

{{Navbox fractional-octave}}

[[Category:26edo]]</text>
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  </page>
  <page>
    <title>11-limit</title>
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    <revision>
      <id>54</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>54</origin>
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      <text bytes="3762" sha1="filnumlwcefzgbbtmdnvilm13hyi6ak" xml:space="preserve">The ''11-limit'' consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11|14/11]], [[11/8|11/8]], [[27/22|27/22]] and [[99/98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

{| class="wikitable"
|-
| | ratio
| | harmonic solfege
|-
| | 12/11
| | fu-sol
|-
| | 11/10
| | mi-fu
|-
| | 10/9
| | re-mi
|-
| | 9/8
| | do-re
|-
| | 8/7
| | ta-do
|-
| | 7/6
| | sol-ta
|-
| | 6/5
| | mi-sol, ti-re
|-
| | 11/9
| | re-fu
|-
| | 5/4
| | do-mi
|-
| | 14/11
| | fu-ta
|-
| | 9/7
| | ta-re
|-
| | 4/3
| | do-fa
|-
| | 11/8
| | do-fu
|-
| | 7/5
| | mi-ta
|-
| | 10/7
| | ta-mi
|-
| | 16/11
| | fu-do
|-
| | 3/2
| | do-sol
|-
| | 14/9
| | re-ta
|-
| | 11/7
| | ta-fu
|-
| | 8/5
| | mi-do
|-
| | 18/11
| | fu-re
|-
| | 5/3
| | sol-mi
|-
| | 12/7
| | ta-sol
|-
| | 7/4
| | do-ta
|-
| | 16/9
| | re-do
|-
| | 9/5
| | mi-re
|-
| | 20/11
| | fu-mi
|-
| | 11/6
| | sol-fu
|-
| | 2/1
| | do-do
|}

While the [[7-limit|7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo|12edo]], the 11-limit introduces neutral intervals, [[Superfourth|superfourth]]s and [[Subfifth|subfifth]]s, which fall in between major, minor and perfect [[interval_category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[EDO|edo]]s which do (relatively) well in supporting 11-limit intervals are: [[1edo|1edo]], [[2edo|2edo]], [[3edo|3edo]], [[4edo|4edo]], [[5edo|5edo]], [[6edo|6edo]], [[7edo|7edo]], [[9edo|9edo]], [[10edo|10edo]], [[12edo|12edo]], [[15edo|15edo]], [[22edo|22edo]], [[26edo|26edo]], [[31edo|31edo]], [[41edo|41edo]], [[63edo|63edo]], [[72edo|72edo]], [[87edo|87edo]], [[109edo|109edo]], [[161edo|161edo]].

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

==Intervals==
Some of the simplest intervals of 11 include:

{| class="wikitable"
|-
| | Interval
| | Cents Value
|-
| | [[12/11|12/11]]
| | 150.637
|-
| | [[11/10|11/10]]
| | 165.004
|-
| | [[11/9|11/9]]
| | 347.408
|-
| | [[14/11|14/11]]
| | 417.508
|-
| | [[15/11|15/11]]
| | 536.951
|-
| | [[11/8|11/8]]
| | 551.318
|-
| | [[16/11|16/11]]
| | 648.682
|-
| | [[22/15|22/15]]
| | 663.049
|-
| | [[11/7|11/7]]
| | 782.492
|-
| | [[18/11|18/11]]
| | 852.592
|-
| | [[20/11|20/11]]
| | 1034.996
|-
| | [[11/6|11/6]]
| | 1049.363
|}
See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]

=Music=
[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave_Hill|Dave Hill]]

[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill

[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris_Vaisvil|Chris Vaisvil]]

[https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew_Heathwaite|Andrew Heathwaite]]

=See also=
[[Harmonic_Limit|Harmonic Limit]]

[[Category:11-limit]]
[[Category:example]]
[[Category:interval]]
[[Category:limit]]
[[Category:listen]]
[[Category:prime_limit]]</text>
      <sha1>filnumlwcefzgbbtmdnvilm13hyi6ak</sha1>
    </revision>
    <revision>
      <id>4258</id>
      <timestamp>2016-09-08T15:27:10Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 591448544 - Original comment: **</comment>
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      <text bytes="16609" sha1="30h3309c8inm3tvw8hk7f09byogns5y" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-09-08 15:27:10 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;591448544&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

|| ratio || harmonic solfege ||
|| 12/11 || fu-sol ||
|| 11/10 || mi-fu ||
|| 10/9 || re-mi ||
|| 9/8 || do-re ||
|| 8/7 || ta-do ||
|| 7/6 || sol-ta ||
|| 6/5 || mi-sol, ti-re ||
|| 11/9 || re-fu ||
|| 5/4 || do-mi ||
|| 14/11 || fu-ta ||
|| 9/7 || ta-re ||
|| 4/3 || do-fa ||
|| 11/8 || do-fu ||
|| 7/5 || mi-ta ||
|| 10/7 || ta-mi ||
|| 16/11 || fu-do ||
|| 3/2 || do-sol ||
|| 14/9 || re-ta ||
|| 11/7 || ta-fu ||
|| 8/5 || mi-do ||
|| 18/11 || fu-re ||
|| 5/3 || sol-mi ||
|| 12/7 || ta-sol ||
|| 7/4 || do-ta ||
|| 16/9 || re-do ||
|| 9/5 || mi-re ||
|| 20/11 || fu-mi ||
|| 11/6 || sol-fu ||
|| 2/1 || do-do ||

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[1edo]], [[2edo]], [[3edo]], [[4edo]], [[5edo]], [[6edo]], [[7edo]], [[9edo]], [[10edo]], [[12edo]], [[15edo]], [[22edo]], [[26edo]], [[31edo]], [[41edo]], [[63edo]], [[72edo]], [[87edo]], [[109edo]], [[161edo]].

[[image:11-limit_compare.png]]

==Intervals== 
Some of the simplest intervals of 11 include:

|| Interval || Cents Value ||
|| [[12_11|12/11]] || 150.637 ||
|| [[11_10|11/10]] || 165.004 ||
|| [[11_9|11/9]] || 347.408 ||
|| [[14_11|14/11]] || 417.508 ||
|| [[15_11|15/11]] || 536.951 ||
|| [[11_8|11/8]] || 551.318 ||
|| [[16_11|16/11]] || 648.682 ||
|| [[22_15|22/15]] || 663.049 ||
|| [[11_7|11/7]] || 782.492 ||
|| [[18_11|18/11]] || 852.592 ||
|| [[20_11|20/11]] || 1034.996 ||
|| [[11_6|11/6]] || 1049.363 ||
See: [[Gallery of Just Intervals]]

=Music= 
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] [[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]] by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Brief 11-ratio composition]] [[http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3|play]] by Dave Hill
[[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3|11 Limit Piano]] by [[Chris Vaisvil]]
[[https://soundcloud.com/andrew_heathwaite/11-limit-singtervals|11-limit singtervals]] by [[Andrew Heathwaite]]

=See also= 
[[Harmonic Limit]]
[[media type="custom" key="12438854"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;ratio&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;harmonic solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-sol&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sol-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-sol, ti-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-fa&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-sol&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sol-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-sol&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sol-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
While the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, the 11-limit introduces neutral intervals, &amp;lt;a class="wiki_link" href="/superfourth"&amp;gt;superfourth&amp;lt;/a&amp;gt;s and &amp;lt;a class="wiki_link" href="/subfifth"&amp;gt;subfifth&amp;lt;/a&amp;gt;s, which fall in between major, minor and perfect &amp;lt;a class="wiki_link" href="/interval%20category"&amp;gt;interval categories&amp;lt;/a&amp;gt; and thus demand new distinctions. It is thus inescapably xenharmonic.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/1edo"&amp;gt;1edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/4edo"&amp;gt;4edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/6edo"&amp;gt;6edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/9edo"&amp;gt;9edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/10edo"&amp;gt;10edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/26edo"&amp;gt;26edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/63edo"&amp;gt;63edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/87edo"&amp;gt;87edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/109edo"&amp;gt;109edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:269:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-limit_compare.png/354832066/11-limit_compare.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-limit_compare.png/354832066/11-limit_compare.png" alt="11-limit_compare.png" title="11-limit_compare.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:269 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h2&amp;gt;
 Some of the simplest intervals of 11 include:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents Value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/12_11"&amp;gt;12/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;150.637&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;165.004&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;347.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;417.508&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;536.951&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.318&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648.682&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;663.049&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;782.492&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;852.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/20_11"&amp;gt;20/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1034.996&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_6"&amp;gt;11/6&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1049.363&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Brief 11-ratio composition&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Dave Hill&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3" rel="nofollow"&amp;gt;11 Limit Piano&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/andrew_heathwaite/11-limit-singtervals" rel="nofollow"&amp;gt;11-limit singtervals&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="See also"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;See also&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/12438854?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;12438854&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>30h3309c8inm3tvw8hk7f09byogns5y</sha1>
    </revision>
    <revision>
      <id>4259</id>
      <timestamp>2016-09-08T15:26:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 591448464 - Original comment: **</comment>
      <origin>4259</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16609" sha1="4wtyp902txuw5c35neul9xpptq0w6vb" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-09-08 15:26:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;591448464&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

|| ratio || harmonic solfege ||
|| 12/11 || fu-sol ||
|| 11/10 || mi-fu ||
|| 10/9 || re-mi ||
|| 9/8 || do-re ||
|| 8/7 || ta-do ||
|| 7/6 || sol-ta ||
|| 6/5 || mi-sol, ti-re ||
|| 11/9 || re-fu ||
|| 5/4 || do-mi ||
|| 14/11 || fu-ta ||
|| 9/7 || ta-re ||
|| 4/3 || do-fa ||
|| 11/8 || do-fu ||
|| 7/5 || mi-ta ||
|| 10/7 || ta-mi ||
|| 16/11 || fu-do ||
|| 3/2 || do-sol ||
|| 14/9 || re-ta ||
|| 11/7 || ta-fu ||
|| 8/5 || mi-do ||
|| 18/11 || fu-re ||
|| 5/3 || sol-mi ||
|| 12/7 || ta-sol ||
|| 7/4 || do-ta ||
|| 16/9 || re-do ||
|| 9/5 || mi-re ||
|| 20/11 || fu-mi ||
|| 11/6 || sol-fu ||
|| 2/1 || do-do ||

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[1edo]], [[2edo]], [[3edo]], [[4edo]], [[5edo]], [[6edo]], [[7edo]], [[9edo]], [[10edo]], [[12edo]], [[15edo]], [[22edo]], [[26edo]], [[31edo]], [[41edo]], [[63edo]], [[72edo]], [[87edo]], [[109edo]], [[161edo]].

[[image:11-limit_compare.png]]

==Intervals== 
Some of the simplest intervals of 11 include:

|| Interval || Cents Value ||
|| [[12_11|12/11]] || 150.637 ||
|| [[11_10|11/10]] || 165.004 ||
|| [[11_9|11/9]] || 347.408 ||
|| [[14_11|14/11]] || 417.508 ||
|| [[15_11|15/11]] || 536.951 ||
|| [[11_8|11/8]] || 551.318 ||
|| [[16_11|16/11]] || 648.682 ||
|| [[22_15|22/15]] || 663.049 ||
|| [[11_7|11/7]] || 782.492 ||
|| [[18_11|18/11]] || 852.592 ||
|| [[20_11|20/11]] || 1034.996 ||
|| [[11_6|11/6]] || 1049.363 ||
See: [[Gallery of Just Intervals]]

=Music= 
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] [[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]] by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Brief 11-ratio composition]] [[http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3|play]] by Dave Hill
[[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3|11 Limit Piano]] by [[Chris Vaisvil]]
[[https://soundcloud.com/andrew_heathwaite/11-limit-singtervals|11-limit singtervals]] by [[Andrew Heathwaite]]

=See also= 
[[Harmonic Limit]]
[[media type="custom" key="12438854"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;ratio&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;harmonic solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-sol&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sol-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-sol, ti-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-fa&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-sol&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sol-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-sol&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sol-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
While the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, the 11-limit introduces neutral intervals, &amp;lt;a class="wiki_link" href="/superfourth"&amp;gt;superfourth&amp;lt;/a&amp;gt;s and &amp;lt;a class="wiki_link" href="/subfifth"&amp;gt;subfifth&amp;lt;/a&amp;gt;s, which fall in between major, minor and perfect &amp;lt;a class="wiki_link" href="/interval%20category"&amp;gt;interval categories&amp;lt;/a&amp;gt; and thus demand new distinctions. It is thus inescapably xenharmonic.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/1edo"&amp;gt;1edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/4edo"&amp;gt;4edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/6edo"&amp;gt;6edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/9edo"&amp;gt;9edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/10edo"&amp;gt;10edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/26edo"&amp;gt;26edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/63edo"&amp;gt;63edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/87edo"&amp;gt;87edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/109edo"&amp;gt;109edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:269:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-limit_compare.png/354832066/11-limit_compare.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-limit_compare.png/354832066/11-limit_compare.png" alt="11-limit_compare.png" title="11-limit_compare.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:269 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h2&amp;gt;
 Some of the simplest intervals of 11 include:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents Value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/12_11"&amp;gt;12/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;150.637&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;165.004&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;347.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;417.508&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;536.951&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.318&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648.682&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;663.049&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;782.492&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;852.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/20_11"&amp;gt;20/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1034.996&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_6"&amp;gt;11/6&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1049.363&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Brief 11-ratio composition&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Dave Hill&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3" rel="nofollow"&amp;gt;11 Limit Piano&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/andrew_heathwaite/11-limit-singtervals" rel="nofollow"&amp;gt;11-limit singtervals&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="See also"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;See also&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/12438854?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;12438854&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>4wtyp902txuw5c35neul9xpptq0w6vb</sha1>
    </revision>
    <revision>
      <id>4260</id>
      <timestamp>2013-10-17T03:32:33Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 460616054 - Original comment: another listening example found**</comment>
      <origin>4260</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16056" sha1="chd1y3yq3bjnqkf3aesiq4kb7fsrex8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2013-10-17 03:32:33 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;460616054&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;another listening example found&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

||ratio||harmonic solfege||
|| 12/11 || fu-sol ||
|| 11/10 || mi-fu ||
|| 10/9 || re-mi ||
|| 9/8 || do-re ||
|| 8/7 || ta-do ||
|| 7/6 || sol-ta ||
|| 6/5 || mi-sol, ti-re ||
|| 11/9 || re-fu ||
|| 5/4 || do-mi ||
|| 14/11 || fu-ta ||
|| 9/7 || ta-re ||
|| 4/3 || do-fa ||
|| 11/8 || do-fu ||
|| 7/5 || mi-ta ||
|| 10/7 || ta-mi ||
|| 16/11 || fu-do ||
|| 3/2 || do-sol ||
|| 14/9 || re-ta ||
|| 11/7 || ta-fu ||
|| 8/5 || mi-do ||
|| 18/11 || fu-re ||
|| 5/3 || sol-mi ||
|| 12/7 || ta-sol ||
|| 7/4 || do-ta ||
|| 16/9 || re-do ||
|| 9/5 || mi-re ||
|| 20/11 || fu-mi ||
|| 11/6 || sol-fu ||
|| 2/1 || do-do ||

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

[[image:11-limit_compare.png]]

==Intervals== 
Some of the simplest intervals of 11 include:

|| Interval || Cents Value ||
|| [[12_11|12/11]] || 150.637 ||
|| [[11_10|11/10]] || 165.004 ||
|| [[11_9|11/9]] || 347.408 ||
|| [[14_11|14/11]] || 417.508 ||
|| [[15_11|15/11]] || 536.951 ||
|| [[11_8|11/8]] || 551.318 ||
|| [[16_11|16/11]] || 648.682 ||
|| [[22_15|22/15]] || 663.049 ||
|| [[11_7|11/7]] || 782.492 ||
|| [[18_11|18/11]] || 852.592 ||
|| [[20_11|20/11]] || 1034.996 ||
|| [[11_6|11/6]] || 1049.363 ||
See: [[Gallery of Just Intervals]]

=Music= 
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] ////[[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]]//// by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Brief 11-ratio composition]] ////[[http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3|play]]//// by Dave Hill
////[[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3|11 Limit Piano]]//// by [[Chris Vaisvil]]
[[https://soundcloud.com/andrew_heathwaite/11-limit-singtervals|11-limit singtervals]] by [[Andrew Heathwaite]]

=See also= 
[[Harmonic Limit]]
[[media type="custom" key="12438854"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;ratio&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;harmonic solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-sol&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sol-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-sol, ti-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-fa&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-sol&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sol-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-sol&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sol-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
While the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, the 11-limit introduces neutral intervals, &amp;lt;a class="wiki_link" href="/superfourth"&amp;gt;superfourth&amp;lt;/a&amp;gt;s and &amp;lt;a class="wiki_link" href="/subfifth"&amp;gt;subfifth&amp;lt;/a&amp;gt;s, which fall in between major, minor and perfect &amp;lt;a class="wiki_link" href="/interval%20category"&amp;gt;interval categories&amp;lt;/a&amp;gt; and thus demand new distinctions. It is thus inescapably xenharmonic.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:269:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-limit_compare.png/354832066/11-limit_compare.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-limit_compare.png/354832066/11-limit_compare.png" alt="11-limit_compare.png" title="11-limit_compare.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:269 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h2&amp;gt;
 Some of the simplest intervals of 11 include:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents Value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/12_11"&amp;gt;12/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;150.637&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;165.004&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;347.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;417.508&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;536.951&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.318&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648.682&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;663.049&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;782.492&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;852.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/20_11"&amp;gt;20/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1034.996&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_6"&amp;gt;11/6&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1049.363&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Brief 11-ratio composition&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Dave Hill&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3" rel="nofollow"&amp;gt;11 Limit Piano&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/andrew_heathwaite/11-limit-singtervals" rel="nofollow"&amp;gt;11-limit singtervals&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="See also"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;See also&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/12438854?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;12438854&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>chd1y3yq3bjnqkf3aesiq4kb7fsrex8</sha1>
    </revision>
    <revision>
      <id>4261</id>
      <timestamp>2012-07-25T23:47:57Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenjacob</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 354855404 - Original comment: **</comment>
      <origin>4261</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15671" sha1="m0ddjz78z9c9z697lkxjl5n5tihbalj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenjacob|xenjacob]] and made on &lt;tt&gt;2012-07-25 23:47:57 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;354855404&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

||ratio||harmonic solfege||
|| 12/11 || fu-sol ||
|| 11/10 || mi-fu ||
|| 10/9 || re-mi ||
|| 9/8 || do-re ||
|| 8/7 || ta-do ||
|| 7/6 || sol-ta ||
|| 6/5 || mi-sol, ti-re ||
|| 11/9 || re-fu ||
|| 5/4 || do-mi ||
|| 14/11 || fu-ta ||
|| 9/7 || ta-re ||
|| 4/3 || do-fa ||
|| 11/8 || do-fu ||
|| 7/5 || mi-ta ||
|| 10/7 || ta-mi ||
|| 16/11 || fu-do ||
|| 3/2 || do-sol ||
|| 14/9 || re-ta ||
|| 11/7 || ta-fu ||
|| 8/5 || mi-do ||
|| 18/11 || fu-re ||
|| 5/3 || sol-mi ||
|| 12/7 || ta-sol ||
|| 7/4 || do-ta ||
|| 16/9 || re-do ||
|| 9/5 || mi-re ||
|| 20/11 || fu-mi ||
|| 11/6 || sol-fu ||
|| 2/1 || do-do ||

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

[[image:11-limit_compare.png]]

==Intervals== 
Some of the simplest intervals of 11 include:

|| Interval || Cents Value ||
|| [[12_11|12/11]] || 150.637 ||
|| [[11_10|11/10]] || 165.004 ||
|| [[11_9|11/9]] || 347.408 ||
|| [[14_11|14/11]] || 417.508 ||
|| [[15_11|15/11]] || 536.951 ||
|| [[11_8|11/8]] || 551.318 ||
|| [[16_11|16/11]] || 648.682 ||
|| [[22_15|22/15]] || 663.049 ||
|| [[11_7|11/7]] || 782.492 ||
|| [[18_11|18/11]] || 852.592 ||
|| [[20_11|20/11]] || 1034.996 ||
|| [[11_6|11/6]] || 1049.363 ||
See: [[Gallery of Just Intervals]]

=Music= 
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] ////[[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]]//// by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Brief 11-ratio composition]] ////[[http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3|play]]//// by Dave Hill
////[[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3|11 Limit Piano]]//// by [[Chris Vaisvil]]

=See also= 
[[Harmonic Limit]]
[[media type="custom" key="12438854"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;ratio&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;harmonic solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-sol&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sol-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-sol, ti-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-fa&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-sol&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;18/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sol-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ta-sol&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7/4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-ta&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;re-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9/5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;mi-re&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;20/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;fu-mi&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11/6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;sol-fu&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;do-do&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
While the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, the 11-limit introduces neutral intervals, &amp;lt;a class="wiki_link" href="/superfourth"&amp;gt;superfourth&amp;lt;/a&amp;gt;s and &amp;lt;a class="wiki_link" href="/subfifth"&amp;gt;subfifth&amp;lt;/a&amp;gt;s, which fall in between major, minor and perfect &amp;lt;a class="wiki_link" href="/interval%20category"&amp;gt;interval categories&amp;lt;/a&amp;gt; and thus demand new distinctions. It is thus inescapably xenharmonic.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:269:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-limit_compare.png/354832066/11-limit_compare.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-limit_compare.png/354832066/11-limit_compare.png" alt="11-limit_compare.png" title="11-limit_compare.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:269 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h2&amp;gt;
 Some of the simplest intervals of 11 include:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents Value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/12_11"&amp;gt;12/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;150.637&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;165.004&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;347.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;417.508&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;536.951&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.318&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648.682&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;663.049&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;782.492&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;852.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/20_11"&amp;gt;20/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1034.996&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_6"&amp;gt;11/6&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1049.363&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Brief 11-ratio composition&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Dave Hill&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3" rel="nofollow"&amp;gt;11 Limit Piano&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="See also"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;See also&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/12438854?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;12438854&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>m0ddjz78z9c9z697lkxjl5n5tihbalj</sha1>
    </revision>
    <revision>
      <id>4262</id>
      <timestamp>2012-07-25T20:52:24Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 354832096 - Original comment: **</comment>
      <origin>4262</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11261" sha1="em8u5piaem98l4bpyu4maqbqehjl75b" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2012-07-25 20:52:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;354832096&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

[[image:11-limit_compare.png]]

==Intervals== 
Some of the simplest intervals of 11 include:

|| Interval || Cents Value ||
|| [[12_11|12/11]] || 150.637 ||
|| [[11_10|11/10]] || 165.004 ||
|| [[11_9|11/9]] || 347.408 ||
|| [[14_11|14/11]] || 417.508 ||
|| [[15_11|15/11]] || 536.951 ||
|| [[11_8|11/8]] || 551.318 ||
|| [[16_11|16/11]] || 648.682 ||
|| [[22_15|22/15]] || 663.049 ||
|| [[11_7|11/7]] || 782.492 ||
|| [[18_11|18/11]] || 852.592 ||
|| [[20_11|20/11]] || 1034.996 ||
|| [[11_6|11/6]] || 1049.363 ||
See: [[Gallery of Just Intervals]]

=Music= 
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] //[[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]]// by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Brief 11-ratio composition]] //[[http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3|play]]// by Dave Hill
//[[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3|11 Limit Piano]]// by [[Chris Vaisvil]]

=See also= 
[[Harmonic Limit]]
[[media type="custom" key="12438854"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
While the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, the 11-limit introduces neutral intervals, &amp;lt;a class="wiki_link" href="/superfourth"&amp;gt;superfourth&amp;lt;/a&amp;gt;s and &amp;lt;a class="wiki_link" href="/subfifth"&amp;gt;subfifth&amp;lt;/a&amp;gt;s, which fall in between major, minor and perfect &amp;lt;a class="wiki_link" href="/interval%20category"&amp;gt;interval categories&amp;lt;/a&amp;gt; and thus demand new distinctions. It is thus inescapably xenharmonic.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:87:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-limit_compare.png/354832066/11-limit_compare.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-limit_compare.png/354832066/11-limit_compare.png" alt="11-limit_compare.png" title="11-limit_compare.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:87 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h2&amp;gt;
 Some of the simplest intervals of 11 include:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents Value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/12_11"&amp;gt;12/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;150.637&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;165.004&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;347.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;417.508&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;536.951&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.318&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648.682&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;663.049&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;782.492&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;852.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/20_11"&amp;gt;20/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1034.996&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_6"&amp;gt;11/6&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1049.363&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Brief 11-ratio composition&amp;lt;/a&amp;gt; &amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by Dave Hill&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3" rel="nofollow"&amp;gt;11 Limit Piano&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="See also"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;See also&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/12438854?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;12438854&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>em8u5piaem98l4bpyu4maqbqehjl75b</sha1>
    </revision>
    <revision>
      <id>4263</id>
      <timestamp>2012-02-10T20:02:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 300685118 - Original comment: **</comment>
      <origin>4263</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10713" sha1="7i4rc8wof8mugwjvpuk3ae2rh5409zy" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2012-02-10 20:02:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;300685118&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

==Intervals== 
Some of the simplest intervals of 11 include:

|| Interval || Cents Value ||
|| [[12_11|12/11]] || 150.637 ||
|| [[11_10|11/10]] || 165.004 ||
|| [[11_9|11/9]] || 347.408 ||
|| [[14_11|14/11]] || 417.508 ||
|| [[15_11|15/11]] || 536.951 ||
|| [[11_8|11/8]] || 551.318 ||
|| [[16_11|16/11]] || 648.682 ||
|| [[22_15|22/15]] || 663.049 ||
|| [[11_7|11/7]] || 782.492 ||
|| [[18_11|18/11]] || 852.592 ||
|| [[20_11|20/11]] || 1034.996 ||
|| [[11_6|11/6]] || 1049.363 ||
See: [[Gallery of Just Intervals]]

=Music= 
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] [[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]] by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Brief 11-ratio composition]] [[http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3|play]] by Dave Hill
[[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3|11 Limit Piano]] by [[Chris Vaisvil]]

=See also= 
[[Harmonic Limit]]
[[media type="custom" key="12438854"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
While the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, the 11-limit introduces neutral intervals, &amp;lt;a class="wiki_link" href="/superfourth"&amp;gt;superfourth&amp;lt;/a&amp;gt;s and &amp;lt;a class="wiki_link" href="/subfifth"&amp;gt;subfifth&amp;lt;/a&amp;gt;s, which fall in between major, minor and perfect &amp;lt;a class="wiki_link" href="/interval%20category"&amp;gt;interval categories&amp;lt;/a&amp;gt; and thus demand new distinctions. It is thus inescapably xenharmonic.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Intervals&amp;lt;/h2&amp;gt;
 Some of the simplest intervals of 11 include:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents Value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/12_11"&amp;gt;12/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;150.637&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;165.004&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;347.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;417.508&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;536.951&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.318&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648.682&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;663.049&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;782.492&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;852.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/20_11"&amp;gt;20/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1034.996&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_6"&amp;gt;11/6&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1049.363&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Brief 11-ratio composition&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Dave Hill&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3" rel="nofollow"&amp;gt;11 Limit Piano&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="See also"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;See also&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/12438854?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;12438854&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>7i4rc8wof8mugwjvpuk3ae2rh5409zy</sha1>
    </revision>
    <revision>
      <id>4264</id>
      <timestamp>2011-10-10T12:56:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 263287012 - Original comment: **</comment>
      <origin>4264</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9808" sha1="qqgg0ctlckjns8q63r5jds4j1g2n83x" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-10-10 12:56:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;263287012&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

==Intervals== 
Some of the simplest intervals of 11 include:

|| Interval || Cents Value ||
|| [[12_11|12/11]] || 150.637 ||
|| [[11_10|11/10]] || 165.004 ||
|| [[11_9|11/9]] || 347.408 ||
|| [[14_11|14/11]] || 417.508 ||
|| [[15_11|15/11]] || 536.951 ||
|| [[11_8|11/8]] || 551.318 ||
|| [[16_11|16/11]] || 648.682 ||
|| [[22_15|22/15]] || 663.049 ||
|| [[11_7|11/7]] || 782.492 ||
|| [[18_11|18/11]] || 852.592 ||
|| [[20_11|20/11]] || 1034.996 ||
|| [[11_6|11/6]] || 1049.363 ||
See: [[Gallery of Just Intervals]]

=Music= 
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] [[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]] by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Brief 11-ratio composition]] [[http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3|play]] by Dave Hill

=See also= 
[[Harmonic Limit]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
While the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, the 11-limit introduces neutral intervals, &amp;lt;a class="wiki_link" href="/superfourth"&amp;gt;superfourth&amp;lt;/a&amp;gt;s and &amp;lt;a class="wiki_link" href="/subfifth"&amp;gt;subfifth&amp;lt;/a&amp;gt;s, which fall in between major, minor and perfect &amp;lt;a class="wiki_link" href="/interval%20category"&amp;gt;interval categories&amp;lt;/a&amp;gt; and thus demand new distinctions. It is thus inescapably xenharmonic.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Intervals&amp;lt;/h2&amp;gt;
 Some of the simplest intervals of 11 include:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents Value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/12_11"&amp;gt;12/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;150.637&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;165.004&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;347.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;417.508&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;536.951&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.318&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648.682&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;663.049&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;782.492&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;852.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/20_11"&amp;gt;20/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1034.996&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_6"&amp;gt;11/6&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1049.363&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Brief 11-ratio composition&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Dave Hill&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="See also"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;See also&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>qqgg0ctlckjns8q63r5jds4j1g2n83x</sha1>
    </revision>
    <revision>
      <id>4265</id>
      <timestamp>2011-10-10T10:58:22Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 263241198 - Original comment: **</comment>
      <origin>4265</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9818" sha1="m7zm9a586rit6tc958tk6jvu4to88j3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-10-10 10:58:22 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;263241198&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourths]] and [[subfifths]], which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

==Intervals== 
Some of the simplest intervals of 11 include:

|| Interval || Cents Value ||
|| [[12_11|12/11]] || 150.637 ||
|| [[11_10|11/10]] || 165.004 ||
|| [[11_9|11/9]] || 347.408 ||
|| [[14_11|14/11]] || 417.508 ||
|| [[15_11|15/11]] || 536.951 ||
|| [[11_8|11/8]] || 551.318 ||
|| [[16_11|16/11]] || 648.682 ||
|| [[22_15|22/15]] || 663.049 ||
|| [[11_7|11/7]] || 782.492 ||
|| [[18_11|18/11]] || 852.592 ||
|| [[20_11|20/11]] || 1034.996 ||
|| [[11_6|11/6]] || 1049.363 ||
See: [[Gallery of Just Intervals]]

=Music= 
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] [[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]] by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Brief 11-ratio composition]] [[http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3|play]] by Dave Hill

=See also= 
[[Harmonic Limit]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
While the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, the 11-limit introduces neutral intervals, &amp;lt;a class="wiki_link" href="/superfourths"&amp;gt;superfourths&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/subfifths"&amp;gt;subfifths&amp;lt;/a&amp;gt;, which fall in between major, minor and perfect &amp;lt;a class="wiki_link" href="/interval%20category"&amp;gt;interval categories&amp;lt;/a&amp;gt; and thus demand new distinctions. It is thus inescapably xenharmonic.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Intervals&amp;lt;/h2&amp;gt;
 Some of the simplest intervals of 11 include:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents Value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/12_11"&amp;gt;12/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;150.637&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;165.004&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;347.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;417.508&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;536.951&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.318&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648.682&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;663.049&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;782.492&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;852.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/20_11"&amp;gt;20/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1034.996&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_6"&amp;gt;11/6&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1049.363&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Brief 11-ratio composition&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Dave Hill&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="See also"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;See also&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>m7zm9a586rit6tc958tk6jvu4to88j3</sha1>
    </revision>
    <revision>
      <id>4266</id>
      <timestamp>2011-10-09T12:15:26Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 262980320 - Original comment: **</comment>
      <origin>4266</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9486" sha1="q8orob6e7dku6w3ktuhidc1lx3iykbl" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-10-09 12:15:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;262980320&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval classes of [[12edo]], the 11-limit introduces neutral intervals, which fall in between major and minor and thus demand new categories. It is thus inescapably xenharmonic.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

==Intervals== 
Some of the simplest intervals of 11 include:

|| Interval || Cents Value ||
|| [[12_11|12/11]] || 150.637 ||
|| [[11_10|11/10]] || 165.004 ||
|| [[11_9|11/9]] || 347.408 ||
|| [[14_11|14/11]] || 417.508 ||
|| [[15_11|15/11]] || 536.951 ||
|| [[11_8|11/8]] || 551.318 ||
|| [[16_11|16/11]] || 648.682 ||
|| [[22_15|22/15]] || 663.049 ||
|| [[11_7|11/7]] || 782.492 ||
|| [[18_11|18/11]] || 852.592 ||
|| [[20_11|20/11]] || 1034.996 ||
|| [[11_6|11/6]] || 1049.363 ||
See: [[Gallery of Just Intervals]]

=Music= 
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] [[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]] by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Brief 11-ratio composition]] [[http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3|play]] by Dave Hill

=See also= 
[[Harmonic Limit]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
While the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval classes of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, the 11-limit introduces neutral intervals, which fall in between major and minor and thus demand new categories. It is thus inescapably xenharmonic.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Intervals&amp;lt;/h2&amp;gt;
 Some of the simplest intervals of 11 include:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Cents Value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/12_11"&amp;gt;12/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;150.637&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;165.004&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;347.408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;417.508&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;536.951&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.318&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;648.682&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;663.049&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;782.492&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;852.592&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/20_11"&amp;gt;20/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1034.996&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/11_6"&amp;gt;11/6&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1049.363&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Brief 11-ratio composition&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Dave Hill&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="See also"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;See also&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>q8orob6e7dku6w3ktuhidc1lx3iykbl</sha1>
    </revision>
    <revision>
      <id>4267</id>
      <timestamp>2011-09-14T20:37:27Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 254162614 - Original comment: **</comment>
      <origin>4267</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6236" sha1="7tvb1llbk9v4r46ba2opdybqfrd9gvb" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-09-14 20:37:27 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;254162614&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval classes of [[12edo]], the 11-limit introduces neutral intervals, which fall in between major and minor and thus demand new categories. It is thus inescapably xenharmonic.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

=Music= 
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] [[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]] by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Brief 11-ratio composition]] [[http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3|play]] by Dave Hill

=See also= 
[[Harmonic Limit]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
While the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval classes of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, the 11-limit introduces neutral intervals, which fall in between major and minor and thus demand new categories. It is thus inescapably xenharmonic.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Brief 11-ratio composition&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Dave Hill&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="See also"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;See also&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>7tvb1llbk9v4r46ba2opdybqfrd9gvb</sha1>
    </revision>
    <revision>
      <id>4268</id>
      <timestamp>2011-07-01T11:00:27Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 239656049 - Original comment: **</comment>
      <origin>4268</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5281" sha1="4j1ryaliwajl1lgoaqj9nxsfbsg0bvf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-07-01 11:00:27 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239656049&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

=Music=
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] [[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]] by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Brief 11-ratio composition]] [[http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3|play]] by Dave Hill

=See also=
[[Harmonic Limit]]
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Brief 11-ratio composition&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Dave Hill&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="See also"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;See also&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>4j1ryaliwajl1lgoaqj9nxsfbsg0bvf</sha1>
    </revision>
    <revision>
      <id>4269</id>
      <timestamp>2011-06-14T16:44:11Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 236663426 - Original comment: **</comment>
      <origin>4269</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5109" sha1="15cx4qblwbp6e5cg04cwo031w8whuyy" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-14 16:44:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;236663426&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

See [[Harmonic Limit]].

=Music=
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] [[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]] by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Brief 11-ratio composition]] [[http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3|play]] by Dave Hill&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Brief 11-ratio composition&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Dave Hill&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>15cx4qblwbp6e5cg04cwo031w8whuyy</sha1>
    </revision>
    <revision>
      <id>4270</id>
      <timestamp>2011-06-14T16:40:27Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 236662552 - Original comment: **</comment>
      <origin>4270</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4590" sha1="818z314n7fjqdk7fgywpv5i3qtdrkbt" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-14 16:40:27 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;236662552&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

See [[Harmonic Limit]].

=Music=
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] [[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]] by [[Dave Hill]]
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>818z314n7fjqdk7fgywpv5i3qtdrkbt</sha1>
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    <revision>
      <id>4271</id>
      <timestamp>2011-06-14T16:36:30Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 236661682 - Original comment: **</comment>
      <origin>4271</origin>
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      <text bytes="5175" sha1="nboaqyl8s42whw0yghz8v0eo4gd05xi" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-14 16:36:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;236661682&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

See [[Harmonic Limit]].

=Music=
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] [[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]] by [[Dave Hill]]
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Chord Progression on the Harmonic Overtone Series]] [[http://sonic-arts.org/hill/10-passages-ji/06_hill_chord-progression-on-harmonic-series.mp3|play]] by Dave Hill&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Chord Progression on the Harmonic Overtone Series&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/06_hill_chord-progression-on-harmonic-series.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Dave Hill&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>nboaqyl8s42whw0yghz8v0eo4gd05xi</sha1>
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    <revision>
      <id>4272</id>
      <timestamp>2011-06-14T16:33:46Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 236661058 - Original comment: **</comment>
      <origin>4272</origin>
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      <format>text/x-wiki</format>
      <text bytes="4590" sha1="p78d2c7kqnq1f511g7it66uejey10ib" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-14 16:33:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;236661058&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

See [[Harmonic Limit]].

=Music=
[[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm|Study #3]] [[http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3|play]] by [[Dave Hill]]
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm" rel="nofollow"&amp;gt;Study #3&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Dave%20Hill"&amp;gt;Dave Hill&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>p78d2c7kqnq1f511g7it66uejey10ib</sha1>
    </revision>
    <revision>
      <id>4273</id>
      <timestamp>2011-05-27T12:41:23Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 232431954 - Original comment: **</comment>
      <origin>4273</origin>
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      <text bytes="3885" sha1="a9i2b4hnhufsg0ahpp478on9n8c5tsq" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-05-27 12:41:23 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;232431954&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

See [[Harmonic Limit]].
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/JustIntonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4274</id>
      <timestamp>2011-05-27T12:38:53Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 232431280 - Original comment: **</comment>
      <origin>4274</origin>
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      <text bytes="3889" sha1="6ucu91nq93oubpi2gqvwobyxprxaeyb" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-05-27 12:38:53 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;232431280&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //11-limit// consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]] and [[99_98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.

Relative to their size, [[edo]]s which do (relatively) well in supporting 11-limit intervals are: [[12edo]], [[15edo]], [[22edo]], [[31edo]], [[41edo]], [[46edo]], [[58edo]], [[72edo]], [[118edo]], [[130edo]] and [[152edo]].

See [[Harmonic Limit]].
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;11-limit&amp;lt;/em&amp;gt; consists of all &amp;lt;a class="wiki_link" href="/Just%20intonation"&amp;gt;justly tuned&amp;lt;/a&amp;gt; intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Relative to their size, &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s which do (relatively) well in supporting 11-limit intervals are: &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/46edo"&amp;gt;46edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/118edo"&amp;gt;118edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>6ucu91nq93oubpi2gqvwobyxprxaeyb</sha1>
    </revision>
    <revision>
      <id>4275</id>
      <timestamp>2011-03-31T03:25:06Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 215737622 - Original comment: **</comment>
      <origin>4275</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2046" sha1="hl63imsyddsv8768xrcvvrhcadzgshn" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-03-31 03:25:06 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;215737622&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;Some examples for 11-limit intervalls are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]], [[99_98|99/98]]

Some examples [[edo]]s with good support of 11-limit intervals are: [[15edo]], tbc.

For a definition see [[Harmonic Limit]]
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;Some examples for 11-limit intervalls are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Some examples &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s with good support of 11-limit intervals are: &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, tbc.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For a definition see &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>hl63imsyddsv8768xrcvvrhcadzgshn</sha1>
    </revision>
    <revision>
      <id>4276</id>
      <timestamp>2011-03-31T03:23:29Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 215737390 - Original comment: **</comment>
      <origin>4276</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1977" sha1="3ewb62dftym8pc2ufgor7w16sbd34k7" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-03-31 03:23:29 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;215737390&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;Some examples for 11-limit intervalls are [[14_11|14/11]], [[11_8|11/8]], [[27_22|27/22]], [[99_98|99/98]]

Some examples [[edo]]s with good support of 11-limit intervals are: todo

For a definition see [[Harmonic Limit]]
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11-limit&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;Some examples for 11-limit intervalls are &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Some examples &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;s with good support of 11-limit intervals are: todo&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For a definition see &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;Harmonic Limit&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>3ewb62dftym8pc2ufgor7w16sbd34k7</sha1>
    </revision>
    <revision>
      <id>38240</id>
      <parentid>54</parentid>
      <timestamp>2018-10-31T22:39:27Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>added color names</comment>
      <origin>38240</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4424" sha1="o6h48q0ycc4k0e1mmmzri3vtqp2xksw" xml:space="preserve">The ''11-limit'' consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11|14/11]], [[11/8|11/8]], [[27/22|27/22]] and [[99/98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

{| class="wikitable"
|-
| colspan="2" |[[Kite's color notation|interval name]]
| | ratio
| | harmonic solfege
|-
|1u2
|lu 2nd
| | 12/11
| | fu-sol
|-
|1og2
|logu 2nd
| | 11/10
| | mi-fu
|-
|y2
|yo 2nd
| | 10/9
| | re-mi
|-
|w2
|wa 2nd
| | 9/8
| | do-re
|-
|r2
|ru 2nd
| | 8/7
| | ta-do
|-
|z3
|zo 3rd
| | 7/6
| | sol-ta
|-
|g3
|gu 3rd
| | 6/5
| | mi-sol, ti-re
|-
|1o3
|ilo 3rd
| | 11/9
| | re-fu
|-
|y3
|yo 3rd
| | 5/4
| | do-mi
|-
|1uz4
|luzo 4th
| | 14/11
| | fu-ta
|-
|r3
|ru 3rd
| | 9/7
| | ta-re
|-
|w4
|wa 4th
| | 4/3
| | do-fa
|-
|1o4
|ilo 4th
| | 11/8
| | do-fu
|-
|zg5
|zogu 5th
| | 7/5
| | mi-ta
|-
|ry4
|ruyo 4th
| | 10/7
| | ta-mi
|-
|1u5
|lu 5th
| | 16/11
| | fu-do
|-
|w5
|wa 5th
| | 3/2
| | do-sol
|-
|z6
|zo 6th
| | 14/9
| | re-ta
|-
|1or5
|loru 5th
| | 11/7
| | ta-fu
|-
|g6
|gu 6th
| | 8/5
| | mi-do
|-
|1u6
|lu 6th
| | 18/11
| | fu-re
|-
|y6
|yo 6th
| | 5/3
| | sol-mi
|-
|r6
|ru 6th
| | 12/7
| | ta-sol
|-
|z7
|zo 7th
| | 7/4
| | do-ta
|-
|w7
|wa 7th
| | 16/9
| | re-do
|-
|g7
|gu 7th
| | 9/5
| | mi-re
|-
|1uy7
|luyo 7th
| | 20/11
| | fu-mi
|-
|1o7
|ilo 7th
| | 11/6
| | sol-fu
|-
|w8
|wa 8ve
| | 2/1
| | do-do
|}

While the [[7-limit|7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo|12edo]], the 11-limit introduces neutral intervals, [[Superfourth|superfourth]]s and [[Subfifth|subfifth]]s, which fall in between major, minor and perfect [[interval_category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[EDO|edo]]s which do (relatively) well in supporting 11-limit intervals are: [[1edo|1edo]], [[2edo|2edo]], [[3edo|3edo]], [[4edo|4edo]], [[5edo|5edo]], [[6edo|6edo]], [[7edo|7edo]], [[9edo|9edo]], [[10edo|10edo]], [[12edo|12edo]], [[15edo|15edo]], [[22edo|22edo]], [[26edo|26edo]], [[31edo|31edo]], [[41edo|41edo]], [[63edo|63edo]], [[72edo|72edo]], [[87edo|87edo]], [[109edo|109edo]], [[161edo|161edo]].

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

==Intervals==
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable"
|-
| colspan="2" |[[Kite's color notation|Interval name]]
| | Ratio
| | Cents Value
|-
|1u2
|lu 2nd
| | [[12/11|12/11]]
| | 150.637
|-
|1og2
|logu 2nd
| | [[11/10|11/10]]
| | 165.004
|-
|1o3
|ilo 3rd
| | [[11/9|11/9]]
| | 347.408
|-
|1uz4
|lu 4th
| | [[14/11|14/11]]
| | 417.508
|-
|1uy4
|luyo 4th
| | [[15/11|15/11]]
| | 536.951
|-
|1o4
|ilo 4th
| | [[11/8|11/8]]
| | 551.318
|-
|1u5
|lu 5th
| | [[16/11|16/11]]
| | 648.682
|-
|1og5
|logu 5th
| | [[22/15|22/15]]
| | 663.049
|-
|1or5
|loru 5th
| | [[11/7|11/7]]
| | 782.492
|-
|1u6
|lu 6th
| | [[18/11|18/11]]
| | 852.592
|-
|1uy7
|luyo 7th
| | [[20/11|20/11]]
| | 1034.996
|-
|1o7
|ilo 7th
| | [[11/6|11/6]]
| | 1049.363
|}
See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]

=Music=
[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave_Hill|Dave Hill]]

[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill

[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris_Vaisvil|Chris Vaisvil]]

[https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew_Heathwaite|Andrew Heathwaite]]

=See also=
[[Harmonic_Limit|Harmonic Limit]]

[[Category:11-limit]]
[[Category:example]]
[[Category:interval]]
[[Category:limit]]
[[Category:listen]]
[[Category:prime_limit]]</text>
      <sha1>o6h48q0ycc4k0e1mmmzri3vtqp2xksw</sha1>
    </revision>
    <revision>
      <id>38247</id>
      <parentid>38240</parentid>
      <timestamp>2018-10-31T23:04:10Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <origin>38247</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4424" sha1="e1nyx95bipwqgz2mpijf0x1np15cm3q" xml:space="preserve">The ''11-limit'' consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11|14/11]], [[11/8|11/8]], [[27/22|27/22]] and [[99/98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

{| class="wikitable"
|-
! colspan="2" |[[Kite's color notation|interval name]]
! | ratio
! | harmonic solfege
|-
|1u2
|lu 2nd
| | 12/11
| | fu-sol
|-
|1og2
|logu 2nd
| | 11/10
| | mi-fu
|-
|y2
|yo 2nd
| | 10/9
| | re-mi
|-
|w2
|wa 2nd
| | 9/8
| | do-re
|-
|r2
|ru 2nd
| | 8/7
| | ta-do
|-
|z3
|zo 3rd
| | 7/6
| | sol-ta
|-
|g3
|gu 3rd
| | 6/5
| | mi-sol, ti-re
|-
|1o3
|ilo 3rd
| | 11/9
| | re-fu
|-
|y3
|yo 3rd
| | 5/4
| | do-mi
|-
|1uz4
|luzo 4th
| | 14/11
| | fu-ta
|-
|r3
|ru 3rd
| | 9/7
| | ta-re
|-
|w4
|wa 4th
| | 4/3
| | do-fa
|-
|1o4
|ilo 4th
| | 11/8
| | do-fu
|-
|zg5
|zogu 5th
| | 7/5
| | mi-ta
|-
|ry4
|ruyo 4th
| | 10/7
| | ta-mi
|-
|1u5
|lu 5th
| | 16/11
| | fu-do
|-
|w5
|wa 5th
| | 3/2
| | do-sol
|-
|z6
|zo 6th
| | 14/9
| | re-ta
|-
|1or5
|loru 5th
| | 11/7
| | ta-fu
|-
|g6
|gu 6th
| | 8/5
| | mi-do
|-
|1u6
|lu 6th
| | 18/11
| | fu-re
|-
|y6
|yo 6th
| | 5/3
| | sol-mi
|-
|r6
|ru 6th
| | 12/7
| | ta-sol
|-
|z7
|zo 7th
| | 7/4
| | do-ta
|-
|w7
|wa 7th
| | 16/9
| | re-do
|-
|g7
|gu 7th
| | 9/5
| | mi-re
|-
|1uy7
|luyo 7th
| | 20/11
| | fu-mi
|-
|1o7
|ilo 7th
| | 11/6
| | sol-fu
|-
|w8
|wa 8ve
| | 2/1
| | do-do
|}

While the [[7-limit|7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo|12edo]], the 11-limit introduces neutral intervals, [[Superfourth|superfourth]]s and [[Subfifth|subfifth]]s, which fall in between major, minor and perfect [[interval_category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[EDO|edo]]s which do (relatively) well in supporting 11-limit intervals are: [[1edo|1edo]], [[2edo|2edo]], [[3edo|3edo]], [[4edo|4edo]], [[5edo|5edo]], [[6edo|6edo]], [[7edo|7edo]], [[9edo|9edo]], [[10edo|10edo]], [[12edo|12edo]], [[15edo|15edo]], [[22edo|22edo]], [[26edo|26edo]], [[31edo|31edo]], [[41edo|41edo]], [[63edo|63edo]], [[72edo|72edo]], [[87edo|87edo]], [[109edo|109edo]], [[161edo|161edo]].

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

==Intervals==
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable"
|-
| colspan="2" |[[Kite's color notation|Interval name]]
| | Ratio
| | Cents Value
|-
|1u2
|lu 2nd
| | [[12/11|12/11]]
| | 150.637
|-
|1og2
|logu 2nd
| | [[11/10|11/10]]
| | 165.004
|-
|1o3
|ilo 3rd
| | [[11/9|11/9]]
| | 347.408
|-
|1uz4
|lu 4th
| | [[14/11|14/11]]
| | 417.508
|-
|1uy4
|luyo 4th
| | [[15/11|15/11]]
| | 536.951
|-
|1o4
|ilo 4th
| | [[11/8|11/8]]
| | 551.318
|-
|1u5
|lu 5th
| | [[16/11|16/11]]
| | 648.682
|-
|1og5
|logu 5th
| | [[22/15|22/15]]
| | 663.049
|-
|1or5
|loru 5th
| | [[11/7|11/7]]
| | 782.492
|-
|1u6
|lu 6th
| | [[18/11|18/11]]
| | 852.592
|-
|1uy7
|luyo 7th
| | [[20/11|20/11]]
| | 1034.996
|-
|1o7
|ilo 7th
| | [[11/6|11/6]]
| | 1049.363
|}
See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]

=Music=
[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave_Hill|Dave Hill]]

[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill

[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris_Vaisvil|Chris Vaisvil]]

[https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew_Heathwaite|Andrew Heathwaite]]

=See also=
[[Harmonic_Limit|Harmonic Limit]]

[[Category:11-limit]]
[[Category:example]]
[[Category:interval]]
[[Category:limit]]
[[Category:listen]]
[[Category:prime_limit]]</text>
      <sha1>e1nyx95bipwqgz2mpijf0x1np15cm3q</sha1>
    </revision>
    <revision>
      <id>38344</id>
      <parentid>38247</parentid>
      <timestamp>2018-11-02T22:15:48Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <origin>38344</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4424" sha1="krdi5alllgide0g74wr7pwx1fx0iof4" xml:space="preserve">The ''11-limit'' consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11|14/11]], [[11/8|11/8]], [[27/22|27/22]] and [[99/98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

{| class="wikitable"
|-
! colspan="2" |[[Kite's color notation|interval name]]
! | ratio
! | harmonic solfege
|-
|1u2
|lu 2nd
| | 12/11
| | fu-sol
|-
|1og2
|logu 2nd
| | 11/10
| | mi-fu
|-
|y2
|yo 2nd
| | 10/9
| | re-mi
|-
|w2
|wa 2nd
| | 9/8
| | do-re
|-
|r2
|ru 2nd
| | 8/7
| | ta-do
|-
|z3
|zo 3rd
| | 7/6
| | sol-ta
|-
|g3
|gu 3rd
| | 6/5
| | mi-sol, ti-re
|-
|1o3
|ilo 3rd
| | 11/9
| | re-fu
|-
|y3
|yo 3rd
| | 5/4
| | do-mi
|-
|1uz4
|luzo 4th
| | 14/11
| | fu-ta
|-
|r3
|ru 3rd
| | 9/7
| | ta-re
|-
|w4
|wa 4th
| | 4/3
| | do-fa
|-
|1o4
|ilo 4th
| | 11/8
| | do-fu
|-
|zg5
|zogu 5th
| | 7/5
| | mi-ta
|-
|ry4
|ruyo 4th
| | 10/7
| | ta-mi
|-
|1u5
|lu 5th
| | 16/11
| | fu-do
|-
|w5
|wa 5th
| | 3/2
| | do-sol
|-
|z6
|zo 6th
| | 14/9
| | re-ta
|-
|1or5
|loru 5th
| | 11/7
| | ta-fu
|-
|g6
|gu 6th
| | 8/5
| | mi-do
|-
|1u6
|lu 6th
| | 18/11
| | fu-re
|-
|y6
|yo 6th
| | 5/3
| | sol-mi
|-
|r6
|ru 6th
| | 12/7
| | ta-sol
|-
|z7
|zo 7th
| | 7/4
| | do-ta
|-
|w7
|wa 7th
| | 16/9
| | re-do
|-
|g7
|gu 7th
| | 9/5
| | mi-re
|-
|1uy7
|luyo 7th
| | 20/11
| | fu-mi
|-
|1o7
|ilo 7th
| | 11/6
| | sol-fu
|-
|w8
|wa 8ve
| | 2/1
| | do-do
|}

While the [[7-limit|7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo|12edo]], the 11-limit introduces neutral intervals, [[Superfourth|superfourth]]s and [[Subfifth|subfifth]]s, which fall in between major, minor and perfect [[interval_category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[EDO|edo]]s which do (relatively) well in supporting 11-limit intervals are: [[1edo|1edo]], [[2edo|2edo]], [[3edo|3edo]], [[4edo|4edo]], [[5edo|5edo]], [[6edo|6edo]], [[7edo|7edo]], [[9edo|9edo]], [[10edo|10edo]], [[12edo|12edo]], [[15edo|15edo]], [[22edo|22edo]], [[26edo|26edo]], [[31edo|31edo]], [[41edo|41edo]], [[63edo|63edo]], [[72edo|72edo]], [[87edo|87edo]], [[109edo|109edo]], [[161edo|161edo]].

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

==Intervals==
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable"
|-
! colspan="2" |[[Kite's color notation|Interval name]]
! | Ratio
! | Cents Value
|-
|1u2
|lu 2nd
| | [[12/11|12/11]]
| | 150.637
|-
|1og2
|logu 2nd
| | [[11/10|11/10]]
| | 165.004
|-
|1o3
|ilo 3rd
| | [[11/9|11/9]]
| | 347.408
|-
|1uz4
|lu 4th
| | [[14/11|14/11]]
| | 417.508
|-
|1uy4
|luyo 4th
| | [[15/11|15/11]]
| | 536.951
|-
|1o4
|ilo 4th
| | [[11/8|11/8]]
| | 551.318
|-
|1u5
|lu 5th
| | [[16/11|16/11]]
| | 648.682
|-
|1og5
|logu 5th
| | [[22/15|22/15]]
| | 663.049
|-
|1or5
|loru 5th
| | [[11/7|11/7]]
| | 782.492
|-
|1u6
|lu 6th
| | [[18/11|18/11]]
| | 852.592
|-
|1uy7
|luyo 7th
| | [[20/11|20/11]]
| | 1034.996
|-
|1o7
|ilo 7th
| | [[11/6|11/6]]
| | 1049.363
|}
See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]

=Music=
[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave_Hill|Dave Hill]]

[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill

[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris_Vaisvil|Chris Vaisvil]]

[https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew_Heathwaite|Andrew Heathwaite]]

=See also=
[[Harmonic_Limit|Harmonic Limit]]

[[Category:11-limit]]
[[Category:example]]
[[Category:interval]]
[[Category:limit]]
[[Category:listen]]
[[Category:prime_limit]]</text>
      <sha1>krdi5alllgide0g74wr7pwx1fx0iof4</sha1>
    </revision>
    <revision>
      <id>38345</id>
      <parentid>38344</parentid>
      <timestamp>2018-11-02T22:20:05Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>changed the order of the columns in the 1st table</comment>
      <origin>38345</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4424" sha1="5cmtbxi8kfaehru1cs6uankil1jmo1t" xml:space="preserve">The ''11-limit'' consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11|14/11]], [[11/8|11/8]], [[27/22|27/22]] and [[99/98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

{| class="wikitable"
|-
! | Ratio
! colspan="2" |[[Kite's color notation|interval name]]
! | harmonic solfege
|-
| | 12/11
|1u2
|lu 2nd
| | fu-sol
|-
| | 11/10
|1og2
|logu 2nd
| | mi-fu
|-
| | 10/9
|y2
|yo 2nd
| | re-mi
|-
| | 9/8
|w2
|wa 2nd
| | do-re
|-
| | 8/7
|r2
|ru 2nd
| | ta-do
|-
| | 7/6
|z3
|zo 3rd
| | sol-ta
|-
| | 6/5
|g3
|gu 3rd
| | mi-sol, ti-re
|-
| | 11/9
|1o3
|ilo 3rd
| | re-fu
|-
| | 5/4
|y3
|yo 3rd
| | do-mi
|-
| | 14/11
|1uz4
|luzo 4th
| | fu-ta
|-
| | 9/7
|r3
|ru 3rd
| | ta-re
|-
| | 4/3
|w4
|wa 4th
| | do-fa
|-
| | 11/8
|1o4
|ilo 4th
| | do-fu
|-
| | 7/5
|zg5
|zogu 5th
| | mi-ta
|-
| | 10/7
|ry4
|ruyo 4th
| | ta-mi
|-
| | 16/11
|1u5
|lu 5th
| | fu-do
|-
| | 3/2
|w5
|wa 5th
| | do-sol
|-
| | 14/9
|z6
|zo 6th
| | re-ta
|-
| | 11/7
|1or5
|loru 5th
| | ta-fu
|-
| | 8/5
|g6
|gu 6th
| | mi-do
|-
| | 18/11
|1u6
|lu 6th
| | fu-re
|-
| | 5/3
|y6
|yo 6th
| | sol-mi
|-
| | 12/7
|r6
|ru 6th
| | ta-sol
|-
| | 7/4
|z7
|zo 7th
| | do-ta
|-
| | 16/9
|w7
|wa 7th
| | re-do
|-
| | 9/5
|g7
|gu 7th
| | mi-re
|-
| | 20/11
|1uy7
|luyo 7th
| | fu-mi
|-
| | 11/6
|1o7
|ilo 7th
| | sol-fu
|-
| | 2/1
|w8
|wa 8ve
| | do-do
|}

While the [[7-limit|7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo|12edo]], the 11-limit introduces neutral intervals, [[Superfourth|superfourth]]s and [[Subfifth|subfifth]]s, which fall in between major, minor and perfect [[interval_category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[EDO|edo]]s which do (relatively) well in supporting 11-limit intervals are: [[1edo|1edo]], [[2edo|2edo]], [[3edo|3edo]], [[4edo|4edo]], [[5edo|5edo]], [[6edo|6edo]], [[7edo|7edo]], [[9edo|9edo]], [[10edo|10edo]], [[12edo|12edo]], [[15edo|15edo]], [[22edo|22edo]], [[26edo|26edo]], [[31edo|31edo]], [[41edo|41edo]], [[63edo|63edo]], [[72edo|72edo]], [[87edo|87edo]], [[109edo|109edo]], [[161edo|161edo]].

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

==Intervals==
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable"
|-
! colspan="2" |[[Kite's color notation|Interval name]]
! | Ratio
! | Cents Value
|-
|1u2
|lu 2nd
| | [[12/11|12/11]]
| | 150.637
|-
|1og2
|logu 2nd
| | [[11/10|11/10]]
| | 165.004
|-
|1o3
|ilo 3rd
| | [[11/9|11/9]]
| | 347.408
|-
|1uz4
|lu 4th
| | [[14/11|14/11]]
| | 417.508
|-
|1uy4
|luyo 4th
| | [[15/11|15/11]]
| | 536.951
|-
|1o4
|ilo 4th
| | [[11/8|11/8]]
| | 551.318
|-
|1u5
|lu 5th
| | [[16/11|16/11]]
| | 648.682
|-
|1og5
|logu 5th
| | [[22/15|22/15]]
| | 663.049
|-
|1or5
|loru 5th
| | [[11/7|11/7]]
| | 782.492
|-
|1u6
|lu 6th
| | [[18/11|18/11]]
| | 852.592
|-
|1uy7
|luyo 7th
| | [[20/11|20/11]]
| | 1034.996
|-
|1o7
|ilo 7th
| | [[11/6|11/6]]
| | 1049.363
|}
See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]

=Music=
[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave_Hill|Dave Hill]]

[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill

[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris_Vaisvil|Chris Vaisvil]]

[https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew_Heathwaite|Andrew Heathwaite]]

=See also=
[[Harmonic_Limit|Harmonic Limit]]

[[Category:11-limit]]
[[Category:example]]
[[Category:interval]]
[[Category:limit]]
[[Category:listen]]
[[Category:prime_limit]]</text>
      <sha1>5cmtbxi8kfaehru1cs6uankil1jmo1t</sha1>
    </revision>
    <revision>
      <id>38346</id>
      <parentid>38345</parentid>
      <timestamp>2018-11-03T03:32:25Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <origin>38346</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4416" sha1="f90h8l4uq35uda9woqq1bqar95eivjy" xml:space="preserve">The ''11-limit'' consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11|14/11]], [[11/8|11/8]], [[27/22|27/22]] and [[99/98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

{| class="wikitable"
|-
! | Ratio
! colspan="2" |[[Kite's color notation|interval name]]
! | harmonic solfege
|-
| | 12/11
|1u2
|lu 2nd
| | fu-sol
|-
| | 11/10
|1og2
|logu 2nd
| | mi-fu
|-
| | 10/9
|y2
|yo 2nd
| | re-mi
|-
| | 9/8
|w2
|wa 2nd
| | do-re
|-
| | 8/7
|r2
|ru 2nd
| | ta-do
|-
| | 7/6
|z3
|zo 3rd
| | sol-ta
|-
| | 6/5
|g3
|gu 3rd
| | mi-sol, ti-re
|-
| | 11/9
|1o3
|ilo 3rd
| | re-fu
|-
| | 5/4
|y3
|yo 3rd
| | do-mi
|-
| | 14/11
|1uz4
|luzo 4th
| | fu-ta
|-
| | 9/7
|r3
|ru 3rd
| | ta-re
|-
| | 4/3
|w4
|wa 4th
| | do-fa
|-
| | 11/8
|1o4
|ilo 4th
| | do-fu
|-
| | 7/5
|zg5
|zogu 5th
| | mi-ta
|-
| | 10/7
|ry4
|ruyo 4th
| | ta-mi
|-
| | 16/11
|1u5
|lu 5th
| | fu-do
|-
| | 3/2
|w5
|wa 5th
| | do-sol
|-
| | 14/9
|z6
|zo 6th
| | re-ta
|-
| | 11/7
|1or5
|loru 5th
| | ta-fu
|-
| | 8/5
|g6
|gu 6th
| | mi-do
|-
| | 18/11
|1u6
|lu 6th
| | fu-re
|-
| | 5/3
|y6
|yo 6th
| | sol-mi
|-
| | 12/7
|r6
|ru 6th
| | ta-sol
|-
| | 7/4
|z7
|zo 7th
| | do-ta
|-
| | 16/9
|w7
|wa 7th
| | re-do
|-
| | 9/5
|g7
|gu 7th
| | mi-re
|-
| | 20/11
|1uy7
|luyo 7th
| | fu-mi
|-
| | 11/6
|1o7
|ilo 7th
| | sol-fu
|-
| | 2/1
|w8
|wa 8ve
| | do-do
|}

While the [[7-limit|7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo|12edo]], the 11-limit introduces neutral intervals, [[Superfourth|superfourth]]s and [[Subfifth|subfifth]]s, which fall in between major, minor and perfect [[interval_category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[EDO|edo]]s which do (relatively) well in supporting 11-limit intervals are: [[1edo|1edo]], [[2edo|2edo]], [[3edo|3edo]], [[4edo|4edo]], [[5edo|5edo]], [[6edo|6edo]], [[7edo|7edo]], [[9edo|9edo]], [[10edo|10edo]], [[12edo|12edo]], [[15edo|15edo]], [[22edo|22edo]], [[26edo|26edo]], [[31edo|31edo]], [[41edo|41edo]], [[63edo|63edo]], [[72edo|72edo]], [[87edo|87edo]], [[109edo|109edo]], [[161edo|161edo]].

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

==Intervals==
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable"
|-
! |[[Kite's color notation|Interval name]]
! |
! | Ratio
! | Cents Value
|-
|1u2
|lu 2nd
| | [[12/11|12/11]]
| | 150.637
|-
|1og2
|logu 2nd
| | [[11/10|11/10]]
| | 165.004
|-
|1o3
|ilo 3rd
| | [[11/9|11/9]]
| | 347.408
|-
|1uz4
|lu 4th
| | [[14/11|14/11]]
| | 417.508
|-
|1uy4
|luyo 4th
| | [[15/11|15/11]]
| | 536.951
|-
|1o4
|ilo 4th
| | [[11/8|11/8]]
| | 551.318
|-
|1u5
|lu 5th
| | [[16/11|16/11]]
| | 648.682
|-
|1og5
|logu 5th
| | [[22/15|22/15]]
| | 663.049
|-
|1or5
|loru 5th
| | [[11/7|11/7]]
| | 782.492
|-
|1u6
|lu 6th
| | [[18/11|18/11]]
| | 852.592
|-
|1uy7
|luyo 7th
| | [[20/11|20/11]]
| | 1034.996
|-
|1o7
|ilo 7th
| | [[11/6|11/6]]
| | 1049.363
|}
See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]

=Music=
[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave_Hill|Dave Hill]]

[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill

[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris_Vaisvil|Chris Vaisvil]]

[https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew_Heathwaite|Andrew Heathwaite]]

=See also=
[[Harmonic_Limit|Harmonic Limit]]

[[Category:11-limit]]
[[Category:example]]
[[Category:interval]]
[[Category:limit]]
[[Category:listen]]
[[Category:prime_limit]]</text>
      <sha1>f90h8l4uq35uda9woqq1bqar95eivjy</sha1>
    </revision>
    <revision>
      <id>38347</id>
      <parentid>38346</parentid>
      <timestamp>2018-11-03T03:33:23Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <origin>38347</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4424" sha1="8atd8e059oc5c9je50lj3wtw4wc8pzx" xml:space="preserve">The ''11-limit'' consists of all [[JustIntonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11|14/11]], [[11/8|11/8]], [[27/22|27/22]] and [[99/98|99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

{| class="wikitable"
|-
! | Ratio
! colspan="2" |[[Kite's color notation|interval name]]
! | harmonic solfege
|-
| | 12/11
|1u2
|lu 2nd
| | fu-sol
|-
| | 11/10
|1og2
|logu 2nd
| | mi-fu
|-
| | 10/9
|y2
|yo 2nd
| | re-mi
|-
| | 9/8
|w2
|wa 2nd
| | do-re
|-
| | 8/7
|r2
|ru 2nd
| | ta-do
|-
| | 7/6
|z3
|zo 3rd
| | sol-ta
|-
| | 6/5
|g3
|gu 3rd
| | mi-sol, ti-re
|-
| | 11/9
|1o3
|ilo 3rd
| | re-fu
|-
| | 5/4
|y3
|yo 3rd
| | do-mi
|-
| | 14/11
|1uz4
|luzo 4th
| | fu-ta
|-
| | 9/7
|r3
|ru 3rd
| | ta-re
|-
| | 4/3
|w4
|wa 4th
| | do-fa
|-
| | 11/8
|1o4
|ilo 4th
| | do-fu
|-
| | 7/5
|zg5
|zogu 5th
| | mi-ta
|-
| | 10/7
|ry4
|ruyo 4th
| | ta-mi
|-
| | 16/11
|1u5
|lu 5th
| | fu-do
|-
| | 3/2
|w5
|wa 5th
| | do-sol
|-
| | 14/9
|z6
|zo 6th
| | re-ta
|-
| | 11/7
|1or5
|loru 5th
| | ta-fu
|-
| | 8/5
|g6
|gu 6th
| | mi-do
|-
| | 18/11
|1u6
|lu 6th
| | fu-re
|-
| | 5/3
|y6
|yo 6th
| | sol-mi
|-
| | 12/7
|r6
|ru 6th
| | ta-sol
|-
| | 7/4
|z7
|zo 7th
| | do-ta
|-
| | 16/9
|w7
|wa 7th
| | re-do
|-
| | 9/5
|g7
|gu 7th
| | mi-re
|-
| | 20/11
|1uy7
|luyo 7th
| | fu-mi
|-
| | 11/6
|1o7
|ilo 7th
| | sol-fu
|-
| | 2/1
|w8
|wa 8ve
| | do-do
|}

While the [[7-limit|7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo|12edo]], the 11-limit introduces neutral intervals, [[Superfourth|superfourth]]s and [[Subfifth|subfifth]]s, which fall in between major, minor and perfect [[interval_category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[EDO|edo]]s which do (relatively) well in supporting 11-limit intervals are: [[1edo|1edo]], [[2edo|2edo]], [[3edo|3edo]], [[4edo|4edo]], [[5edo|5edo]], [[6edo|6edo]], [[7edo|7edo]], [[9edo|9edo]], [[10edo|10edo]], [[12edo|12edo]], [[15edo|15edo]], [[22edo|22edo]], [[26edo|26edo]], [[31edo|31edo]], [[41edo|41edo]], [[63edo|63edo]], [[72edo|72edo]], [[87edo|87edo]], [[109edo|109edo]], [[161edo|161edo]].

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

==Intervals==
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable"
|-
! | Ratio
! | Cents Value
!  colspan="2"|[[Kite's color notation|Interval name]]
|-
| | [[12/11|12/11]]
| | 150.637
|1u2
|lu 2nd
|-
| | [[11/10|11/10]]
| | 165.004
|1og2
|logu 2nd
|-
| | [[11/9|11/9]]
| | 347.408
|1o3
|ilo 3rd
|-
| | [[14/11|14/11]]
| | 417.508
|1uz4
|lu 4th
|-
| | [[15/11|15/11]]
| | 536.951
|1uy4
|luyo 4th
|-
| | [[11/8|11/8]]
| | 551.318
|1o4
|ilo 4th
|-
| | [[16/11|16/11]]
| | 648.682
|1u5
|lu 5th
|-
| | [[22/15|22/15]]
| | 663.049
|1og5
|logu 5th
|-
| | [[11/7|11/7]]
| | 782.492
|1or5
|loru 5th
|-
| | [[18/11|18/11]]
| | 852.592
|1u6
|lu 6th
|-
| | [[20/11|20/11]]
| | 1034.996
|1uy7
|luyo 7th
|-
| | [[11/6|11/6]]
| | 1049.363
|1o7
|ilo 7th
|}
See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]

=Music=
[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave_Hill|Dave Hill]]

[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill

[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris_Vaisvil|Chris Vaisvil]]

[https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew_Heathwaite|Andrew Heathwaite]]

=See also=
[[Harmonic_Limit|Harmonic Limit]]

[[Category:11-limit]]
[[Category:example]]
[[Category:interval]]
[[Category:limit]]
[[Category:listen]]
[[Category:prime_limit]]</text>
      <sha1>8atd8e059oc5c9je50lj3wtw4wc8pzx</sha1>
    </revision>
    <revision>
      <id>46403</id>
      <parentid>38347</parentid>
      <timestamp>2020-06-10T20:07:45Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>reworked (some intersections with 11-odd-limit found -&gt; todo)</comment>
      <origin>46403</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3948" sha1="49edb1hootops7xtfu5qwwhngk8rdz9" xml:space="preserve">The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

== Intervals 1 ==

{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name]]
! harmonic solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Relative to their size, [[EDO]]s which do (relatively) well in supporting 11-limit intervals are: {{EDOs| 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 15, 22, 26, 31, 41, 63, 72, 87, 109, and 161edo }}.

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

== Intervals 2 ==

Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
! Ratio
! Site ([[cents|¢]])
! colspan="2" | [[Color name]]
|-
| [[12/11]]
| 150.637
| 1u2
| lu 2nd
|-
| [[11/10]]
| 165.004
| 1og2
| logu 2nd
|-
| [[11/9]]
| 347.408
| 1o3
| ilo 3rd
|-
| [[14/11]]
| 417.508
| 1uz4
| lu 4th
|-
| [[15/11]]
| 536.951
| 1uy4
| luyo 4th
|-
| [[11/8]]
| 551.318
| 1o4
| ilo 4th
|-
| [[16/11]]
| 648.682
| 1u5
| lu 5th
|-
| [[22/15]]
| 663.049
| 1og5
| logu 5th
|-
| [[11/7]]
| 782.492
| 1or5
| loru 5th
|-
| [[18/11]]
| 852.592
| 1u6
| lu 6th
|-
| [[20/11]]
| 1034.996
| 1uy7
| luyo 7th
|-
| [[11/6]]
| 1049.363
| 1o7
| ilo 7th
|}

== Music ==

* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris Vaisvil]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew Heathwaite]]

== See also ==

* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:11-limit]]
[[Category:example]]
[[Category:interval]]
[[Category:limit]]
[[Category:listen]]
[[Category:prime_limit]]

[[Category:Todo]]</text>
      <sha1>49edb1hootops7xtfu5qwwhngk8rdz9</sha1>
    </revision>
    <revision>
      <id>79856</id>
      <parentid>46403</parentid>
      <timestamp>2021-10-24T02:46:55Z</timestamp>
      <contributor>
        <username>Morion</username>
        <id>322</id>
      </contributor>
      <comment>It is best to include EDOs which represent 11-limit consistently.</comment>
      <origin>79856</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3880" sha1="cljiaukaqg9oljmxeghvwdfbl74axkw" xml:space="preserve">The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

== Intervals 1 ==

{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name]]
! harmonic solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Examples of [[EDO]]s which are consistent in 11-limit include: {{EDOs|22, 26, 31, 41, 46, 63, 72, 87, 109, and 161edo }}.

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

== Intervals 2 ==

Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
! Ratio
! Site ([[cents|¢]])
! colspan="2" | [[Color name]]
|-
| [[12/11]]
| 150.637
| 1u2
| lu 2nd
|-
| [[11/10]]
| 165.004
| 1og2
| logu 2nd
|-
| [[11/9]]
| 347.408
| 1o3
| ilo 3rd
|-
| [[14/11]]
| 417.508
| 1uz4
| lu 4th
|-
| [[15/11]]
| 536.951
| 1uy4
| luyo 4th
|-
| [[11/8]]
| 551.318
| 1o4
| ilo 4th
|-
| [[16/11]]
| 648.682
| 1u5
| lu 5th
|-
| [[22/15]]
| 663.049
| 1og5
| logu 5th
|-
| [[11/7]]
| 782.492
| 1or5
| loru 5th
|-
| [[18/11]]
| 852.592
| 1u6
| lu 6th
|-
| [[20/11]]
| 1034.996
| 1uy7
| luyo 7th
|-
| [[11/6]]
| 1049.363
| 1o7
| ilo 7th
|}

== Music ==

* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris Vaisvil]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew Heathwaite]]

== See also ==

* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:11-limit]]
[[Category:example]]
[[Category:interval]]
[[Category:limit]]
[[Category:listen]]
[[Category:prime_limit]]

[[Category:Todo]]</text>
      <sha1>cljiaukaqg9oljmxeghvwdfbl74axkw</sha1>
    </revision>
    <revision>
      <id>79857</id>
      <parentid>79856</parentid>
      <timestamp>2021-10-24T02:55:01Z</timestamp>
      <contributor>
        <username>Morion</username>
        <id>322</id>
      </contributor>
      <comment>/* Intervals 1 */ Corrected information as it was previously referring to 11-odd-limit.</comment>
      <origin>79857</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3839" sha1="hsu2ikoftiu6b0ruvu54zc9g62kavgw" xml:space="preserve">The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The 11 odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

== Intervals 1 ==

{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name]]
! harmonic solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Examples of [[EDO]]s which represent 11-limit intervals well include: {{EDOs|22, 26, 31, 41, 46, 63, 72, 87, 109, 118, 152, and 161edo }}.

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

==Intervals 2==

Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

==Music==

*[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave Hill]]
*[http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill
*[http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris Vaisvil]]
*[https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew Heathwaite]]

==See also==

*[[Gallery of just intervals]]
*[[Harmonic limit]]
*[[11-odd-limit]]

[[Category:11-limit]]
[[Category:example]]
[[Category:interval]]
[[Category:limit]]
[[Category:listen]]
[[Category:prime_limit]]

[[Category:Todo]]</text>
      <sha1>hsu2ikoftiu6b0ruvu54zc9g62kavgw</sha1>
    </revision>
    <revision>
      <id>79866</id>
      <parentid>79857</parentid>
      <timestamp>2021-10-24T10:36:05Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>79866</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3892" sha1="eid9grko6y50k4bgttoog4lmudtc481" xml:space="preserve">The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

Examples of [[EDO]]s which represent 11-limit intervals well include: {{EDOs|22, 26, 31, 41, 46, 63, 72, 87, 109, 118, 152, and 161edo }}.

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name]]
! harmonic solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris Vaisvil]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew Heathwaite]]

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:Just intonation]]
[[Category:11-limit]]
[[Category:Interval]]
[[Category:Limit]]
[[Category:Listen]]
[[Category:Prime limit]]</text>
      <sha1>eid9grko6y50k4bgttoog4lmudtc481</sha1>
    </revision>
    <revision>
      <id>79867</id>
      <parentid>79866</parentid>
      <timestamp>2021-10-24T10:44:14Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Adopt more objective measures in edo lists</comment>
      <origin>79867</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4225" sha1="rop3f14yl4fxh2hei3fga20nfhhkf2m" xml:space="preserve">The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name]]
! harmonic solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris Vaisvil]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew Heathwaite]]

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:Just intonation]]
[[Category:11-limit]]
[[Category:Interval]]
[[Category:Limit]]
[[Category:Listen]]
[[Category:Prime limit]]</text>
      <sha1>rop3f14yl4fxh2hei3fga20nfhhkf2m</sha1>
    </revision>
    <revision>
      <id>79873</id>
      <parentid>79867</parentid>
      <timestamp>2021-10-24T11:15:17Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Improve categories</comment>
      <origin>79873</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4280" sha1="bd7xohj0wqrzp5yuue59kg1olzl1nnz" xml:space="preserve">The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name]]
! harmonic solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris Vaisvil]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew Heathwaite]]

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:Just intonation]]
[[Category:Limit]]
[[Category:Prime limit]]
[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Interval collection]]
[[Category:Rank 5]]
[[Category:Listen]]</text>
      <sha1>bd7xohj0wqrzp5yuue59kg1olzl1nnz</sha1>
    </revision>
    <revision>
      <id>82008</id>
      <parentid>79873</parentid>
      <timestamp>2021-12-16T15:30:35Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Categories</comment>
      <origin>82008</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4232" sha1="akwgmcgrtpromeqnrq20hwm0uojn4ec" xml:space="preserve">The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name]]
! harmonic solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris Vaisvil]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew Heathwaite]]

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Prime limit]]
[[Category:Interval collection]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>akwgmcgrtpromeqnrq20hwm0uojn4ec</sha1>
    </revision>
    <revision>
      <id>99356</id>
      <parentid>82008</parentid>
      <timestamp>2022-11-29T14:11:55Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+navigation bar</comment>
      <origin>99356</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4262" sha1="1dmztmqwkreeagmesldm5t7ti83blnc" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name]]
! harmonic solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris Vaisvil]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew Heathwaite]]

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Prime limit]]
[[Category:Interval collection]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>1dmztmqwkreeagmesldm5t7ti83blnc</sha1>
    </revision>
    <revision>
      <id>100852</id>
      <parentid>99356</parentid>
      <timestamp>2023-01-03T08:22:29Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Prime limit]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>100852</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4237" sha1="8mk5g9zuulzn8kn16ntbaslcspmz173" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name]]
! harmonic solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Study #3] [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play] by [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm Brief 11-ratio composition] [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play] by Dave Hill
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 11 Limit Piano] by [[Chris Vaisvil]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals 11-limit singtervals] by [[Andrew Heathwaite]]

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Interval collection]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>8mk5g9zuulzn8kn16ntbaslcspmz173</sha1>
    </revision>
    <revision>
      <id>105557</id>
      <parentid>100852</parentid>
      <timestamp>2023-03-14T08:29:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Music */ cleanup</comment>
      <origin>105557</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4291" sha1="k0bux736hrr45ieolg48ryj9rziap0v" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name]]
! harmonic solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals'']

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Interval collection]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>k0bux736hrr45ieolg48ryj9rziap0v</sha1>
    </revision>
    <revision>
      <id>107523</id>
      <parentid>105557</parentid>
      <timestamp>2023-04-13T22:04:36Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Interval collection]] to [[Category:Lists of intervals]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>107523</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4290" sha1="rt7oqtzedk4gnigjcx9icde8uok3ax1" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name]]
! harmonic solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals'']

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>rt7oqtzedk4gnigjcx9icde8uok3ax1</sha1>
    </revision>
    <revision>
      <id>121315</id>
      <parentid>107523</parentid>
      <timestamp>2023-08-01T07:46:26Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+note on what that "e" in "27e" means</comment>
      <origin>121315</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4393" sha1="mip8i2nemu6ru7qvvtu1k970de0hqn8" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

Note: [[wart notation]] is used to specify the [[val]] chosen for the edo. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name]]
! harmonic solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals'']

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>mip8i2nemu6ru7qvvtu1k970de0hqn8</sha1>
    </revision>
    <revision>
      <id>121318</id>
      <parentid>121315</parentid>
      <timestamp>2023-08-01T07:53:44Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Style</comment>
      <origin>121318</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4405" sha1="s6o276pnnmd5slggo9bs0gpsjmptazr" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximations ==
A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

Note: [[wart notation]] is used to specify the [[val]] chosen for the edo. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals'']

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>s6o276pnnmd5slggo9bs0gpsjmptazr</sha1>
    </revision>
    <revision>
      <id>124864</id>
      <parentid>121318</parentid>
      <timestamp>2023-09-28T18:19:23Z</timestamp>
      <contributor>
        <username>Red Opal</username>
        <id>403</id>
      </contributor>
      <minor/>
      <comment>Deleted a "e" being after 27- a small typing mistake that I needed to fix due to my OCD</comment>
      <origin>124864</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4404" sha1="okrw3sxshtkigjbpmrs3ehee1z8nimi" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximations ==
A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

Note: [[wart notation]] is used to specify the [[val]] chosen for the edo. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals'']

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>okrw3sxshtkigjbpmrs3ehee1z8nimi</sha1>
    </revision>
    <revision>
      <id>124865</id>
      <parentid>124864</parentid>
      <timestamp>2023-09-28T18:27:20Z</timestamp>
      <contributor>
        <username>Red Opal</username>
        <id>403</id>
      </contributor>
      <comment>Undo revision 124864 by [[Special:Contributions/Red Opal|Red Opal]] ([[User talk:Red Opal|talk]])</comment>
      <origin>124865</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4405" sha1="s6o276pnnmd5slggo9bs0gpsjmptazr" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] intervals whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximations ==
A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

Note: [[wart notation]] is used to specify the [[val]] chosen for the edo. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals'']

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>s6o276pnnmd5slggo9bs0gpsjmptazr</sha1>
    </revision>
    <revision>
      <id>124866</id>
      <parentid>124865</parentid>
      <timestamp>2023-09-29T09:41:22Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>More specific explanation since "27e" repeatedly confused readers and editors</comment>
      <origin>124866</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4504" sha1="694rq0tqkulyuo4j4yp6qhphekacpm0" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] [[interval]]s whose numerators and denominators are both products of the primes 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential consonances.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximations ==
A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals'']

== See also ==
* [[Gallery of just intervals]]
* [[Harmonic limit]]
* [[11-odd-limit]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>694rq0tqkulyuo4j4yp6qhphekacpm0</sha1>
    </revision>
    <revision>
      <id>124885</id>
      <parentid>124866</parentid>
      <timestamp>2023-09-29T11:03:49Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+subsets and supersets</comment>
      <origin>124885</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4619" sha1="ansnpawly12l307s7qo7x9es7ehu0ao" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] [[interval]]s whose [[Ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The 11-limit is the 5th [[prime limit]] and is thus a superset of the [[7-limit]] and a subset of the [[13-limit]]. 

The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximations ==
A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals'']

== See also ==
* [[11-odd-limit]]
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>ansnpawly12l307s7qo7x9es7ehu0ao</sha1>
    </revision>
    <revision>
      <id>131063</id>
      <parentid>124885</parentid>
      <timestamp>2024-01-07T10:19:06Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Music */ unify with "music in just intonation article"</comment>
      <origin>131063</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5562" sha1="cbtqor4935bfakk2m8yoseptyck7pq7" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] [[interval]]s whose [[Ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The 11-limit is the 5th [[prime limit]] and is thus a superset of the [[7-limit]] and a subset of the [[13-limit]]. 

The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximations ==
A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[11-odd-limit]]
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>cbtqor4935bfakk2m8yoseptyck7pq7</sha1>
    </revision>
    <revision>
      <id>138614</id>
      <parentid>131063</parentid>
      <timestamp>2024-03-08T19:49:33Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Music */ +music</comment>
      <origin>138614</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5829" sha1="dps0krxhulpccjntaw634ka2pc1srdf" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] [[interval]]s whose [[Ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The 11-limit is the 5th [[prime limit]] and is thus a superset of the [[7-limit]] and a subset of the [[13-limit]]. 

The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximations ==
A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[11-odd-limit]]
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>dps0krxhulpccjntaw634ka2pc1srdf</sha1>
    </revision>
    <revision>
      <id>140681</id>
      <parentid>138614</parentid>
      <timestamp>2024-04-04T11:48:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Music */ +ben johnston's work</comment>
      <origin>140681</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6067" sha1="qn1vzo0ljdtfwqiggclzsjvmfzseiwt" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] [[interval]]s whose [[Ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The 11-limit is the 5th [[prime limit]] and is thus a superset of the [[7-limit]] and a subset of the [[13-limit]]. 

The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximations ==
A list of [[edo]]s which represent 11-limit intervals with better accuracy: {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Another list of edos which tunes the 11-limit well relative to their size ([[TE relative error|relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11. 

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[11-odd-limit]]
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>qn1vzo0ljdtfwqiggclzsjvmfzseiwt</sha1>
    </revision>
    <revision>
      <id>153734</id>
      <parentid>140681</parentid>
      <timestamp>2024-09-02T10:39:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Edo approximations */ wording</comment>
      <origin>153734</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6086" sha1="82jkme4b499fuox8zvforl0yqyxncfe" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] [[interval]]s whose [[Ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The 11-limit is the 5th [[prime limit]] and is thus a superset of the [[7-limit]] and a subset of the [[13-limit]]. 

The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy (decreasing [[TE error]]): {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[11-odd-limit]]
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>82jkme4b499fuox8zvforl0yqyxncfe</sha1>
    </revision>
    <revision>
      <id>157335</id>
      <parentid>153734</parentid>
      <timestamp>2024-10-01T12:24:24Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Music */ +music</comment>
      <origin>157335</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6357" sha1="tkv6zdzfsx8zx7a27uv04qsl7t3yq0d" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] [[interval]]s whose [[Ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The 11-limit is the 5th [[prime limit]] and is thus a superset of the [[7-limit]] and a subset of the [[13-limit]]. 

The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy (decreasing [[TE error]]): {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[11-odd-limit]]
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>tkv6zdzfsx8zx7a27uv04qsl7t3yq0d</sha1>
    </revision>
    <revision>
      <id>166951</id>
      <parentid>157335</parentid>
      <timestamp>2024-11-19T17:27:03Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Music */ +music</comment>
      <origin>166951</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6446" sha1="ag2xkpyqybkheklparqx486hjbavpct" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[Just intonation|justly tuned]] [[interval]]s whose [[Ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The 11-limit is the 5th [[prime limit]] and is thus a superset of the [[7-limit]] and a subset of the [[13-limit]]. 

The [[11-odd-limit]] consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy (decreasing [[TE error]]): {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[11-odd-limit]]
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>ag2xkpyqybkheklparqx486hjbavpct</sha1>
    </revision>
    <revision>
      <id>171408</id>
      <parentid>166951</parentid>
      <timestamp>2024-12-17T15:51:40Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+relation to odd limits and harmonic/subharmonic modes</comment>
      <origin>171408</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6559" sha1="iob8ntylalttfpvq7yn9q9o2fkebco6" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. 

These things are contained by the 11-limit, but not the 7-limit: 
* The [[11-odd-limit]]; 
* Mode 6 of the harmonic or subharmonic series. 

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy (decreasing [[TE error]]): {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>iob8ntylalttfpvq7yn9q9o2fkebco6</sha1>
    </revision>
    <revision>
      <id>171411</id>
      <parentid>171408</parentid>
      <timestamp>2024-12-17T16:08:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <origin>171411</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6559" sha1="dd4fbuv5nooamyqjsp4v8v74giau9k8" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. 

These things are contained by the 11-limit, but not the 7-limit: 
* The [[11-odd-limit]]; 
* Mode 6 of the harmonic or subharmonic series. 

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy (decreasing [[TE error]]): {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>dd4fbuv5nooamyqjsp4v8v74giau9k8</sha1>
    </revision>
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      <timestamp>2025-04-23T04:13:01Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
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      <text bytes="6703" sha1="eozoqtq53jnyzeut9poq5n63vnet75l" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. 

These things are contained by the 11-limit, but not the 7-limit: 
* The [[11-odd-limit]]; 
* Mode 6 of the harmonic or subharmonic series. 

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy (decreasing [[TE error]]): {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== 2.3.5.11 subgroup ==

== 2.3.7.11 subgroup ==

== 2.5.7.11 subgroup ==

== 2.3.11 subgroup ==

== 2.5.11 subgroup ==

== 2.7.11 subgroup ==

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>eozoqtq53jnyzeut9poq5n63vnet75l</sha1>
    </revision>
    <revision>
      <id>193674</id>
      <parentid>193606</parentid>
      <timestamp>2025-04-23T10:09:56Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Undo revision [[Special:Diff/193606|193606]] by [[Special:Contributions/VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]). The sections are empty</comment>
      <origin>193674</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6559" sha1="dd4fbuv5nooamyqjsp4v8v74giau9k8" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. 

These things are contained by the 11-limit, but not the 7-limit: 
* The [[11-odd-limit]]; 
* Mode 6 of the harmonic or subharmonic series. 

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy (decreasing [[TE error]]): {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>dd4fbuv5nooamyqjsp4v8v74giau9k8</sha1>
    </revision>
    <revision>
      <id>197218</id>
      <parentid>193674</parentid>
      <timestamp>2025-05-21T10:36:52Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Fill in some info that's present in the other limits but not here somehow</comment>
      <origin>197218</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6904" sha1="eudhka4blrx2sd5oykysk8c4qz0d0ih" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. 

The 11-limit is a [[rank and codimension|rank-5]] system, and can be modeled in a 4-dimensional [[lattice]], with the primes 3, 5, 7, and 11 represented by each dimension. The prime 2 does not appear in the typical 11-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a fifth dimension is needed.

These things are contained by the 11-limit, but not the 7-limit: 
* The [[11-odd-limit]]; 
* Mode 6 of the harmonic or subharmonic series. 

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy (decreasing [[TE error]]): {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Study #3'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/04_hill_study-3.mp3 play]{{dead link}}
* [http://sonic-arts.org/hill/10-passages-ji/10-passages-ji.htm ''Brief 11-ratio composition'']{{dead link}} [http://sonic-arts.org/hill/10-passages-ji/09_hill_brief-11-ratio-composition.mp3 play]{{dead link}}

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>eudhka4blrx2sd5oykysk8c4qz0d0ih</sha1>
    </revision>
    <revision>
      <id>205089</id>
      <parentid>197218</parentid>
      <timestamp>2025-07-25T20:13:03Z</timestamp>
      <contributor>
        <username>Xenjacob</username>
        <id>19</id>
      </contributor>
      <comment>/* Music */ new links for Dave Hill</comment>
      <origin>205089</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6698" sha1="f5732qf5ty386fb5fknukof7coyqofv" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. 

The 11-limit is a [[rank and codimension|rank-5]] system, and can be modeled in a 4-dimensional [[lattice]], with the primes 3, 5, 7, and 11 represented by each dimension. The prime 2 does not appear in the typical 11-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a fifth dimension is needed.

These things are contained by the 11-limit, but not the 7-limit: 
* The [[11-odd-limit]]; 
* Mode 6 of the harmonic or subharmonic series. 

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy (decreasing [[TE error]]): {{EDOs| 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [https://ralphdavidhill.bandcamp.com/track/study-3 ''Study #3'']
* [https://ralphdavidhill.bandcamp.com/track/brief-11-limit-ratio-composition ''Brief 11-ratio composition'']

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>f5732qf5ty386fb5fknukof7coyqofv</sha1>
    </revision>
    <revision>
      <id>207777</id>
      <parentid>205089</parentid>
      <timestamp>2025-08-20T16:29:49Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Edo approximation */ calibrate the list according to my research results</comment>
      <origin>207777</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6824" sha1="bjwpx5dqg945s6260lfdgdli5df8zng" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. 

The 11-limit is a [[rank and codimension|rank-5]] system, and can be modeled in a 4-dimensional [[lattice]], with the primes 3, 5, 7, and 11 represented by each dimension. The prime 2 does not appear in the typical 11-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a fifth dimension is needed.

These things are contained by the 11-limit, but not the 7-limit: 
* The [[11-odd-limit]]; 
* Mode 6 of the harmonic or subharmonic series. 

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy ([[monotonicity limit]] ≥ 11 and decreasing [[TE error]]): {{EDOs| 12, 15, 19, 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]]. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

: '''Note''': [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11.

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [https://ralphdavidhill.bandcamp.com/track/study-3 ''Study #3'']
* [https://ralphdavidhill.bandcamp.com/track/brief-11-limit-ratio-composition ''Brief 11-ratio composition'']

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>bjwpx5dqg945s6260lfdgdli5df8zng</sha1>
    </revision>
    <revision>
      <id>220050</id>
      <parentid>207777</parentid>
      <timestamp>2025-12-27T17:42:04Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Edo approximation */ adopt template</comment>
      <origin>220050</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6821" sha1="tg04ileql93yvwkwwen1r4dav2svyzz" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. 

The 11-limit is a [[rank and codimension|rank-5]] system, and can be modeled in a 4-dimensional [[lattice]], with the primes 3, 5, 7, and 11 represented by each dimension. The prime 2 does not appear in the typical 11-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a fifth dimension is needed.

These things are contained by the 11-limit, but not the 7-limit: 
* The [[11-odd-limit]]; 
* Mode 6 of the harmonic or subharmonic series. 

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy ([[monotonicity limit]] ≥ 11 and decreasing [[TE error]]): {{EDOs| 12, 15, 19, 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]]. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

{{Note| [[wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11. }}

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [https://ralphdavidhill.bandcamp.com/track/study-3 ''Study #3'']
* [https://ralphdavidhill.bandcamp.com/track/brief-11-limit-ratio-composition ''Brief 11-ratio composition'']

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>tg04ileql93yvwkwwen1r4dav2svyzz</sha1>
    </revision>
    <revision>
      <id>221356</id>
      <parentid>220050</parentid>
      <timestamp>2026-01-12T07:35:23Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <minor/>
      <comment>/* Edo approximation */ caps</comment>
      <origin>221356</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6821" sha1="t9yck4lgsaxpqls1kjh58m4hfw3n70p" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. 

The 11-limit is a [[rank and codimension|rank-5]] system, and can be modeled in a 4-dimensional [[lattice]], with the primes 3, 5, 7, and 11 represented by each dimension. The prime 2 does not appear in the typical 11-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a fifth dimension is needed.

These things are contained by the 11-limit, but not the 7-limit: 
* The [[11-odd-limit]]; 
* Mode 6 of the harmonic or subharmonic series. 

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy ([[monotonicity limit]] ≥ 11 and decreasing [[TE error]]): {{EDOs| 12, 15, 19, 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]]. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

{{Note| [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11. }}

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [https://ralphdavidhill.bandcamp.com/track/study-3 ''Study #3'']
* [https://ralphdavidhill.bandcamp.com/track/brief-11-limit-ratio-composition ''Brief 11-ratio composition'']

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Lists of intervals]]
[[Category:Listen]]
[[Category:Rank 5]]</text>
      <sha1>t9yck4lgsaxpqls1kjh58m4hfw3n70p</sha1>
    </revision>
    <revision>
      <id>223403</id>
      <parentid>221356</parentid>
      <timestamp>2026-02-06T08:27:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Recategorize</comment>
      <origin>223403</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6834" sha1="4j1n14fm6lyl9ixrr7gqd5k0tjr0305" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. 

The 11-limit is a [[rank and codimension|rank-5]] system, and can be modeled in a 4-dimensional [[lattice]], with the primes 3, 5, 7, and 11 represented by each dimension. The prime 2 does not appear in the typical 11-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a fifth dimension is needed.

These things are contained by the 11-limit, but not the 7-limit: 
* The [[11-odd-limit]]; 
* Mode 6 of the harmonic or subharmonic series. 

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy ([[monotonicity limit]] ≥ 11 and decreasing [[TE error]]): {{EDOs| 12, 15, 19, 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]]. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

{{Note| [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11. }}

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [https://ralphdavidhill.bandcamp.com/track/study-3 ''Study #3'']
* [https://ralphdavidhill.bandcamp.com/track/brief-11-limit-ratio-composition ''Brief 11-ratio composition'']

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Rank-5 temperaments]]
[[Category:Lists of intervals]]
[[Category:Listen]]</text>
      <sha1>4j1n14fm6lyl9ixrr7gqd5k0tjr0305</sha1>
    </revision>
    <revision>
      <id>228022</id>
      <parentid>223403</parentid>
      <timestamp>2026-04-19T09:43:53Z</timestamp>
      <contributor>
        <username>Eufalesio</username>
        <id>581</id>
      </contributor>
      <minor/>
      <comment>color notation</comment>
      <origin>228022</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6872" sha1="3suxj2do1z6kjbjrcwm6c2xh6soiccf" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' (AKA ''yazala'' in [[color notation]])consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. 

The 11-limit is a [[rank and codimension|rank-5]] system, and can be modeled in a 4-dimensional [[lattice]], with the primes 3, 5, 7, and 11 represented by each dimension. The prime 2 does not appear in the typical 11-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a fifth dimension is needed.

These things are contained by the 11-limit, but not the 7-limit: 
* The [[11-odd-limit]]; 
* Mode 6 of the harmonic or subharmonic series. 

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy ([[monotonicity limit]] ≥ 11 and decreasing [[TE error]]): {{EDOs| 12, 15, 19, 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]]. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

{{Note| [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11. }}

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [https://ralphdavidhill.bandcamp.com/track/study-3 ''Study #3'']
* [https://ralphdavidhill.bandcamp.com/track/brief-11-limit-ratio-composition ''Brief 11-ratio composition'']

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Rank-5 temperaments]]
[[Category:Lists of intervals]]
[[Category:Listen]]</text>
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      <text bytes="6876" sha1="moutxocn5cvo06bi7w0e6bqyi0am825" xml:space="preserve">{{Prime limit navigation|11}}
The '''11-limit''' (a.k.a. ''yazala'' in [[color notation]]) consists of all [[just intonation|justly tuned]] [[interval]]s whose [[ratio|numerators and denominators]] are both products of the [[prime]]s 2, 3, 5, 7 and 11. The 11-limit is the 5th [[prime limit]] and is a superset of the [[7-limit]] and a subset of the [[13-limit]]. Some examples of 11-limit intervals are [[14/11]], [[11/8]], [[27/22]] and [[99/98]]. 

The 11-limit is a [[rank and codimension|rank-5]] system, and can be modeled in a 4-dimensional [[lattice]], with the primes 3, 5, 7, and 11 represented by each dimension. The prime 2 does not appear in the typical 11-limit lattice because [[octave equivalence]] is presumed. If octave equivalence is not presumed, a fifth dimension is needed.

These things are contained by the 11-limit, but not the 7-limit: 
* The [[11-odd-limit]]; 
* Mode 6 of the harmonic or subharmonic series. 

The 11-odd-limit consists of intervals whose numerators and denominators, when all factors of two have been removed, are less than or equal to 11. Reduced to an octave, these are the ratios 1/1, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 6/5, 11/9, 5/4, 14/11, 9/7, 4/3, 11/8, 7/5, 10/7, 16/11, 3/2, 14/9, 11/7, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 2/1. In an 11-limit system, all the ratios of the 11 odd-limit can be treated as potential [[consonance]]s.

While the [[7-limit]] introduces subminor and supermajor intervals, which can sound like dramatic inflections of the familiar interval categories of [[12edo]], the 11-limit introduces neutral intervals, [[superfourth]]s and [[subfifth]]s, which fall in between major, minor and perfect [[interval category|interval categories]] and thus demand new distinctions. It is thus inescapably xenharmonic.

== Edo approximation ==
Here is a list of [[edo]]s which represent 11-limit intervals with better accuracy ([[monotonicity limit]] ≥ 11 and decreasing [[TE error]]): {{EDOs| 12, 15, 19, 22, 27e, 31, 41, 53, 58, 72, 118, 130, 152, 224, 270, 342, 612 }} and so on. For a more comprehensive list, see [[Sequence of equal temperaments by error]]. 

Here is a list of edos which tunes the 11-limit well relative to their size ([[TE relative error]] &lt; 5%): {{EDOs| 31, 41, 58, 72, 87, 118, 130, 152, 183, 190, 198, 212, 224, 239, 255, 270, 301, 311, 342, 369, 373, 400, 414, 422, 441, 453, 460, 463, 472, 494, 525, 552, 566, 581, 612 }} and so on. 

{{Note| [[Wart notation]] is used to specify the [[val]] chosen for the edo. In the above list, "27e" means taking the second closest approximation of harmonic 11. }}

== Intervals ==
=== 11-odd-limit intervals ===
{| class="wikitable center-all"
! Ratio
! colspan="2" | [[Color name|Color Name]]
! Harmonic Solfege
|-
| 12/11
| 1u2
| lu 2nd
| fu-sol
|-
| 11/10
| 1og2
| logu 2nd
| mi-fu
|-
| 10/9
| y2
| yo 2nd
| re-mi
|-
| 9/8
| w2
| wa 2nd
| do-re
|-
| 8/7
| r2
| ru 2nd
| ta-do
|-
| 7/6
| z3
| zo 3rd
| sol-ta
|-
| 6/5
| g3
| gu 3rd
| mi-sol, ti-re
|-
| 11/9
| 1o3
| ilo 3rd
| re-fu
|-
| 5/4
| y3
| yo 3rd
| do-mi
|-
| 14/11
| 1uz4
| luzo 4th
| fu-ta
|-
| 9/7
| r3
| ru 3rd
| ta-re
|-
| 4/3
| w4
| wa 4th
| do-fa
|-
| 11/8
| 1o4
| ilo 4th
| do-fu
|-
| 7/5
| zg5
| zogu 5th
| mi-ta
|-
| 10/7
| ry4
| ruyo 4th
| ta-mi
|-
| 16/11
| 1u5
| lu 5th
| fu-do
|-
| 3/2
| w5
| wa 5th
| do-sol
|-
| 14/9
| z6
| zo 6th
| re-ta
|-
| 11/7
| 1or5
| loru 5th
| ta-fu
|-
| 8/5
| g6
| gu 6th
| mi-do
|-
| 18/11
| 1u6
| lu 6th
| fu-re
|-
| 5/3
| y6
| yo 6th
| sol-mi
|-
| 12/7
| r6
| ru 6th
| ta-sol
|-
| 7/4
| z7
| zo 7th
| do-ta
|-
| 16/9
| w7
| wa 7th
| re-do
|-
| 9/5
| g7
| gu 7th
| mi-re
|-
| 20/11
| 1uy7
| luyo 7th
| fu-mi
|-
| 11/6
| 1o7
| ilo 7th
| sol-fu
|-
| 2/1
| w8
| wa 8ve
| do-do
|}

[[File:11-limit_compare.png|alt=11-limit_compare.png|11-limit_compare.png]]

=== Selected 15-odd-limit intervals ===
Here are all the 15-odd-limit intervals of 11:

{| class="wikitable center-all right-2"
!Ratio
!Site ([[cents|¢]])
! colspan="2" |[[Color name]]
|-
|[[12/11]]
|150.637
|1u2
|lu 2nd
|-
|[[11/10]]
|165.004
| 1og2
|logu 2nd
|-
|[[11/9]]
|347.408
|1o3
|ilo 3rd
|-
|[[14/11]]
|417.508
| 1uz4
|lu 4th
|-
|[[15/11]]
|536.951
| 1uy4
|luyo 4th
|-
|[[11/8]]
|551.318
|1o4
|ilo 4th
|-
|[[16/11]]
|648.682
|1u5
|lu 5th
|-
|[[22/15]]
|663.049
| 1og5
|logu 5th
|-
|[[11/7]]
|782.492
| 1or5
|loru 5th
|-
|[[18/11]]
|852.592
|1u6
|lu 6th
|-
|[[20/11]]
|1034.996
| 1uy7
|luyo 7th
|-
|[[11/6]]
|1049.363
|1o7
|ilo 7th
|}

== Music ==
; [[Brody Bigwood]]
* [https://www.youtube.com/watch?v=i-FokV8dicQ ''Presence''] (2024)

; [[birdshite stalactite]]
* "swelter" from ''tropical nosebleed'' (2023) – [https://open.spotify.com/track/6EysxRhdzwhpebjjk5j0hg Spotify] | [https://birdshitestalactite.bandcamp.com/track/swelter Bandcamp] | [https://www.youtube.com/watch?v=gv8ouzpHzTU YouTube]

; [[Francium]]
* "I Forgot My Line" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/5UAphCjwDnNeIxP4xg7a75 Spotify] | [https://francium223.bandcamp.com/track/i-forgot-my-line Bandcamp] | [https://www.youtube.com/watch?v=khMcdyqRmPA YouTube]

; [[Andrew Heathwaite]]
* [https://soundcloud.com/andrew_heathwaite/11-limit-singtervals ''11-limit singtervals''] (2012)

; [[Dave Hill]]
* [https://ralphdavidhill.bandcamp.com/track/study-3 ''Study #3'']
* [https://ralphdavidhill.bandcamp.com/track/brief-11-limit-ratio-composition ''Brief 11-ratio composition'']

; [[Ben Johnston]]
* ''String Quartet No. 6'' (1980) – [https://newworldrecords.bandcamp.com/track/string-quartet-no-6-legato-espressivo Bandcamp] | [https://www.youtube.com/watch?v=ApOa8c0dZdA YouTube] – performed by Kepler Quartet

; [[Claudi Meneghin]]
* [http://web.archive.org/web/20191230113642/http://soonlabel.com:80/xenharmonic/archives/1201 ''Blue Canon''] (2013)
* [http://web.archive.org/web/20191230113723/http://soonlabel.com:80/xenharmonic/archives/1158 ''11-limit Canon on Elgar's Enigma Theme''] (2013)
* [http://web.archive.org/web/20191230033820/http://soonlabel.com:80/xenharmonic/archives/1175 ''El Cant dels Ocells'' ("The Song of the Birds")] – Catalan traditional, arranged by Claudi Meneghin (2013)

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/just/11-limit/20120210-piano-11-limit.mp3 ''11 Limit Piano'']

; [[Randy Wells]]
* [https://www.youtube.com/watch?v=k9yMpCbwvEc ''A Keepsake Found After So Many Years''] (2021)
* [https://www.youtube.com/watch?v=0CzBl22R3TI ''Eros''] (2021)
* [https://www.youtube.com/watch?v=0IaUmGT0RYk ''Music for Liminal Spaces''] (2021)
* [https://www.youtube.com/watch?v=1xjE3YVnlHY ''Marshmallow Beatdown''] (2022)
* [https://www.youtube.com/watch?v=V7X4gHgs0Xo ''A Compendium of Things That Molecules Do''] (2022)

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit| ]] &lt;!-- main article --&gt;
[[Category:Rank-5 temperaments]]
[[Category:Lists of intervals]]
[[Category:Listen]]</text>
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  <page>
    <title>1106edo</title>
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      <text bytes="540" sha1="gulr0hf92rv2b0jc4iozo6b3bh1loa5" xml:space="preserve">The 1106 division is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] which divides the octave into 1106 parts of 1.805 cents each. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] being 171, 270, 342, 441 and 612. It is even stronger in the 11-limit; the only ones being it out now being 270, 342 and 612. It is less strong in the 13 and 17 limits, but even so is distinctly consistent through the 17 limit.</text>
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      <comment>**Imported revision 556759881 - Original comment: **</comment>
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      <text bytes="2406" sha1="h8iuzk6jzdpaq5gqi7ef5y9y19q7r5a" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-08-16 12:05:10 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;556759881&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 1106 division is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak edo]] which divides the octave into 1106 parts of 1.805 cents each. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]] being 171, 270, 342, 441 and 612. It is even stronger in the 11-limit; the only ones being it out now being 270, 342 and 612. It is less strong in the 13 and 17 limits, but even so is distinctly consistent through the 17 limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1106edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 1106 division is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt; which divides the octave into 1106 parts of 1.805 cents each. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit &amp;lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&amp;gt;relative error&amp;lt;/a&amp;gt; being 171, 270, 342, 441 and 612. It is even stronger in the 11-limit; the only ones being it out now being 270, 342 and 612. It is less strong in the 13 and 17 limits, but even so is distinctly consistent through the 17 limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <text bytes="619" sha1="gn5fe2m4udod59jtl1a11fofxvoc7km" xml:space="preserve">The '''1106 division''' is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] which divides the octave into 1106 parts of 1.805 cents each. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] being 171, 270, 342, 441 and 612. It is even stronger in the 11-limit; the only ones being it out now being 270, 342 and 612. It is less strong in the 13 and 17 limits, but even so is distinctly consistent through the 17 limit.

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
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      <timestamp>2022-10-04T22:07:31Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
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      <text bytes="634" sha1="at2ok29fvtxbel0b4x2iqfoyepzxu65" xml:space="preserve">{{Infobox ET}}
The '''1106 division''' is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] which divides the octave into 1106 parts of 1.805 cents each. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] being 171, 270, 342, 441 and 612. It is even stronger in the 11-limit; the only ones being it out now being 270, 342 and 612. It is less strong in the 13 and 17 limits, but even so is distinctly consistent through the 17 limit.

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
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      <id>101082</id>
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      <timestamp>2023-01-08T03:48:47Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Rework and expansion</comment>
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      <text bytes="709" sha1="cqjyrzi5ej6h11zsbf8jdomdu9p7fj2" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13 and 17 limits, but even so is distinctly [[consistent]] through the [[17-odd-limit]].

=== Prime harmonics ===
{{Harmonics in equal|1106}}

=== Divisors ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.</text>
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      <contributor>
        <username>FloraC</username>
        <id>186</id>
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      <comment>+some temps</comment>
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      <text bytes="842" sha1="jwvse1mdebslm0z3tefvf4bwo014jqu" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13 and 17 limits, but even so is distinctly [[consistent]] through the [[17-odd-limit]]. 

It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. 

=== Prime harmonics ===
{{Harmonics in equal|1106}}

=== Divisors ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.</text>
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      <timestamp>2023-07-03T20:01:01Z</timestamp>
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        <username>Eliora</username>
        <id>274</id>
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      <text bytes="909" sha1="hywmf6bqjg6q9ayvoxbomrgq91d8orq" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13 and 17 limits, but even so is distinctly [[consistent]] through the [[17-odd-limit]]. 

It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. In higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106}}

=== Divisors ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.</text>
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      <id>118727</id>
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      <timestamp>2023-07-06T00:34:54Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
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      <comment>fill in the temperament table</comment>
      <origin>118727</origin>
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      <text bytes="1452" sha1="fjhkar0i5jnszeagpeuuj12s8yyj6jq" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}
== Theory ==
1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13 and 17 limits, but even so is distinctly [[consistent]] through the [[17-odd-limit]]. 

It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. In higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106}}

=== Divisors ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]]
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 401\1106
| 435.081
| 9/7
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(325/324)
| [[Brahmagupta]]
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}</text>
      <sha1>fjhkar0i5jnszeagpeuuj12s8yyj6jq</sha1>
    </revision>
    <revision>
      <id>118735</id>
      <parentid>118727</parentid>
      <timestamp>2023-07-06T00:48:31Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>completed the regular temperament table</comment>
      <origin>118735</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2431" sha1="nybfw1a59b9656eif497yzj8ik7ooi5" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}
== Theory ==
1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13 and 17 limits, but even so is distinctly [[consistent]] through the [[17-odd-limit]]. 

It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. In higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106}}

=== Divisors ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal
8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|[1753, -1106⟩
|1106 1753
| -0.010
|0.010
|0.99
|-
|2.3.5
|&lt;nowiki&gt;-53  10 16,  40 -56 21&lt;/nowiki&gt;
|1106 1753 2568
| +0.001
|0.019
|1.73
|-
|2.3.5.7
|4375/4374, 52734375/52706752, 46 -14 -3 -6
|1106 1753 2568 3105
|&lt;nowiki&gt;-0.006&lt;/nowiki&gt;
|0.020
|1.83
|-
|2.3.5.7.11
|3025/3024, 4375/4374, 5767168:5764801, 35156250:35153041
|1106 1753 2568 3105 3826
| +0.004
|0.026
|2.38
|-
|2.3.5.7.11.13
|3025/3024, 4096/4095, 4375/4374, 456533/456300, 928125/927472
|1106 1753 2568 3105 3826 4093
|&lt;nowiki&gt;-0.012&lt;/nowiki&gt;
|0.043
|3.94
|-
|2.3.5.7.11.13.17
|2500/2499, 3025/3024, 4096/4095, 8624/8619, 9801/9800, 14875/14572
|1106 1753 2568 3105 3826 4093 4521
|&lt;nowiki&gt;-0.021&lt;/nowiki&gt;
|0.045
|4.11
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]]
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 401\1106
| 435.081
| 9/7
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(325/324)
| [[Brahmagupta]]
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}</text>
      <sha1>nybfw1a59b9656eif497yzj8ik7ooi5</sha1>
    </revision>
    <revision>
      <id>118736</id>
      <parentid>118735</parentid>
      <timestamp>2023-07-06T00:50:01Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */</comment>
      <origin>118736</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2495" sha1="qnk50s7fo7ot2n0s0aspm261rde9t80" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}
== Theory ==
1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13 and 17 limits, but even so is distinctly [[consistent]] through the [[17-odd-limit]]. 

It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. In higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106}}

=== Divisors ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal
8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo|1753 -1106}}
|{{val|1106 1753}}
| -0.010
|0.010
|0.99
|-
|2.3.5
|{{monzo|-53 10 16}}, {{monzo|40 -56 21}}
|{{val|1106 1753 2568}}
| +0.001
|0.019
|1.73
|-
|2.3.5.7
|4375/4374, 52734375/52706752, {{monzo|46 -14 -3 -6}}
|{{val|1106 1753 2568 3105}}
|&lt;nowiki&gt;-0.006&lt;/nowiki&gt;
|0.020
|1.83
|-
|2.3.5.7.11
|3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
|{{val|1106 1753 2568 3105 3826}}
| +0.004
|0.026
|2.38
|-
|2.3.5.7.11.13
|3025/3024, 4096/4095, 4375/4374, 456533/456300, 928125/927472
|{{val|1106 1753 2568 3105 3826 4093}}
|&lt;nowiki&gt;-0.012&lt;/nowiki&gt;
|0.043
|3.94
|-
|2.3.5.7.11.13.17
|2500/2499, 3025/3024, 4096/4095, 8624/8619, 9801/9800, 14875/14572
|{{val|1106 1753 2568 3105 3826 4093 4521}}
|&lt;nowiki&gt;-0.021&lt;/nowiki&gt;
|0.045
|4.11
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]]
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 401\1106
| 435.081
| 9/7
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(325/324)
| [[Brahmagupta]]
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}</text>
      <sha1>qnk50s7fo7ot2n0s0aspm261rde9t80</sha1>
    </revision>
    <revision>
      <id>118745</id>
      <parentid>118736</parentid>
      <timestamp>2023-07-06T01:29:26Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Formatting</comment>
      <origin>118745</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2525" sha1="tvtilqgzd80y15svkkr920ox7m9gbbl" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

== Theory ==
1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13- and 17-limit, but even so is distinctly [[consistent]] through the [[17-odd-limit]]. 

It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106}}

=== Subsets and supersets ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 1753 -1106 }}
| {{val| 1106 1753 }}
| -0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| 40 -56 21 }}
| {{val| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{val| 1106 1753 2568 3105 }}
| -0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{val| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
|2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 456533/456300, 928125/927472
| {{val| 1106 1753 2568 3105 3826 4093 }}
| -0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 8624/8619, 9801/9800, 14875/14572
| {{val| 1106 1753 2568 3105 3826 4093 4521 }}
| -0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]]
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 401\1106
| 435.081
| 9/7
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(325/324)
| [[Brahmagupta]]
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}</text>
      <sha1>tvtilqgzd80y15svkkr920ox7m9gbbl</sha1>
    </revision>
    <revision>
      <id>118751</id>
      <parentid>118745</parentid>
      <timestamp>2023-07-06T01:38:35Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ comma bases; reduced gen for semisupermajor and note on orga and brahmagupta</comment>
      <origin>118751</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2583" sha1="008do24mihfjg7ktk6xbr8b9qxfit26" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

== Theory ==
1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13- and 17-limit, but even so is distinctly [[consistent]] through the [[17-odd-limit]]. 

It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106}}

=== Subsets and supersets ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 1753 -1106 }}
| {{val| 1106 1753 }}
| -0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| 40 -56 21 }}
| {{val| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{val| 1106 1753 2568 3105 }}
| -0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{val| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{val| 1106 1753 2568 3105 3826 4093 }}
| -0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{val| 1106 1753 2568 3105 3826 4093 4521 }}
| -0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 401\1106&lt;br&gt;(152\1106)
| 435.081&lt;br&gt;(164.919)
| 9/7&lt;br&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}</text>
      <sha1>008do24mihfjg7ktk6xbr8b9qxfit26</sha1>
    </revision>
    <revision>
      <id>119384</id>
      <parentid>118751</parentid>
      <timestamp>2023-07-09T05:21:10Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119384</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2602" sha1="mpphrues2zlcqu0virs3ywhboon5f79" xml:space="preserve">{{novelty}}{{stub}}{{Infobox ET}}
{{EDO intro|1106}}

== Theory ==
1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13- and 17-limit, but even so is distinctly [[consistent]] through the [[17-odd-limit]]. 

It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106}}

=== Subsets and supersets ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 1753 -1106 }}
| {{val| 1106 1753 }}
| -0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| 40 -56 21 }}
| {{val| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{val| 1106 1753 2568 3105 }}
| -0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{val| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{val| 1106 1753 2568 3105 3826 4093 }}
| -0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{val| 1106 1753 2568 3105 3826 4093 4521 }}
| -0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 401\1106&lt;br&gt;(152\1106)
| 435.081&lt;br&gt;(164.919)
| 9/7&lt;br&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}</text>
      <sha1>mpphrues2zlcqu0virs3ywhboon5f79</sha1>
    </revision>
    <revision>
      <id>119604</id>
      <parentid>119384</parentid>
      <timestamp>2023-07-09T06:50:17Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119604</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2583" sha1="008do24mihfjg7ktk6xbr8b9qxfit26" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

== Theory ==
1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13- and 17-limit, but even so is distinctly [[consistent]] through the [[17-odd-limit]]. 

It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106}}

=== Subsets and supersets ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 1753 -1106 }}
| {{val| 1106 1753 }}
| -0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| 40 -56 21 }}
| {{val| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{val| 1106 1753 2568 3105 }}
| -0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{val| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{val| 1106 1753 2568 3105 3826 4093 }}
| -0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{val| 1106 1753 2568 3105 3826 4093 4521 }}
| -0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 401\1106&lt;br&gt;(152\1106)
| 435.081&lt;br&gt;(164.919)
| 9/7&lt;br&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}</text>
      <sha1>008do24mihfjg7ktk6xbr8b9qxfit26</sha1>
    </revision>
    <revision>
      <id>121880</id>
      <parentid>119604</parentid>
      <timestamp>2023-08-10T16:13:38Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Prime harmonics */</comment>
      <origin>121880</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2594" sha1="nvxovdlwy2n12vur7g10vo7pwy4x6n5" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

== Theory ==
1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13- and 17-limit, but even so is distinctly [[consistent]] through the [[17-odd-limit]]. 

It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106|columns=12}}

=== Subsets and supersets ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 1753 -1106 }}
| {{val| 1106 1753 }}
| -0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| 40 -56 21 }}
| {{val| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{val| 1106 1753 2568 3105 }}
| -0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{val| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{val| 1106 1753 2568 3105 3826 4093 }}
| -0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{val| 1106 1753 2568 3105 3826 4093 4521 }}
| -0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 401\1106&lt;br&gt;(152\1106)
| 435.081&lt;br&gt;(164.919)
| 9/7&lt;br&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}</text>
      <sha1>nvxovdlwy2n12vur7g10vo7pwy4x6n5</sha1>
    </revision>
    <revision>
      <id>126112</id>
      <parentid>121880</parentid>
      <timestamp>2023-10-17T13:16:35Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Clarify the title row of the rank-2 temp table</comment>
      <origin>126112</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2728" sha1="joue2e51io57ovnojrae2udjfj8rcf9" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

== Theory ==
1106edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13- and 17-limit, but even so is distinctly [[consistent]] through the [[17-odd-limit]]. 

It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106|columns=12}}

=== Subsets and supersets ===
Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 1753 -1106 }}
| {{val| 1106 1753 }}
| -0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| 40 -56 21 }}
| {{val| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{val| 1106 1753 2568 3105 }}
| -0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{val| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{val| 1106 1753 2568 3105 3826 4093 }}
| -0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{val| 1106 1753 2568 3105 3826 4093 4521 }}
| -0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 401\1106&lt;br&gt;(152\1106)
| 435.081&lt;br&gt;(164.919)
| 9/7&lt;br&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct</text>
      <sha1>joue2e51io57ovnojrae2udjfj8rcf9</sha1>
    </revision>
    <revision>
      <id>126666</id>
      <parentid>126112</parentid>
      <timestamp>2023-10-31T10:29:50Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Adopt template: Factorization; misc. cleanup</comment>
      <origin>126666</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2769" sha1="fhv6wkuur1h3eur3dztlya0dfdx9co0" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

== Theory ==
1106edo is a [[zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13- and 17-limit, but even so is [[consistency|distinctly consistent]] through the [[17-odd-limit]]. 

It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and notably [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106|columns=12}}

=== Subsets and supersets ===
Since 1106 factors into {{factorization|1106}}, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 1753 -1106 }}
| {{mapping| 1106 1753 }}
| -0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| 40 -56 21 }}
| {{mapping| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{mapping| 1106 1753 2568 3105 }}
| -0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{mapping| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{mapping| 1106 1753 2568 3105 3826 4093 }}
| -0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| -0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 401\1106&lt;br&gt;(152\1106)
| 435.081&lt;br&gt;(164.919)
| 9/7&lt;br&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct</text>
      <sha1>fhv6wkuur1h3eur3dztlya0dfdx9co0</sha1>
    </revision>
    <revision>
      <id>126667</id>
      <parentid>126666</parentid>
      <timestamp>2023-10-31T10:38:40Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Expand on theory</comment>
      <origin>126667</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3189" sha1="7e70eqt3xx3xftagzf4jw29uw1nytu8" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

== Theory ==
1106edo is a [[zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13- and 17-limit, but even so is [[consistency|distinctly consistent]] through the [[17-odd-limit]]. 

The equal temperament [[tempering out|tempers out]] {{monzo| -53 10 16 }} (kwazy comma) and {{monzo| -13 -46 37 }} (supermajor comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit; [[4096/4095]], 78125/78078, and 105644/105625 in the 13-limit; [[2500/2499]], [[4914/4913]], and 8624/8619 in the 17-limit. It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106|columns=12}}

=== Subsets and supersets ===
Since 1106 factors into {{factorization|1106}}, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 1753 -1106 }}
| {{mapping| 1106 1753 }}
| -0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| -13 -46 37 }}
| {{mapping| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{mapping| 1106 1753 2568 3105 }}
| -0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{mapping| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{mapping| 1106 1753 2568 3105 3826 4093 }}
| -0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| -0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 150\1106
| 162.749
| 1125/1024
| [[Kwazy]]
|-
| 2
| 401\1106&lt;br&gt;(152\1106)
| 435.081&lt;br&gt;(164.919)
| 9/7&lt;br&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct</text>
      <sha1>7e70eqt3xx3xftagzf4jw29uw1nytu8</sha1>
    </revision>
    <revision>
      <id>165894</id>
      <parentid>126667</parentid>
      <timestamp>2024-11-16T04:23:42Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>165894</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2693" sha1="25x7m6vs4maxr1sqrz114sbkdjr4553" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

== Theory ==
1106edo is a [[zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13- and 17-limit, but even so is [[consistency|distinctly consistent]] through the [[17-odd-limit]]. 

The equal temperament [[tempering out|tempers out]] {{monzo| -53 10 16 }} (kwazy comma) and {{monzo| -13 -46 37 }} (supermajor comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit; [[4096/4095]], 78125/78078, and 105644/105625 in the 13-limit; [[2500/2499]], [[4914/4913]], and 8624/8619 in the 17-limit. It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106|columns=12}}

=== Subsets and supersets ===
Since 1106 factors into {{factorization|1106}}, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{{comma basis begin}}
|-
| 2.3
| {{monzo| 1753 -1106 }}
| {{mapping| 1106 1753 }}
| &amp;minus;0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| -13 -46 37 }}
| {{mapping| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{mapping| 1106 1753 2568 3105 }}
| &amp;minus;0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{mapping| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{mapping| 1106 1753 2568 3105 3826 4093 }}
| &amp;minus;0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| &amp;minus;0.021
| 0.045
| 4.11
{{comma basis end}}

=== Rank-2 temperaments ===
{{rank-2 begin}}
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 150\1106
| 162.749
| 1125/1024
| [[Kwazy]]
|-
| 2
| 401\1106&lt;br /&gt;(152\1106)
| 435.081&lt;br /&gt;(164.919)
| 9/7&lt;br /&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br /&gt;(5\1106)
| 519.711&lt;br /&gt;(5.424)
| 27/20&lt;br /&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br /&gt;(11\1106)
| 498.011&lt;br /&gt;(11.935)
| 4/3&lt;br /&gt;(?)
| [[Gold]]
{{rank-2 end}}
{{orf}}</text>
      <sha1>25x7m6vs4maxr1sqrz114sbkdjr4553</sha1>
    </revision>
    <revision>
      <id>166184</id>
      <parentid>165894</parentid>
      <timestamp>2024-11-16T12:34:46Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Partial undo</comment>
      <origin>166184</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3250" sha1="kf37spaowidv67p780nfs83u3e6b04g" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

== Theory ==
1106edo is a [[zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13- and 17-limit, but even so is [[consistency|distinctly consistent]] through the [[17-odd-limit]]. 

The equal temperament [[tempering out|tempers out]] {{monzo| -53 10 16 }} (kwazy comma) and {{monzo| -13 -46 37 }} (supermajor comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit; [[4096/4095]], 78125/78078, and 105644/105625 in the 13-limit; [[2500/2499]], [[4914/4913]], and 8624/8619 in the 17-limit. It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106|columns=12}}

=== Subsets and supersets ===
Since 1106 factors into {{factorization|1106}}, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 1753 -1106 }}
| {{mapping| 1106 1753 }}
| &amp;minus;0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| -13 -46 37 }}
| {{mapping| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{mapping| 1106 1753 2568 3105 }}
| &amp;minus;0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{mapping| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{mapping| 1106 1753 2568 3105 3826 4093 }}
| &amp;minus;0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| &amp;minus;0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 150\1106
| 162.749
| 1125/1024
| [[Kwazy]]
|-
| 2
| 401\1106&lt;br /&gt;(152\1106)
| 435.081&lt;br /&gt;(164.919)
| 9/7&lt;br /&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br /&gt;(5\1106)
| 519.711&lt;br /&gt;(5.424)
| 27/20&lt;br /&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br /&gt;(11\1106)
| 498.011&lt;br /&gt;(11.935)
| 4/3&lt;br /&gt;(?)
| [[Gold]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct</text>
      <sha1>kf37spaowidv67p780nfs83u3e6b04g</sha1>
    </revision>
    <revision>
      <id>176266</id>
      <parentid>166184</parentid>
      <timestamp>2025-01-15T18:16:28Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176266</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3230" sha1="a4gbcudve1t1gx0mimgl6b1wtx2dgjq" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1106}}

== Theory ==
1106edo is a [[zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441, and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342, and 612 }}. It is less strong in the 13- and 17-limit, but even so is [[consistency|distinctly consistent]] through the [[17-odd-limit]]. 

The equal temperament [[tempering out|tempers out]] {{monzo| -53 10 16 }} (kwazy comma) and {{monzo| -13 -46 37 }} (supermajor comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit; [[4096/4095]], 78125/78078, and 105644/105625 in the 13-limit; [[2500/2499]], [[4914/4913]], and 8624/8619 in the 17-limit. It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106|columns=12}}

=== Subsets and supersets ===
Since 1106 factors into {{factorization|1106}}, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 1753 -1106 }}
| {{mapping| 1106 1753 }}
| −0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| -13 -46 37 }}
| {{mapping| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{mapping| 1106 1753 2568 3105 }}
| −0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{mapping| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{mapping| 1106 1753 2568 3105 3826 4093 }}
| −0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| −0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 150\1106
| 162.749
| 1125/1024
| [[Kwazy]]
|-
| 2
| 401\1106&lt;br /&gt;(152\1106)
| 435.081&lt;br /&gt;(164.919)
| 9/7&lt;br /&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br /&gt;(5\1106)
| 519.711&lt;br /&gt;(5.424)
| 27/20&lt;br /&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br /&gt;(11\1106)
| 498.011&lt;br /&gt;(11.935)
| 4/3&lt;br /&gt;(?)
| [[Gold]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct</text>
      <sha1>a4gbcudve1t1gx0mimgl6b1wtx2dgjq</sha1>
    </revision>
    <revision>
      <id>181316</id>
      <parentid>176266</parentid>
      <timestamp>2025-02-18T16:50:34Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181316</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3224" sha1="q79ipksme76634d40wsyk1036dvtr10" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1106edo is a [[zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441, and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342, and 612 }}. It is less strong in the 13- and 17-limit, but even so is [[consistency|distinctly consistent]] through the [[17-odd-limit]]. 

The equal temperament [[tempering out|tempers out]] {{monzo| -53 10 16 }} (kwazy comma) and {{monzo| -13 -46 37 }} (supermajor comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit; [[4096/4095]], 78125/78078, and 105644/105625 in the 13-limit; [[2500/2499]], [[4914/4913]], and 8624/8619 in the 17-limit. It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106|columns=12}}

=== Subsets and supersets ===
Since 1106 factors into {{factorization|1106}}, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 1753 -1106 }}
| {{mapping| 1106 1753 }}
| −0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{monzo| -53 10 16 }}, {{monzo| -13 -46 37 }}
| {{mapping| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{mapping| 1106 1753 2568 3105 }}
| −0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{mapping| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{mapping| 1106 1753 2568 3105 3826 4093 }}
| −0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| −0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 150\1106
| 162.749
| 1125/1024
| [[Kwazy]]
|-
| 2
| 401\1106&lt;br /&gt;(152\1106)
| 435.081&lt;br /&gt;(164.919)
| 9/7&lt;br /&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br /&gt;(5\1106)
| 519.711&lt;br /&gt;(5.424)
| 27/20&lt;br /&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br /&gt;(11\1106)
| 498.011&lt;br /&gt;(11.935)
| 4/3&lt;br /&gt;(?)
| [[Gold]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct</text>
      <sha1>q79ipksme76634d40wsyk1036dvtr10</sha1>
    </revision>
    <revision>
      <id>184816</id>
      <parentid>181316</parentid>
      <timestamp>2025-03-06T11:31:20Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ crazy update</comment>
      <origin>184816</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3199" sha1="0fpup1zhdtk5nk2cckg9gn5nitanq1x" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1106edo is a [[zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441, and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342, and 612 }}. It is less strong in the 13- and 17-limit, but even so is [[consistency|distinctly consistent]] through the [[17-odd-limit]]. 

The equal temperament [[tempering out|tempers out]] {{monzo| -53 10 16 }} (kwazy comma) and {{monzo| -13 -46 37 }} (supermajor comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit; [[4096/4095]], 78125/78078, and 105644/105625 in the 13-limit; [[2500/2499]], [[4914/4913]], and 8624/8619 in the 17-limit. It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106|columns=12}}

=== Subsets and supersets ===
Since 1106 factors into {{factorization|1106}}, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 1753 -1106 }}
| {{Mapping| 1106 1753 }}
| −0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{Monzo| -53 10 16 }}, {{monzo| -13 -46 37 }}
| {{Mapping| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{Mapping| 1106 1753 2568 3105 }}
| −0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{Mapping| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{Mapping| 1106 1753 2568 3105 3826 4093 }}
| −0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{Mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| −0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 150\1106
| 162.749
| 1125/1024
| [[Crazy]]
|-
| 2
| 401\1106&lt;br&gt;(152\1106)
| 435.081&lt;br&gt;(164.919)
| 9/7&lt;br&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct</text>
      <sha1>0fpup1zhdtk5nk2cckg9gn5nitanq1x</sha1>
    </revision>
    <revision>
      <id>214789</id>
      <parentid>184816</parentid>
      <timestamp>2025-10-27T13:13:35Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Linking &amp; cleanup</comment>
      <origin>214789</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3227" sha1="9zoodx6rnc3r1g437fprj0v3tu01qw7" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1106edo is a [[zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441, and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342, and 612 }}. It is less strong in the 13- and 17-limit, but even so is [[consistency|distinctly consistent]] through the [[17-odd-limit]]. 

The equal temperament [[tempering out|tempers out]] {{monzo| -53 10 16 }} (kwazy comma) and {{monzo| -13 -46 37 }} (supermajor comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit; [[4096/4095]], 78125/78078, and 105644/105625 in the 13-limit; [[2500/2499]], [[4914/4913]], and 8624/8619 in the 17-limit. It notably supports [[supermajor (temperament)|supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106|columns=11}}

=== Subsets and supersets ===
Since 1106 factors into {{nowrap| 2 × 7 × 79 }}, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 1753 -1106 }}
| {{Mapping| 1106 1753 }}
| −0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{Monzo| -53 10 16 }}, {{monzo| -13 -46 37 }}
| {{Mapping| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{Mapping| 1106 1753 2568 3105 }}
| −0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{Mapping| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{Mapping| 1106 1753 2568 3105 3826 4093 }}
| −0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{Mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| −0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor]]
|-
| 2
| 150\1106
| 162.749
| 1125/1024
| [[Crazy]]
|-
| 2
| 401\1106&lt;br&gt;(152\1106)
| 435.081&lt;br&gt;(164.919)
| 9/7&lt;br&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}
&lt;nowiki/&gt;* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct</text>
      <sha1>9zoodx6rnc3r1g437fprj0v3tu01qw7</sha1>
    </revision>
    <revision>
      <id>214790</id>
      <parentid>214789</parentid>
      <timestamp>2025-10-27T13:14:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+1</comment>
      <origin>214790</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3252" sha1="n8e2k6wxj2dxc90o9or9r9hie2u9vp0" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1106edo is a [[zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441, and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342, and 612 }}. It is less strong in the 13- and 17-limit, but even so is [[consistency|distinctly consistent]] through the [[17-odd-limit]]. 

The equal temperament [[tempering out|tempers out]] {{monzo| -53 10 16 }} (kwazy comma) and {{monzo| -13 -46 37 }} (supermajor comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit; [[4096/4095]], 78125/78078, and 105644/105625 in the 13-limit; [[2500/2499]], [[4914/4913]], and 8624/8619 in the 17-limit. It notably supports [[supermajor (temperament)|supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106|columns=11}}

=== Subsets and supersets ===
Since 1106 factors into {{nowrap| 2 × 7 × 79 }}, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 1753 -1106 }}
| {{Mapping| 1106 1753 }}
| −0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{Monzo| -53 10 16 }}, {{monzo| -13 -46 37 }}
| {{Mapping| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{Mapping| 1106 1753 2568 3105 }}
| −0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{Mapping| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{Mapping| 1106 1753 2568 3105 3826 4093 }}
| −0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{Mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| −0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor (temperament)|Supermajor]]
|-
| 2
| 150\1106
| 162.749
| 1125/1024
| [[Crazy]]
|-
| 2
| 401\1106&lt;br&gt;(152\1106)
| 435.081&lt;br&gt;(164.919)
| 9/7&lt;br&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}
&lt;nowiki/&gt;* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct</text>
      <sha1>n8e2k6wxj2dxc90o9or9r9hie2u9vp0</sha1>
    </revision>
    <revision>
      <id>225706</id>
      <parentid>214790</parentid>
      <timestamp>2026-03-13T13:30:00Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct" to "[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct"</comment>
      <origin>225706</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3307" sha1="6t1w8myotdal0bt7mjgx8s2b1mirrii" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1106edo is a [[zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441, and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342, and 612 }}. It is less strong in the 13- and 17-limit, but even so is [[consistency|distinctly consistent]] through the [[17-odd-limit]]. 

The equal temperament [[tempering out|tempers out]] {{monzo| -53 10 16 }} (kwazy comma) and {{monzo| -13 -46 37 }} (supermajor comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit; [[4096/4095]], 78125/78078, and 105644/105625 in the 13-limit; [[2500/2499]], [[4914/4913]], and 8624/8619 in the 17-limit. It notably supports [[supermajor (temperament)|supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].

=== Prime harmonics ===
{{Harmonics in equal|1106|columns=11}}

=== Subsets and supersets ===
Since 1106 factors into {{nowrap| 2 × 7 × 79 }}, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 1753 -1106 }}
| {{Mapping| 1106 1753 }}
| −0.010
| 0.010
| 0.99
|-
| 2.3.5
| {{Monzo| -53 10 16 }}, {{monzo| -13 -46 37 }}
| {{Mapping| 1106 1753 2568 }}
| +0.001
| 0.019
| 1.73
|-
| 2.3.5.7
| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
| {{Mapping| 1106 1753 2568 3105 }}
| −0.006
| 0.020
| 1.83
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
| {{Mapping| 1106 1753 2568 3105 3826 }}
| +0.004
| 0.026
| 2.38
|-
| 2.3.5.7.11.13
| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
| {{Mapping| 1106 1753 2568 3105 3826 4093 }}
| −0.012
| 0.043
| 3.94
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
| {{Mapping| 1106 1753 2568 3105 3826 4093 4521 }}
| −0.021
| 0.045
| 4.11
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 213\1106
| 231.103
| 8/7
| [[Orga]] (11-limit)
|-
| 1
| 401\1106
| 435.081
| 9/7
| [[Supermajor (temperament)|Supermajor]]
|-
| 2
| 150\1106
| 162.749
| 1125/1024
| [[Crazy]]
|-
| 2
| 401\1106&lt;br&gt;(152\1106)
| 435.081&lt;br&gt;(164.919)
| 9/7&lt;br&gt;(11/10)
| [[Semisupermajor]]
|-
| 7
| 479\1106&lt;br&gt;(5\1106)
| 519.711&lt;br&gt;(5.424)
| 27/20&lt;br&gt;(5120/5103)
| [[Brahmagupta]] (7-limit)
|-
| 79
| 459\1106&lt;br&gt;(11\1106)
| 498.011&lt;br&gt;(11.935)
| 4/3&lt;br&gt;(?)
| [[Gold]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct</text>
      <sha1>6t1w8myotdal0bt7mjgx8s2b1mirrii</sha1>
    </revision>
  </page>
  <page>
    <title>111edo</title>
    <ns>0</ns>
    <id>56</id>
    <revision>
      <id>56</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>56</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1537" sha1="dps14kx6w9p6pzxf7fa18ybhuautajr" xml:space="preserve">'''111edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 111 parts, each of size 10.81 [[cent|cent]]s. It is [[consistent|consistent]] through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[Optimal_patent_val|optimal patent val]] tuning for the corresponding [[11-limit|11-limit]] rank four temperament. In fact in the [[7-limit|7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and buzzard, the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic_family|guanyin temperament]], the [[Planar_Temperament|planar temperament]] [[tempering_out|tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is

&lt;math&gt;111 = 3 \cdot 37&lt;/math&gt;

== Music ==

[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene_Ward_Smith|Gene Ward Smith]]      [[Category:buzzard]]
[[Category:edo]]
[[Category:listen]]
[[Category:semisept]]
[[Category:theory]]</text>
      <sha1>dps14kx6w9p6pzxf7fa18ybhuautajr</sha1>
    </revision>
    <revision>
      <id>4281</id>
      <timestamp>2011-08-06T03:34:07Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 244584911 - Original comment: **</comment>
      <origin>4281</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4981" sha1="kbtla7esg10t9ooh2f9o7kxscplnhoc" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-06 03:34:07 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;244584911&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**111edo** is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent]]s. It is [[consistent]] through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and buzzard, the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic family|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is
[[math]]
111 = 3 \cdot 37
[[math]]

== Music ==

[[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297|Trio for SoftSaturn, NebulaSing and TromBonehead]] [[http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3|play]] by [[Gene Ward Smith]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;111edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;111edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 111 parts, each of size 10.81 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is &amp;lt;a class="wiki_link" href="/consistent"&amp;gt;consistent&amp;lt;/a&amp;gt; through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; tuning for the corresponding &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; rank four temperament. In fact in the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;amp;111 temperament, and buzzard, the 58&amp;amp;amp;111 temperament. The Trio piece below is in &amp;lt;a class="wiki_link" href="/Orwellismic%20family"&amp;gt;guanyin temperament&amp;lt;/a&amp;gt;, the &amp;lt;a class="wiki_link" href="/planar%20temperament"&amp;gt;planar temperament&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempering out&amp;lt;/a&amp;gt; 176/175 and 540/539, for which 111 also provides the optimal patent val.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The prime factorization is&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMathRule:0:
[[math]]&amp;amp;lt;br/&amp;amp;gt;
111 = 3 \cdot 37&amp;amp;lt;br/&amp;amp;gt;[[math]]
 --&amp;gt;&amp;lt;script type="math/tex"&amp;gt;111 = 3 \cdot 37&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMathRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; Music &amp;lt;/h2&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297" rel="nofollow"&amp;gt;Trio for SoftSaturn, NebulaSing and TromBonehead&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&amp;gt;Gene Ward Smith&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>kbtla7esg10t9ooh2f9o7kxscplnhoc</sha1>
    </revision>
    <revision>
      <id>4282</id>
      <timestamp>2011-06-29T09:38:20Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 239311847 - Original comment: **</comment>
      <origin>4282</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5079" sha1="gwhghf7o81i3lkxh9dbn5myqytgjthz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 09:38:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239311847&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**111edo** is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent]]s. It is [[consistent]] through to the 21 odd limit, and the [[tonality diamond]] is distinct through to the [[15-limit]], marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and buzzard, the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic family|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is
[[math]]
111 = 3 \cdot 37
[[math]]

== Music ==

[[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297|Trio for SoftSaturn, NebulaSing and TromBonehead]] [[http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3|play]] by [[Gene Ward Smith]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;111edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;111edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 111 parts, each of size 10.81 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is &amp;lt;a class="wiki_link" href="/consistent"&amp;gt;consistent&amp;lt;/a&amp;gt; through to the 21 odd limit, and the &amp;lt;a class="wiki_link" href="/tonality%20diamond"&amp;gt;tonality diamond&amp;lt;/a&amp;gt; is distinct through to the &amp;lt;a class="wiki_link" href="/15-limit"&amp;gt;15-limit&amp;lt;/a&amp;gt;, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; tuning for the corresponding &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; rank four temperament. In fact in the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;amp;111 temperament, and buzzard, the 58&amp;amp;amp;111 temperament. The Trio piece below is in &amp;lt;a class="wiki_link" href="/Orwellismic%20family"&amp;gt;guanyin temperament&amp;lt;/a&amp;gt;, the &amp;lt;a class="wiki_link" href="/planar%20temperament"&amp;gt;planar temperament&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempering out&amp;lt;/a&amp;gt; 176/175 and 540/539, for which 111 also provides the optimal patent val.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The prime factorization is&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMathRule:0:
[[math]]&amp;amp;lt;br/&amp;amp;gt;
111 = 3 \cdot 37&amp;amp;lt;br/&amp;amp;gt;[[math]]
 --&amp;gt;&amp;lt;script type="math/tex"&amp;gt;111 = 3 \cdot 37&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMathRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; Music &amp;lt;/h2&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297" rel="nofollow"&amp;gt;Trio for SoftSaturn, NebulaSing and TromBonehead&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&amp;gt;Gene Ward Smith&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <timestamp>2011-06-28T02:41:35Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 239085907 - Original comment: **</comment>
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      <text bytes="4515" sha1="d9365rjunyt03uge2ze0up32jwf2ooh" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2011-06-28 02:41:35 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239085907&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //111 equal temperament// divides the [[octave]] into 111 equal parts, each of size 10.81 [[cent]]s. It is [[consistent]] through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and buzzard, the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic family|guanyin temperament]], the [[planar temperament]] tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.

=Music= 

[[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297|Trio for SoftSaturn, NebulaSing and TromBonehead]] [[http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3|play]] by [[Gene Ward Smith]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;111edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;111 equal temperament&amp;lt;/em&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 111 equal parts, each of size 10.81 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is &amp;lt;a class="wiki_link" href="/consistent"&amp;gt;consistent&amp;lt;/a&amp;gt; through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; tuning for the corresponding &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; rank four temperament. In fact in the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;amp;111 temperament, and buzzard, the 58&amp;amp;amp;111 temperament. The Trio piece below is in &amp;lt;a class="wiki_link" href="/Orwellismic%20family"&amp;gt;guanyin temperament&amp;lt;/a&amp;gt;, the &amp;lt;a class="wiki_link" href="/planar%20temperament"&amp;gt;planar temperament&amp;lt;/a&amp;gt; tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297" rel="nofollow"&amp;gt;Trio for SoftSaturn, NebulaSing and TromBonehead&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&amp;gt;Gene Ward Smith&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <timestamp>2011-06-11T11:48:05Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 235936714 - Original comment: some links**</comment>
      <origin>4284</origin>
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      <text bytes="4465" sha1="2e1x02s3r8jc2lfvhpsgh912hcghic4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-11 11:48:05 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;235936714&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;some links&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //111 equal temperament// divides the [[octave]] into 111 equal parts, each of size 10.81 [[cent]]s. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and buzzard, the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic family|guanyin temperament]], the [[planar temperament]] tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.

=Music=

[[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297|Trio for SoftSaturn, NebulaSing and TromBonehead]]  [[http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3|play]] by [[Gene Ward Smith]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;111edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;111 equal temperament&amp;lt;/em&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 111 equal parts, each of size 10.81 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; tuning for the corresponding &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; rank four temperament. In fact in the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;amp;111 temperament, and buzzard, the 58&amp;amp;amp;111 temperament. The Trio piece below is in &amp;lt;a class="wiki_link" href="/Orwellismic%20family"&amp;gt;guanyin temperament&amp;lt;/a&amp;gt;, the &amp;lt;a class="wiki_link" href="/planar%20temperament"&amp;gt;planar temperament&amp;lt;/a&amp;gt; tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297" rel="nofollow"&amp;gt;Trio for SoftSaturn, NebulaSing and TromBonehead&amp;lt;/a&amp;gt;  &amp;lt;a class="wiki_link_ext" href="http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&amp;gt;Gene Ward Smith&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 234952346 - Original comment: **</comment>
      <origin>4285</origin>
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      <text bytes="4242" sha1="3ga85kqwoskcz2nk60bntfdrr9wy2rd" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-07 15:57:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234952346&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //111 equal temperament// divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and buzzard, the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic family|guanyin temperament]], the [[planar temperament]] tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.

=Music=

[[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297|Trio for SoftSaturn, NebulaSing and TromBonehead]]  [[http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3|play]] by [[Gene Ward Smith]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;111edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;111 equal temperament&amp;lt;/em&amp;gt; divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;amp;111 temperament, and buzzard, the 58&amp;amp;amp;111 temperament. The Trio piece below is in &amp;lt;a class="wiki_link" href="/Orwellismic%20family"&amp;gt;guanyin temperament&amp;lt;/a&amp;gt;, the &amp;lt;a class="wiki_link" href="/planar%20temperament"&amp;gt;planar temperament&amp;lt;/a&amp;gt; tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297" rel="nofollow"&amp;gt;Trio for SoftSaturn, NebulaSing and TromBonehead&amp;lt;/a&amp;gt;  &amp;lt;a class="wiki_link_ext" href="http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&amp;gt;Gene Ward Smith&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 234944316 - Original comment: **</comment>
      <origin>4286</origin>
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      <text bytes="3767" sha1="a7gp7jmsw3rk8qmbdvsizqg3z099qey" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-07 15:22:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234944316&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //111 equal temperament// divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and buzzard, the 58&amp;amp;111 temperament.

=Music=

[[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297|Trio for SoftSaturn, NebulaSing and TromBonehead]]  [[http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3|play]] by [[Gene Ward Smith]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;111edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;111 equal temperament&amp;lt;/em&amp;gt; divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;amp;111 temperament, and buzzard, the 58&amp;amp;amp;111 temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297" rel="nofollow"&amp;gt;Trio for SoftSaturn, NebulaSing and TromBonehead&amp;lt;/a&amp;gt;  &amp;lt;a class="wiki_link_ext" href="http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&amp;gt;Gene Ward Smith&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 234944092 - Original comment: **</comment>
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      <text bytes="3581" sha1="1qfj997x18o94xg87vzqy3qwe01bnxf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-07 15:21:40 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234944092&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //111 equal temperament// divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and buzzard, the 58&amp;amp;111 temperament.

=Music=[[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297|Trio for SoftSaturn, NebulaSing and TromBonehead]] [[http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3|play]] by [[Gene Ward Smith]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;111edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;111 equal temperament&amp;lt;/em&amp;gt; divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;amp;111 temperament, and buzzard, the 58&amp;amp;amp;111 temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
=Music=&amp;lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297" rel="nofollow"&amp;gt;Trio for SoftSaturn, NebulaSing and TromBonehead&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Gene%20Ward%20Smith"&amp;gt;Gene Ward Smith&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 213359042 - Original comment: **</comment>
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      <text bytes="3209" sha1="s4alp91tcazurz4hzrny5p1bt0tzfdp" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-23 17:14:37 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;213359042&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //111 equal temperament// divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and buzzard, the 58&amp;amp;111 temperament.

Music: [[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297|Trio for SoftSaturn, NebulaSing and TromBonehead]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;111edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;111 equal temperament&amp;lt;/em&amp;gt; divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;amp;111 temperament, and buzzard, the 58&amp;amp;amp;111 temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Music: &amp;lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297" rel="nofollow"&amp;gt;Trio for SoftSaturn, NebulaSing and TromBonehead&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 199492864 - Original comment: **</comment>
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      <text bytes="3099" sha1="22xrkif4c653ygrf3nyx12809e5phvz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-02-07 16:38:35 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;199492864&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //111 equal temperament// divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and buzzard, the 58&amp;amp;111 temperament.

Music: [[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297|Trio for SoftSaturn, NebulaSing and TromBonehead]]&lt;/pre&gt;&lt;/div&gt;
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&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;111edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;111 equal temperament&amp;lt;/em&amp;gt; divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;amp;111 temperament, and buzzard, the 58&amp;amp;amp;111 temperament.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Music: &amp;lt;a class="wiki_link_ext" href="http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297" rel="nofollow"&amp;gt;Trio for SoftSaturn, NebulaSing and TromBonehead&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 190297104 - Original comment: **</comment>
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      <text bytes="2738" sha1="mpkbz9bfrayetr93bxts9bpx960gkb8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2010-12-30 14:46:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;190297104&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //111 equal temperament// divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and buzzard, the 58&amp;amp;111 temperament.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;111edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;111 equal temperament&amp;lt;/em&amp;gt; divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;amp;111 temperament, and buzzard, the 58&amp;amp;amp;111 temperament.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 155516399 - Original comment: **</comment>
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      <text bytes="2738" sha1="in78ccjjr1eizck3gz2scitx07gf8us" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2010-08-07 02:23:03 UTC&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //111 equal temperament// divides the octave into 111 equal parts, each of size 10.81 cents. It is consistent through to the 21 odd limit, and the tonality diamond is distinct through to the 15 limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent tuning for the corresponding 11-limit rank four temperament. In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372 and 540/539. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and vulture, the 58&amp;amp;111 temperament.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>49131</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1559" sha1="eepfr8urnhloyslpp0622e4yhbgno34" xml:space="preserve">'''111edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 111 parts, each of size 10.81 [[cent|cent]]s. It is [[consistent|consistent]] through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[Optimal_patent_val|optimal patent val]] tuning for the corresponding [[11-limit|11-limit]] rank four temperament. In fact in the [[7-limit|7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372, 540/539 and 117440512/117406179. It is a particularly good tuning for the 11- or 13- versions of semisept, the 31&amp;amp;111 temperament, and buzzard, the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic_family|guanyin temperament]], the [[Planar_Temperament|planar temperament]] [[tempering_out|tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is

&lt;math&gt;111 = 3 \cdot 37&lt;/math&gt;

== Music ==

[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene_Ward_Smith|Gene Ward Smith]]      
[[Category:buzzard]]
[[Category:edo]]
[[Category:listen]]
[[Category:semisept]]
[[Category:theory]]</text>
      <sha1>eepfr8urnhloyslpp0622e4yhbgno34</sha1>
    </revision>
    <revision>
      <id>49173</id>
      <parentid>49131</parentid>
      <timestamp>2020-09-11T10:03:06Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Cleanup</comment>
      <origin>49173</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1517" sha1="i4kdgf83wqu53lbaz540khwndybx6f6" xml:space="preserve">'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent|cents]]. 

111edo is [[consistent]] through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372, 540/539 and 117440512/117406179. It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic_family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is

&lt;math&gt;111 = 3 \cdot 37&lt;/math&gt;

== Music ==

[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Edo]]
[[Category:111edo]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>i4kdgf83wqu53lbaz540khwndybx6f6</sha1>
    </revision>
    <revision>
      <id>49216</id>
      <parentid>49173</parentid>
      <timestamp>2020-09-11T18:29:25Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>I removed the singleton category. Don't hesitate to re-add it if there is more than this page to collect!</comment>
      <origin>49216</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1497" sha1="6uhhcs493wc060tj83iermini876mm6" xml:space="preserve">'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent|cents]]. 

111edo is [[consistent]] through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372, 540/539 and 117440512/117406179. It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic_family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is

&lt;math&gt;111 = 3 \cdot 37&lt;/math&gt;

== Music ==

[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>6uhhcs493wc060tj83iermini876mm6</sha1>
    </revision>
    <revision>
      <id>49443</id>
      <parentid>49216</parentid>
      <timestamp>2020-09-13T20:27:12Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>49443</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1511" sha1="szl229dei7nj81xjhkf1yqe1qcx6wjo" xml:space="preserve">'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent|cents]]. 

111edo is [[consistent]] through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372, 540/539 and [[Quartisma|117440512/117406179]]. It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic_family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is

&lt;math&gt;111 = 3 \cdot 37&lt;/math&gt;

== Music ==

[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>szl229dei7nj81xjhkf1yqe1qcx6wjo</sha1>
    </revision>
    <revision>
      <id>54957</id>
      <parentid>49443</parentid>
      <timestamp>2020-12-04T23:13:05Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54957</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1537" sha1="20veq6w9p19p5sbfq0sv8m3peci6lye" xml:space="preserve">'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent|cents]]. 

111edo is [[consistent]] through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372, 540/539 and [[Quartisma|117440512/117406179]]. It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic_family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is

&lt;math&gt;111 = 3 \cdot 37&lt;/math&gt;

== Music ==

[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>20veq6w9p19p5sbfq0sv8m3peci6lye</sha1>
    </revision>
    <revision>
      <id>68087</id>
      <parentid>54957</parentid>
      <timestamp>2021-04-21T04:28:37Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68087</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3404" sha1="haempddb0ewxbsh3mkr7u5yqg93hw30" xml:space="preserve">'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent|cents]]. 

111edo is [[consistent]] through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372, 540/539 and [[Quartisma|117440512/117406179]]. It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic_family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is

&lt;math&gt;111 = 3 \cdot 37&lt;/math&gt;


Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments. As 10*[[11edo]], it is the first edo which allows one to use consecutive smaller edos as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[11L 3s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}
== Music ==

[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>haempddb0ewxbsh3mkr7u5yqg93hw30</sha1>
    </revision>
    <revision>
      <id>68089</id>
      <parentid>68087</parentid>
      <timestamp>2021-04-21T04:49:45Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68089</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3404" sha1="5tnt7esyafshqwirkgavh0xpw50bxmu" xml:space="preserve">'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent|cents]]. 

111edo is [[consistent]] through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372, 540/539 and [[Quartisma|117440512/117406179]]. It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic_family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is

&lt;math&gt;111 = 3 \cdot 37&lt;/math&gt;


Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments. As 10*[[11edo]], it is the first edo which allows one to use consecutive smaller edos as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}
== Music ==

[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>5tnt7esyafshqwirkgavh0xpw50bxmu</sha1>
    </revision>
    <revision>
      <id>70405</id>
      <parentid>68089</parentid>
      <timestamp>2021-05-23T20:40:29Z</timestamp>
      <contributor>
        <username>Keenan Pepper</username>
        <id>25</id>
      </contributor>
      <comment>111 does not equal 10 times 11... no idea what this means, so removing it</comment>
      <origin>70405</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3289" sha1="db8pzgz9otgqiwrzwawtapj2z1791v8" xml:space="preserve">'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent|cents]]. 

111edo is [[consistent]] through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372, 540/539 and [[Quartisma|117440512/117406179]]. It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic_family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is

&lt;math&gt;111 = 3 \cdot 37&lt;/math&gt;


Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}
== Music ==

[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>db8pzgz9otgqiwrzwawtapj2z1791v8</sha1>
    </revision>
    <revision>
      <id>70990</id>
      <parentid>70405</parentid>
      <timestamp>2021-05-29T12:25:10Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>added prime errors</comment>
      <origin>70990</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3328" sha1="92fxd8lxm1iy7vk9jbbvhh7jocmahjb" xml:space="preserve">'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent|cents]]. 

111edo is [[consistent]] through to the 21 odd limit, and is the smallest edo uniquely consistent through the 15 odd limit, marking it as an important higher limit temperament. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out 176/175 and gives an excellent [[optimal patent val]] tuning for the corresponding [[11-limit]] rank four temperament. In fact in the [[7-limit]] it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 1331/1323, 176/175, 1375/1372, 540/539 and [[Quartisma|117440512/117406179]]. It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The Trio piece below is in [[Orwellismic_family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is

&lt;math&gt;111 = 3 \cdot 37&lt;/math&gt;
{{Primes in edo|111|columns=11|prec=2}}

Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}
== Music ==

[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>92fxd8lxm1iy7vk9jbbvhh7jocmahjb</sha1>
    </revision>
    <revision>
      <id>70992</id>
      <parentid>70990</parentid>
      <timestamp>2021-05-29T12:33:37Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Cleanup</comment>
      <origin>70992</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3338" sha1="5prbphytfthqy0uhq6bwssr92ysjt66" xml:space="preserve">'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent]]s. 

111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. It is also significant for lower limits, especially in terms of what it tempers out; for example it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] rank-4 temperament. In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, 1375/1372, and the [[quartisma]] 117440512/117406179. It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The trio piece below is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is 111 = 3 × 37. 

{{Primes in edo|111|columns=11|prec=2}}

Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}
== Music ==

[http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>5prbphytfthqy0uhq6bwssr92ysjt66</sha1>
    </revision>
    <revision>
      <id>72241</id>
      <parentid>70992</parentid>
      <timestamp>2021-06-09T07:45:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Some paragraphing</comment>
      <origin>72241</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3364" sha1="o68rl8tv5h52jtqapzarnq70j77jjxy" xml:space="preserve">'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. 

It is also significant for lower limits, especially in terms of what it tempers out; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] rank-4 temperament. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is 111 = 3 × 37. 

{{Primes in edo|111|columns=11|prec=2}}

Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>o68rl8tv5h52jtqapzarnq70j77jjxy</sha1>
    </revision>
    <revision>
      <id>72245</id>
      <parentid>72241</parentid>
      <timestamp>2021-06-09T08:56:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+table of rank-2 temperaments</comment>
      <origin>72245</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4146" sha1="re60362cniwmrb2a5988hfh9d97ufx0" xml:space="preserve">'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. 

It is also significant for lower limits, especially in terms of what it tempers out; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] rank-4 temperament. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is 111 = 3 × 37. 

{{Primes in edo|111|columns=11|prec=2}}

Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}

== Rank-2 temperaments ==
{| class="wikitable center-all"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]]/[[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>re60362cniwmrb2a5988hfh9d97ufx0</sha1>
    </revision>
    <revision>
      <id>74124</id>
      <parentid>72245</parentid>
      <timestamp>2021-07-11T11:57:58Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ the same prec is now estimated by EDO magnitude</comment>
      <origin>74124</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4139" sha1="i8ghcdtc8huzeshjiw19m23aam8abfj" xml:space="preserve">'''111edo''' is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. 

It is also significant for lower limits, especially in terms of what it tempers out; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] rank-4 temperament. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is 111 = 3 × 37. 

{{Primes in edo|111|columns=11}}

Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}

== Rank-2 temperaments ==
{| class="wikitable center-all"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]]/[[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>i8ghcdtc8huzeshjiw19m23aam8abfj</sha1>
    </revision>
    <revision>
      <id>75221</id>
      <parentid>74124</parentid>
      <timestamp>2021-07-29T10:40:39Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Improve intro</comment>
      <origin>75221</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4312" sha1="m3fda0iqn4lvmr8icft6ab75es4o4kq" xml:space="preserve">The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. 

It is also significant for lower limits, especially in terms of what it tempers out; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] rank-4 temperament. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is 111 = 3 × 37. 

{{Primes in edo|111|columns=11}}

Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}

== Rank-2 temperaments ==
{| class="wikitable center-all"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]]/[[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>m3fda0iqn4lvmr8icft6ab75es4o4kq</sha1>
    </revision>
    <revision>
      <id>75222</id>
      <parentid>75221</parentid>
      <timestamp>2021-07-29T10:43:20Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Improve sectioning</comment>
      <origin>75222</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4397" sha1="aueobr2dmqzmfj0jyxa99jjk3whk8fi" xml:space="preserve">The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. 

It is also significant for lower limits, especially in terms of what it tempers out; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] rank-4 temperament. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is 111 = 3 × 37. 

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>aueobr2dmqzmfj0jyxa99jjk3whk8fi</sha1>
    </revision>
    <revision>
      <id>75229</id>
      <parentid>75222</parentid>
      <timestamp>2021-07-29T13:09:49Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ +RTT table</comment>
      <origin>75229</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5539" sha1="6dgzn131kz47h2vzcsv4sbhuyq125fb" xml:space="preserve">The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size 10.81 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. 

It is also significant for lower limits, especially in terms of what it tempers out; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] rank-4 temperament. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is 111 = 3 × 37. 

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]</text>
      <sha1>6dgzn131kz47h2vzcsv4sbhuyq125fb</sha1>
    </revision>
    <revision>
      <id>76595</id>
      <parentid>75229</parentid>
      <timestamp>2021-08-20T07:07:11Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+infobox</comment>
      <origin>76595</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5761" sha1="r1c0q0age96qyjundgtspyz5pp2inwx" xml:space="preserve">{{Infobox ET
| Prime factorization = 3 × 37
| Step size = 10.81081¢
| Fifth = 65\111 (702.70¢)
| Major 2nd = 19\111 (205¢)
| Minor 2nd = 8\111 (86¢)
| Augmented 1sn = 11\111  (119¢)
}}

The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.8 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. 

It is also significant for lower limits, especially in terms of what it tempers out; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] rank-4 temperament. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;111 temperament, and [[buzzard]], the 58&amp;amp;111 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

The prime factorization is 111 = 3 × 37. 

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>r1c0q0age96qyjundgtspyz5pp2inwx</sha1>
    </revision>
    <revision>
      <id>76596</id>
      <parentid>76595</parentid>
      <timestamp>2021-08-20T07:37:40Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Expansion on theory</comment>
      <origin>76596</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5879" sha1="krr9s26ctpu6xd2ctoqffkz5xu95yvu" xml:space="preserve">{{Infobox ET
| Prime factorization = 3 × 37
| Step size = 10.81081¢
| Fifth = 65\111 (702.70¢)
| Major 2nd = 19\111 (205¢)
| Minor 2nd = 8\111 (86¢)
| Augmented 1sn = 11\111  (119¢)
}}

The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.8 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>krr9s26ctpu6xd2ctoqffkz5xu95yvu</sha1>
    </revision>
    <revision>
      <id>78458</id>
      <parentid>76596</parentid>
      <timestamp>2021-09-23T21:13:41Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ +subsedia</comment>
      <origin>78458</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6003" sha1="l9asioa6ysyuejps1lvpnxj1kppnscm" xml:space="preserve">{{Infobox ET
| Prime factorization = 3 × 37
| Step size = 10.81081¢
| Fifth = 65\111 (702.70¢)
| Major 2nd = 19\111 (205¢)
| Minor 2nd = 8\111 (86¢)
| Augmented 1sn = 11\111  (119¢)
}}

The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.8 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37et are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111edo has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111edo
!Tones
!Pattern
!L:s
|-
|5
| [[1L 4s]]
|23:22
|-
|6
|[[3L 3s]]
|19:18
|-
|7
|[[6L 1s]]
|16:15
|-
|8
| [[7L 1s]]
|14:13
|-
|9
|[[3L 6s]]
|13:12
|-
| 10
|[[1L 9s]]
|12:11
|-
| 11
|[[1L 10s]]
|11:10
|-
| 12
|[[3L 9s]]
|10:9
|-
|13
| [[6L 7s]]
|9:8
|-
|14
|[[13L 1s]]
| rowspan="2" |8:7
|-
|15
|[[6L 9s]]
|-
|16
|[[15L 1s]]
| rowspan="3" |7:6
|-
|17
|[[9L 8s]]
|-
|18
|3L 15s
|-
|19
|[[16L 3s]]
| rowspan="4" |6:5
|-
|20
|11L 9s
|-
|21
|6L 15s
|-
|22
|1L 21s
|-
|23
|19L 4s
| rowspan="5" | 5:4
|-
|24
|15L 9s
|-
|25
| 11L 14s
|-
|26
|7L 19s
|-
|27
| 3L 24s
|-
|28
|27L 1s
| rowspan="9" |4:3
|-
|29
|24L 5s
|-
|30
|21L 9s
|-
|31
|18L 13s
|-
| 32
|15L 17s
|-
|33
|12L 21s
|-
|34
|9L 25s
|-
|35
|6L 29s
|-
|36
|3L 33s
|-
|37
|[[37edo]]
|equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>l9asioa6ysyuejps1lvpnxj1kppnscm</sha1>
    </revision>
    <revision>
      <id>79320</id>
      <parentid>78458</parentid>
      <timestamp>2021-10-10T13:06:09Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>79320</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6132" sha1="tatewd175k2ppiubv9ce9k5k8yuwk2q" xml:space="preserve">{{Infobox ET
| Prime factorization = 3 × 37
| Step size = 10.81081¢
| Fifth = 65\111 (702.7¢)
| Major 2nd = 19\111 (205.4¢)
| Semitones = 11:8 (118.9¢ : 86.5¢)
| Consistency = 21
}}

The '''111 equal divisions of the octave''' ('''111EDO'''), or the '''111(-tone) equal temperament''' ('''111TET''', '''111ET''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s. 

== Theory ==
111EDO is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo|37EDO]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37EDO are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111EDO has a step of 10.81 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111EDO
!Tones
!Pattern
!L:s
|-
| 5
| [[1L 4s]]
|23:22
|-
| 6
| [[3L 3s]]
| 19:18
|-
| 7
| [[6L 1s]]
| 16:15
|-
| 8
| [[7L 1s]]
| 14:13
|-
| 9
| [[3L 6s]]
| 13:12
|-
| 10
| [[1L 9s]]
| 12:11
|-
| 11
| [[1L 10s]]
| 11:10
|-
| 12
| [[3L 9s]]
| 10:9
|-
| 13
| [[6L 7s]]
| 9:8
|-
| 14
| [[13L 1s]]
| rowspan="2" |8:7
|-
| 15
| [[6L 9s]]
|-
| 16
| [[15L 1s]]
| rowspan="3" |7:6
|-
| 17
| [[9L 8s]]
|-
| 18
| 3L 15s
|-
| 19
| [[16L 3s]]
| rowspan="4" |6:5
|-
| 20
| 11L 9s
|-
| 21
| 6L 15s
|-
| 22
| 1L 21s
|-
| 23
| 19L 4s
| rowspan="5" | 5:4
|-
| 24
| 15L 9s
|-
| 25
| 11L 14s
|-
| 26
| 7L 19s
|-
| 27
| 3L 24s
|-
| 28
| 27L 1s
| rowspan="9" |4:3
|-
| 29
| 24L 5s
|-
| 30
| 21L 9s
|-
| 31
| 18L 13s
|-
| 32
| 15L 17s
|-
| 33
| 12L 21s
|-
| 34
| 9L 25s
|-
| 35
| 6L 29s
|-
| 36
| 3L 33s
|-
| 37
| [[37edo|37EDO]]
| equal
|-
|38
|35L 3s
| rowspan="18" |3:2
|-
|39
|33L 6s
|-
| 40
|31L 9s
|-
|41
|29L 12s
|-
|42
|27L 15s
|-
|43
|25L 18s
|-
|44
|23L 21s
|-
|45
|21L 24s
|-
|46
|19L 27s
|-
|47
|17L 30s
|-
|48
|15L 33s
|-
|49
|13L 36s
|-
|50
|11L 39s
|-
|51
|9L 42s
|-
|52
|7L 45s
|-
|53
|5L 48s
|-
|54
|3L 51s
|-
|55
|1L 54s
|-
|56
|55L 1s
| rowspan="33" |2:1
|-
|57
|54L 3s
|-
|58
|53L 5s
|-
|59
|52L 7s
|-
|60
|51L 9s
|-
|61
|50L 11s
|-
|62
|49L 13s
|-
|63
|48L 15s
|-
|64
|47L 17s
|-
|65
|46L 19s
|-
|66
|45L 21s
|-
|67
|44L 23s
|-
|68
|43L 25s
|-
|69
|42L 27s
|-
|70
|41L 29s
|-
|71
|40L 31s
|-
|72
|39L 33s
|-
|73
|38L 35s
|-
|74
|37L 37s
|-
|75
|36L 39s
|-
|76
|35L 41s
|-
|77
|34L 43s
|-
|78
|33L 45s
|-
|79
|32L 47s
|-
|80
|31L 49s
|-
|81
|30L 51s
|-
|82
|29L 53s
|-
|83
|28L 55s
|-
|84
|27L 57s
|-
|85
|26L 59s
|-
|86
|25L 61s
|-
|87
|24L 63s
|-
|88
|23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>tatewd175k2ppiubv9ce9k5k8yuwk2q</sha1>
    </revision>
    <revision>
      <id>79321</id>
      <parentid>79320</parentid>
      <timestamp>2021-10-10T13:16:43Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <minor/>
      <origin>79321</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6235" sha1="f697lcmgzk4guz9ukksy45m35oxc9dg" xml:space="preserve">{{Infobox ET
| Prime factorization = 3 × 37
| Step size = 10.81081¢
| Fifth = 65\111 (702.7¢)
| Major 2nd = 19\111 (205.4¢)
| Semitones = 11:8 (118.9¢ : 86.5¢)
| Consistency = 21
}}

The '''111 equal divisions of the octave''' ('''111EDO'''), or the '''111(-tone) equal temperament''' ('''111TET''', '''111ET''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s. 

== Theory ==
111EDO is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111EDO is somewhat related to [[37edo|37EDO]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37EDO are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111EDO has a step of 10.811 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111EDO
!Tones
!Pattern
!L:s
|-
| 5
| [[1L 4s]]
| 23:22
|-
| 6
| [[3L 3s]]
| 19:18
|-
| 7
| [[6L 1s]]
| 16:15
|-
| 8
| [[7L 1s]]
| 14:13
|-
| 9
| [[3L 6s]]
| 13:12
|-
| 10
| [[1L 9s]]
| 12:11
|-
| 11
| [[1L 10s]]
| 11:10
|-
| 12
| [[3L 9s]]
| 10:9
|-
| 13
| [[6L 7s]]
| 9:8
|-
| 14
| [[13L 1s]]
| rowspan="2" |8:7
|-
| 15
| [[6L 9s]]
|-
| 16
| [[15L 1s]]
| rowspan="3" |7:6
|-
| 17
| [[9L 8s]]
|-
| 18
| 3L 15s
|-
| 19
| [[16L 3s]]
| rowspan="4" |6:5
|-
| 20
| 11L 9s
|-
| 21
| 6L 15s
|-
| 22
| 1L 21s
|-
| 23
| 19L 4s
| rowspan="5" | 5:4
|-
| 24
| 15L 9s
|-
| 25
| 11L 14s
|-
| 26
| 7L 19s
|-
| 27
| 3L 24s
|-
| 28
| 27L 1s
| rowspan="9" |4:3
|-
| 29
| 24L 5s
|-
| 30
| 21L 9s
|-
| 31
| 18L 13s
|-
| 32
| 15L 17s
|-
| 33
| 12L 21s
|-
| 34
| 9L 25s
|-
| 35
| 6L 29s
|-
| 36
| 3L 33s
|-
| 37
| [[37edo|37EDO]]
| equal
|-
| 38
| 35L 3s
| rowspan="18" |3:2
|-
| 39
| 33L 6s
|-
| 40
| 31L 9s
|-
| 41
| 29L 12s
|-
| 42
| 27L 15s
|-
| 43
| 25L 18s
|-
| 44
| 23L 21s
|-
| 45
| 21L 24s
|-
| 46
| 19L 27s
|-
| 47
| 17L 30s
|-
| 48
| 15L 33s
|-
| 49
| 13L 36s
|-
| 50
| 11L 39s
|-
| 51
| 9L 42s
|-
| 52
| 7L 45s
|-
| 53
| 5L 48s
|-
| 54
| 3L 51s
|-
| 55
| 1L 54s
|-
| 56
| 55L 1s
| rowspan="33" |2:1
|-
| 57
| 54L 3s
|-
| 58
| 53L 5s
|-
| 59
| 52L 7s
|-
| 60
| 51L 9s
|-
| 61
| 50L 11s
|-
| 62
| 49L 13s
|-
| 63
| 48L 15s
|-
| 64
| 47L 17s
|-
| 65
| 46L 19s
|-
| 66
| 45L 21s
|-
| 67
| 44L 23s
|-
| 68
| 43L 25s
|-
| 69
| 42L 27s
|-
| 70
| 41L 29s
|-
| 71
| 40L 31s
|-
| 72
| 39L 33s
|-
| 73
| 38L 35s
|-
| 74
| 37L 37s
|-
| 75
| 36L 39s
|-
| 76
| 35L 41s
|-
| 77
| 34L 43s
|-
| 78
| 33L 45s
|-
| 79
| 32L 47s
|-
| 80
| 31L 49s
|-
| 81
| 30L 51s
|-
| 82
| 29L 53s
|-
| 83
| 28L 55s
|-
| 84
| 27L 57s
|-
| 85
| 26L 59s
|-
| 86
| 25L 61s
|-
| 87
| 24L 63s
|-
| 88
| 23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>f697lcmgzk4guz9ukksy45m35oxc9dg</sha1>
    </revision>
    <revision>
      <id>84179</id>
      <parentid>79321</parentid>
      <timestamp>2022-01-18T09:47:08Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>84179</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6300" sha1="dju26liruz2nys7icw9zsr7tgc44ddj" xml:space="preserve">{{Infobox ET
| Prime factorization = 3 × 37
| Step size = 10.81081¢
| Fifth = 65\111 (702.7¢)
| Major 2nd = 19\111 (205.4¢)
| Semitones = 11:8 (118.9¢ : 86.5¢)
| Consistency = 21
}}

The '''111 equal divisions of the octave''' ('''111EDO'''), or the '''111(-tone) equal temperament''' ('''111TET''', '''111ET''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s. 

== Theory ==
111EDO is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111EDO is somewhat related to [[37edo|37EDO]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37EDO are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111EDO has a step of 10.811 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111EDO
!Tones
!Pattern
!L:s
|-
| 5
| [[1L 4s]]
| 23:22
|-
| 6
| [[3L 3s]]
| 19:18
|-
| 7
| [[6L 1s]]
| 16:15
|-
| 8
| [[7L 1s]]
| 14:13
|-
| 9
| [[3L 6s]]
| 13:12
|-
| 10
| [[1L 9s]]
| 12:11
|-
| 11
| [[1L 10s]]
| 11:10
|-
| 12
| [[3L 9s]]
| 10:9
|-
| 13
| [[6L 7s]]
| 9:8
|-
| 14
| [[13L 1s]]
| rowspan="2" |8:7
|-
| 15
| [[6L 9s]]
|-
| 16
| [[15L 1s]]
| rowspan="3" |7:6
|-
| 17
| [[9L 8s]]
|-
| 18
| 3L 15s
|-
| 19
| [[16L 3s]]
| rowspan="4" |6:5
|-
| 20
| 11L 9s
|-
| 21
| 6L 15s
|-
| 22
| 1L 21s
|-
| 23
| 19L 4s
| rowspan="5" | 5:4
|-
| 24
| 15L 9s
|-
| 25
| 11L 14s
|-
| 26
| 7L 19s
|-
| 27
| 3L 24s
|-
| 28
| 27L 1s
| rowspan="9" |4:3
|-
| 29
| 24L 5s
|-
| 30
| 21L 9s
|-
| 31
| 18L 13s
|-
| 32
| 15L 17s
|-
| 33
| 12L 21s
|-
| 34
| 9L 25s
|-
| 35
| 6L 29s
|-
| 36
| 3L 33s
|-
| 37
| [[37edo|37EDO]]
| equal
|-
| 38
| 35L 3s
| rowspan="18" |3:2
|-
| 39
| 33L 6s
|-
| 40
| 31L 9s
|-
| 41
| 29L 12s
|-
| 42
| 27L 15s
|-
| 43
| 25L 18s
|-
| 44
| 23L 21s
|-
| 45
| 21L 24s
|-
| 46
| 19L 27s
|-
| 47
| 17L 30s
|-
| 48
| 15L 33s
|-
| 49
| 13L 36s
|-
| 50
| 11L 39s
|-
| 51
| 9L 42s
|-
| 52
| 7L 45s
|-
| 53
| 5L 48s
|-
| 54
| 3L 51s
|-
| 55
| 1L 54s
|-
| 56
| 55L 1s
| rowspan="33" |2:1
|-
| 57
| 54L 3s
|-
| 58
| 53L 5s
|-
| 59
| 52L 7s
|-
| 60
| 51L 9s
|-
| 61
| 50L 11s
|-
| 62
| 49L 13s
|-
| 63
| 48L 15s
|-
| 64
| 47L 17s
|-
| 65
| 46L 19s
|-
| 66
| 45L 21s
|-
| 67
| 44L 23s
|-
| 68
| 43L 25s
|-
| 69
| 42L 27s
|-
| 70
| 41L 29s
|-
| 71
| 40L 31s
|-
| 72
| 39L 33s
|-
| 73
| 38L 35s
|-
| 74
| 37L 37s
|-
| 75
| 36L 39s
|-
| 76
| 35L 41s
|-
| 77
| 34L 43s
|-
| 78
| 33L 45s
|-
| 79
| 32L 47s
|-
| 80
| 31L 49s
|-
| 81
| 30L 51s
|-
| 82
| 29L 53s
|-
| 83
| 28L 55s
|-
| 84
| 27L 57s
|-
| 85
| 26L 59s
|-
| 86
| 25L 61s
|-
| 87
| 24L 63s
|-
| 88
| 23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>dju26liruz2nys7icw9zsr7tgc44ddj</sha1>
    </revision>
    <revision>
      <id>84452</id>
      <parentid>84179</parentid>
      <timestamp>2022-01-22T08:28:11Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>84452</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6451" sha1="712xu9bz7nvjx11ux821h85u24dn4vm" xml:space="preserve">{{Infobox ET
| Prime factorization = 3 × 37
| Step size = 10.81081¢
| Fifth = 65\111 (702.7¢)
| Major 2nd = 19\111 (205.4¢)
| Semitones = 11:8 (118.9¢ : 86.5¢)
| Consistency = 21
}}

The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37EDO are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111EDO has a step of 10.811 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111EDO
!Tones
!Pattern
!L:s
|-
| 5
| [[1L 4s]]
| 23:22
|-
| 6
| [[3L 3s]]
| 19:18
|-
| 7
| [[6L 1s]]
| 16:15
|-
| 8
| [[7L 1s]]
| 14:13
|-
| 9
| [[3L 6s]]
| 13:12
|-
| 10
| [[1L 9s]]
| 12:11
|-
| 11
| [[1L 10s]]
| 11:10
|-
| 12
| [[3L 9s]]
| 10:9
|-
| 13
| [[6L 7s]]
| 9:8
|-
| 14
| [[13L 1s]]
| rowspan="2" |8:7
|-
| 15
| [[6L 9s]]
|-
| 16
| [[15L 1s]]
| rowspan="3" |7:6
|-
| 17
| [[9L 8s]]
|-
| 18
| 3L 15s
|-
| 19
| [[16L 3s]]
| rowspan="4" |6:5
|-
| 20
| 11L 9s
|-
| 21
| 6L 15s
|-
| 22
| 1L 21s
|-
| 23
| 19L 4s
| rowspan="5" | 5:4
|-
| 24
| 15L 9s
|-
| 25
| 11L 14s
|-
| 26
| 7L 19s
|-
| 27
| 3L 24s
|-
| 28
| 27L 1s
| rowspan="9" |4:3
|-
| 29
| 24L 5s
|-
| 30
| 21L 9s
|-
| 31
| 18L 13s
|-
| 32
| 15L 17s
|-
| 33
| 12L 21s
|-
| 34
| 9L 25s
|-
| 35
| 6L 29s
|-
| 36
| 3L 33s
|-
| 37
| [[37edo|37EDO]]
| equal
|-
| 38
| 35L 3s
| rowspan="18" |3:2
|-
| 39
| 33L 6s
|-
| 40
| 31L 9s
|-
| 41
| 29L 12s
|-
| 42
| 27L 15s
|-
| 43
| 25L 18s
|-
| 44
| 23L 21s
|-
| 45
| 21L 24s
|-
| 46
| 19L 27s
|-
| 47
| 17L 30s
|-
| 48
| 15L 33s
|-
| 49
| 13L 36s
|-
| 50
| 11L 39s
|-
| 51
| 9L 42s
|-
| 52
| 7L 45s
|-
| 53
| 5L 48s
|-
| 54
| 3L 51s
|-
| 55
| 1L 54s
|-
| 56
| 55L 1s
| rowspan="33" |2:1
|-
| 57
| 54L 3s
|-
| 58
| 53L 5s
|-
| 59
| 52L 7s
|-
| 60
| 51L 9s
|-
| 61
| 50L 11s
|-
| 62
| 49L 13s
|-
| 63
| 48L 15s
|-
| 64
| 47L 17s
|-
| 65
| 46L 19s
|-
| 66
| 45L 21s
|-
| 67
| 44L 23s
|-
| 68
| 43L 25s
|-
| 69
| 42L 27s
|-
| 70
| 41L 29s
|-
| 71
| 40L 31s
|-
| 72
| 39L 33s
|-
| 73
| 38L 35s
|-
| 74
| 37L 37s
|-
| 75
| 36L 39s
|-
| 76
| 35L 41s
|-
| 77
| 34L 43s
|-
| 78
| 33L 45s
|-
| 79
| 32L 47s
|-
| 80
| 31L 49s
|-
| 81
| 30L 51s
|-
| 82
| 29L 53s
|-
| 83
| 28L 55s
|-
| 84
| 27L 57s
|-
| 85
| 26L 59s
|-
| 86
| 25L 61s
|-
| 87
| 24L 63s
|-
| 88
| 23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Theory]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>712xu9bz7nvjx11ux821h85u24dn4vm</sha1>
    </revision>
    <revision>
      <id>87434</id>
      <parentid>84452</parentid>
      <timestamp>2022-02-26T16:48:48Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87434</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6431" sha1="oip539h1co9byuj3x2sfqfqdtr8odo7" xml:space="preserve">{{Infobox ET
| Prime factorization = 3 × 37
| Step size = 10.81081¢
| Fifth = 65\111 (702.7¢)
| Major 2nd = 19\111 (205.4¢)
| Semitones = 11:8 (118.9¢ : 86.5¢)
| Consistency = 21
}}

The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37EDO are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111EDO has a step of 10.811 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111EDO
!Tones
!Pattern
!L:s
|-
| 5
| [[1L 4s]]
| 23:22
|-
| 6
| [[3L 3s]]
| 19:18
|-
| 7
| [[6L 1s]]
| 16:15
|-
| 8
| [[7L 1s]]
| 14:13
|-
| 9
| [[3L 6s]]
| 13:12
|-
| 10
| [[1L 9s]]
| 12:11
|-
| 11
| [[1L 10s]]
| 11:10
|-
| 12
| [[3L 9s]]
| 10:9
|-
| 13
| [[6L 7s]]
| 9:8
|-
| 14
| [[13L 1s]]
| rowspan="2" |8:7
|-
| 15
| [[6L 9s]]
|-
| 16
| [[15L 1s]]
| rowspan="3" |7:6
|-
| 17
| [[9L 8s]]
|-
| 18
| 3L 15s
|-
| 19
| [[16L 3s]]
| rowspan="4" |6:5
|-
| 20
| 11L 9s
|-
| 21
| 6L 15s
|-
| 22
| 1L 21s
|-
| 23
| 19L 4s
| rowspan="5" | 5:4
|-
| 24
| 15L 9s
|-
| 25
| 11L 14s
|-
| 26
| 7L 19s
|-
| 27
| 3L 24s
|-
| 28
| 27L 1s
| rowspan="9" |4:3
|-
| 29
| 24L 5s
|-
| 30
| 21L 9s
|-
| 31
| 18L 13s
|-
| 32
| 15L 17s
|-
| 33
| 12L 21s
|-
| 34
| 9L 25s
|-
| 35
| 6L 29s
|-
| 36
| 3L 33s
|-
| 37
| [[37edo|37EDO]]
| equal
|-
| 38
| 35L 3s
| rowspan="18" |3:2
|-
| 39
| 33L 6s
|-
| 40
| 31L 9s
|-
| 41
| 29L 12s
|-
| 42
| 27L 15s
|-
| 43
| 25L 18s
|-
| 44
| 23L 21s
|-
| 45
| 21L 24s
|-
| 46
| 19L 27s
|-
| 47
| 17L 30s
|-
| 48
| 15L 33s
|-
| 49
| 13L 36s
|-
| 50
| 11L 39s
|-
| 51
| 9L 42s
|-
| 52
| 7L 45s
|-
| 53
| 5L 48s
|-
| 54
| 3L 51s
|-
| 55
| 1L 54s
|-
| 56
| 55L 1s
| rowspan="33" |2:1
|-
| 57
| 54L 3s
|-
| 58
| 53L 5s
|-
| 59
| 52L 7s
|-
| 60
| 51L 9s
|-
| 61
| 50L 11s
|-
| 62
| 49L 13s
|-
| 63
| 48L 15s
|-
| 64
| 47L 17s
|-
| 65
| 46L 19s
|-
| 66
| 45L 21s
|-
| 67
| 44L 23s
|-
| 68
| 43L 25s
|-
| 69
| 42L 27s
|-
| 70
| 41L 29s
|-
| 71
| 40L 31s
|-
| 72
| 39L 33s
|-
| 73
| 38L 35s
|-
| 74
| 37L 37s
|-
| 75
| 36L 39s
|-
| 76
| 35L 41s
|-
| 77
| 34L 43s
|-
| 78
| 33L 45s
|-
| 79
| 32L 47s
|-
| 80
| 31L 49s
|-
| 81
| 30L 51s
|-
| 82
| 29L 53s
|-
| 83
| 28L 55s
|-
| 84
| 27L 57s
|-
| 85
| 26L 59s
|-
| 86
| 25L 61s
|-
| 87
| 24L 63s
|-
| 88
| 23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>oip539h1co9byuj3x2sfqfqdtr8odo7</sha1>
    </revision>
    <revision>
      <id>89050</id>
      <parentid>87434</parentid>
      <timestamp>2022-03-06T04:37:03Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>89050</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6496" sha1="556d70e6jp03srjy1lncf121gow754h" xml:space="preserve">{{Infobox ET
| Prime factorization = 3 × 37
| Step size = 10.81081¢
| Fifth = 65\111 (702.7¢)
| Major 2nd = 19\111 (205.4¢)
| Semitones = 11:8 (118.9¢ : 86.5¢)
| Consistency = 21
}}

The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37EDO are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111EDO has a step of 10.811 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111EDO
! Tones
! Pattern
! L:s
|-
| 5
| [[1L 4s]]
| 23:22
|-
| 6
| [[3L 3s]]
| 19:18
|-
| 7
| [[6L 1s]]
| 16:15
|-
| 8
| [[7L 1s]]
| 14:13
|-
| 9
| [[3L 6s]]
| 13:12
|-
| 10
| [[1L 9s]]
| 12:11
|-
| 11
| [[1L 10s]]
| 11:10
|-
| 12
| [[3L 9s]]
| 10:9
|-
| 13
| [[6L 7s]]
| 9:8
|-
| 14
| [[13L 1s]]
| rowspan="2" |8:7
|-
| 15
| [[6L 9s]]
|-
| 16
| [[15L 1s]]
| rowspan="3" |7:6
|-
| 17
| [[9L 8s]]
|-
| 18
| 3L 15s
|-
| 19
| [[16L 3s]]
| rowspan="4" |6:5
|-
| 20
| 11L 9s
|-
| 21
| 6L 15s
|-
| 22
| 1L 21s
|-
| 23
| 19L 4s
| rowspan="5" | 5:4
|-
| 24
| 15L 9s
|-
| 25
| 11L 14s
|-
| 26
| 7L 19s
|-
| 27
| 3L 24s
|-
| 28
| 27L 1s
| rowspan="9" |4:3
|-
| 29
| 24L 5s
|-
| 30
| 21L 9s
|-
| 31
| 18L 13s
|-
| 32
| 15L 17s
|-
| 33
| 12L 21s
|-
| 34
| 9L 25s
|-
| 35
| 6L 29s
|-
| 36
| 3L 33s
|-
| 37
| [[37edo|37EDO]]
| equal
|-
| 38
| 35L 3s
| rowspan="18" |3:2
|-
| 39
| 33L 6s
|-
| 40
| 31L 9s
|-
| 41
| 29L 12s
|-
| 42
| 27L 15s
|-
| 43
| 25L 18s
|-
| 44
| 23L 21s
|-
| 45
| 21L 24s
|-
| 46
| 19L 27s
|-
| 47
| 17L 30s
|-
| 48
| 15L 33s
|-
| 49
| 13L 36s
|-
| 50
| 11L 39s
|-
| 51
| 9L 42s
|-
| 52
| 7L 45s
|-
| 53
| 5L 48s
|-
| 54
| 3L 51s
|-
| 55
| 1L 54s
|-
| 56
| 55L 1s
| rowspan="33" |2:1
|-
| 57
| 54L 3s
|-
| 58
| 53L 5s
|-
| 59
| 52L 7s
|-
| 60
| 51L 9s
|-
| 61
| 50L 11s
|-
| 62
| 49L 13s
|-
| 63
| 48L 15s
|-
| 64
| 47L 17s
|-
| 65
| 46L 19s
|-
| 66
| 45L 21s
|-
| 67
| 44L 23s
|-
| 68
| 43L 25s
|-
| 69
| 42L 27s
|-
| 70
| 41L 29s
|-
| 71
| 40L 31s
|-
| 72
| 39L 33s
|-
| 73
| 38L 35s
|-
| 74
| 37L 37s
|-
| 75
| 36L 39s
|-
| 76
| 35L 41s
|-
| 77
| 34L 43s
|-
| 78
| 33L 45s
|-
| 79
| 32L 47s
|-
| 80
| 31L 49s
|-
| 81
| 30L 51s
|-
| 82
| 29L 53s
|-
| 83
| 28L 55s
|-
| 84
| 27L 57s
|-
| 85
| 26L 59s
|-
| 86
| 25L 61s
|-
| 87
| 24L 63s
|-
| 88
| 23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>556d70e6jp03srjy1lncf121gow754h</sha1>
    </revision>
    <revision>
      <id>89053</id>
      <parentid>89050</parentid>
      <timestamp>2022-03-06T06:11:06Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <minor/>
      <origin>89053</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6589" sha1="oehunmh32whw72zc25dmq3hv6f555mj" xml:space="preserve">{{Infobox ET
| Prime factorization = 3 × 37
| Step size = 10.81081¢
| Fifth = 65\111 (702.7¢)
| Major 2nd = 19\111 (205.4¢)
| Semitones = 11:8 (118.9¢ : 86.5¢)
| Consistency = 21
}}

The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37EDO are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111EDO has a step of 10.811 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111EDO
! Tones
! Pattern
! L:s
|-
| 5
| [[1L 4s]]
| 23:22
|-
| 6
| [[3L 3s]]
| 19:18
|-
| 7
| [[6L 1s]]
| 16:15
|-
| 8
| [[7L 1s]]
| 14:13
|-
| 9
| [[3L 6s]]
| 13:12
|-
| 10
| [[1L 9s]]
| 12:11
|-
| 11
| [[1L 10s]]
| 11:10
|-
| 12
| [[3L 9s]]
| 10:9
|-
| 13
| [[6L 7s]]
| 9:8
|-
| 14
| [[13L 1s]]
| rowspan="2" |8:7
|-
| 15
| [[6L 9s]]
|-
| 16
| [[15L 1s]]
| rowspan="3" |7:6
|-
| 17
| [[9L 8s]]
|-
| 18
| 3L 15s
|-
| 19
| [[16L 3s]]
| rowspan="4" |6:5
|-
| 20
| 11L 9s
|-
| 21
| 6L 15s
|-
| 22
| 1L 21s
|-
| 23
| 19L 4s
| rowspan="5" | 5:4
|-
| 24
| 15L 9s
|-
| 25
| 11L 14s
|-
| 26
| 7L 19s
|-
| 27
| 3L 24s
|-
| 28
| 27L 1s
| rowspan="9" |4:3
|-
| 29
| 24L 5s
|-
| 30
| 21L 9s
|-
| 31
| 18L 13s
|-
| 32
| 15L 17s
|-
| 33
| 12L 21s
|-
| 34
| 9L 25s
|-
| 35
| 6L 29s
|-
| 36
| 3L 33s
|-
| 37
| [[37edo|37EDO]]
| equal
|-
| 38
| 35L 3s
| rowspan="18" |3:2
|-
| 39
| 33L 6s
|-
| 40
| 31L 9s
|-
| 41
| 29L 12s
|-
| 42
| 27L 15s
|-
| 43
| 25L 18s
|-
| 44
| 23L 21s
|-
| 45
| 21L 24s
|-
| 46
| 19L 27s
|-
| 47
| 17L 30s
|-
| 48
| 15L 33s
|-
| 49
| 13L 36s
|-
| 50
| 11L 39s
|-
| 51
| 9L 42s
|-
| 52
| 7L 45s
|-
| 53
| 5L 48s
|-
| 54
| 3L 51s
|-
| 55
| 1L 54s
|-
| 56
| 55L 1s
| rowspan="33" |2:1
|-
| 57
| 54L 3s
|-
| 58
| 53L 5s
|-
| 59
| 52L 7s
|-
| 60
| 51L 9s
|-
| 61
| 50L 11s
|-
| 62
| 49L 13s
|-
| 63
| 48L 15s
|-
| 64
| 47L 17s
|-
| 65
| 46L 19s
|-
| 66
| 45L 21s
|-
| 67
| 44L 23s
|-
| 68
| 43L 25s
|-
| 69
| 42L 27s
|-
| 70
| 41L 29s
|-
| 71
| 40L 31s
|-
| 72
| 39L 33s
|-
| 73
| 38L 35s
|-
| 74
| 37L 37s
|-
| 75
| 36L 39s
|-
| 76
| 35L 41s
|-
| 77
| 34L 43s
|-
| 78
| 33L 45s
|-
| 79
| 32L 47s
|-
| 80
| 31L 49s
|-
| 81
| 30L 51s
|-
| 82
| 29L 53s
|-
| 83
| 28L 55s
|-
| 84
| 27L 57s
|-
| 85
| 26L 59s
|-
| 86
| 25L 61s
|-
| 87
| 24L 63s
|-
| 88
| 23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>oehunmh32whw72zc25dmq3hv6f555mj</sha1>
    </revision>
    <revision>
      <id>92921</id>
      <parentid>89053</parentid>
      <timestamp>2022-07-02T16:21:35Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92921</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6617" sha1="k77zi38tumntfun9vuemga17jov5whm" xml:space="preserve">{{Infobox ET
| Prime factorization = 3 × 37
| Step size = 10.81081¢
| Fifth = 65\111 (702.7¢)
| Major 2nd = 19\111 (205.4¢)
| Semitones = 11:8 (118.9¢ : 86.5¢)
| Consistency = 21
}}

The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37EDO are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111EDO has a step of 10.811 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111EDO
! Tones
! Pattern
! L:s
|-
| 5
| [[1L 4s]]
| 23:22
|-
| 6
| [[3L 3s]]
| 19:18
|-
| 7
| [[6L 1s]]
| 16:15
|-
| 8
| [[7L 1s]]
| 14:13
|-
| 9
| [[3L 6s]]
| 13:12
|-
| 10
| [[1L 9s]]
| 12:11
|-
| 11
| [[1L 10s]]
| 11:10
|-
| 12
| [[3L 9s]]
| 10:9
|-
| 13
| [[6L 7s]]
| 9:8
|-
| 14
| [[13L 1s]]
| rowspan="2" |8:7
|-
| 15
| [[6L 9s]]
|-
| 16
| [[15L 1s]]
| rowspan="3" |7:6
|-
| 17
| [[9L 8s]]
|-
| 18
| 3L 15s
|-
| 19
| [[16L 3s]]
| rowspan="4" |6:5
|-
| 20
| 11L 9s
|-
| 21
| 6L 15s
|-
| 22
| 1L 21s
|-
| 23
| 19L 4s
| rowspan="5" | 5:4
|-
| 24
| 15L 9s
|-
| 25
| 11L 14s
|-
| 26
| 7L 19s
|-
| 27
| 3L 24s
|-
| 28
| 27L 1s
| rowspan="9" |4:3
|-
| 29
| 24L 5s
|-
| 30
| 21L 9s
|-
| 31
| 18L 13s
|-
| 32
| 15L 17s
|-
| 33
| 12L 21s
|-
| 34
| 9L 25s
|-
| 35
| 6L 29s
|-
| 36
| 3L 33s
|-
| 37
| [[37edo|37EDO]]
| equal
|-
| 38
| 35L 3s
| rowspan="18" |3:2
|-
| 39
| 33L 6s
|-
| 40
| 31L 9s
|-
| 41
| 29L 12s
|-
| 42
| 27L 15s
|-
| 43
| 25L 18s
|-
| 44
| 23L 21s
|-
| 45
| 21L 24s
|-
| 46
| 19L 27s
|-
| 47
| 17L 30s
|-
| 48
| 15L 33s
|-
| 49
| 13L 36s
|-
| 50
| 11L 39s
|-
| 51
| 9L 42s
|-
| 52
| 7L 45s
|-
| 53
| 5L 48s
|-
| 54
| 3L 51s
|-
| 55
| 1L 54s
|-
| 56
| 55L 1s
| rowspan="33" |2:1
|-
| 57
| 54L 3s
|-
| 58
| 53L 5s
|-
| 59
| 52L 7s
|-
| 60
| 51L 9s
|-
| 61
| 50L 11s
|-
| 62
| 49L 13s
|-
| 63
| 48L 15s
|-
| 64
| 47L 17s
|-
| 65
| 46L 19s
|-
| 66
| 45L 21s
|-
| 67
| 44L 23s
|-
| 68
| 43L 25s
|-
| 69
| 42L 27s
|-
| 70
| 41L 29s
|-
| 71
| 40L 31s
|-
| 72
| 39L 33s
|-
| 73
| 38L 35s
|-
| 74
| 37L 37s
|-
| 75
| 36L 39s
|-
| 76
| 35L 41s
|-
| 77
| 34L 43s
|-
| 78
| 33L 45s
|-
| 79
| 32L 47s
|-
| 80
| 31L 49s
|-
| 81
| 30L 51s
|-
| 82
| 29L 53s
|-
| 83
| 28L 55s
|-
| 84
| 27L 57s
|-
| 85
| 26L 59s
|-
| 86
| 25L 61s
|-
| 87
| 24L 63s
|-
| 88
| 23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>k77zi38tumntfun9vuemga17jov5whm</sha1>
    </revision>
    <revision>
      <id>96083</id>
      <parentid>92921</parentid>
      <timestamp>2022-10-04T16:02:21Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET now computes most parameters automatically</comment>
      <origin>96083</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6444" sha1="1092ckgjk45yywy7jgs2ocjtas6c55b" xml:space="preserve">{{Infobox ET}}

The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Primes in edo|111|columns=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37EDO are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111EDO has a step of 10.811 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111EDO
! Tones
! Pattern
! L:s
|-
| 5
| [[1L 4s]]
| 23:22
|-
| 6
| [[3L 3s]]
| 19:18
|-
| 7
| [[6L 1s]]
| 16:15
|-
| 8
| [[7L 1s]]
| 14:13
|-
| 9
| [[3L 6s]]
| 13:12
|-
| 10
| [[1L 9s]]
| 12:11
|-
| 11
| [[1L 10s]]
| 11:10
|-
| 12
| [[3L 9s]]
| 10:9
|-
| 13
| [[6L 7s]]
| 9:8
|-
| 14
| [[13L 1s]]
| rowspan="2" |8:7
|-
| 15
| [[6L 9s]]
|-
| 16
| [[15L 1s]]
| rowspan="3" |7:6
|-
| 17
| [[9L 8s]]
|-
| 18
| 3L 15s
|-
| 19
| [[16L 3s]]
| rowspan="4" |6:5
|-
| 20
| 11L 9s
|-
| 21
| 6L 15s
|-
| 22
| 1L 21s
|-
| 23
| 19L 4s
| rowspan="5" | 5:4
|-
| 24
| 15L 9s
|-
| 25
| 11L 14s
|-
| 26
| 7L 19s
|-
| 27
| 3L 24s
|-
| 28
| 27L 1s
| rowspan="9" |4:3
|-
| 29
| 24L 5s
|-
| 30
| 21L 9s
|-
| 31
| 18L 13s
|-
| 32
| 15L 17s
|-
| 33
| 12L 21s
|-
| 34
| 9L 25s
|-
| 35
| 6L 29s
|-
| 36
| 3L 33s
|-
| 37
| [[37edo|37EDO]]
| equal
|-
| 38
| 35L 3s
| rowspan="18" |3:2
|-
| 39
| 33L 6s
|-
| 40
| 31L 9s
|-
| 41
| 29L 12s
|-
| 42
| 27L 15s
|-
| 43
| 25L 18s
|-
| 44
| 23L 21s
|-
| 45
| 21L 24s
|-
| 46
| 19L 27s
|-
| 47
| 17L 30s
|-
| 48
| 15L 33s
|-
| 49
| 13L 36s
|-
| 50
| 11L 39s
|-
| 51
| 9L 42s
|-
| 52
| 7L 45s
|-
| 53
| 5L 48s
|-
| 54
| 3L 51s
|-
| 55
| 1L 54s
|-
| 56
| 55L 1s
| rowspan="33" |2:1
|-
| 57
| 54L 3s
|-
| 58
| 53L 5s
|-
| 59
| 52L 7s
|-
| 60
| 51L 9s
|-
| 61
| 50L 11s
|-
| 62
| 49L 13s
|-
| 63
| 48L 15s
|-
| 64
| 47L 17s
|-
| 65
| 46L 19s
|-
| 66
| 45L 21s
|-
| 67
| 44L 23s
|-
| 68
| 43L 25s
|-
| 69
| 42L 27s
|-
| 70
| 41L 29s
|-
| 71
| 40L 31s
|-
| 72
| 39L 33s
|-
| 73
| 38L 35s
|-
| 74
| 37L 37s
|-
| 75
| 36L 39s
|-
| 76
| 35L 41s
|-
| 77
| 34L 43s
|-
| 78
| 33L 45s
|-
| 79
| 32L 47s
|-
| 80
| 31L 49s
|-
| 81
| 30L 51s
|-
| 82
| 29L 53s
|-
| 83
| 28L 55s
|-
| 84
| 27L 57s
|-
| 85
| 26L 59s
|-
| 86
| 25L 61s
|-
| 87
| 24L 63s
|-
| 88
| 23L 65s
|}

== Music ==
* [http://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [http://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>1092ckgjk45yywy7jgs2ocjtas6c55b</sha1>
    </revision>
    <revision>
      <id>97840</id>
      <parentid>96083</parentid>
      <timestamp>2022-10-23T12:33:34Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Prime harmonics */ harmonics in equal</comment>
      <origin>97840</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6440" sha1="4z4wk28ngjeh4d1ht52bdgi7217e959" xml:space="preserve">{{Infobox ET}}

The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37EDO are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111EDO has a step of 10.811 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111EDO
! Tones
! Pattern
! L:s
|-
| 5
| [[1L 4s]]
| 23:22
|-
| 6
| [[3L 3s]]
| 19:18
|-
| 7
| [[6L 1s]]
| 16:15
|-
| 8
| [[7L 1s]]
| 14:13
|-
| 9
| [[3L 6s]]
| 13:12
|-
| 10
| [[1L 9s]]
| 12:11
|-
| 11
| [[1L 10s]]
| 11:10
|-
| 12
| [[3L 9s]]
| 10:9
|-
| 13
| [[6L 7s]]
| 9:8
|-
| 14
| [[13L 1s]]
| rowspan="2" |8:7
|-
| 15
| [[6L 9s]]
|-
| 16
| [[15L 1s]]
| rowspan="3" |7:6
|-
| 17
| [[9L 8s]]
|-
| 18
| 3L 15s
|-
| 19
| [[16L 3s]]
| rowspan="4" |6:5
|-
| 20
| 11L 9s
|-
| 21
| 6L 15s
|-
| 22
| 1L 21s
|-
| 23
| 19L 4s
| rowspan="5" | 5:4
|-
| 24
| 15L 9s
|-
| 25
| 11L 14s
|-
| 26
| 7L 19s
|-
| 27
| 3L 24s
|-
| 28
| 27L 1s
| rowspan="9" |4:3
|-
| 29
| 24L 5s
|-
| 30
| 21L 9s
|-
| 31
| 18L 13s
|-
| 32
| 15L 17s
|-
| 33
| 12L 21s
|-
| 34
| 9L 25s
|-
| 35
| 6L 29s
|-
| 36
| 3L 33s
|-
| 37
| [[37edo|37EDO]]
| equal
|-
| 38
| 35L 3s
| rowspan="18" |3:2
|-
| 39
| 33L 6s
|-
| 40
| 31L 9s
|-
| 41
| 29L 12s
|-
| 42
| 27L 15s
|-
| 43
| 25L 18s
|-
| 44
| 23L 21s
|-
| 45
| 21L 24s
|-
| 46
| 19L 27s
|-
| 47
| 17L 30s
|-
| 48
| 15L 33s
|-
| 49
| 13L 36s
|-
| 50
| 11L 39s
|-
| 51
| 9L 42s
|-
| 52
| 7L 45s
|-
| 53
| 5L 48s
|-
| 54
| 3L 51s
|-
| 55
| 1L 54s
|-
| 56
| 55L 1s
| rowspan="33" |2:1
|-
| 57
| 54L 3s
|-
| 58
| 53L 5s
|-
| 59
| 52L 7s
|-
| 60
| 51L 9s
|-
| 61
| 50L 11s
|-
| 62
| 49L 13s
|-
| 63
| 48L 15s
|-
| 64
| 47L 17s
|-
| 65
| 46L 19s
|-
| 66
| 45L 21s
|-
| 67
| 44L 23s
|-
| 68
| 43L 25s
|-
| 69
| 42L 27s
|-
| 70
| 41L 29s
|-
| 71
| 40L 31s
|-
| 72
| 39L 33s
|-
| 73
| 38L 35s
|-
| 74
| 37L 37s
|-
| 75
| 36L 39s
|-
| 76
| 35L 41s
|-
| 77
| 34L 43s
|-
| 78
| 33L 45s
|-
| 79
| 32L 47s
|-
| 80
| 31L 49s
|-
| 81
| 30L 51s
|-
| 82
| 29L 53s
|-
| 83
| 28L 55s
|-
| 84
| 27L 57s
|-
| 85
| 26L 59s
|-
| 86
| 25L 61s
|-
| 87
| 24L 63s
|-
| 88
| 23L 65s
|}

== Music ==
* [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>4z4wk28ngjeh4d1ht52bdgi7217e959</sha1>
    </revision>
    <revision>
      <id>113151</id>
      <parentid>97840</parentid>
      <timestamp>2023-05-12T12:41:05Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Rank-2 temperaments */</comment>
      <origin>113151</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6437" sha1="9c4iq6msngyxu7nsa05demkmidcb9ih" xml:space="preserve">{{Infobox ET}}

The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Scales ==
Since 111EDO has a step of 10.811 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}.
{| class="wikitable"
|+Circulating temperaments in 111EDO
! Tones
! Pattern
! L:s
|-
| 5
| [[1L 4s]]
| 23:22
|-
| 6
| [[3L 3s]]
| 19:18
|-
| 7
| [[6L 1s]]
| 16:15
|-
| 8
| [[7L 1s]]
| 14:13
|-
| 9
| [[3L 6s]]
| 13:12
|-
| 10
| [[1L 9s]]
| 12:11
|-
| 11
| [[1L 10s]]
| 11:10
|-
| 12
| [[3L 9s]]
| 10:9
|-
| 13
| [[6L 7s]]
| 9:8
|-
| 14
| [[13L 1s]]
| rowspan="2" |8:7
|-
| 15
| [[6L 9s]]
|-
| 16
| [[15L 1s]]
| rowspan="3" |7:6
|-
| 17
| [[9L 8s]]
|-
| 18
| 3L 15s
|-
| 19
| [[16L 3s]]
| rowspan="4" |6:5
|-
| 20
| 11L 9s
|-
| 21
| 6L 15s
|-
| 22
| 1L 21s
|-
| 23
| 19L 4s
| rowspan="5" | 5:4
|-
| 24
| 15L 9s
|-
| 25
| 11L 14s
|-
| 26
| 7L 19s
|-
| 27
| 3L 24s
|-
| 28
| 27L 1s
| rowspan="9" |4:3
|-
| 29
| 24L 5s
|-
| 30
| 21L 9s
|-
| 31
| 18L 13s
|-
| 32
| 15L 17s
|-
| 33
| 12L 21s
|-
| 34
| 9L 25s
|-
| 35
| 6L 29s
|-
| 36
| 3L 33s
|-
| 37
| [[37edo|37EDO]]
| equal
|-
| 38
| 35L 3s
| rowspan="18" |3:2
|-
| 39
| 33L 6s
|-
| 40
| 31L 9s
|-
| 41
| 29L 12s
|-
| 42
| 27L 15s
|-
| 43
| 25L 18s
|-
| 44
| 23L 21s
|-
| 45
| 21L 24s
|-
| 46
| 19L 27s
|-
| 47
| 17L 30s
|-
| 48
| 15L 33s
|-
| 49
| 13L 36s
|-
| 50
| 11L 39s
|-
| 51
| 9L 42s
|-
| 52
| 7L 45s
|-
| 53
| 5L 48s
|-
| 54
| 3L 51s
|-
| 55
| 1L 54s
|-
| 56
| 55L 1s
| rowspan="33" |2:1
|-
| 57
| 54L 3s
|-
| 58
| 53L 5s
|-
| 59
| 52L 7s
|-
| 60
| 51L 9s
|-
| 61
| 50L 11s
|-
| 62
| 49L 13s
|-
| 63
| 48L 15s
|-
| 64
| 47L 17s
|-
| 65
| 46L 19s
|-
| 66
| 45L 21s
|-
| 67
| 44L 23s
|-
| 68
| 43L 25s
|-
| 69
| 42L 27s
|-
| 70
| 41L 29s
|-
| 71
| 40L 31s
|-
| 72
| 39L 33s
|-
| 73
| 38L 35s
|-
| 74
| 37L 37s
|-
| 75
| 36L 39s
|-
| 76
| 35L 41s
|-
| 77
| 34L 43s
|-
| 78
| 33L 45s
|-
| 79
| 32L 47s
|-
| 80
| 31L 49s
|-
| 81
| 30L 51s
|-
| 82
| 29L 53s
|-
| 83
| 28L 55s
|-
| 84
| 27L 57s
|-
| 85
| 26L 59s
|-
| 86
| 25L 61s
|-
| 87
| 24L 63s
|-
| 88
| 23L 65s
|}

== Music ==
* [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>9c4iq6msngyxu7nsa05demkmidcb9ih</sha1>
    </revision>
    <revision>
      <id>115196</id>
      <parentid>113151</parentid>
      <timestamp>2023-05-30T21:15:31Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Scales */</comment>
      <origin>115196</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4483" sha1="dj3hb41ezd69duxwr2tcikdm9cwdvpk" xml:space="preserve">{{Infobox ET}}

The '''111 equal divisions of the octave''' ('''111edo'''), or the '''111(-tone) equal temperament''' ('''111tet''', '''111et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 111 parts, each of size about 10.811 [[cent]]s. 

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest EDO uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it tempers out in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31&amp;amp;80 temperament, and [[buzzard]], the 53&amp;amp;58 temperament. The trio piece in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| [{{val| 111 176 }}]
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| [{{val| 111 176 258 }}]
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| [{{val| 111 176 258 312 }}]
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| [{{val| 111 176 258 312 384 }}]
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| [{{val| 111 176 258 312 384 411 }}]
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| [{{val| 111 176 258 312 384 411 454 }}]
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| [{{val| 111 176 258 312 384 411 454 472 }}]
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated ratio&lt;br&gt;(reduced)
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}

== Music ==
* [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 Trio for SoftSaturn, NebulaSing and TromBonehead] [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] by [[Gene Ward Smith]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>dj3hb41ezd69duxwr2tcikdm9cwdvpk</sha1>
    </revision>
    <revision>
      <id>126367</id>
      <parentid>115196</parentid>
      <timestamp>2023-10-24T08:57:43Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Adopt template: EDO intro; cleanup; clarify the title row of the rank-2 temp table; -redundant categories</comment>
      <origin>126367</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4470" sha1="480y673sdc0u6xhavlm6guslp24makw" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With [[harmonic]]s 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>480y673sdc0u6xhavlm6guslp24makw</sha1>
    </revision>
    <revision>
      <id>132299</id>
      <parentid>126367</parentid>
      <timestamp>2024-01-21T07:12:03Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added interval table</comment>
      <origin>132299</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4506" sha1="m6q5hpwboj4fxg44cf32wk2pntrfu5p" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With [[harmonic]]s 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>m6q5hpwboj4fxg44cf32wk2pntrfu5p</sha1>
    </revision>
    <revision>
      <id>138219</id>
      <parentid>132299</parentid>
      <timestamp>2024-03-04T19:47:12Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Added scales section</comment>
      <origin>138219</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4639" sha1="kb3yerq47postj0bmzdja502y7aos2r" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With [[harmonic]]s 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==

Hypersakura (subset of [[Sensi]][19], approximated from [[27edo]])
* 54.0541
* 497.297
* 702.703
* 756.757
* 1200.000

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>kb3yerq47postj0bmzdja502y7aos2r</sha1>
    </revision>
    <revision>
      <id>138221</id>
      <parentid>138219</parentid>
      <timestamp>2024-03-04T19:48:24Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Scales */</comment>
      <origin>138221</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4638" sha1="70cul8zng4yx0bn8mm9epee36yyrrkk" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With [[harmonic]]s 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==

Hypersakura (subset of [[Sensi]][19], approximated from [[27edo]])
* 54.054
* 497.297
* 702.703
* 756.757
* 1200.000

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>70cul8zng4yx0bn8mm9epee36yyrrkk</sha1>
    </revision>
    <revision>
      <id>138226</id>
      <parentid>138221</parentid>
      <timestamp>2024-03-04T19:53:25Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Scales */</comment>
      <origin>138226</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4773" sha1="h1uxhh673zutbmoejf0qx8zxpupm9xj" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With [[harmonic]]s 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==

Direct sunlight (subset of [[Sensi]][19], approximated from [[27edo]])
* 54.054
* 129.730
* 497.297
* 702.703
* 756.757
* 1145.946
* 1200.000

Hypersakura (subset of Sensi[19], approximated from 27edo)
* 54.054
* 497.297
* 702.703
* 756.757
* 1200.000

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>h1uxhh673zutbmoejf0qx8zxpupm9xj</sha1>
    </revision>
    <revision>
      <id>145694</id>
      <parentid>138226</parentid>
      <timestamp>2024-06-04T12:54:50Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Scales */ cleanup (there's only one reasonable sensi scale in 111edo so "approximated from 27edo" is redundant)</comment>
      <origin>145694</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4695" sha1="1vzguh66qoxtlm5qx6okedrj9twqtwe" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With [[harmonic]]s 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 162/125
| [[Sensipent]] / [[warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Vulture]] / [[buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 231/200&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>1vzguh66qoxtlm5qx6okedrj9twqtwe</sha1>
    </revision>
    <revision>
      <id>145695</id>
      <parentid>145694</parentid>
      <timestamp>2024-06-04T13:00:23Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ start with nullity-2 temp if applicable</comment>
      <origin>145695</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4661" sha1="3jmf54v9u3ydqyvkcrded69qyb2b9yb" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. With [[harmonic]]s 3 through 19 all tuned sharp, 111edo is somewhat related to [[37edo]], with which it shares the mappings for 5, 7, 11, and 13. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>3jmf54v9u3ydqyvkcrded69qyb2b9yb</sha1>
    </revision>
    <revision>
      <id>145700</id>
      <parentid>145695</parentid>
      <timestamp>2024-06-04T13:21:41Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ rework; +subsets and supersets</comment>
      <origin>145700</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4824" sha1="s82spz5doudasq68bcksqiilmv0x9sm" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. 111 = 3 × 37, and 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
[[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| -0.740
| 0.521
| 4.83
|}

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>s82spz5doudasq68bcksqiilmv0x9sm</sha1>
    </revision>
    <revision>
      <id>145702</id>
      <parentid>145700</parentid>
      <timestamp>2024-06-04T13:32:32Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ 13- to 23-limit notability</comment>
      <origin>145702</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5016" sha1="5i0owjp3eid32z486rv4bq2d5rkfqaw" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. 111 = 3 × 37, and 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
[[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| -0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>5i0owjp3eid32z486rv4bq2d5rkfqaw</sha1>
    </revision>
    <revision>
      <id>145703</id>
      <parentid>145702</parentid>
      <timestamp>2024-06-04T13:32:56Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */</comment>
      <origin>145703</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5017" sha1="1b3hnd1ymbrc3uj9qejkwpkmpz8gpyf" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. 111 = 3 × 37, and 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
[[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| -0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| -0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| -0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| -0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| -0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| -0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| -0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]
[[Category:Listen]]</text>
      <sha1>1b3hnd1ymbrc3uj9qejkwpkmpz8gpyf</sha1>
    </revision>
    <revision>
      <id>165974</id>
      <parentid>145703</parentid>
      <timestamp>2024-11-16T05:52:08Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>165974</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4533" sha1="rcjo8y3v3sfd8c79a8zxqy169hd6aj1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. 111 = 3 × 37, and 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
[[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{{comma basis begin}}
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| &amp;minus;0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| &amp;minus;0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| &amp;minus;0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| &amp;minus;0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| &amp;minus;0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| &amp;minus;0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| &amp;minus;0.740
| 0.521
| 4.83
{{comma basis end}}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{{rank-2 begin}}
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
{{rank-2 end}}
{{orf}}

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) &amp;ndash; [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] &amp;ndash; guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>rcjo8y3v3sfd8c79a8zxqy169hd6aj1</sha1>
    </revision>
    <revision>
      <id>166103</id>
      <parentid>165974</parentid>
      <timestamp>2024-11-16T11:48:33Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Partial undo</comment>
      <origin>166103</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5089" sha1="coyml0qqbe1cel0zv137n8s1kzmcxov" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo uniquely consistent through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. 111 = 3 × 37, and 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
[[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| &amp;minus;0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| &amp;minus;0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| &amp;minus;0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| &amp;minus;0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| &amp;minus;0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| &amp;minus;0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| &amp;minus;0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) &amp;ndash; [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] &amp;ndash; guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>coyml0qqbe1cel0zv137n8s1kzmcxov</sha1>
    </revision>
    <revision>
      <id>175840</id>
      <parentid>166103</parentid>
      <timestamp>2025-01-13T08:39:30Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ Add link</comment>
      <origin>175840</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5095" sha1="s1n21b1ei3n1exbshvjqfvafgrlurog" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. 111 = 3 × 37, and 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]] and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
[[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| &amp;minus;0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| &amp;minus;0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| &amp;minus;0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| &amp;minus;0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| &amp;minus;0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| &amp;minus;0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| &amp;minus;0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) &amp;ndash; [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] &amp;ndash; guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>s1n21b1ei3n1exbshvjqfvafgrlurog</sha1>
    </revision>
    <revision>
      <id>176482</id>
      <parentid>175840</parentid>
      <timestamp>2025-01-16T14:53:53Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176482</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5094" sha1="2f9vwk4r2gjrtjzg5c8smsp96gvk0ef" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since 111&amp;nbsp;=&amp;nbsp;{{factorisation|111}}}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the 31 &amp;amp; 80 temperament, and [[buzzard]], the 53 &amp;amp; 58 temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
[[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br /&gt;(14\111)
| 248.65&lt;br /&gt;(151.35)
| 15/13&lt;br /&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br /&gt;(9\111)
| 497.30&lt;br /&gt;(97.30)
| 4/3&lt;br /&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>2f9vwk4r2gjrtjzg5c8smsp96gvk0ef</sha1>
    </revision>
    <revision>
      <id>176483</id>
      <parentid>176482</parentid>
      <timestamp>2025-01-16T14:54:45Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176483</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5116" sha1="jtq0xr9bcm0d80r22ab1g0p0usbg68d" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since 111&amp;nbsp;=&amp;nbsp;{{factorisation|111}}}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap|31 &amp;amp; 80}} temperament, and [[buzzard]], the {{nowrap|53 &amp;amp; 58}} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
[[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br /&gt;(14\111)
| 248.65&lt;br /&gt;(151.35)
| 15/13&lt;br /&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br /&gt;(9\111)
| 497.30&lt;br /&gt;(97.30)
| 4/3&lt;br /&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>jtq0xr9bcm0d80r22ab1g0p0usbg68d</sha1>
    </revision>
    <revision>
      <id>176976</id>
      <parentid>176483</parentid>
      <timestamp>2025-01-18T23:11:50Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <origin>176976</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5513" sha1="3sot16m5qdvoq0g62occdb1i5mo2bxn" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since 111&amp;nbsp;=&amp;nbsp;{{factorisation|111}}}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap|31 &amp;amp; 80}} temperament, and [[buzzard]], the {{nowrap|53 &amp;amp; 58}} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
[[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br /&gt;(14\111)
| 248.65&lt;br /&gt;(151.35)
| 15/13&lt;br /&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br /&gt;(9\111)
| 497.30&lt;br /&gt;(97.30)
| 4/3&lt;br /&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Zeta properties ==
===Zeta peak index===
{| class="wikitable"
! colspan="3" |Tuning
! colspan="3" |Strength
! colspan="2" |Closest EDO
! colspan="2" |Integer limit
|-
!ZPI
!Steps per octave
!Step size (cents)
!Height
!Integral
!Gap
!EDO
!Octave (cents)
!Consistent
!Distinct
|-
|[[655zpi]]
|111.059577998833
|10.8050113427643
|9.038544
|1.394739
|18.041165
|111edo
|1199.35625904684
|22
|16
|}
== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>3sot16m5qdvoq0g62occdb1i5mo2bxn</sha1>
    </revision>
    <revision>
      <id>177059</id>
      <parentid>176976</parentid>
      <timestamp>2025-01-19T02:43:57Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Formatting</comment>
      <origin>177059</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5543" sha1="az25lh2r59fz5u0stogwy4fotb4vmi4" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since 111&amp;nbsp;=&amp;nbsp;{{factorisation|111}}}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap|31 &amp;amp; 80}} temperament, and [[buzzard]], the {{nowrap|53 &amp;amp; 58}} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
[[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br /&gt;(14\111)
| 248.65&lt;br /&gt;(151.35)
| 15/13&lt;br /&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br /&gt;(9\111)
| 497.30&lt;br /&gt;(97.30)
| 4/3&lt;br /&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Zeta properties ==
=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[655zpi]]
| 111.059577998833
| 10.8050113427643
| 9.038544
| 1.394739
| 18.041165
| 111edo
| 1199.35625904684
| 22
| 16
|}

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>az25lh2r59fz5u0stogwy4fotb4vmi4</sha1>
    </revision>
    <revision>
      <id>177135</id>
      <parentid>177059</parentid>
      <timestamp>2025-01-19T10:12:29Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Consolidate sections</comment>
      <origin>177135</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5556" sha1="g60o3lwiwlpe9gul5uv786li2c8d1zd" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since 111&amp;nbsp;=&amp;nbsp;{{factorisation|111}}}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap|31 &amp;amp; 80}} temperament, and [[buzzard]], the {{nowrap|53 &amp;amp; 58}} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
[[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[655zpi]]
| 111.059577998833
| 10.8050113427643
| 9.038544
| 1.394739
| 18.041165
| 111edo
| 1199.35625904684
| 22
| 16
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br /&gt;(14\111)
| 248.65&lt;br /&gt;(151.35)
| 15/13&lt;br /&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br /&gt;(9\111)
| 497.30&lt;br /&gt;(97.30)
| 4/3&lt;br /&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>g60o3lwiwlpe9gul5uv786li2c8d1zd</sha1>
    </revision>
    <revision>
      <id>179295</id>
      <parentid>177135</parentid>
      <timestamp>2025-02-03T07:14:58Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Subsets and supersets */ Add subsets.</comment>
      <origin>179295</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5774" sha1="hfqcuhyftrttsh87tofuhbzd3gknix5" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since 111&amp;nbsp;=&amp;nbsp;{{factorisation|111}}}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap|31 &amp;amp; 80}} temperament, and [[buzzard]], the {{nowrap|53 &amp;amp; 58}} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
111edo has [[3edo|3]] and [[37edo|37]] as its divisors, of which 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3s [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[655zpi]]
| 111.059577998833
| 10.8050113427643
| 9.038544
| 1.394739
| 18.041165
| 111edo
| 1199.35625904684
| 22
| 16
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br /&gt;(14\111)
| 248.65&lt;br /&gt;(151.35)
| 15/13&lt;br /&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br /&gt;(9\111)
| 497.30&lt;br /&gt;(97.30)
| 4/3&lt;br /&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>hfqcuhyftrttsh87tofuhbzd3gknix5</sha1>
    </revision>
    <revision>
      <id>179301</id>
      <parentid>179295</parentid>
      <timestamp>2025-02-03T10:23:56Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ fix formatting</comment>
      <origin>179301</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5763" sha1="3w8lkcb5sdr6n1gdta01pg7y7f2s3ox" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap|111 {{=}} 3 × 37}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap|31 &amp;amp; 80}} temperament, and [[buzzard]], the {{nowrap|53 &amp;amp; 58}} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
111edo has [[3edo|3]] and [[37edo|37]] as its divisors, of which 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3s [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[655zpi]]
| 111.059577998833
| 10.8050113427643
| 9.038544
| 1.394739
| 18.041165
| 111edo
| 1199.35625904684
| 22
| 16
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br /&gt;(14\111)
| 248.65&lt;br /&gt;(151.35)
| 15/13&lt;br /&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br /&gt;(9\111)
| 497.30&lt;br /&gt;(97.30)
| 4/3&lt;br /&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>3w8lkcb5sdr6n1gdta01pg7y7f2s3ox</sha1>
    </revision>
    <revision>
      <id>179432</id>
      <parentid>179301</parentid>
      <timestamp>2025-02-04T07:04:48Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Subsets and supersets */ Attempt to improve wording.</comment>
      <origin>179432</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5795" sha1="epvx7r92db1ro2rhypwn4uysmpz26px" xml:space="preserve">{{Infobox ET}}
{{EDO intro|111}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap|111 {{=}} 3 × 37}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap|31 &amp;amp; 80}} temperament, and [[buzzard]], the {{nowrap|53 &amp;amp; 58}} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
Since 111 factors into 3 × 37, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3s [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[655zpi]]
| 111.059577998833
| 10.8050113427643
| 9.038544
| 1.394739
| 18.041165
| 111edo
| 1199.35625904684
| 22
| 16
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br /&gt;(14\111)
| 248.65&lt;br /&gt;(151.35)
| 15/13&lt;br /&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br /&gt;(9\111)
| 497.30&lt;br /&gt;(97.30)
| 4/3&lt;br /&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>epvx7r92db1ro2rhypwn4uysmpz26px</sha1>
    </revision>
    <revision>
      <id>181317</id>
      <parentid>179432</parentid>
      <timestamp>2025-02-18T16:50:44Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181317</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5790" sha1="f9eotfkxplm41wo7kwa5i83eq6x7g79" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap|111 {{=}} 3 × 37}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap|31 &amp;amp; 80}} temperament, and [[buzzard]], the {{nowrap|53 &amp;amp; 58}} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
Since 111 factors into 3 × 37, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3s [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{{Interval table}}

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[655zpi]]
| 111.059577998833
| 10.8050113427643
| 9.038544
| 1.394739
| 18.041165
| 111edo
| 1199.35625904684
| 22
| 16
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br /&gt;(14\111)
| 248.65&lt;br /&gt;(151.35)
| 15/13&lt;br /&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br /&gt;(9\111)
| 497.30&lt;br /&gt;(97.30)
| 4/3&lt;br /&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>f9eotfkxplm41wo7kwa5i83eq6x7g79</sha1>
    </revision>
    <revision>
      <id>182900</id>
      <parentid>181317</parentid>
      <timestamp>2025-02-24T09:47:06Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ rework into a manually curated table due to complaints on FB</comment>
      <origin>182900</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9145" sha1="tb4h9ha16yjgm94oozbp5qrtk50l5su" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap|111 {{=}} 3 × 37}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap|31 &amp;amp; 80}} temperament, and [[buzzard]], the {{nowrap|53 &amp;amp; 58}} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[Orwellismic family #Guanyin|guanyin temperament]], the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111}}

=== Subsets and supersets ===
Since 111 factors into 3 × 37, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3s [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
| [[22/19]]
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], ''[[28/23]]'', [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[655zpi]]
| 111.059577998833
| 10.8050113427643
| 9.038544
| 1.394739
| 18.041165
| 111edo
| 1199.35625904684
| 22
| 16
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br /&gt;(14\111)
| 248.65&lt;br /&gt;(151.35)
| 15/13&lt;br /&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br /&gt;(9\111)
| 497.30&lt;br /&gt;(97.30)
| 4/3&lt;br /&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>tb4h9ha16yjgm94oozbp5qrtk50l5su</sha1>
    </revision>
    <revision>
      <id>182901</id>
      <parentid>182900</parentid>
      <timestamp>2025-02-24T09:54:08Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ extend the harmonics table to the 61-limit. Misc. cleanup</comment>
      <origin>182901</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9269" sha1="igmhtiy2317zoq2vsfr9kqm838kzecb" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap|111 {{=}} 3 × 37}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
| [[22/19]]
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], ''[[28/23]]'', [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[655zpi]]
| 111.059577998833
| 10.8050113427643
| 9.038544
| 1.394739
| 18.041165
| 111edo
| 1199.35625904684
| 22
| 16
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br /&gt;(14\111)
| 248.65&lt;br /&gt;(151.35)
| 15/13&lt;br /&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br /&gt;(9\111)
| 497.30&lt;br /&gt;(97.30)
| 4/3&lt;br /&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – guanyin[22] in 111edo tuning

[[Category:Buzzard]]
[[Category:Guanyin]]
[[Category:Listen]]
[[Category:Orwellismic]]
[[Category:Semisept]]
[[Category:Valinorsmic]]</text>
      <sha1>igmhtiy2317zoq2vsfr9kqm838kzecb</sha1>
    </revision>
    <revision>
      <id>182902</id>
      <parentid>182901</parentid>
      <timestamp>2025-02-24T09:55:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Music */ cleanup</comment>
      <origin>182902</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9270" sha1="n0j1d6ysx602oed1jblslfyy6ogl1rx" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap|111 {{=}} 3 × 37}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
| [[22/19]]
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], ''[[28/23]]'', [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[655zpi]]
| 111.059577998833
| 10.8050113427643
| 9.038544
| 1.394739
| 18.041165
| 111edo
| 1199.35625904684
| 22
| 16
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 176 -111 }}
| {{mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br /&gt;(14\111)
| 248.65&lt;br /&gt;(151.35)
| 15/13&lt;br /&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br /&gt;(9\111)
| 497.30&lt;br /&gt;(97.30)
| 4/3&lt;br /&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>n0j1d6ysx602oed1jblslfyy6ogl1rx</sha1>
    </revision>
    <revision>
      <id>182903</id>
      <parentid>182902</parentid>
      <timestamp>2025-02-24T10:02:38Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ +23-limit</comment>
      <origin>182903</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9432" sha1="29n9br28hs25ew62moy6zk4sknq91xx" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap|111 {{=}} 3 × 37}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
| [[22/19]]
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], ''[[28/23]]'', [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[655zpi]]
| 111.059577998833
| 10.8050113427643
| 9.038544
| 1.394739
| 18.041165
| 111edo
| 1199.35625904684
| 22
| 16
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>29n9br28hs25ew62moy6zk4sknq91xx</sha1>
    </revision>
    <revision>
      <id>185405</id>
      <parentid>182903</parentid>
      <timestamp>2025-03-10T20:25:36Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>+15-odd-limit interval table</comment>
      <origin>185405</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9493" sha1="sni6hpy4vsyvcohaqdcqajshvy9t7kx" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap|111 {{=}} 3 × 37}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp;amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp;amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
| [[22/19]]
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], ''[[28/23]]'', [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[655zpi]]
| 111.059577998833
| 10.8050113427643
| 9.038544
| 1.394739
| 18.041165
| 111edo
| 1199.35625904684
| 22
| 16
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>sni6hpy4vsyvcohaqdcqajshvy9t7kx</sha1>
    </revision>
    <revision>
      <id>187316</id>
      <parentid>185405</parentid>
      <timestamp>2025-03-20T15:12:27Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Zeta peak index */</comment>
      <origin>187316</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9337" sha1="f939j2qmhzv52j7jk9i64ip06t3oaxx" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap|111 {{=}} 3 × 37}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp;amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp;amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
| [[22/19]]
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], ''[[28/23]]'', [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 655
| steps = 111.059577998833
| step size = 10.8050113427643
| tempered height = 9.038544
| pure height = 6.131468
| integral = 1.394739
| gap = 18.041165
| octave = 1199.35625904684
| consistent = 22
| distinct = 16
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>f939j2qmhzv52j7jk9i64ip06t3oaxx</sha1>
    </revision>
    <revision>
      <id>191036</id>
      <parentid>187316</parentid>
      <timestamp>2025-04-11T07:41:32Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ (''See [[regular temperament]] for more about what all this means and how to use it.'')</comment>
      <origin>191036</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9425" sha1="kbrsdca7co0gt8s6sfrciy7zlj243by" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap|111 {{=}} 3 × 37}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp;amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp;amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val. (''See [[regular temperament]] for more about what all this means and how to use it.'')

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
| [[22/19]]
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], ''[[28/23]]'', [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 655
| steps = 111.059577998833
| step size = 10.8050113427643
| tempered height = 9.038544
| pure height = 6.131468
| integral = 1.394739
| gap = 18.041165
| octave = 1199.35625904684
| consistent = 22
| distinct = 16
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>kbrsdca7co0gt8s6sfrciy7zlj243by</sha1>
    </revision>
    <revision>
      <id>191151</id>
      <parentid>191036</parentid>
      <timestamp>2025-04-11T11:17:04Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>No need to remind readers of what a regular temperament is everywhere</comment>
      <origin>191151</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9337" sha1="f939j2qmhzv52j7jk9i64ip06t3oaxx" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap|111 {{=}} 3 × 37}}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp;amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp;amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[planar temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
| [[22/19]]
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], ''[[28/23]]'', [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 655
| steps = 111.059577998833
| step size = 10.8050113427643
| tempered height = 9.038544
| pure height = 6.131468
| integral = 1.394739
| gap = 18.041165
| octave = 1199.35625904684
| consistent = 22
| distinct = 16
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>f939j2qmhzv52j7jk9i64ip06t3oaxx</sha1>
    </revision>
    <revision>
      <id>191352</id>
      <parentid>191151</parentid>
      <timestamp>2025-04-12T15:59:28Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ +octave stretch</comment>
      <origin>191352</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9755" sha1="o50tw9c1zry4emf0j977fzxw2cxtoip" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[harmonic]]s 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. 

In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], 1331/1323, [[1375/1372]], and notably the [[quartisma]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Octave stretch ===
111edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[176edt]] or [[287ed6]]. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit. 

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
| [[22/19]]
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], ''[[28/23]]'', [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 655
| steps = 111.059577998833
| step size = 10.8050113427643
| tempered height = 9.038544
| pure height = 6.131468
| integral = 1.394739
| gap = 18.041165
| octave = 1199.35625904684
| consistent = 22
| distinct = 16
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>o50tw9c1zry4emf0j977fzxw2cxtoip</sha1>
    </revision>
    <revision>
      <id>192135</id>
      <parentid>191352</parentid>
      <timestamp>2025-04-14T16:59:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ 23-limit notability</comment>
      <origin>192135</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10290" sha1="21s66moxvgil2q72i0vks2kkwgs05d1" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], [[1331/1323]], [[1375/1372]]. 

It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Octave stretch ===
111edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[176edt]] or [[287ed6]]. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit. 

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
| [[22/19]]
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], ''[[28/23]]'', [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 655
| steps = 111.059577998833
| step size = 10.8050113427643
| tempered height = 9.038544
| pure height = 6.131468
| integral = 1.394739
| gap = 18.041165
| octave = 1199.35625904684
| consistent = 22
| distinct = 16
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>21s66moxvgil2q72i0vks2kkwgs05d1</sha1>
    </revision>
    <revision>
      <id>192140</id>
      <parentid>192135</parentid>
      <timestamp>2025-04-14T17:06:32Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ 28/23 isn't inconsistent. Note 45/34</comment>
      <origin>192140</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10344" sha1="fdnaaxgew0nsmkv5qvtdiq0wurkbq6a" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], [[1331/1323]], [[1375/1372]]. 

It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Octave stretch ===
111edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[176edt]] or [[287ed6]]. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit. 

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
| [[22/19]]
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], [[28/23]], [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[45/34]], [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament, inconsistently mapped intervals in ''italic''

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 655
| steps = 111.059577998833
| step size = 10.8050113427643
| tempered height = 9.038544
| pure height = 6.131468
| integral = 1.394739
| gap = 18.041165
| octave = 1199.35625904684
| consistent = 22
| distinct = 16
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>fdnaaxgew0nsmkv5qvtdiq0wurkbq6a</sha1>
    </revision>
    <revision>
      <id>206123</id>
      <parentid>192140</parentid>
      <timestamp>2025-08-02T00:04:35Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Add instruments section</comment>
      <origin>206123</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10397" sha1="sstxukpxoas69bpvzj7pddb45pjzz3y" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], [[1331/1323]], [[1375/1372]]. 

It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Octave stretch ===
111edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[176edt]] or [[287ed6]]. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit. 

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
| [[22/19]]
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], [[28/23]], [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[45/34]], [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament, inconsistently mapped intervals in ''italic''

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 655
| steps = 111.059577998833
| step size = 10.8050113427643
| tempered height = 9.038544
| pure height = 6.131468
| integral = 1.394739
| gap = 18.041165
| octave = 1199.35625904684
| consistent = 22
| distinct = 16
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Instruments ==
* [[Lumatone mapping for 111edo]]

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>sstxukpxoas69bpvzj7pddb45pjzz3y</sha1>
    </revision>
    <revision>
      <id>216260</id>
      <parentid>206123</parentid>
      <timestamp>2025-11-06T05:05:39Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Intervals */ 22/19 was in the wrong place</comment>
      <origin>216260</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10398" sha1="egfspnar97fltqavt4a93pi2die8lvx" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], [[1331/1323]], [[1375/1372]]. 

It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Octave stretch ===
111edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[176edt]] or [[287ed6]]. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit. 

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]], [[22/19]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
|
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], [[28/23]], [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[45/34]], [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament, inconsistently mapped intervals in ''italic''

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 655
| steps = 111.059577998833
| step size = 10.8050113427643
| tempered height = 9.038544
| pure height = 6.131468
| integral = 1.394739
| gap = 18.041165
| octave = 1199.35625904684
| consistent = 22
| distinct = 16
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Instruments ==
* [[Lumatone mapping for 111edo]]

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>egfspnar97fltqavt4a93pi2die8lvx</sha1>
    </revision>
    <revision>
      <id>220880</id>
      <parentid>216260</parentid>
      <timestamp>2026-01-05T08:14:09Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Rank-2 temperaments */ Add Rubidium; also add 161/160 to ratios</comment>
      <origin>220880</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10534" sha1="rqt3u6ec71nfgapxojvu0m6nct8yc3w" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], [[1331/1323]], [[1375/1372]]. 

It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Octave stretch ===
111edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[176edt]] or [[287ed6]]. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit. 

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[161/160]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]], [[22/19]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
|
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], [[28/23]], [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[45/34]], [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament, inconsistently mapped intervals in ''italic''

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 655
| steps = 111.059577998833
| step size = 10.8050113427643
| tempered height = 9.038544
| pure height = 6.131468
| integral = 1.394739
| gap = 18.041165
| octave = 1199.35625904684
| consistent = 22
| distinct = 16
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|-
| 37
| 46\111&lt;br&gt;(1\111)
| 497.30&lt;br&gt;(10.81)
| 4/3&lt;br&gt;(161/160~169/168)
| [[7th-octave temperaments#Rubidium|Rubidium]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Instruments ==
* [[Lumatone mapping for 111edo]]

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>rqt3u6ec71nfgapxojvu0m6nct8yc3w</sha1>
    </revision>
    <revision>
      <id>220898</id>
      <parentid>220880</parentid>
      <timestamp>2026-01-05T13:52:24Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */</comment>
      <origin>220898</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10493" sha1="m3359le742h66xp7i9vlj5la632dii2" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], [[1331/1323]], [[1375/1372]]. 

It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Octave stretch ===
111edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[176edt]] or [[287ed6]]. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit. 

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[161/160]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]], [[22/19]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
|
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], [[28/23]], [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[45/34]], [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament, inconsistently mapped intervals in ''italic''

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

=== Zeta peak index ===
{{ZPI
| zpi = 655
| steps = 111.059577998833
| step size = 10.8050113427643
| tempered height = 9.038544
| pure height = 6.131468
| integral = 1.394739
| gap = 18.041165
| octave = 1199.35625904684
| consistent = 22
| distinct = 16
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|-
| 37
| 46\111&lt;br&gt;(1\111)
| 497.30&lt;br&gt;(10.81)
| 4/3&lt;br&gt;(169/168)
| [[Rubidium]]
|}
&lt;nowiki/&gt;* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Instruments ==
* [[Lumatone mapping for 111edo]]

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>m3359le742h66xp7i9vlj5la632dii2</sha1>
    </revision>
    <revision>
      <id>220899</id>
      <parentid>220898</parentid>
      <timestamp>2026-01-05T13:52:38Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Approximation to JI */ - zeta table</comment>
      <origin>220899</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10233" sha1="p9se31wk4wt4oui242z3085l478noda" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], [[1331/1323]], [[1375/1372]]. 

It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Octave stretch ===
111edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[176edt]] or [[287ed6]]. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit. 

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[161/160]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]], [[22/19]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
|
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], [[28/23]], [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[45/34]], [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament, inconsistently mapped intervals in ''italic''

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|-
| 37
| 46\111&lt;br&gt;(1\111)
| 497.30&lt;br&gt;(10.81)
| 4/3&lt;br&gt;(169/168)
| [[Rubidium]]
|}
&lt;nowiki/&gt;* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Instruments ==
* [[Lumatone mapping for 111edo]]

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>p9se31wk4wt4oui242z3085l478noda</sha1>
    </revision>
    <revision>
      <id>220906</id>
      <parentid>220899</parentid>
      <timestamp>2026-01-05T16:43:01Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Rank-2 temperaments */ Fix fractional octave &amp;mdash; half-octave was probably copied/pasted from something that actually has half-octaves</comment>
      <origin>220906</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10239" sha1="iws7go3h8pzu3mjp05cxfigi8pa7w3y" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], [[1331/1323]], [[1375/1372]]. 

It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Octave stretch ===
111edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[176edt]] or [[287ed6]]. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit. 

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[161/160]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]], [[22/19]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
|
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], [[28/23]], [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[45/34]], [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament, inconsistently mapped intervals in ''italic''

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|-
| 37
| 46\111&lt;br&gt;(1\111)
| 497.30&lt;br&gt;(10.81)
| 4/3&lt;br&gt;(169/168)
| [[Rubidium]]
|}
&lt;nowiki/&gt;* [[Normal forms|Octave-reduced form]], reduced to the first fractional octave, and [[Normal forms|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Instruments ==
* [[Lumatone mapping for 111edo]]

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>iws7go3h8pzu3mjp05cxfigi8pa7w3y</sha1>
    </revision>
    <revision>
      <id>221024</id>
      <parentid>220906</parentid>
      <timestamp>2026-01-06T14:05:26Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Was not wrong</comment>
      <origin>221024</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10233" sha1="hromdxzl3jo7n2ll8sc0clr585picd1" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], [[1331/1323]], [[1375/1372]]. 

It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Octave stretch ===
111edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[176edt]] or [[287ed6]]. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit. 

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[161/160]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]], [[22/19]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
|
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], [[28/23]], [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[45/34]], [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament, inconsistently mapped intervals in ''italic''

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|-
| 37
| 46\111&lt;br&gt;(1\111)
| 497.30&lt;br&gt;(10.81)
| 4/3&lt;br&gt;(169/168)
| [[Rubidium]]
|}
&lt;nowiki/&gt;* [[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Instruments ==
* [[Lumatone mapping for 111edo]]

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>hromdxzl3jo7n2ll8sc0clr585picd1</sha1>
    </revision>
    <revision>
      <id>225713</id>
      <parentid>221024</parentid>
      <timestamp>2026-03-13T13:30:04Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Normal forms|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct" to "[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct"</comment>
      <origin>225713</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10288" sha1="3882i9qgfgsxc765ck0n4i48fd6sbm4" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
111edo is [[consistent]] through to the [[21-odd-limit]], and is the smallest edo [[distinctly consistent]] through the [[15-odd-limit]], marking it as an important higher limit tuning. It has a sharp tendency, with [[prime harmonic|primes]] 3 through 19 all tuned sharp. Since {{nowrap| 111 {{=}} 3 × 37 }}, 111edo shares the mappings for [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]] with [[37edo]]. 

It is also significant for lower limits, especially in terms of what it [[tempering out|tempers out]] in its [[patent val]]; for example, it tempers out [[176/175]] and gives an excellent [[optimal patent val]] for the corresponding [[11-limit]] [[rank-4 temperament]]. In fact in the [[7-limit]] it tempers out [[1728/1715]], [[3136/3125]], and [[5120/5103]], and in the 11-limit, 176/175, [[540/539]], [[1331/1323]], [[1375/1372]]. 

It further tempers out among others [[351/350]], [[352/351]], [[640/637]], [[676/675]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]] in the 13-limit; [[256/255]], [[325/324]], [[442/441]] in the 17-limit; [[286/285]], [[400/399]], [[476/475]] in the 19-limit. It excels as a full [[23-limit]] temperament, tempering out [[253/252]] and [[276/275]]. The [[23/1|23]] is tuned a little flat, unlike the lower primes. [[23/19]], [[23/21]] and their [[octave complement]]s are the only inconsistently mapped intervals in the [[23-odd-limit]]. 

It is a particularly good tuning for the 11- or 13-limit versions of [[semisept]], the {{nowrap| 31 &amp; 80 }} temperament, and [[buzzard]], the {{nowrap| 53 &amp; 58 }} temperament. [[Gene Ward Smith]]'s trio in [[#Music]] section is in [[guanyin]] temperament, the [[rank-3 temperament]] [[tempering out]] 176/175 and 540/539, for which 111 also provides the optimal patent val.

=== Prime harmonics ===
{{Harmonics in equal|111|columns=9}}
{{Harmonics in equal|111|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 111edo (continued)}}

=== Octave stretch ===
111edo can benefit from slightly [[stretched and compressed tuning|compressing the octave]] if that is acceptable, using tunings such as [[176edt]] or [[287ed6]]. This improves the approximated harmonics 3, 5, 7, 13, 17 and 19; the 11 becomes less accurate as it is quite spot-on already. 23 also gets worse on compression, so the compression should be very mild if the target is the full 23-limit. 

=== Subsets and supersets ===
Since 111 factors into primes as {{nowrap| 3 × 37 }}, 111edo contains [[3edo]] and [[37edo]] as its subsets. Of these, 37edo has the same approximations of several prime harmonics, notably 5, 7, 11, and 13, and thus offers the same accuracy in the no-3's [[13-odd-limit]]. [[333edo]], which slices the step of 111edo in three, is a significant tuning.

== Intervals ==
{| class="wikitable center-1 right-2 center-4"
|-
! #
! Cents
! Approximated ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| [[1/1]]
| {{UDnote|step=0}}
|-
| 1
| 10.8
| [[121/120]], [[126/125]], [[144/143]], [[161/160]], [[169/168]], [[196/195]], [[225/224]]
| {{UDnote|step=1}}
|-
| 2
| 21.6
| ''[[64/63]]'', [[81/80]], [[91/90]], [[100/99]], [[105/104]]
| {{UDnote|step=2}}
|-
| 3
| 32.4
| ''[[46/45]]'', [[50/49]], [[55/54]], [[56/55]], [[57/56]], ''[[65/64]]''
| {{UDnote|step=3}}
|-
| 4
| 43.2
| [[36/35]], [[39/38]], [[40/39]], [[45/44]], ''[[49/48]]''
| {{UDnote|step=4}}
|-
| 5
| 54.1
| [[33/32]], [[34/33]], [[35/34]]
| {{UDnote|step=5}}
|-
| 6
| 64.9
| [[26/25]], [[27/26]], [[28/27]]
| {{UDnote|step=6}}
|-
| 7
| 75.7
| [[22/21]], [[23/22]], [[24/23]], [[25/24]]
| {{UDnote|step=7}}
|-
| 8
| 86.5
| [[20/19]], [[21/20]]
| {{UDnote|step=8}}
|-
| 9
| 97.3
| [[18/17]], [[19/18]]
| {{UDnote|step=9}}
|-
| 10
| 108.1
| [[16/15]], [[17/16]]
| {{UDnote|step=10}}
|-
| 11
| 118.9
| [[15/14]]
| {{UDnote|step=11}}
|-
| 12
| 129.7
| [[14/13]]
| {{UDnote|step=12}}
|-
| 13
| 140.5
| [[13/12]]
| {{UDnote|step=13}}
|-
| 14
| 151.4
| [[12/11]]
| {{UDnote|step=14}}
|-
| 15
| 162.2
| [[11/10]]
| {{UDnote|step=15}}
|-
| 16
| 173.0
| [[21/19]]
| {{UDnote|step=16}}
|-
| 17
| 183.8
| [[10/9]]
| {{UDnote|step=17}}
|-
| 18
| 194.6
| [[19/17]], [[28/25]]
| {{UDnote|step=18}}
|-
| 19
| 205.4
| [[9/8]]
| {{UDnote|step=19}}
|-
| 20
| 216.2
| [[17/15]], [[26/23]]
| {{UDnote|step=20}}
|-
| 21
| 227.0
| [[8/7]]
| {{UDnote|step=21}}
|-
| 22
| 237.8
| [[23/20]]
| {{UDnote|step=22}}
|-
| 23
| 248.6
| [[15/13]], [[22/19]]
| {{UDnote|step=23}}
|-
| 24
| 259.5
|
| {{UDnote|step=24}}
|-
| 25
| 270.3
| [[7/6]]
| {{UDnote|step=25}}
|-
| 26
| 281.1
| [[20/17]]
| {{UDnote|step=26}}
|-
| 27
| 291.9
| [[13/11]]
| {{UDnote|step=27}}
|-
| 28
| 302.7
| [[19/16]], [[25/21]]
| {{UDnote|step=28}}
|-
| 29
| 313.5
| [[6/5]]
| {{UDnote|step=29}}
|-
| 30
| 324.3
| ''[[23/19]]'', [[77/64]]
| {{UDnote|step=30}}
|-
| 31
| 335.1
| [[17/14]], [[40/33]]
| {{UDnote|step=31}}
|-
| 32
| 345.9
| [[11/9]], [[28/23]], [[39/32]]
| {{UDnote|step=32}}
|-
| 33
| 356.8
| [[16/13]], [[27/22]]
| {{UDnote|step=33}}
|-
| 34
| 367.6
| [[21/17]], [[26/21]]
| {{UDnote|step=34}}
|-
| 35
| 378.4
| [[56/45]]
| {{UDnote|step=35}}
|-
| 36
| 389.2
| [[5/4]]
| {{UDnote|step=36}}
|-
| 37
| 400.0
| [[24/19]], [[34/27]]
| {{UDnote|step=37}}
|-
| 38
| 410.8
| [[19/15]]
| {{UDnote|step=38}}
|-
| 39
| 421.6
| [[14/11]], [[23/18]]
| {{UDnote|step=39}}
|-
| 40
| 432.4
| [[9/7]]
| {{UDnote|step=40}}
|-
| 41
| 443.2
| [[22/17]]
| {{UDnote|step=41}}
|-
| 42
| 454.1
| [[13/10]]
| {{UDnote|step=42}}
|-
| 43
| 464.9
| [[17/13]]
| {{UDnote|step=43}}
|-
| 44
| 475.7
| [[21/16]], [[25/19]]
| {{UDnote|step=44}}
|-
| 45
| 486.5
| [[45/34]], [[65/49]]
| {{UDnote|step=45}}
|-
| 46
| 497.3
| [[4/3]]
| {{UDnote|step=46}}
|-
| 47
| 508.1
| [[51/38]]
| {{UDnote|step=47}}
|-
| 48
| 518.9
| [[23/17]], [[27/20]]
| {{UDnote|step=48}}
|-
| 49
| 529.7
| [[19/14]]
| {{UDnote|step=49}}
|-
| 50
| 540.5
| [[15/11]], [[26/19]]
| {{UDnote|step=50}}
|-
| 51
| 551.4
| [[11/8]]
| {{UDnote|step=51}}
|-
| 52
| 562.2
| [[18/13]]
| {{UDnote|step=52}}
|-
| 53
| 573.0
| [[32/23]]
| {{UDnote|step=53}}
|-
| 54
| 583.8
| [[7/5]]
| {{UDnote|step=54}}
|-
| 55
| 594.6
| [[24/17]]
| {{UDnote|step=55}}
|-
| …
| …
| …
| …
|}
&lt;nowiki/&gt;* As a 23-limit temperament, inconsistently mapped intervals in ''italic''

== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 176 -111 }}
| {{Mapping| 111 176 }}
| −0.236
| 0.236
| 2.18
|-
| 2.3.5
| 78732/78125, 67108864/66430125
| {{Mapping| 111 176 258 }}
| −0.570
| 0.510
| 4.72
|-
| 2.3.5.7
| 1728/1715, 3136/3125, 5120/5103
| {{Mapping| 111 176 258 312 }}
| −0.797
| 0.591
| 5.47
|-
| 2.3.5.7.11
| 176/175, 540/539, 1331/1323, 5120/5103
| {{Mapping| 111 176 258 312 384 }}
| −0.639
| 0.615
| 5.69
|-
| 2.3.5.7.11.13
| 176/175, 351/350, 540/539, 676/675, 1331/1323
| {{Mapping| 111 176 258 312 384 411 }}
| −0.655
| 0.562
| 5.21
|-
| 2.3.5.7.11.13.17
| 176/175, 256/255, 351/350, 442/441, 540/539, 715/714
| {{Mapping| 111 176 258 312 384 411 454 }}
| −0.672
| 0.523
| 4.84
|-
| 2.3.5.7.11.13.17.19
| 176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475
| {{Mapping| 111 176 258 312 384 411 454 472 }}
| −0.740
| 0.521
| 4.83
|-
| 2.3.5.7.11.13.17.19.23
| 176/175, 253/252, 256/255, 276/275, 286/285, 324/323, 351/350, 400/399
| {{Mapping| 111 176 258 312 384 411 454 472 502 }}
| −0.628
| 0.586
| 5.43
|}
* 111et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[94edo|94]] and [[103edo|103h]] before being superseded by [[121edo|121i]]. 

=== Rank-2 temperaments ===
Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed. 

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\111
| 118.92
| 15/14
| [[Subsedia]]
|-
| 1
| 13\111
| 140.54
| 13/12
| [[Quanic]]
|-
| 1
| 14\111
| 151.35
| 12/11
| [[Browser]]
|-
| 1
| 16\111
| 172.97
| 400/363
| [[Undetrita]]
|-
| 1
| 20\111
| 216.22
| 17/15
| [[Tremka]]
|-
| 1
| 23\111
| 248.65
| 15/13
| [[Hemikwai]]
|-
| 1
| 31\111
| 335.14
| 17/14
| [[Cohemimabila]]
|-
| 1
| 35\111
| 378.38
| 56/45
| [[Subpental]]
|-
| 1
| 41\111
| 443.24
| 22/17
| [[Warrior]]
|-
| 1
| 43\111
| 464.86
| 17/13
| [[Semisept]]
|-
| 1
| 44\111
| 475.68
| 21/16
| [[Buzzard]]
|-
| 1
| 46\111
| 497.30
| 4/3
| [[Kwai]]
|-
| 1
| 49\111
| 529.73
| 19/14
| [[Tuskaloosa]]
|-
| 1
| 55\111
| 594.59
| 55/39
| [[Gaster temperament|Gaster]]
|-
| 3
| 7\111
| 75.68
| 24/23
| [[Terture]]
|-
| 3
| 12\111
| 129.73
| 14/13
| [[Trimabila]]
|-
| 3
| 13\111
| 140.54
| 243/224
| [[Septichrome]]
|-
| 3
| 17\111
| 183.55
| 10/9
| [[Mirkat]]
|-
| 3
| 23\111&lt;br&gt;(14\111)
| 248.65&lt;br&gt;(151.35)
| 15/13&lt;br&gt;(12/11)
| [[Hemimist]]
|-
| 3
| 46\111&lt;br&gt;(9\111)
| 497.30&lt;br&gt;(97.30)
| 4/3&lt;br&gt;(18/17~19/18)
| [[Misty]]
|-
| 37
| 46\111&lt;br&gt;(1\111)
| 497.30&lt;br&gt;(10.81)
| 4/3&lt;br&gt;(169/168)
| [[Rubidium]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Scales ==
* Direct sunlight (subset of [[Sensi]][19]): 5 7 34 19 5 36 5 ((5, 12, 46, 65, 70, 106, 111)\111)
* Hypersakura (subset of Sensi[19]): 5 41 19 5 41 ((5, 46, 65, 70, 111)\111)

== Instruments ==
* [[Lumatone mapping for 111edo]]

== Music ==
; [[Gene Ward Smith]]
* ''Trio for SoftSaturn, NebulaSing and TromBonehead'' (archived 2010) – [https://soundcloud.com/genewardsmith/trio-gorts SoundCloud] | [https://www.archive.org/details/TrioForSoftsaturnNebulasingAndTrombonehead_297 details] | [https://www.archive.org/download/TrioForSoftsaturnNebulasingAndTrombonehead_297/trio-gorts.mp3 play] – in Guanyin[22], 111edo tuning

[[Category:Listen]]
[[Category:Buzzard]]
[[Category:Semisept]]
[[Category:Orwellismic]]
[[Category:Guanyin]]
[[Category:Valinorsmic]]</text>
      <sha1>3882i9qgfgsxc765ck0n4i48fd6sbm4</sha1>
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  <page>
    <title>112/75</title>
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      <id>57</id>
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      <text bytes="646" sha1="r3vrs2qc31fqt5gej0hnvr5ed6fx52i" xml:space="preserve">'''112/75'''
|4 -1 -2 1&amp;gt;

694.2435 cents

[[File:jid_112_75_pluck_adu_dr220.mp3]] [[:File:jid_112_75_pluck_adu_dr220.mp3|sound sample]]

112/75, the marvelous fifth (694.7¢), is 225/224 (7.7¢) below the just fifth [[3/2|3/2]]. It often arises in just intonation scales, for instance as the interval from 15/14 to 8/5, or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo|19edo]] fifth of 11\19. Considered as a [[Meantone|meantone]] fifth, it is approximately 5/14-comma meantone.
[[Category:interval]]
[[Category:just_interval]]</text>
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      <comment>**Imported revision 515320664 - Original comment: **</comment>
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      <text bytes="3824" sha1="o5lxre0wu44s9x6oob4ri6psn2p9ljk" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-30 21:55:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;515320664&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**112/75**
|4 -1 -2 1&amp;gt;
694.2435 cents
[[media type="file" key="jid_112_75_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_112_75_pluck_adu_dr220.mp3|sound sample]]

112/75, the marvelous fifth (694.7¢), is 225/224 (7.7¢) below the just fifth [[3_2|3/2]]. It often arises in just intonation scales, for instance as the interval from 15/14 to 8/5, or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo]] fifth of 11\19. Considered as a [[meantone]] fifth, it is approximately 5/14-comma meantone.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;112_75&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;112/75&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|4 -1 -2 1&amp;amp;gt;&amp;lt;br /&amp;gt;
694.2435 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_112_75_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_112_75_pluck_adu_dr220.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_112_75_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_112_75_pluck_adu_dr220.mp3/515315532/jid_112_75_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_112_75_pluck_adu_dr220.mp3/515315532/jid_112_75_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
112/75, the marvelous fifth (694.7¢), is 225/224 (7.7¢) below the just fifth &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. It often arises in just intonation scales, for instance as the interval from 15/14 to 8/5, or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt; fifth of 11\19. Considered as a &amp;lt;a class="wiki_link" href="/meantone"&amp;gt;meantone&amp;lt;/a&amp;gt; fifth, it is approximately 5/14-comma meantone.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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        <id>0</id>
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      <comment>**Imported revision 515247040 - Original comment: **</comment>
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      <text bytes="2297" sha1="o4iv4t8oy9jah6p6fgvemvg6ys95i68" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2014-06-30 02:05:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;515247040&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**112/75**
|4 -1 -2 1&amp;gt;
694.7368 cents

112/75, the marvelous fifth (694.7¢), is 225/224 (7.7¢) below the just fifth [[3_2|3/2]]. It often arises in just intonation scales, for instance as the interval from 15/14 to 8/5, or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the 19et fifth of 11\19. Considered as a meantone fifth, it is approximately 5/14-comma meantone.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;112_75&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;112/75&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|4 -1 -2 1&amp;amp;gt;&amp;lt;br /&amp;gt;
694.7368 cents&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
112/75, the marvelous fifth (694.7¢), is 225/224 (7.7¢) below the just fifth &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. It often arises in just intonation scales, for instance as the interval from 15/14 to 8/5, or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the 19et fifth of 11\19. Considered as a meantone fifth, it is approximately 5/14-comma meantone.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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        <id>0</id>
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      <comment>**Imported revision 515165508 - Original comment: **</comment>
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      <text bytes="2167" sha1="2v813wvgvejhc1ro8aeosmso12c9zfs" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2014-06-28 11:56:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;515165508&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**112/75**
|4 -1 -2 1&amp;gt;
694.7368 cents

112/75, the marvelous fifth (694.7¢), is 225/224 (7.7¢) below the just fifth [[3_2|3/2]]. It often arises in just intonation scales. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the 19et fifth of 11\19. Considered as a meantone fifth, it is approximately 5/14-comma meantone.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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|4 -1 -2 1&amp;amp;gt;&amp;lt;br /&amp;gt;
694.7368 cents&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
112/75, the marvelous fifth (694.7¢), is 225/224 (7.7¢) below the just fifth &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;. It often arises in just intonation scales. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the 19et fifth of 11\19. Considered as a meantone fifth, it is approximately 5/14-comma meantone.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2018-10-11T22:51:04Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
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      <comment>switched to infobox interval</comment>
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      <text bytes="658" sha1="pxg7zecz6mrvmc5ah4rgazobdspglcw" xml:space="preserve">{{Infobox Interval
| Ratio = 112/75
| Monzo = 4 -1 -2 1
| Cents = 694.2435
| Name =  marvelous fifth
| Sound = jid_112_75_pluck_adu_dr220.mp3
}}

'''112/75''', the marvelous fifth (694.7¢), is 225/224 (7.7¢) below the just fifth [[3/2|3/2]]. It often arises in just intonation scales, for instance as the interval from 15/14 to 8/5, or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo|19edo]] fifth of 11\19. Considered as a [[Meantone|meantone]] fifth, it is approximately 5/14-comma meantone.
[[Category:interval]]
[[Category:just_interval]]</text>
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    <revision>
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      <timestamp>2018-10-23T03:23:02Z</timestamp>
      <contributor>
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        <id>23</id>
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      <text bytes="690" sha1="5smzo4r3bai5a12bpxrakow4qez8gvb" xml:space="preserve">{{Infobox Interval
| Ratio = 112/75
| Monzo = 4 -1 -2 1
| Cents = 694.2435
| Name =  marvelous fifth
| Color name = zgg6, zogugu 6th
| Sound = jid_112_75_pluck_adu_dr220.mp3
}}

'''112/75''', the marvelous fifth (694.7¢), is 225/224 (7.7¢) below the just fifth [[3/2|3/2]]. It often arises in just intonation scales, for instance as the interval from 15/14 to 8/5, or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo|19edo]] fifth of 11\19. Considered as a [[Meantone|meantone]] fifth, it is approximately 5/14-comma meantone.
[[Category:interval]]
[[Category:just_interval]]</text>
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    <revision>
      <id>37992</id>
      <parentid>37582</parentid>
      <timestamp>2018-10-27T14:14:51Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>added categories and see-also section</comment>
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      <text bytes="866" sha1="mk9xwuph7b8bl84d2iox2yvvfn4gug1" xml:space="preserve">{{Infobox Interval
| Ratio = 112/75
| Monzo = 4 -1 -2 1
| Cents = 694.24348
| Name =  marvelous fifth
| Color name = zgg6, zogugu 6th
| Sound = jid_112_75_pluck_adu_dr220.mp3
}}

'''112/75''', the '''marvelous fifth''' (694.7¢), is [[225/224]] (7.7¢) below the just fifth [[3/2]]. It often arises in just intonation scales, for instance as the interval from [[15/14]] to [[8/5]], or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo]] fifth of 11\19. Considered as a [[meantone]] fifth, it is approximately 5/14-comma meantone.

== See also ==
* [[75/56]] - its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Just interval]]
[[Category:Ratio]]
[[Category:Sound example]]
[[Category:Fourth]]</text>
      <sha1>mk9xwuph7b8bl84d2iox2yvvfn4gug1</sha1>
    </revision>
    <revision>
      <id>38000</id>
      <parentid>37992</parentid>
      <timestamp>2018-10-27T18:38:32Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>added another sound example; moved the former into the see also section</comment>
      <origin>38000</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="912" sha1="rskrzh843rr5q8vtrxwb7xfnynwz4rt" xml:space="preserve">{{Infobox Interval
| Ratio = 112/75
| Monzo = 4 -1 -2 1
| Cents = 694.24348
| Name =  marvelous fifth
| Color name = zgg6, zogugu 6th
| Sound = Ji 112-75-csound-foscil-220hz.mp3
}}

'''112/75''', the '''marvelous fifth''' (694.7¢), is [[225/224]] (7.7¢) below the just fifth [[3/2]]. It often arises in just intonation scales, for instance as the interval from [[15/14]] to [[8/5]], or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo]] fifth of 11\19. Considered as a [[meantone]] fifth, it is approximately 5/14-comma meantone.

== See also ==
* [[:File:jid_112_75_pluck_adu_dr220.mp3]]
* [[75/56]] - its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Just interval]]
[[Category:Ratio]]
[[Category:Sound example]]
[[Category:Fourth]]</text>
      <sha1>rskrzh843rr5q8vtrxwb7xfnynwz4rt</sha1>
    </revision>
    <revision>
      <id>50360</id>
      <parentid>38000</parentid>
      <timestamp>2020-09-27T13:37:31Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>recat</comment>
      <origin>50360</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="942" sha1="dkjq471420nxx1mceuuce0qxvytcdkw" xml:space="preserve">{{Infobox Interval
| Ratio = 112/75
| Monzo = 4 -1 -2 1
| Cents = 694.24348
| Name =  marvelous fifth
| Color name = zgg6, zogugu 6th
| Sound = Ji 112-75-csound-foscil-220hz.mp3
}}

'''112/75''', the '''marvelous fifth''' (694.7¢), is [[225/224]] (7.7¢) below the just fifth [[3/2]]. It often arises in just intonation scales, for instance as the interval from [[15/14]] to [[8/5]], or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo]] fifth of 11\19. Considered as a [[meantone]] fifth, it is approximately 5/14-comma meantone.

== See also ==

* [[:File:jid_112_75_pluck_adu_dr220.mp3]] – another sound example
* [[75/56]] - its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Fourth]]
[[Category:Listen]]
[[Category:Marvelous]]</text>
      <sha1>dkjq471420nxx1mceuuce0qxvytcdkw</sha1>
    </revision>
    <revision>
      <id>50366</id>
      <parentid>50360</parentid>
      <timestamp>2020-09-27T13:49:00Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>recat</comment>
      <origin>50366</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="939" sha1="kppc4dqgvs1xc72b015uf621c354ymm" xml:space="preserve">{{Infobox Interval
| Ratio = 112/75
| Monzo = 4 -1 -2 1
| Cents = 694.24348
| Name =  marvelous fifth
| Color name = zgg6, zogugu 6th
| Sound = Ji 112-75-csound-foscil-220hz.mp3
}}

'''112/75''', the '''marvelous fifth''' (694.7¢), is [[225/224]] (7.7¢) below the just fifth [[3/2]]. It often arises in just intonation scales, for instance as the interval from [[15/14]] to [[8/5]], or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo]] fifth of 11\19. Considered as a [[meantone]] fifth, it is approximately 5/14-comma meantone.

== See also ==

* [[:File:jid_112_75_pluck_adu_dr220.mp3]] – another sound example
* [[75/56]] - its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Fourth]]
[[Category:Listen]]
[[Category:Marvel]]</text>
      <sha1>kppc4dqgvs1xc72b015uf621c354ymm</sha1>
    </revision>
    <revision>
      <id>64575</id>
      <parentid>50366</parentid>
      <timestamp>2021-03-27T17:09:38Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64575</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="987" sha1="r74xaxe8cn6f50rcaozaqxkq9w8yze1" xml:space="preserve">{{Infobox Interval
| Ratio = 112/75
| Monzo = 4 -1 -2 1
| Cents = 694.24348
| Name =  marvelous fifth
| Color name = zgg6, zogugu 6th
| Sound = Ji 112-75-csound-foscil-220hz.mp3
}}

'''112/75''', the '''marvelous fifth''' (694.7¢), is [[225/224]] (7.7¢) below the just fifth [[3/2]]. It often arises in just intonation scales, for instance as the interval from [[15/14]] to [[8/5]], or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo]] fifth of 11\19. Considered as a [[meantone]] fifth, it is approximately 5/14-comma meantone.

== See also ==

* [[:File:jid_112_75_pluck_adu_dr220.mp3]] – another sound example
* [[75/56]] - its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Fourth]]
[[Category:Listen]]
[[Category:Marvel]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>r74xaxe8cn6f50rcaozaqxkq9w8yze1</sha1>
    </revision>
    <revision>
      <id>86293</id>
      <parentid>64575</parentid>
      <timestamp>2022-02-22T23:46:39Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86293</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="939" sha1="kppc4dqgvs1xc72b015uf621c354ymm" xml:space="preserve">{{Infobox Interval
| Ratio = 112/75
| Monzo = 4 -1 -2 1
| Cents = 694.24348
| Name =  marvelous fifth
| Color name = zgg6, zogugu 6th
| Sound = Ji 112-75-csound-foscil-220hz.mp3
}}

'''112/75''', the '''marvelous fifth''' (694.7¢), is [[225/224]] (7.7¢) below the just fifth [[3/2]]. It often arises in just intonation scales, for instance as the interval from [[15/14]] to [[8/5]], or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo]] fifth of 11\19. Considered as a [[meantone]] fifth, it is approximately 5/14-comma meantone.

== See also ==

* [[:File:jid_112_75_pluck_adu_dr220.mp3]] – another sound example
* [[75/56]] - its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Fourth]]
[[Category:Listen]]
[[Category:Marvel]]</text>
      <sha1>kppc4dqgvs1xc72b015uf621c354ymm</sha1>
    </revision>
    <revision>
      <id>86460</id>
      <parentid>86293</parentid>
      <timestamp>2022-02-22T23:51:30Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Listen]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86460</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="919" sha1="c1yhlisifb204xmbwjwo9i3kkkxvk81" xml:space="preserve">{{Infobox Interval
| Ratio = 112/75
| Monzo = 4 -1 -2 1
| Cents = 694.24348
| Name =  marvelous fifth
| Color name = zgg6, zogugu 6th
| Sound = Ji 112-75-csound-foscil-220hz.mp3
}}

'''112/75''', the '''marvelous fifth''' (694.7¢), is [[225/224]] (7.7¢) below the just fifth [[3/2]]. It often arises in just intonation scales, for instance as the interval from [[15/14]] to [[8/5]], or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo]] fifth of 11\19. Considered as a [[meantone]] fifth, it is approximately 5/14-comma meantone.

== See also ==

* [[:File:jid_112_75_pluck_adu_dr220.mp3]] – another sound example
* [[75/56]] - its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:7-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Fourth]]
[[Category:Marvel]]</text>
      <sha1>c1yhlisifb204xmbwjwo9i3kkkxvk81</sha1>
    </revision>
    <revision>
      <id>89344</id>
      <parentid>86460</parentid>
      <timestamp>2022-03-11T17:03:13Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Categories</comment>
      <origin>89344</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="863" sha1="t3yon3m6e9zwqq41hoaov1nqrlj8pnq" xml:space="preserve">{{Infobox Interval
| Ratio = 112/75
| Monzo = 4 -1 -2 1
| Cents = 694.24348
| Name =  marvelous fifth
| Color name = zgg6, zogugu 6th
| Sound = Ji 112-75-csound-foscil-220hz.mp3
}}

'''112/75''', the '''marvelous fifth''' (694.7¢), is [[225/224]] (7.7¢) below the just fifth [[3/2]]. It often arises in just intonation scales, for instance as the interval from [[15/14]] to [[8/5]], or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo]] fifth of 11\19. Considered as a [[meantone]] fifth, it is approximately 5/14-comma meantone.

== See also ==

* [[:File:jid_112_75_pluck_adu_dr220.mp3]] – another sound example
* [[75/56]] - its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:7-limit]]
[[Category:Fifth]]
[[Category:Marvel]]</text>
      <sha1>t3yon3m6e9zwqq41hoaov1nqrlj8pnq</sha1>
    </revision>
    <revision>
      <id>97944</id>
      <parentid>89344</parentid>
      <timestamp>2022-10-25T11:30:28Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97944</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="785" sha1="mydh2ozojca8uv5gxll1clnh09qyw0a" xml:space="preserve">{{Infobox Interval
| Name =  marvelous fifth
| Color name = zgg6, zogugu 6th
| Sound = Ji 112-75-csound-foscil-220hz.mp3
}}

'''112/75''', the '''marvelous fifth''' (694.7¢), is [[225/224]] (7.7¢) below the just fifth [[3/2]]. It often arises in just intonation scales, for instance as the interval from [[15/14]] to [[8/5]], or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo]] fifth of 11\19. Considered as a [[meantone]] fifth, it is approximately 5/14-comma meantone.

== See also ==

* [[:File:jid_112_75_pluck_adu_dr220.mp3]] – another sound example
* [[75/56]] - its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:Fifth]]
[[Category:Marvel]]</text>
      <sha1>mydh2ozojca8uv5gxll1clnh09qyw0a</sha1>
    </revision>
    <revision>
      <id>102539</id>
      <parentid>97944</parentid>
      <timestamp>2023-02-08T16:24:57Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>102539</origin>
      <model>wikitext</model>
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      <text bytes="784" sha1="ltneevupilil3u0lqo6krtl8af6q8i8" xml:space="preserve">{{Infobox Interval
| Name = marvelous fifth
| Color name = zgg6, zogugu 6th
| Sound = Ji 112-75-csound-foscil-220hz.mp3
}}

'''112/75''', the '''marvelous fifth''' (694.7¢), is [[225/224]] (7.7¢) below the just fifth [[3/2]]. It often arises in just intonation scales, for instance as the interval from [[15/14]] to [[8/5]], or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo]] fifth of 11\19. Considered as a [[meantone]] fifth, it is approximately 5/14-comma meantone.

== See also ==

* [[:File:jid_112_75_pluck_adu_dr220.mp3]] – another sound example
* [[75/56]] - its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:Fifth]]
[[Category:Marvel]]</text>
      <sha1>ltneevupilil3u0lqo6krtl8af6q8i8</sha1>
    </revision>
    <revision>
      <id>102540</id>
      <parentid>102539</parentid>
      <timestamp>2023-02-08T16:32:54Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>102540</origin>
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      <text bytes="784" sha1="rhqatm817je6sqpmg7mx0e1rrw8q6cw" xml:space="preserve">{{Infobox Interval
| Name = marvelous fifth
| Color name = zgg6, zogugu 6th
| Sound = Ji 112-75-csound-foscil-220hz.mp3
}}

'''112/75''', the '''marvelous fifth''' (694.2¢), is [[225/224]] (7.7¢) below the just fifth [[3/2]]. It often arises in just intonation scales, for instance as the interval from [[15/14]] to [[8/5]], or 5/7 to 16/15. Tempering out 225/224 from the scale can be used to make it much less flat, but it is usable as it is, being only half a cent flat from the [[19edo]] fifth of 11\19. Considered as a [[meantone]] fifth, it is approximately 5/14-comma meantone.

== See also ==

* [[:File:jid_112_75_pluck_adu_dr220.mp3]] – another sound example
* [[75/56]] - its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:Fifth]]
[[Category:Marvel]]</text>
      <sha1>rhqatm817je6sqpmg7mx0e1rrw8q6cw</sha1>
    </revision>
  </page>
  <page>
    <title>112edo</title>
    <ns>0</ns>
    <id>59</id>
    <revision>
      <id>59</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>59</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="812" sha1="0zif7i0kd3oniv892koebfihqiahyx6" xml:space="preserve">112EDO has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which is identical to the perfect fifth of [[56edo|56edo]], a great inverse gentle fifth where +5 fifths gives a near-just 27:28 while -8 fifths gives a near-just 32:39 (identical to 2 degrees of [[7edo|7edo]]) and +9 fifths gives a close approximation to 17:21.

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo|17edo]], but sharing a similar structure.

=Music in 112EDO=

[https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by Cam Taylor
[[Category:31edo]]
[[Category:what_is]]
[[Category:wiki]]</text>
      <sha1>0zif7i0kd3oniv892koebfihqiahyx6</sha1>
    </revision>
    <revision>
      <id>4295</id>
      <timestamp>2017-03-14T21:27:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;iamcamtaylor</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 608844543 - Original comment: **</comment>
      <origin>4295</origin>
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      <format>text/x-wiki</format>
      <text bytes="3296" sha1="l4eq4w9u9wvrwo1cyvagan39fyesqo8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:iamcamtaylor|iamcamtaylor]] and made on &lt;tt&gt;2017-03-14 21:27:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;608844543&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;112EDO has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which is identical to the perfect fifth of [[xenharmonic/56edo|56edo]], a great inverse gentle fifth where +5 fifths gives a near-just 27:28 while -8 fifths gives a near-just 32:39 (identical to 2 degrees of [[xenharmonic/7edo|7edo]]) and +9 fifths gives a close approximation to 17:21.

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[xenharmonic/17edo|17edo]], but sharing a similar structure.

=Music in 112EDO= 

[[https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv|Circulating 2.3.7.11.13 Floaty Piano Improv]] by Cam Taylor&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;112edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;112EDO has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which is identical to the perfect fifth of &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/56edo"&amp;gt;56edo&amp;lt;/a&amp;gt;, a great inverse gentle fifth where +5 fifths gives a near-just 27:28 while -8 fifths gives a near-just 32:39 (identical to 2 degrees of &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;) and +9 fifths gives a close approximation to 17:21.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, but sharing a similar structure.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music in 112EDO"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music in 112EDO&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv" rel="nofollow"&amp;gt;Circulating 2.3.7.11.13 Floaty Piano Improv&amp;lt;/a&amp;gt; by Cam Taylor&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>l4eq4w9u9wvrwo1cyvagan39fyesqo8</sha1>
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    <revision>
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      <timestamp>2016-09-15T03:27:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 592121694 - Original comment: **</comment>
      <origin>4296</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1295" sha1="a9xve87xcw0yay27fjhjrvwo8tharj9" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-09-15 03:27:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;592121694&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;Approximates 1/4 comma meantone&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;112edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;Approximates 1/4 comma meantone&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>a9xve87xcw0yay27fjhjrvwo8tharj9</sha1>
    </revision>
    <revision>
      <id>45270</id>
      <parentid>59</parentid>
      <timestamp>2020-05-29T17:11:26Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>reworked structure, added links</comment>
      <origin>45270</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="859" sha1="82p7hvmjowxyd4ee3v5atumaxstdwr3" xml:space="preserve">'''112EDO''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

== Music in 112EDO ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:edo]]
[[Category:31edo]]
[[Category:what_is]]
[[Category:wiki]]</text>
      <sha1>82p7hvmjowxyd4ee3v5atumaxstdwr3</sha1>
    </revision>
    <revision>
      <id>50556</id>
      <parentid>45270</parentid>
      <timestamp>2020-10-02T18:45:42Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>50556</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="840" sha1="c9zbsfii7f5gzh0u28rj90swzs6v7ss" xml:space="preserve">'''112EDO''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

== Music in 112EDO ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:edo]]
[[Category:what_is]]
[[Category:wiki]]</text>
      <sha1>c9zbsfii7f5gzh0u28rj90swzs6v7ss</sha1>
    </revision>
    <revision>
      <id>54960</id>
      <parentid>50556</parentid>
      <timestamp>2020-12-04T23:13:06Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54960</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="866" sha1="lj6knmiijkrfd2sun5ktetgzi1oasav" xml:space="preserve">'''112EDO''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

== Music in 112EDO ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave]]
[[Category:what_is]]
[[Category:wiki]]</text>
      <sha1>lj6knmiijkrfd2sun5ktetgzi1oasav</sha1>
    </revision>
    <revision>
      <id>59496</id>
      <parentid>54960</parentid>
      <timestamp>2021-01-19T04:45:23Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>59496</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="889" sha1="57s56hq796fvaduq2vzeeuj0q28l5cm" xml:space="preserve">'''112EDO''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

== Music in 112EDO ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave]]
[[Category:what_is]]
[[Category:wiki]]</text>
      <sha1>57s56hq796fvaduq2vzeeuj0q28l5cm</sha1>
    </revision>
    <revision>
      <id>68096</id>
      <parentid>59496</parentid>
      <timestamp>2021-04-21T05:43:17Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68096</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2661" sha1="1kksq1q4yxcqhz2a18gsx8ln076pant" xml:space="preserve">'''112EDO''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.


Since 112edo has a step of 10.714 cents, it also allows one to use its MOS scales as circulating temperaments. 
{| class="wikitable"
|+Circulating temperaments in 112edo
!Tones
!Pattern
!L:s
|-
|5
|[[2L 3s]]
|23:22
|-
|6
|[[4L 2s]]
|19:18
|-
|7
|[[7edo]]
| rowspan="2" |equal
|-
|8
|[[8edo]]
|-
|9
|[[4L 5s]]
|13:12
|-
|10
|[[2L 8s]]
|12:11
|-
|11
|[[2L 9s]]
|11:10
|-
|12
|[[4L 8s]]
|10:9
|-
|13
|[[8L 5s]]
|9:8
|-
|14
|[[14edo]]
|equal
|-
|15
|[[6L 9s]]
|8:7
|-
|16
|[[16edo]]
|equal
|-
|17
|[[10L 7s]]
| rowspan="2" |7:6
|-
|18
|4L 14s
|-
|19
|[[17L 2s]]
| rowspan="4" |6:5
|-
|20
|12L 8s
|-
|21
|7L 14s
|-
|22
|2L 20s
|-
|23
|20L 3s
| rowspan="5" |5:4
|-
|24
|16L 8s
|-
|25
|12L 13s
|-
|26
|8L 18s
|-
|27
|4L 23s
|-
|28
|[[28edo]]
|equal
|-
|29
|25L 4s
| rowspan="9" |4:3
|-
|30
|22L 8s
|-
|31
|19L 12s
|-
|32
|16L 16s
|-
|33
|13L 20s
|-
|34
|10L 24s
|-
|35
|7L 28s
|-
|36
|4L 32s
|-
|37
|1L 36s
|-
|38
|36L 2s
| rowspan="18" |3:2
|-
|39
|34L 5s
|-
|40
|32L 8s
|-
|41
|30L 11s
|-
|42
|28L 14s
|-
|43
|26L 17s
|-
|44
|24L 20s
|-
|45
|22L 23s
|-
|46
|20L 26s
|-
|47
|18L 29s
|-
|48
|16L 32s
|-
|49
|14L 35s
|-
|50
|12L 38s
|-
|51
|10L 41s
|-
|52
|8L 44s
|-
|53
|6L 47s
|-
|54
|4L 50s
|-
|55
|2L 53s
|-
|56
|[[56edo]]
|equal
|-
|57
|55L 2s
| rowspan="33" |2:1
|-
|58
|54L 4s
|-
|59
|53L 6s
|-
|60
|52L 8s
|-
|61
|51L 10s
|-
|62
|50L 12s
|-
|63
|49L 14s
|-
|64
|48L 16s
|-
|65
|47L 18s
|-
|66
|46L 20s
|-
|67
|45L 22s
|-
|68
|44L 24s
|-
|69
|43L 26s
|-
|70
|42L 28s
|-
|71
|41L 30s
|-
|72
|40L 32s
|-
|73
|39L 34s
|-
|74
|38L 36s
|-
|75
|37L 38s
|-
|76
|36L 40s
|-
|77
|35L 42s
|-
|78
|34L 44s
|-
|79
|33L 46s
|-
|80
|32L 48s
|-
|81
|31L 50s
|-
|82
|30L 52s
|-
|83
|29L 54s
|-
|84
|28L 56s
|-
|85
|27L 58s
|-
|86
|26L 60s
|-
|87
|25L 62s
|-
|88
|24L 64s
|-
|89
|23L 66s
|}

==Music in 112EDO==

*[https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave]]
[[Category:what is]]
[[Category:wiki]]</text>
      <sha1>1kksq1q4yxcqhz2a18gsx8ln076pant</sha1>
    </revision>
    <revision>
      <id>81459</id>
      <parentid>68096</parentid>
      <timestamp>2021-12-04T12:37:42Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>fix table and cats</comment>
      <origin>81459</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2817" sha1="fadxb9071bin4jbkk4ncjpz040qas3y" xml:space="preserve">'''112EDO''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].


One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.


Since 112edo has a step of 10.714 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 112edo
! Tones
! Pattern
! L:s
|-
| 5
| [[2L 3s]]
| 23:22
|-
| 6
| [[4L 2s]]
| 19:18
|-
| 7
| [[7edo]]
| rowspan="2" | equal
|-
| 8
| [[8edo]]
|-
| 9
| [[4L 5s]]
| 13:12
|-
| 10
| [[2L 8s]]
| 12:11
|-
| 11
| [[2L 9s]]
| 11:10
|-
| 12
| [[4L 8s]]
| 10:9
|-
| 13
| [[8L 5s]]
| 9:8
|-
| 14
| [[14edo]]
| equal
|-
| 15
| [[6L 9s]]
| 8:7
|-
| 16
| [[16edo]]
| equal
|-
| 17
| [[10L 7s]]
| rowspan="2" | 7:6
|-
| 18
| 4L 14s
|-
| 19
| [[17L 2s]]
| rowspan="4" | 6:5
|-
| 20
| 12L 8s
|-
| 21
| 7L 14s
|-
| 22
| 2L 20s
|-
| 23
| 20L 3s
| rowspan="5" | 5:4
|-
| 24
| 16L 8s
|-
| 25
| 12L 13s
|-
| 26
| 8L 18s
|-
| 27
| 4L 23s
|-
| 28
| [[28edo]]
| equal
|-
| 29
| 25L 4s
| rowspan="9" | 4:3
|-
| 30
| 22L 8s
|-
| 31
| 19L 12s
|-
| 32
| 16L 16s
|-
| 33
| 13L 20s
|-
| 34
| 10L 24s
|-
| 35
| 7L 28s
|-
| 36
| 4L 32s
|-
| 37
| 1L 36s
|-
| 38
| 36L 2s
| rowspan="18" | 3:2
|-
| 39
| 34L 5s
|-
| 40
| 32L 8s
|-
| 41
| 30L 11s
|-
| 42
| 28L 14s
|-
| 43
| 26L 17s
|-
| 44
| 24L 20s
|-
| 45
| 22L 23s
|-
| 46
| 20L 26s
|-
| 47
| 18L 29s
|-
| 48
| 16L 32s
|-
| 49
| 14L 35s
|-
| 50
| 12L 38s
|-
| 51
| 10L 41s
|-
| 52
| 8L 44s
|-
| 53
| 6L 47s
|-
| 54
| 4L 50s
|-
| 55
| 2L 53s
|-
| 56
| [[56edo]]
| equal
|-
| 57
| 55L 2s
| rowspan="33" | 2:1
|-
| 58
| 54L 4s
|-
| 59
| 53L 6s
|-
| 60
| 52L 8s
|-
| 61
| 51L 10s
|-
| 62
| 50L 12s
|-
| 63
| 49L 14s
|-
| 64
| 48L 16s
|-
| 65
| 47L 18s
|-
| 66
| 46L 20s
|-
| 67
| 45L 22s
|-
| 68
| 44L 24s
|-
| 69
| 43L 26s
|-
| 70
| 42L 28s
|-
| 71
| 41L 30s
|-
| 72
| 40L 32s
|-
| 73
| 39L 34s
|-
| 74
| 38L 36s
|-
| 75
| 37L 38s
|-
| 76
| 36L 40s
|-
| 77
| 35L 42s
|-
| 78
| 34L 44s
|-
| 79
| 33L 46s
|-
| 80
| 32L 48s
|-
| 81
| 31L 50s
|-
| 82
| 30L 52s
|-
| 83
| 29L 54s
|-
| 84
| 28L 56s
|-
| 85
| 27L 58s
|-
| 86
| 26L 60s
|-
| 87
| 25L 62s
|-
| 88
| 24L 64s
|-
| 89
| 23L 66s
|}

== Music in 112EDO ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave]]</text>
      <sha1>fadxb9071bin4jbkk4ncjpz040qas3y</sha1>
    </revision>
    <revision>
      <id>81460</id>
      <parentid>81459</parentid>
      <timestamp>2021-12-04T12:38:36Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>remove redundant empty line</comment>
      <origin>81460</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2815" sha1="pqx9z77rdrt078r64sg4b6b7zz79azr" xml:space="preserve">'''112EDO''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

Since 112edo has a step of 10.714 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 112edo
! Tones
! Pattern
! L:s
|-
| 5
| [[2L 3s]]
| 23:22
|-
| 6
| [[4L 2s]]
| 19:18
|-
| 7
| [[7edo]]
| rowspan="2" | equal
|-
| 8
| [[8edo]]
|-
| 9
| [[4L 5s]]
| 13:12
|-
| 10
| [[2L 8s]]
| 12:11
|-
| 11
| [[2L 9s]]
| 11:10
|-
| 12
| [[4L 8s]]
| 10:9
|-
| 13
| [[8L 5s]]
| 9:8
|-
| 14
| [[14edo]]
| equal
|-
| 15
| [[6L 9s]]
| 8:7
|-
| 16
| [[16edo]]
| equal
|-
| 17
| [[10L 7s]]
| rowspan="2" | 7:6
|-
| 18
| 4L 14s
|-
| 19
| [[17L 2s]]
| rowspan="4" | 6:5
|-
| 20
| 12L 8s
|-
| 21
| 7L 14s
|-
| 22
| 2L 20s
|-
| 23
| 20L 3s
| rowspan="5" | 5:4
|-
| 24
| 16L 8s
|-
| 25
| 12L 13s
|-
| 26
| 8L 18s
|-
| 27
| 4L 23s
|-
| 28
| [[28edo]]
| equal
|-
| 29
| 25L 4s
| rowspan="9" | 4:3
|-
| 30
| 22L 8s
|-
| 31
| 19L 12s
|-
| 32
| 16L 16s
|-
| 33
| 13L 20s
|-
| 34
| 10L 24s
|-
| 35
| 7L 28s
|-
| 36
| 4L 32s
|-
| 37
| 1L 36s
|-
| 38
| 36L 2s
| rowspan="18" | 3:2
|-
| 39
| 34L 5s
|-
| 40
| 32L 8s
|-
| 41
| 30L 11s
|-
| 42
| 28L 14s
|-
| 43
| 26L 17s
|-
| 44
| 24L 20s
|-
| 45
| 22L 23s
|-
| 46
| 20L 26s
|-
| 47
| 18L 29s
|-
| 48
| 16L 32s
|-
| 49
| 14L 35s
|-
| 50
| 12L 38s
|-
| 51
| 10L 41s
|-
| 52
| 8L 44s
|-
| 53
| 6L 47s
|-
| 54
| 4L 50s
|-
| 55
| 2L 53s
|-
| 56
| [[56edo]]
| equal
|-
| 57
| 55L 2s
| rowspan="33" | 2:1
|-
| 58
| 54L 4s
|-
| 59
| 53L 6s
|-
| 60
| 52L 8s
|-
| 61
| 51L 10s
|-
| 62
| 50L 12s
|-
| 63
| 49L 14s
|-
| 64
| 48L 16s
|-
| 65
| 47L 18s
|-
| 66
| 46L 20s
|-
| 67
| 45L 22s
|-
| 68
| 44L 24s
|-
| 69
| 43L 26s
|-
| 70
| 42L 28s
|-
| 71
| 41L 30s
|-
| 72
| 40L 32s
|-
| 73
| 39L 34s
|-
| 74
| 38L 36s
|-
| 75
| 37L 38s
|-
| 76
| 36L 40s
|-
| 77
| 35L 42s
|-
| 78
| 34L 44s
|-
| 79
| 33L 46s
|-
| 80
| 32L 48s
|-
| 81
| 31L 50s
|-
| 82
| 30L 52s
|-
| 83
| 29L 54s
|-
| 84
| 28L 56s
|-
| 85
| 27L 58s
|-
| 86
| 26L 60s
|-
| 87
| 25L 62s
|-
| 88
| 24L 64s
|-
| 89
| 23L 66s
|}

== Music in 112EDO ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave]]</text>
      <sha1>pqx9z77rdrt078r64sg4b6b7zz79azr</sha1>
    </revision>
    <revision>
      <id>81461</id>
      <parentid>81460</parentid>
      <timestamp>2021-12-04T12:40:20Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>add some space before table and some "space" around table title</comment>
      <origin>81461</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2823" sha1="l6bwb2y7ejj9kmr4lnxi854b9dfqzdf" xml:space="preserve">'''112EDO''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

Since 112edo has a step of 10.714 cents, it also allows one to use its MOS scales as circulating temperaments.

{| class="wikitable"
|-
|+ Circulating temperaments in 112edo
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[2L 3s]]
| 23:22
|-
| 6
| [[4L 2s]]
| 19:18
|-
| 7
| [[7edo]]
| rowspan="2" | equal
|-
| 8
| [[8edo]]
|-
| 9
| [[4L 5s]]
| 13:12
|-
| 10
| [[2L 8s]]
| 12:11
|-
| 11
| [[2L 9s]]
| 11:10
|-
| 12
| [[4L 8s]]
| 10:9
|-
| 13
| [[8L 5s]]
| 9:8
|-
| 14
| [[14edo]]
| equal
|-
| 15
| [[6L 9s]]
| 8:7
|-
| 16
| [[16edo]]
| equal
|-
| 17
| [[10L 7s]]
| rowspan="2" | 7:6
|-
| 18
| 4L 14s
|-
| 19
| [[17L 2s]]
| rowspan="4" | 6:5
|-
| 20
| 12L 8s
|-
| 21
| 7L 14s
|-
| 22
| 2L 20s
|-
| 23
| 20L 3s
| rowspan="5" | 5:4
|-
| 24
| 16L 8s
|-
| 25
| 12L 13s
|-
| 26
| 8L 18s
|-
| 27
| 4L 23s
|-
| 28
| [[28edo]]
| equal
|-
| 29
| 25L 4s
| rowspan="9" | 4:3
|-
| 30
| 22L 8s
|-
| 31
| 19L 12s
|-
| 32
| 16L 16s
|-
| 33
| 13L 20s
|-
| 34
| 10L 24s
|-
| 35
| 7L 28s
|-
| 36
| 4L 32s
|-
| 37
| 1L 36s
|-
| 38
| 36L 2s
| rowspan="18" | 3:2
|-
| 39
| 34L 5s
|-
| 40
| 32L 8s
|-
| 41
| 30L 11s
|-
| 42
| 28L 14s
|-
| 43
| 26L 17s
|-
| 44
| 24L 20s
|-
| 45
| 22L 23s
|-
| 46
| 20L 26s
|-
| 47
| 18L 29s
|-
| 48
| 16L 32s
|-
| 49
| 14L 35s
|-
| 50
| 12L 38s
|-
| 51
| 10L 41s
|-
| 52
| 8L 44s
|-
| 53
| 6L 47s
|-
| 54
| 4L 50s
|-
| 55
| 2L 53s
|-
| 56
| [[56edo]]
| equal
|-
| 57
| 55L 2s
| rowspan="33" | 2:1
|-
| 58
| 54L 4s
|-
| 59
| 53L 6s
|-
| 60
| 52L 8s
|-
| 61
| 51L 10s
|-
| 62
| 50L 12s
|-
| 63
| 49L 14s
|-
| 64
| 48L 16s
|-
| 65
| 47L 18s
|-
| 66
| 46L 20s
|-
| 67
| 45L 22s
|-
| 68
| 44L 24s
|-
| 69
| 43L 26s
|-
| 70
| 42L 28s
|-
| 71
| 41L 30s
|-
| 72
| 40L 32s
|-
| 73
| 39L 34s
|-
| 74
| 38L 36s
|-
| 75
| 37L 38s
|-
| 76
| 36L 40s
|-
| 77
| 35L 42s
|-
| 78
| 34L 44s
|-
| 79
| 33L 46s
|-
| 80
| 32L 48s
|-
| 81
| 31L 50s
|-
| 82
| 30L 52s
|-
| 83
| 29L 54s
|-
| 84
| 28L 56s
|-
| 85
| 27L 58s
|-
| 86
| 26L 60s
|-
| 87
| 25L 62s
|-
| 88
| 24L 64s
|-
| 89
| 23L 66s
|}

== Music in 112EDO ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave]]</text>
      <sha1>l6bwb2y7ejj9kmr4lnxi854b9dfqzdf</sha1>
    </revision>
    <revision>
      <id>92922</id>
      <parentid>81461</parentid>
      <timestamp>2022-07-02T16:22:01Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92922</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2851" sha1="87qhdue1ou15jaqz558bvd0dpwueskt" xml:space="preserve">'''112EDO''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

Since 112edo has a step of 10.714 cents, it also allows one to use its MOS scales as circulating temperaments.

{| class="wikitable"
|-
|+ Circulating temperaments in 112edo
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[2L 3s]]
| 23:22
|-
| 6
| [[4L 2s]]
| 19:18
|-
| 7
| [[7edo]]
| rowspan="2" | equal
|-
| 8
| [[8edo]]
|-
| 9
| [[4L 5s]]
| 13:12
|-
| 10
| [[2L 8s]]
| 12:11
|-
| 11
| [[2L 9s]]
| 11:10
|-
| 12
| [[4L 8s]]
| 10:9
|-
| 13
| [[8L 5s]]
| 9:8
|-
| 14
| [[14edo]]
| equal
|-
| 15
| [[6L 9s]]
| 8:7
|-
| 16
| [[16edo]]
| equal
|-
| 17
| [[10L 7s]]
| rowspan="2" | 7:6
|-
| 18
| 4L 14s
|-
| 19
| [[17L 2s]]
| rowspan="4" | 6:5
|-
| 20
| 12L 8s
|-
| 21
| 7L 14s
|-
| 22
| 2L 20s
|-
| 23
| 20L 3s
| rowspan="5" | 5:4
|-
| 24
| 16L 8s
|-
| 25
| 12L 13s
|-
| 26
| 8L 18s
|-
| 27
| 4L 23s
|-
| 28
| [[28edo]]
| equal
|-
| 29
| 25L 4s
| rowspan="9" | 4:3
|-
| 30
| 22L 8s
|-
| 31
| 19L 12s
|-
| 32
| 16L 16s
|-
| 33
| 13L 20s
|-
| 34
| 10L 24s
|-
| 35
| 7L 28s
|-
| 36
| 4L 32s
|-
| 37
| 1L 36s
|-
| 38
| 36L 2s
| rowspan="18" | 3:2
|-
| 39
| 34L 5s
|-
| 40
| 32L 8s
|-
| 41
| 30L 11s
|-
| 42
| 28L 14s
|-
| 43
| 26L 17s
|-
| 44
| 24L 20s
|-
| 45
| 22L 23s
|-
| 46
| 20L 26s
|-
| 47
| 18L 29s
|-
| 48
| 16L 32s
|-
| 49
| 14L 35s
|-
| 50
| 12L 38s
|-
| 51
| 10L 41s
|-
| 52
| 8L 44s
|-
| 53
| 6L 47s
|-
| 54
| 4L 50s
|-
| 55
| 2L 53s
|-
| 56
| [[56edo]]
| equal
|-
| 57
| 55L 2s
| rowspan="33" | 2:1
|-
| 58
| 54L 4s
|-
| 59
| 53L 6s
|-
| 60
| 52L 8s
|-
| 61
| 51L 10s
|-
| 62
| 50L 12s
|-
| 63
| 49L 14s
|-
| 64
| 48L 16s
|-
| 65
| 47L 18s
|-
| 66
| 46L 20s
|-
| 67
| 45L 22s
|-
| 68
| 44L 24s
|-
| 69
| 43L 26s
|-
| 70
| 42L 28s
|-
| 71
| 41L 30s
|-
| 72
| 40L 32s
|-
| 73
| 39L 34s
|-
| 74
| 38L 36s
|-
| 75
| 37L 38s
|-
| 76
| 36L 40s
|-
| 77
| 35L 42s
|-
| 78
| 34L 44s
|-
| 79
| 33L 46s
|-
| 80
| 32L 48s
|-
| 81
| 31L 50s
|-
| 82
| 30L 52s
|-
| 83
| 29L 54s
|-
| 84
| 28L 56s
|-
| 85
| 27L 58s
|-
| 86
| 26L 60s
|-
| 87
| 25L 62s
|-
| 88
| 24L 64s
|-
| 89
| 23L 66s
|}

== Music in 112EDO ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>87qhdue1ou15jaqz558bvd0dpwueskt</sha1>
    </revision>
    <revision>
      <id>96192</id>
      <parentid>92922</parentid>
      <timestamp>2022-10-04T18:51:19Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96192</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2866" sha1="18tx1snb446y5b95tfzxlugp51xopev" xml:space="preserve">{{Infobox ET}}
'''112EDO''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

Since 112edo has a step of 10.714 cents, it also allows one to use its MOS scales as circulating temperaments.

{| class="wikitable"
|-
|+ Circulating temperaments in 112edo
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[2L 3s]]
| 23:22
|-
| 6
| [[4L 2s]]
| 19:18
|-
| 7
| [[7edo]]
| rowspan="2" | equal
|-
| 8
| [[8edo]]
|-
| 9
| [[4L 5s]]
| 13:12
|-
| 10
| [[2L 8s]]
| 12:11
|-
| 11
| [[2L 9s]]
| 11:10
|-
| 12
| [[4L 8s]]
| 10:9
|-
| 13
| [[8L 5s]]
| 9:8
|-
| 14
| [[14edo]]
| equal
|-
| 15
| [[6L 9s]]
| 8:7
|-
| 16
| [[16edo]]
| equal
|-
| 17
| [[10L 7s]]
| rowspan="2" | 7:6
|-
| 18
| 4L 14s
|-
| 19
| [[17L 2s]]
| rowspan="4" | 6:5
|-
| 20
| 12L 8s
|-
| 21
| 7L 14s
|-
| 22
| 2L 20s
|-
| 23
| 20L 3s
| rowspan="5" | 5:4
|-
| 24
| 16L 8s
|-
| 25
| 12L 13s
|-
| 26
| 8L 18s
|-
| 27
| 4L 23s
|-
| 28
| [[28edo]]
| equal
|-
| 29
| 25L 4s
| rowspan="9" | 4:3
|-
| 30
| 22L 8s
|-
| 31
| 19L 12s
|-
| 32
| 16L 16s
|-
| 33
| 13L 20s
|-
| 34
| 10L 24s
|-
| 35
| 7L 28s
|-
| 36
| 4L 32s
|-
| 37
| 1L 36s
|-
| 38
| 36L 2s
| rowspan="18" | 3:2
|-
| 39
| 34L 5s
|-
| 40
| 32L 8s
|-
| 41
| 30L 11s
|-
| 42
| 28L 14s
|-
| 43
| 26L 17s
|-
| 44
| 24L 20s
|-
| 45
| 22L 23s
|-
| 46
| 20L 26s
|-
| 47
| 18L 29s
|-
| 48
| 16L 32s
|-
| 49
| 14L 35s
|-
| 50
| 12L 38s
|-
| 51
| 10L 41s
|-
| 52
| 8L 44s
|-
| 53
| 6L 47s
|-
| 54
| 4L 50s
|-
| 55
| 2L 53s
|-
| 56
| [[56edo]]
| equal
|-
| 57
| 55L 2s
| rowspan="33" | 2:1
|-
| 58
| 54L 4s
|-
| 59
| 53L 6s
|-
| 60
| 52L 8s
|-
| 61
| 51L 10s
|-
| 62
| 50L 12s
|-
| 63
| 49L 14s
|-
| 64
| 48L 16s
|-
| 65
| 47L 18s
|-
| 66
| 46L 20s
|-
| 67
| 45L 22s
|-
| 68
| 44L 24s
|-
| 69
| 43L 26s
|-
| 70
| 42L 28s
|-
| 71
| 41L 30s
|-
| 72
| 40L 32s
|-
| 73
| 39L 34s
|-
| 74
| 38L 36s
|-
| 75
| 37L 38s
|-
| 76
| 36L 40s
|-
| 77
| 35L 42s
|-
| 78
| 34L 44s
|-
| 79
| 33L 46s
|-
| 80
| 32L 48s
|-
| 81
| 31L 50s
|-
| 82
| 30L 52s
|-
| 83
| 29L 54s
|-
| 84
| 28L 56s
|-
| 85
| 27L 58s
|-
| 86
| 26L 60s
|-
| 87
| 25L 62s
|-
| 88
| 24L 64s
|-
| 89
| 23L 66s
|}

== Music in 112EDO ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>18tx1snb446y5b95tfzxlugp51xopev</sha1>
    </revision>
    <revision>
      <id>111197</id>
      <parentid>96192</parentid>
      <timestamp>2023-05-02T11:08:06Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added harmonics table, moved circulating temperaments table into its own subheading</comment>
      <origin>111197</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2964" sha1="1qb387r3lpewcwzmhbev681edwzhvik" xml:space="preserve">{{Infobox ET}}
'''112edo''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

== Odd harmonics ==
{{Harmonics in equal|112|intervals=odd}}

== Circulating temperaments ==

Since 112edo has a step of 10.714 cents, it also allows one to use its MOS scales as [[circulating temperaments]].

{| class="wikitable"
|-
|+ Circulating temperaments in 112edo
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[2L 3s]]
| 23:22
|-
| 6
| [[4L 2s]]
| 19:18
|-
| 7
| [[7edo]]
| rowspan="2" | equal
|-
| 8
| [[8edo]]
|-
| 9
| [[4L 5s]]
| 13:12
|-
| 10
| [[2L 8s]]
| 12:11
|-
| 11
| [[2L 9s]]
| 11:10
|-
| 12
| [[4L 8s]]
| 10:9
|-
| 13
| [[8L 5s]]
| 9:8
|-
| 14
| [[14edo]]
| equal
|-
| 15
| [[6L 9s]]
| 8:7
|-
| 16
| [[16edo]]
| equal
|-
| 17
| [[10L 7s]]
| rowspan="2" | 7:6
|-
| 18
| 4L 14s
|-
| 19
| [[17L 2s]]
| rowspan="4" | 6:5
|-
| 20
| 12L 8s
|-
| 21
| 7L 14s
|-
| 22
| 2L 20s
|-
| 23
| 20L 3s
| rowspan="5" | 5:4
|-
| 24
| 16L 8s
|-
| 25
| 12L 13s
|-
| 26
| 8L 18s
|-
| 27
| 4L 23s
|-
| 28
| [[28edo]]
| equal
|-
| 29
| 25L 4s
| rowspan="9" | 4:3
|-
| 30
| 22L 8s
|-
| 31
| 19L 12s
|-
| 32
| 16L 16s
|-
| 33
| 13L 20s
|-
| 34
| 10L 24s
|-
| 35
| 7L 28s
|-
| 36
| 4L 32s
|-
| 37
| 1L 36s
|-
| 38
| 36L 2s
| rowspan="18" | 3:2
|-
| 39
| 34L 5s
|-
| 40
| 32L 8s
|-
| 41
| 30L 11s
|-
| 42
| 28L 14s
|-
| 43
| 26L 17s
|-
| 44
| 24L 20s
|-
| 45
| 22L 23s
|-
| 46
| 20L 26s
|-
| 47
| 18L 29s
|-
| 48
| 16L 32s
|-
| 49
| 14L 35s
|-
| 50
| 12L 38s
|-
| 51
| 10L 41s
|-
| 52
| 8L 44s
|-
| 53
| 6L 47s
|-
| 54
| 4L 50s
|-
| 55
| 2L 53s
|-
| 56
| [[56edo]]
| equal
|-
| 57
| 55L 2s
| rowspan="33" | 2:1
|-
| 58
| 54L 4s
|-
| 59
| 53L 6s
|-
| 60
| 52L 8s
|-
| 61
| 51L 10s
|-
| 62
| 50L 12s
|-
| 63
| 49L 14s
|-
| 64
| 48L 16s
|-
| 65
| 47L 18s
|-
| 66
| 46L 20s
|-
| 67
| 45L 22s
|-
| 68
| 44L 24s
|-
| 69
| 43L 26s
|-
| 70
| 42L 28s
|-
| 71
| 41L 30s
|-
| 72
| 40L 32s
|-
| 73
| 39L 34s
|-
| 74
| 38L 36s
|-
| 75
| 37L 38s
|-
| 76
| 36L 40s
|-
| 77
| 35L 42s
|-
| 78
| 34L 44s
|-
| 79
| 33L 46s
|-
| 80
| 32L 48s
|-
| 81
| 31L 50s
|-
| 82
| 30L 52s
|-
| 83
| 29L 54s
|-
| 84
| 28L 56s
|-
| 85
| 27L 58s
|-
| 86
| 26L 60s
|-
| 87
| 25L 62s
|-
| 88
| 24L 64s
|-
| 89
| 23L 66s
|}

== Music in 112EDO ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>1qb387r3lpewcwzmhbev681edwzhvik</sha1>
    </revision>
    <revision>
      <id>111198</id>
      <parentid>111197</parentid>
      <timestamp>2023-05-02T11:08:20Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Circulating temperaments */</comment>
      <origin>111198</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2964" sha1="l66l7zlkt2h3ho2um0t5xtytfhufbw9" xml:space="preserve">{{Infobox ET}}
'''112edo''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

== Odd harmonics ==
{{Harmonics in equal|112|intervals=odd}}

== Circulating temperaments ==

Since 112edo has a step of 10.714 cents, it also allows one to use its MOS scales as [[circulating temperament]]s.

{| class="wikitable"
|-
|+ Circulating temperaments in 112edo
|-
! Tones
! Pattern
! L:s
|-
| 5
| [[2L 3s]]
| 23:22
|-
| 6
| [[4L 2s]]
| 19:18
|-
| 7
| [[7edo]]
| rowspan="2" | equal
|-
| 8
| [[8edo]]
|-
| 9
| [[4L 5s]]
| 13:12
|-
| 10
| [[2L 8s]]
| 12:11
|-
| 11
| [[2L 9s]]
| 11:10
|-
| 12
| [[4L 8s]]
| 10:9
|-
| 13
| [[8L 5s]]
| 9:8
|-
| 14
| [[14edo]]
| equal
|-
| 15
| [[6L 9s]]
| 8:7
|-
| 16
| [[16edo]]
| equal
|-
| 17
| [[10L 7s]]
| rowspan="2" | 7:6
|-
| 18
| 4L 14s
|-
| 19
| [[17L 2s]]
| rowspan="4" | 6:5
|-
| 20
| 12L 8s
|-
| 21
| 7L 14s
|-
| 22
| 2L 20s
|-
| 23
| 20L 3s
| rowspan="5" | 5:4
|-
| 24
| 16L 8s
|-
| 25
| 12L 13s
|-
| 26
| 8L 18s
|-
| 27
| 4L 23s
|-
| 28
| [[28edo]]
| equal
|-
| 29
| 25L 4s
| rowspan="9" | 4:3
|-
| 30
| 22L 8s
|-
| 31
| 19L 12s
|-
| 32
| 16L 16s
|-
| 33
| 13L 20s
|-
| 34
| 10L 24s
|-
| 35
| 7L 28s
|-
| 36
| 4L 32s
|-
| 37
| 1L 36s
|-
| 38
| 36L 2s
| rowspan="18" | 3:2
|-
| 39
| 34L 5s
|-
| 40
| 32L 8s
|-
| 41
| 30L 11s
|-
| 42
| 28L 14s
|-
| 43
| 26L 17s
|-
| 44
| 24L 20s
|-
| 45
| 22L 23s
|-
| 46
| 20L 26s
|-
| 47
| 18L 29s
|-
| 48
| 16L 32s
|-
| 49
| 14L 35s
|-
| 50
| 12L 38s
|-
| 51
| 10L 41s
|-
| 52
| 8L 44s
|-
| 53
| 6L 47s
|-
| 54
| 4L 50s
|-
| 55
| 2L 53s
|-
| 56
| [[56edo]]
| equal
|-
| 57
| 55L 2s
| rowspan="33" | 2:1
|-
| 58
| 54L 4s
|-
| 59
| 53L 6s
|-
| 60
| 52L 8s
|-
| 61
| 51L 10s
|-
| 62
| 50L 12s
|-
| 63
| 49L 14s
|-
| 64
| 48L 16s
|-
| 65
| 47L 18s
|-
| 66
| 46L 20s
|-
| 67
| 45L 22s
|-
| 68
| 44L 24s
|-
| 69
| 43L 26s
|-
| 70
| 42L 28s
|-
| 71
| 41L 30s
|-
| 72
| 40L 32s
|-
| 73
| 39L 34s
|-
| 74
| 38L 36s
|-
| 75
| 37L 38s
|-
| 76
| 36L 40s
|-
| 77
| 35L 42s
|-
| 78
| 34L 44s
|-
| 79
| 33L 46s
|-
| 80
| 32L 48s
|-
| 81
| 31L 50s
|-
| 82
| 30L 52s
|-
| 83
| 29L 54s
|-
| 84
| 28L 56s
|-
| 85
| 27L 58s
|-
| 86
| 26L 60s
|-
| 87
| 25L 62s
|-
| 88
| 24L 64s
|-
| 89
| 23L 66s
|}

== Music in 112EDO ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>l66l7zlkt2h3ho2um0t5xtytfhufbw9</sha1>
    </revision>
    <revision>
      <id>115197</id>
      <parentid>111198</parentid>
      <timestamp>2023-05-30T21:16:18Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Circulating temperaments */</comment>
      <origin>115197</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="981" sha1="8ulo38zhlqy480uvvgpf5jumx19yc9o" xml:space="preserve">{{Infobox ET}}

== Theory ==
'''112edo''' has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

=== Odd harmonics ===
{{Harmonics in equal|112|intervals=odd}}

== Music ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Listen]]</text>
      <sha1>8ulo38zhlqy480uvvgpf5jumx19yc9o</sha1>
    </revision>
    <revision>
      <id>115198</id>
      <parentid>115197</parentid>
      <timestamp>2023-05-30T21:16:47Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>115198</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="992" sha1="otnosrwy3qepgj5w9j13mhmbrui8r4u" xml:space="preserve">{{Infobox ET}}
{{EDO intro|112}}
== Theory ==
112edo has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

=== Odd harmonics ===
{{Harmonics in equal|112|intervals=odd}}

== Music ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Listen]]</text>
      <sha1>otnosrwy3qepgj5w9j13mhmbrui8r4u</sha1>
    </revision>
    <revision>
      <id>132301</id>
      <parentid>115198</parentid>
      <timestamp>2024-01-21T07:13:54Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added interval table</comment>
      <origin>132301</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1028" sha1="dw5vvk66byk13e37gknv0526qhudmd9" xml:space="preserve">{{Infobox ET}}
{{EDO intro|112}}
== Theory ==
112edo has two great perfect fifths, the lower of which approximates 1/4-comma meantone (just a tad lower), and the upper of which- the [[patent fifth]]- is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27|28:27]] while -8 fifths gives a near-just [[39/32|32:39]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17|17:21]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

=== Odd harmonics ===
{{Harmonics in equal|112|intervals=odd}}

== Intervals ==
{{Interval table}}

== Music ==

* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv Circulating 2.3.7.11.13 Floaty Piano Improv] by [[Cam Taylor]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Listen]]</text>
      <sha1>dw5vvk66byk13e37gknv0526qhudmd9</sha1>
    </revision>
    <revision>
      <id>145686</id>
      <parentid>132301</parentid>
      <timestamp>2024-06-04T06:38:52Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>145686</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="964" sha1="cc0vv8vn6ba1iiwfxn651e3weebzmra" xml:space="preserve">{{Infobox ET}}
{{EDO intro|112}}

== Theory ==
112edo has two great [[3/2|perfect fifth]]s, the lower of which approximates [[quarter-comma meantone]] (just a tad lower), and the upper of which – the [[patent fifth]] – is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27]] while -8 fifths gives a near-just [[39/32]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

=== Odd harmonics ===
{{Harmonics in equal|112|intervals=odd}}

== Intervals ==
{{Interval table}}

== Music ==
; [[Cam Taylor]]
* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv ''Circulating 2.3.7.11.13 Floaty Piano Improv'']

[[Category:Listen]]</text>
      <sha1>cc0vv8vn6ba1iiwfxn651e3weebzmra</sha1>
    </revision>
    <revision>
      <id>145687</id>
      <parentid>145686</parentid>
      <timestamp>2024-06-04T06:41:39Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+subsets and supersets</comment>
      <origin>145687</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1167" sha1="4gzaq0t8np9bgu8fw2g0siuuzksll2l" xml:space="preserve">{{Infobox ET}}
{{EDO intro|112}}

== Theory ==
112edo has two great [[3/2|perfect fifth]]s, the lower of which approximates [[quarter-comma meantone]] (just a tad lower), and the upper of which – the [[patent fifth]] – is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27]] while -8 fifths gives a near-just [[39/32]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

=== Odd harmonics ===
{{Harmonics in equal|112|intervals=odd}}

=== Subsets and supersets ===
Since 112 factors into {{factorization|112}}, 112edo has subset edos {{EDOs| 2, 4, 7, 8, 14, 16, 28, and 56 }}. [[224edo]], which doubles it, is a strong 13-limit system. 

== Intervals ==
{{Interval table}}

== Music ==
; [[Cam Taylor]]
* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv ''Circulating 2.3.7.11.13 Floaty Piano Improv'']

[[Category:Listen]]</text>
      <sha1>4gzaq0t8np9bgu8fw2g0siuuzksll2l</sha1>
    </revision>
    <revision>
      <id>181318</id>
      <parentid>145687</parentid>
      <timestamp>2025-02-18T16:51:23Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181318</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1160" sha1="9888vri9yid41z932d6aiz7c8ndljpt" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
112edo has two great [[3/2|perfect fifth]]s, the lower of which approximates [[quarter-comma meantone]] (just a tad lower), and the upper of which—the [[patent fifth]]—is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27]] while −8 fifths gives a near-just [[39/32]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

=== Odd harmonics ===
{{Harmonics in equal|112|intervals=odd}}

=== Subsets and supersets ===
Since 112 factors into {{factorization|112}}, 112edo has subset edos {{EDOs| 2, 4, 7, 8, 14, 16, 28, and 56 }}. [[224edo]], which doubles it, is a strong 13-limit system. 

== Intervals ==
{{Interval table}}

== Music ==
; [[Cam Taylor]]
* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv ''Circulating 2.3.7.11.13 Floaty Piano Improv'']

[[Category:Listen]]</text>
      <sha1>9888vri9yid41z932d6aiz7c8ndljpt</sha1>
    </revision>
    <revision>
      <id>193703</id>
      <parentid>181318</parentid>
      <timestamp>2025-04-23T15:30:27Z</timestamp>
      <contributor>
        <username>MisterShafXen</username>
        <id>509</id>
      </contributor>
      <minor/>
      <origin>193703</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1209" sha1="gwe8rs3zyndvxp50sq1zkmly7sq8rfh" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
112edo has two great [[3/2|perfect fifth]]s, the lower of which approximates [[quarter-comma meantone]] (just a tad lower), and the upper of which—the [[patent fifth]]—is identical to the perfect fifth of [[56edo]], a great inverse gentle fifth where +5 fifths gives a near-just [[28/27]] while −8 fifths gives a near-just [[39/32]] (identical to 2 degrees of [[7edo]]) and +9 fifths gives a close approximation to [[21/17]].

One can form a 17-tone circle by taking 15 large fifths and 2 small fifths, as above, which gives some nice interval shadings a wee bit different from [[17edo]], but sharing a similar structure.

=== Odd harmonics ===
{{Harmonics in equal|112|intervals=odd}}

=== Subsets and supersets ===
Since 112 factors into {{factorization|112}}, 112edo has subset edos {{EDOs| 2, 4, 7, 8, 14, 16, 28, and 56 }}. [[224edo]], which doubles it, is a strong 13-limit system. 

== Intervals ==
{{Interval table}}

== Music ==
; [[Cam Taylor]]
* [https://soundcloud.com/camtaylor-1/17_112edo-circulating-2371113-floaty-piano-improv ''Circulating 2.3.7.11.13 Floaty Piano Improv'']

== See also ==
[[Skip fretting system 112 9 11]]
[[Category:Listen]]</text>
      <sha1>gwe8rs3zyndvxp50sq1zkmly7sq8rfh</sha1>
    </revision>
  </page>
  <page>
    <title>114edo</title>
    <ns>0</ns>
    <id>60</id>
    <revision>
      <id>60</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
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      <format>text/x-wiki</format>
      <text bytes="1632" sha1="64lx76na2cqbdpe014x5ogyfvrig1ya" xml:space="preserve">'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent|cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120 and 176/175, in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

===Period of 19-limit Shrutar===

{| class="wikitable"
|-
! | Degree
! | Cents
|-
| | 2
| | 21.05263
|-
| | 3
| | 31.57895
|-
| | 5
| | 52.63158
|-
| | 7
| | 73.68421
|-
| | 8
| | 84.21053
|-
| | 10
| | 105.26316
|-
| | 12
| | 126.31579
|-
| | 13
| | 136.842105
|-
| | 15
| | 157.89474
|-
| | 17
| | 178.94737
|-
| | 18
| | 189.47369
|-
| | 20
| | 210.52632
|-
| | 22
| | 231.57895
|-
| | 23
| | 242.10526
|-
| | 25
| | 263.157895
|-
| | 27
| | 284.21053
|-
| | 29
| | 305.26316
|-
| | 30
| | 315.78947
|-
| | 32
| | 336.842105
|-
| | 34
| | 357.89474
|-
| | 35
| | 368.42105
|-
| | 37
| | 389.47368
|-
| | 39
| | 410.52632
|-
| | 40
| | 421.05263
|-
| | 42
| | 442.10526
|-
| | 44
| | 463.157895
|-
| | 45
| | 473.68421
|-
| | 47
| | 494.73684
|-
| | 49
| | 515.78947
|-
| | 50
| | 526.31579
|-
| | 52
| | 547.36842
|-
| | 54
| | 568.42105
|-
| | 55
| | 578.94737
|}
[[Category:edo]]
[[Category:shrutar]]
[[Category:theory]]</text>
      <sha1>64lx76na2cqbdpe014x5ogyfvrig1ya</sha1>
    </revision>
    <revision>
      <id>4297</id>
      <timestamp>2016-08-08T15:36:12Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 588922354 - Original comment: **</comment>
      <origin>4297</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8473" sha1="n2zq12pdmmlfoixp02aledp1l7izs7i" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-08-08 15:36:12 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;588922354&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**114edo** is the [[equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit]] it [[tempering out|tempers out]] 2048/2025, in the [[7-limit]] 245/243, in the [[11-limit]] 121/120 and 176/175, in the [[13-limit]] 196/195 and 325/324, in the [[17-limit]] 136/135 and 154/153, in the [[19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic family|shrutar temperament]]; it is in fact the [[optimal patent val]] for [[shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

===Period of 19-limit Shrutar=== 
||~ Degree ||~ Cents ||
|| 2 || 21.05263 ||
|| 3 || 31.57895 ||
|| 5 || 52.63158 ||
|| 7 || 73.68421 ||
|| 8 || 84.21053 ||
|| 10 || 105.26316 ||
|| 12 || 126.31579 ||
|| 13 || 136.842105 ||
|| 15 || 157.89474 ||
|| 17 || 178.94737 ||
|| 18 || 189.47369 ||
|| 20 || 210.52632 ||
|| 22 || 231.57895 ||
|| 23 || 242.10526 ||
|| 25 || 263.157895 ||
|| 27 || 284.21053 ||
|| 29 || 305.26316 ||
|| 30 || 315.78947 ||
|| 32 || 336.842105 ||
|| 34 || 357.89474 ||
|| 35 || 368.42105 ||
|| 37 || 389.47368 ||
|| 39 || 410.52632 ||
|| 40 || 421.05263 ||
|| 42 || 442.10526 ||
|| 44 || 463.157895 ||
|| 45 || 473.68421 ||
|| 47 || 494.73684 ||
|| 49 || 515.78947 ||
|| 50 || 526.31579 ||
|| 52 || 547.36842 ||
|| 54 || 568.42105 ||
|| 55 || 578.94737 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
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      <contributor>
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      <comment>**Imported revision 403741050 - Original comment: **</comment>
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      <text bytes="3055" sha1="momjeblg9v9q2m7ws60gnwmztlqa8ae" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2013-02-03 00:54:09 UTC&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**114edo** is the [[equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit]] it [[tempering out|tempers out]] 2048/2025, in the [[7-limit]] 245/243, in the [[11-limit]] 121/120 and 176/175, in the [[13-limit]] 196/195 and 325/324, in the [[17-limit]] 136/135 and 154/153, in the [[19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic family|shrutar temperament]]; it is in fact the [[optimal patent val]] for [[shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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      <contributor>
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      <comment>**Imported revision 239611245 - Original comment: **</comment>
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      <text bytes="2959" sha1="g9wd0qzo564f63rfe4bft15dyyqfhye" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2011-07-01 00:44:02 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239611245&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**114edo** is the [[equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit]] it [[tempering out|tempers out]] 2048/2025, in the [[7-limit]] 245/243, in the [[11-limit]] 121/120 and 176/175, in the [[13-limit]] 196/195 and 325/324, in the [[17-limit]] 136/135 and 154/153, in the [[19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic family|shrutar temperament]]; it is in fact the [[optimal patent val]] for [[shrutar]] in the 11- 13- 17- and 19-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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      <comment>**Imported revision 239312341 - Original comment: **</comment>
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      <text bytes="2943" sha1="7liixvfhqojgsf8mjj7tcm8jxefit1t" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 09:42:07 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239312341&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**114edo** is the [[equal division of the octave]] into 114 parts, each of 10.526 [[cent]]s. In the [[5-limit]] it [[tempering out|tempers out]] 2048/2025, in the [[7-limit]] 245/243, in the [[11-limit]] 121/120 and 176/175, in the [[13-limit]] 196/195 and 325/324, in the [[17-limit]] 136/135 and 154/153, in the [[19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic family|shrutar temperament]]; it is in fact the [[optimal patent val]] for [[shrutar]] in the 11- 13- 17- and 19-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;114edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;114edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 114 parts, each of 10.526 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 2048/2025, in the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; 245/243, in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; 121/120 and 176/175, in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; 196/195 and 325/324, in the &amp;lt;a class="wiki_link" href="/17-limit"&amp;gt;17-limit&amp;lt;/a&amp;gt; 136/135 and 154/153, in the &amp;lt;a class="wiki_link" href="/19-limit"&amp;gt;19-limit&amp;lt;/a&amp;gt; 286/285 and 343/342. These commas make for 114edo being an excellent tuning for &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;shrutar temperament&amp;lt;/a&amp;gt;; it is in fact the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/shrutar"&amp;gt;shrutar&amp;lt;/a&amp;gt; in the 11- 13- 17- and 19-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4301</id>
      <timestamp>2011-03-25T15:14:02Z</timestamp>
      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 214060474 - Original comment: **</comment>
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      <text bytes="2836" sha1="09d9wpxs1txw6k4bgd94dl0vrmakxss" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-03-25 15:14:02 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214060474&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //114 equal division// divides the [[octave]] into 114 equal parts, each of 10.526 [[cent]]s. In the [[5-limit]] it tempers out 2048/2025, in the [[7-limit]] 245/243, in the [[11-limit]] 121/120 and 176/175, in the [[13-limit]] 196/195 and 325/324, in the [[17-limit]] 136/135 and 154/153, in the [[19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic family|shrutar temperament]]; it is in fact the [[optimal patent val]] for [[shrutar]] in the 11- 13- 17- and 19-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;114edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;114 equal division&amp;lt;/em&amp;gt; divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 114 equal parts, each of 10.526 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; it tempers out 2048/2025, in the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; 245/243, in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; 121/120 and 176/175, in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt; 196/195 and 325/324, in the &amp;lt;a class="wiki_link" href="/17-limit"&amp;gt;17-limit&amp;lt;/a&amp;gt; 136/135 and 154/153, in the &amp;lt;a class="wiki_link" href="/19-limit"&amp;gt;19-limit&amp;lt;/a&amp;gt; 286/285 and 343/342. These commas make for 114edo being an excellent tuning for &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;shrutar temperament&amp;lt;/a&amp;gt;; it is in fact the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/shrutar"&amp;gt;shrutar&amp;lt;/a&amp;gt; in the 11- 13- 17- and 19-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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        <id>0</id>
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      <comment>**Imported revision 213074704 - Original comment: **</comment>
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      <text bytes="2335" sha1="j2hb03umj2wizxdsldw91pv1x41m7o2" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-23 00:14:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;213074704&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //114 equal division// divides the octave into 114 equal parts, each of 10.526 cents. In the 5-limit it tempers out 2048/2025, in the 7-limit 245/243, in the 11-limit 121/120 and 176/175, in the 13-limit 196/195 and 325/324, in the 17-limit 136/135 and 154/153, in the 19-limit 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic family|shrutar temperament]]; it is in fact the [[optimal patent val]] for shrutar in the 11- 13- 17- and 19-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;114edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;114 equal division&amp;lt;/em&amp;gt; divides the octave into 114 equal parts, each of 10.526 cents. In the 5-limit it tempers out 2048/2025, in the 7-limit 245/243, in the 11-limit 121/120 and 176/175, in the 13-limit 196/195 and 325/324, in the 17-limit 136/135 and 154/153, in the 19-limit 286/285 and 343/342. These commas make for 114edo being an excellent tuning for &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;shrutar temperament&amp;lt;/a&amp;gt;; it is in fact the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for shrutar in the 11- 13- 17- and 19-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 213063846 - Original comment: **</comment>
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      <text bytes="2099" sha1="sy0exez81sdjltxwhsfu31r0hh2ixbg" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-22 23:15:34 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;213063846&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //114 equal division// divides the octave into 114 equal parts, each of 10.526 cents. In the 5-limit it tempers out 2048/2025, in the 7-limit 245/243, in the 11-limit 121/120 and 176/175, in the 13-limit 196/195 and 325/324, in the 17-limit 136/135 and 154/153, in the 19-limit 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic family|shrutar temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;114edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;114 equal division&amp;lt;/em&amp;gt; divides the octave into 114 equal parts, each of 10.526 cents. In the 5-limit it tempers out 2048/2025, in the 7-limit 245/243, in the 11-limit 121/120 and 176/175, in the 13-limit 196/195 and 325/324, in the 17-limit 136/135 and 154/153, in the 19-limit 286/285 and 343/342. These commas make for 114edo being an excellent tuning for &amp;lt;a class="wiki_link" href="/Diaschismic%20family"&amp;gt;shrutar temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>49135</id>
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      <timestamp>2020-09-10T22:59:07Z</timestamp>
      <contributor>
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        <id>220</id>
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      <comment>Added a comma that is tempered out by this EDO</comment>
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      <text bytes="1657" sha1="qrsbqfll7xq8wd2xpjlkterqegmmvsy" xml:space="preserve">'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent|cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and and 117440512/117406179, in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

===Period of 19-limit Shrutar===

{| class="wikitable"
|-
! | Degree
! | Cents
|-
| | 2
| | 21.05263
|-
| | 3
| | 31.57895
|-
| | 5
| | 52.63158
|-
| | 7
| | 73.68421
|-
| | 8
| | 84.21053
|-
| | 10
| | 105.26316
|-
| | 12
| | 126.31579
|-
| | 13
| | 136.842105
|-
| | 15
| | 157.89474
|-
| | 17
| | 178.94737
|-
| | 18
| | 189.47369
|-
| | 20
| | 210.52632
|-
| | 22
| | 231.57895
|-
| | 23
| | 242.10526
|-
| | 25
| | 263.157895
|-
| | 27
| | 284.21053
|-
| | 29
| | 305.26316
|-
| | 30
| | 315.78947
|-
| | 32
| | 336.842105
|-
| | 34
| | 357.89474
|-
| | 35
| | 368.42105
|-
| | 37
| | 389.47368
|-
| | 39
| | 410.52632
|-
| | 40
| | 421.05263
|-
| | 42
| | 442.10526
|-
| | 44
| | 463.157895
|-
| | 45
| | 473.68421
|-
| | 47
| | 494.73684
|-
| | 49
| | 515.78947
|-
| | 50
| | 526.31579
|-
| | 52
| | 547.36842
|-
| | 54
| | 568.42105
|-
| | 55
| | 578.94737
|}
[[Category:edo]]
[[Category:shrutar]]
[[Category:theory]]</text>
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    <revision>
      <id>49444</id>
      <parentid>49135</parentid>
      <timestamp>2020-09-13T20:27:51Z</timestamp>
      <contributor>
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        <id>220</id>
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      <text bytes="1671" sha1="52splyetx8pz9gv9bm52j14o187ihex" xml:space="preserve">'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent|cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

===Period of 19-limit Shrutar===

{| class="wikitable"
|-
! | Degree
! | Cents
|-
| | 2
| | 21.05263
|-
| | 3
| | 31.57895
|-
| | 5
| | 52.63158
|-
| | 7
| | 73.68421
|-
| | 8
| | 84.21053
|-
| | 10
| | 105.26316
|-
| | 12
| | 126.31579
|-
| | 13
| | 136.842105
|-
| | 15
| | 157.89474
|-
| | 17
| | 178.94737
|-
| | 18
| | 189.47369
|-
| | 20
| | 210.52632
|-
| | 22
| | 231.57895
|-
| | 23
| | 242.10526
|-
| | 25
| | 263.157895
|-
| | 27
| | 284.21053
|-
| | 29
| | 305.26316
|-
| | 30
| | 315.78947
|-
| | 32
| | 336.842105
|-
| | 34
| | 357.89474
|-
| | 35
| | 368.42105
|-
| | 37
| | 389.47368
|-
| | 39
| | 410.52632
|-
| | 40
| | 421.05263
|-
| | 42
| | 442.10526
|-
| | 44
| | 463.157895
|-
| | 45
| | 473.68421
|-
| | 47
| | 494.73684
|-
| | 49
| | 515.78947
|-
| | 50
| | 526.31579
|-
| | 52
| | 547.36842
|-
| | 54
| | 568.42105
|-
| | 55
| | 578.94737
|}
[[Category:edo]]
[[Category:shrutar]]
[[Category:theory]]</text>
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    <revision>
      <id>49445</id>
      <parentid>49444</parentid>
      <timestamp>2020-09-13T20:28:06Z</timestamp>
      <contributor>
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        <id>220</id>
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      <text bytes="1667" sha1="72vdc1vtuvxyzlrg7rjftxnsfzioq84" xml:space="preserve">'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent|cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

===Period of 19-limit Shrutar===

{| class="wikitable"
|-
! | Degree
! | Cents
|-
| | 2
| | 21.05263
|-
| | 3
| | 31.57895
|-
| | 5
| | 52.63158
|-
| | 7
| | 73.68421
|-
| | 8
| | 84.21053
|-
| | 10
| | 105.26316
|-
| | 12
| | 126.31579
|-
| | 13
| | 136.842105
|-
| | 15
| | 157.89474
|-
| | 17
| | 178.94737
|-
| | 18
| | 189.47369
|-
| | 20
| | 210.52632
|-
| | 22
| | 231.57895
|-
| | 23
| | 242.10526
|-
| | 25
| | 263.157895
|-
| | 27
| | 284.21053
|-
| | 29
| | 305.26316
|-
| | 30
| | 315.78947
|-
| | 32
| | 336.842105
|-
| | 34
| | 357.89474
|-
| | 35
| | 368.42105
|-
| | 37
| | 389.47368
|-
| | 39
| | 410.52632
|-
| | 40
| | 421.05263
|-
| | 42
| | 442.10526
|-
| | 44
| | 463.157895
|-
| | 45
| | 473.68421
|-
| | 47
| | 494.73684
|-
| | 49
| | 515.78947
|-
| | 50
| | 526.31579
|-
| | 52
| | 547.36842
|-
| | 54
| | 568.42105
|-
| | 55
| | 578.94737
|}
[[Category:edo]]
[[Category:shrutar]]
[[Category:theory]]</text>
      <sha1>72vdc1vtuvxyzlrg7rjftxnsfzioq84</sha1>
    </revision>
    <revision>
      <id>54963</id>
      <parentid>49445</parentid>
      <timestamp>2020-12-04T23:13:06Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54963</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1693" sha1="o18spr1z2qhn56m1nb1kokabgrc6fc4" xml:space="preserve">'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent|cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

===Period of 19-limit Shrutar===

{| class="wikitable"
|-
! | Degree
! | Cents
|-
| | 2
| | 21.05263
|-
| | 3
| | 31.57895
|-
| | 5
| | 52.63158
|-
| | 7
| | 73.68421
|-
| | 8
| | 84.21053
|-
| | 10
| | 105.26316
|-
| | 12
| | 126.31579
|-
| | 13
| | 136.842105
|-
| | 15
| | 157.89474
|-
| | 17
| | 178.94737
|-
| | 18
| | 189.47369
|-
| | 20
| | 210.52632
|-
| | 22
| | 231.57895
|-
| | 23
| | 242.10526
|-
| | 25
| | 263.157895
|-
| | 27
| | 284.21053
|-
| | 29
| | 305.26316
|-
| | 30
| | 315.78947
|-
| | 32
| | 336.842105
|-
| | 34
| | 357.89474
|-
| | 35
| | 368.42105
|-
| | 37
| | 389.47368
|-
| | 39
| | 410.52632
|-
| | 40
| | 421.05263
|-
| | 42
| | 442.10526
|-
| | 44
| | 463.157895
|-
| | 45
| | 473.68421
|-
| | 47
| | 494.73684
|-
| | 49
| | 515.78947
|-
| | 50
| | 526.31579
|-
| | 52
| | 547.36842
|-
| | 54
| | 568.42105
|-
| | 55
| | 578.94737
|}
[[Category:Equal divisions of the octave]]
[[Category:shrutar]]
[[Category:theory]]</text>
      <sha1>o18spr1z2qhn56m1nb1kokabgrc6fc4</sha1>
    </revision>
    <revision>
      <id>68167</id>
      <parentid>54963</parentid>
      <timestamp>2021-04-21T19:06:37Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Period of 19-limit Shrutar */</comment>
      <origin>68167</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3971" sha1="9rp8khyofa11e1ama32seclvyqv8cek" xml:space="preserve">'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

===Period of 19-limit Shrutar===

{| class="wikitable"
|-
! |Degree
! |Cents
!Difference from 68edo
|-
| |2
| |21.05263
|3.40557¢
|-
| |3
| |31.57895
| -3.71517¢
|-
| | 5
| | 52.63158
| -0.3096¢
|-
| |7
| |73.68421
|3.096¢
|-
| |8
| |84.21053
| -4.02477¢
|-
| |10
| |105.26316
|  -0.619195¢
|-
| |12
| | 126.31579
|2.78638¢
|-
| |13
| |136.842105
|  -4.334365¢
|-
| |15
| |157.89474
| -0.9288¢
|-
| | 17
| |178.94737
|2.47678¢
|-
| |18
| | 189.47369
| -4.644¢
|-
| |20
| |210.52632
| -1.23839¢
|-
| |22
| |231.57895
|2.16718¢
|-
| |23
| |242.10526
|  -4.953560372
|-
| |25
| |263.157895
| -1.548¢
|-
| |27
| |284.21053
|1.857585¢
|-
| |29
| |305.26316
|5.26316¢
|-
| |30
| |315.78947
| -1.857585¢
|-
| |32
| |336.842105
|1.548¢
|-
| |34
| |357.89474
|4.95356¢
|-
| |35
| | 368.42105
| -2.16718¢
|-
| |37
| |389.47368
|1.23839¢
|-
| |39
| |410.52632
|4.64396¢
|-
| |40
| |421.05263
| -2.47678¢
|-
| |42
| |442.10526
|0.92879¢
|-
| |44
| |463.157895
|4.334365¢
|-
| |45
| |473.68421
| -2.78638¢
|-
| |47
| |494.73684
|0.619195¢
|-
| |49
| |515.78947
|4.02477¢
|-
| |50
| |526.31579
| -3.095975¢
|-
| |52
| |547.36842
|0.3096¢
|-
| |54
| |568.42105
|3.71517¢
|-
| |55
| |578.94737
| -3.40557¢
|}
Since 114edo has a step of 10.52632 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first edo which allows one to use an MOS scale of 90 tones or more as a circulating temperament.
{| class="wikitable"
|+Circulating temperaments in 114edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|23:22
|-
|6
|[[6edo]]
|equal
|-
|7
|[[2L 5s]]
|17:16
|-
|8
|[[2L 6s]]
|15:14
|-
|9
|[[6L 3s]]
|13:12
|-
|10
|[[4L 6s]]
|12:11
|-
|11
|[[4L 7s]]
|11:10
|-
|12
|[[6L 6s]]
|10:9
|-
|13
|[[10L 3s]]
| rowspan="2" |9:8
|-
|14
|[[2L 12s]]
|-
|15
|[[9L 6s]]
| rowspan="2" |8:7
|-
|16
|2L 14s
|-
|17
|[[12L 5s]]
| rowspan="2" |7:6
|-
|18
|6L 12s
|-
|19
|[[19edo]]
|equal
|-
|20
|14L 6s
| rowspan="3" |6:5
|-
|21
|9L 12s
|-
|22
|4L 18s
|-
|23
|22L 1s
| rowspan="6" |5:4
|-
|24
|18L 6s
|-
|25
|14L 11s
|-
|26
|10L 16s
|-
|27
|6L 21s
|-
|28
|2L 26s
|-
|29
|27L 2s
| rowspan="9" |4:3
|-
|30
|24L 6s
|-
|31
|21L 10s
|-
|32
|18L 14s
|-
|33
|15L 18s
|-
|34
|12L 22s
|-
|35
|9L 26s
|-
|36
|6L 30s
|-
|37
|3L 34s
|-
|38
|[[38edo]]
|equal
|-
|39
|36L 3s
| rowspan="18" |3:2
|-
|40
|34L 6s
|-
|41
|32L 9s
|-
|42
|30L 12s
|-
|43
|28L 15s
|-
|44
|26L 18s
|-
|45
|24L 21s
|-
|46
|22L 24s
|-
|47
|20L 27s
|-
|48
|18L 30L
|-
|49
|16L 33s
|-
|50
|14L 36s
|-
|51
|12L 39s
|-
|52
|10L 42s
|-
|53
|8L 45s
|-
|54
|6L 48s
|-
|55
|4L 52s
|-
|56
|2L 54s
|-
|57
|[[57edo]]
|equal
|-
|58
|56L 2s
| rowspan="34" |2:1
|-
|59
|55L 4s
|-
|60
|54L 6s
|-
|61
|53L 8s
|-
|62
|52L 10s
|-
|63
|51L 12s
|-
|64
|50L 14s
|-
|65
|49L 16s
|-
|66
|48L 18s
|-
|67
|47L 20s
|-
|68
|46L 22s
|-
|69
|45L 24s
|-
|70
|44L 26s
|-
|71
|43L 28s
|-
|72
|42L 30s
|-
|73
|41L 32s
|-
|74
|40L 34s
|-
|75
|39L 36s
|-
|76
|38L 38s
|-
|77
|37L 40s
|-
|78
|36L 42s
|-
|79
|35L 44s
|-
|80
|34L 46s
|-
|81
|33L 48s
|-
|82
|32L 50s
|-
|83
|31L 52s
|-
|84
|30L 54s
|-
|85
|29L 56s
|-
|86
|28L 58s
|-
|87
|27L 60s
|-
|88
|26L 62s
|-
|89
|25L 64s
|-
|90
|24L 66s
|-
|91
|23L 68s
|}
[[Category:Equal divisions of the octave]]
[[Category:shrutar]]
[[Category:theory]]</text>
      <sha1>9rp8khyofa11e1ama32seclvyqv8cek</sha1>
    </revision>
    <revision>
      <id>68168</id>
      <parentid>68167</parentid>
      <timestamp>2021-04-21T19:09:23Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>68168</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3864" sha1="hlbpub8i6idla8wlxc0kcsxzkky390j" xml:space="preserve">'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

===Period of 19-limit Shrutar===

{| class="wikitable"
|-
! |Degree
! |Cents
!Difference from 68edo
|-
| |2
| |21.05263
|3.40557¢
|-
| |3
| |31.57895
| -3.71517¢
|-
| | 5
| | 52.63158
| -0.3096¢
|-
| |7
| |73.68421
|3.096¢
|-
| |8
| |84.21053
| -4.02477¢
|-
| |10
| |105.26316
|  -0.619195¢
|-
| |12
| | 126.31579
|2.78638¢
|-
| |13
| |136.842105
|  -4.334365¢
|-
| |15
| |157.89474
| -0.9288¢
|-
| | 17
| |178.94737
|2.47678¢
|-
| |18
| | 189.47369
| -4.644¢
|-
| |20
| |210.52632
| -1.23839¢
|-
| |22
| |231.57895
|2.16718¢
|-
| |23
| |242.10526
|  -4.953560372
|-
| |25
| |263.157895
| -1.548¢
|-
| |27
| |284.21053
|1.857585¢
|-
| |29
| |305.26316
|5.26316¢
|-
| |30
| |315.78947
| -1.857585¢
|-
| |32
| |336.842105
|1.548¢
|-
| |34
| |357.89474
|4.95356¢
|-
| |35
| | 368.42105
| -2.16718¢
|-
| |37
| |389.47368
|1.23839¢
|-
| |39
| |410.52632
|4.64396¢
|-
| |40
| |421.05263
| -2.47678¢
|-
| |42
| |442.10526
|0.92879¢
|-
| |44
| |463.157895
|4.334365¢
|-
| |45
| |473.68421
| -2.78638¢
|-
| |47
| |494.73684
|0.619195¢
|-
| |49
| |515.78947
|4.02477¢
|-
| |50
| |526.31579
| -3.095975¢
|-
| |52
| |547.36842
|0.3096¢
|-
| |54
| |568.42105
|3.71517¢
|-
| |55
| |578.94737
| -3.40557¢
|}
Since 114edo has a step of 10.52632 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 114edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|23:22
|-
|6
|[[6edo]]
|equal
|-
|7
|[[2L 5s]]
|17:16
|-
|8
|[[2L 6s]]
|15:14
|-
|9
|[[6L 3s]]
|13:12
|-
|10
|[[4L 6s]]
|12:11
|-
|11
|[[4L 7s]]
|11:10
|-
|12
|[[6L 6s]]
|10:9
|-
|13
|[[10L 3s]]
| rowspan="2" |9:8
|-
|14
|[[2L 12s]]
|-
|15
|[[9L 6s]]
| rowspan="2" |8:7
|-
|16
|2L 14s
|-
|17
|[[12L 5s]]
| rowspan="2" |7:6
|-
|18
|6L 12s
|-
|19
|[[19edo]]
|equal
|-
|20
|14L 6s
| rowspan="3" |6:5
|-
|21
|9L 12s
|-
|22
|4L 18s
|-
|23
|22L 1s
| rowspan="6" |5:4
|-
|24
|18L 6s
|-
|25
|14L 11s
|-
|26
|10L 16s
|-
|27
|6L 21s
|-
|28
|2L 26s
|-
|29
|27L 2s
| rowspan="9" |4:3
|-
|30
|24L 6s
|-
|31
|21L 10s
|-
|32
|18L 14s
|-
|33
|15L 18s
|-
|34
|12L 22s
|-
|35
|9L 26s
|-
|36
|6L 30s
|-
|37
|3L 34s
|-
|38
|[[38edo]]
|equal
|-
|39
|36L 3s
| rowspan="18" |3:2
|-
|40
|34L 6s
|-
|41
|32L 9s
|-
|42
|30L 12s
|-
|43
|28L 15s
|-
|44
|26L 18s
|-
|45
|24L 21s
|-
|46
|22L 24s
|-
|47
|20L 27s
|-
|48
|18L 30L
|-
|49
|16L 33s
|-
|50
|14L 36s
|-
|51
|12L 39s
|-
|52
|10L 42s
|-
|53
|8L 45s
|-
|54
|6L 48s
|-
|55
|4L 52s
|-
|56
|2L 54s
|-
|57
|[[57edo]]
|equal
|-
|58
|56L 2s
| rowspan="34" |2:1
|-
|59
|55L 4s
|-
|60
|54L 6s
|-
|61
|53L 8s
|-
|62
|52L 10s
|-
|63
|51L 12s
|-
|64
|50L 14s
|-
|65
|49L 16s
|-
|66
|48L 18s
|-
|67
|47L 20s
|-
|68
|46L 22s
|-
|69
|45L 24s
|-
|70
|44L 26s
|-
|71
|43L 28s
|-
|72
|42L 30s
|-
|73
|41L 32s
|-
|74
|40L 34s
|-
|75
|39L 36s
|-
|76
|38L 38s
|-
|77
|37L 40s
|-
|78
|36L 42s
|-
|79
|35L 44s
|-
|80
|34L 46s
|-
|81
|33L 48s
|-
|82
|32L 50s
|-
|83
|31L 52s
|-
|84
|30L 54s
|-
|85
|29L 56s
|-
|86
|28L 58s
|-
|87
|27L 60s
|-
|88
|26L 62s
|-
|89
|25L 64s
|-
|90
|24L 66s
|-
|91
|23L 68s
|}
[[Category:Equal divisions of the octave]]
[[Category:shrutar]]
[[Category:theory]]</text>
      <sha1>hlbpub8i6idla8wlxc0kcsxzkky390j</sha1>
    </revision>
    <revision>
      <id>87430</id>
      <parentid>68168</parentid>
      <timestamp>2022-02-26T16:48:48Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87430</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3844" sha1="awhu9ghoje98foac3ucud3vm9tkurhp" xml:space="preserve">'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

===Period of 19-limit Shrutar===

{| class="wikitable"
|-
! |Degree
! |Cents
!Difference from 68edo
|-
| |2
| |21.05263
|3.40557¢
|-
| |3
| |31.57895
| -3.71517¢
|-
| | 5
| | 52.63158
| -0.3096¢
|-
| |7
| |73.68421
|3.096¢
|-
| |8
| |84.21053
| -4.02477¢
|-
| |10
| |105.26316
|  -0.619195¢
|-
| |12
| | 126.31579
|2.78638¢
|-
| |13
| |136.842105
|  -4.334365¢
|-
| |15
| |157.89474
| -0.9288¢
|-
| | 17
| |178.94737
|2.47678¢
|-
| |18
| | 189.47369
| -4.644¢
|-
| |20
| |210.52632
| -1.23839¢
|-
| |22
| |231.57895
|2.16718¢
|-
| |23
| |242.10526
|  -4.953560372
|-
| |25
| |263.157895
| -1.548¢
|-
| |27
| |284.21053
|1.857585¢
|-
| |29
| |305.26316
|5.26316¢
|-
| |30
| |315.78947
| -1.857585¢
|-
| |32
| |336.842105
|1.548¢
|-
| |34
| |357.89474
|4.95356¢
|-
| |35
| | 368.42105
| -2.16718¢
|-
| |37
| |389.47368
|1.23839¢
|-
| |39
| |410.52632
|4.64396¢
|-
| |40
| |421.05263
| -2.47678¢
|-
| |42
| |442.10526
|0.92879¢
|-
| |44
| |463.157895
|4.334365¢
|-
| |45
| |473.68421
| -2.78638¢
|-
| |47
| |494.73684
|0.619195¢
|-
| |49
| |515.78947
|4.02477¢
|-
| |50
| |526.31579
| -3.095975¢
|-
| |52
| |547.36842
|0.3096¢
|-
| |54
| |568.42105
|3.71517¢
|-
| |55
| |578.94737
| -3.40557¢
|}
Since 114edo has a step of 10.52632 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 114edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|23:22
|-
|6
|[[6edo]]
|equal
|-
|7
|[[2L 5s]]
|17:16
|-
|8
|[[2L 6s]]
|15:14
|-
|9
|[[6L 3s]]
|13:12
|-
|10
|[[4L 6s]]
|12:11
|-
|11
|[[4L 7s]]
|11:10
|-
|12
|[[6L 6s]]
|10:9
|-
|13
|[[10L 3s]]
| rowspan="2" |9:8
|-
|14
|[[2L 12s]]
|-
|15
|[[9L 6s]]
| rowspan="2" |8:7
|-
|16
|2L 14s
|-
|17
|[[12L 5s]]
| rowspan="2" |7:6
|-
|18
|6L 12s
|-
|19
|[[19edo]]
|equal
|-
|20
|14L 6s
| rowspan="3" |6:5
|-
|21
|9L 12s
|-
|22
|4L 18s
|-
|23
|22L 1s
| rowspan="6" |5:4
|-
|24
|18L 6s
|-
|25
|14L 11s
|-
|26
|10L 16s
|-
|27
|6L 21s
|-
|28
|2L 26s
|-
|29
|27L 2s
| rowspan="9" |4:3
|-
|30
|24L 6s
|-
|31
|21L 10s
|-
|32
|18L 14s
|-
|33
|15L 18s
|-
|34
|12L 22s
|-
|35
|9L 26s
|-
|36
|6L 30s
|-
|37
|3L 34s
|-
|38
|[[38edo]]
|equal
|-
|39
|36L 3s
| rowspan="18" |3:2
|-
|40
|34L 6s
|-
|41
|32L 9s
|-
|42
|30L 12s
|-
|43
|28L 15s
|-
|44
|26L 18s
|-
|45
|24L 21s
|-
|46
|22L 24s
|-
|47
|20L 27s
|-
|48
|18L 30L
|-
|49
|16L 33s
|-
|50
|14L 36s
|-
|51
|12L 39s
|-
|52
|10L 42s
|-
|53
|8L 45s
|-
|54
|6L 48s
|-
|55
|4L 52s
|-
|56
|2L 54s
|-
|57
|[[57edo]]
|equal
|-
|58
|56L 2s
| rowspan="34" |2:1
|-
|59
|55L 4s
|-
|60
|54L 6s
|-
|61
|53L 8s
|-
|62
|52L 10s
|-
|63
|51L 12s
|-
|64
|50L 14s
|-
|65
|49L 16s
|-
|66
|48L 18s
|-
|67
|47L 20s
|-
|68
|46L 22s
|-
|69
|45L 24s
|-
|70
|44L 26s
|-
|71
|43L 28s
|-
|72
|42L 30s
|-
|73
|41L 32s
|-
|74
|40L 34s
|-
|75
|39L 36s
|-
|76
|38L 38s
|-
|77
|37L 40s
|-
|78
|36L 42s
|-
|79
|35L 44s
|-
|80
|34L 46s
|-
|81
|33L 48s
|-
|82
|32L 50s
|-
|83
|31L 52s
|-
|84
|30L 54s
|-
|85
|29L 56s
|-
|86
|28L 58s
|-
|87
|27L 60s
|-
|88
|26L 62s
|-
|89
|25L 64s
|-
|90
|24L 66s
|-
|91
|23L 68s
|}
[[Category:Equal divisions of the octave]]
[[Category:shrutar]]</text>
      <sha1>awhu9ghoje98foac3ucud3vm9tkurhp</sha1>
    </revision>
    <revision>
      <id>92924</id>
      <parentid>87430</parentid>
      <timestamp>2022-07-02T16:22:28Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92924</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3873" sha1="bk3ez0o9x2x20kmdqftwtqna2ylczmm" xml:space="preserve">'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

===Period of 19-limit Shrutar===

{| class="wikitable"
|-
! |Degree
! |Cents
!Difference from 68edo
|-
| |2
| |21.05263
|3.40557¢
|-
| |3
| |31.57895
| -3.71517¢
|-
| | 5
| | 52.63158
| -0.3096¢
|-
| |7
| |73.68421
|3.096¢
|-
| |8
| |84.21053
| -4.02477¢
|-
| |10
| |105.26316
|  -0.619195¢
|-
| |12
| | 126.31579
|2.78638¢
|-
| |13
| |136.842105
|  -4.334365¢
|-
| |15
| |157.89474
| -0.9288¢
|-
| | 17
| |178.94737
|2.47678¢
|-
| |18
| | 189.47369
| -4.644¢
|-
| |20
| |210.52632
| -1.23839¢
|-
| |22
| |231.57895
|2.16718¢
|-
| |23
| |242.10526
|  -4.953560372
|-
| |25
| |263.157895
| -1.548¢
|-
| |27
| |284.21053
|1.857585¢
|-
| |29
| |305.26316
|5.26316¢
|-
| |30
| |315.78947
| -1.857585¢
|-
| |32
| |336.842105
|1.548¢
|-
| |34
| |357.89474
|4.95356¢
|-
| |35
| | 368.42105
| -2.16718¢
|-
| |37
| |389.47368
|1.23839¢
|-
| |39
| |410.52632
|4.64396¢
|-
| |40
| |421.05263
| -2.47678¢
|-
| |42
| |442.10526
|0.92879¢
|-
| |44
| |463.157895
|4.334365¢
|-
| |45
| |473.68421
| -2.78638¢
|-
| |47
| |494.73684
|0.619195¢
|-
| |49
| |515.78947
|4.02477¢
|-
| |50
| |526.31579
| -3.095975¢
|-
| |52
| |547.36842
|0.3096¢
|-
| |54
| |568.42105
|3.71517¢
|-
| |55
| |578.94737
| -3.40557¢
|}
Since 114edo has a step of 10.52632 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 114edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|23:22
|-
|6
|[[6edo]]
|equal
|-
|7
|[[2L 5s]]
|17:16
|-
|8
|[[2L 6s]]
|15:14
|-
|9
|[[6L 3s]]
|13:12
|-
|10
|[[4L 6s]]
|12:11
|-
|11
|[[4L 7s]]
|11:10
|-
|12
|[[6L 6s]]
|10:9
|-
|13
|[[10L 3s]]
| rowspan="2" |9:8
|-
|14
|[[2L 12s]]
|-
|15
|[[9L 6s]]
| rowspan="2" |8:7
|-
|16
|2L 14s
|-
|17
|[[12L 5s]]
| rowspan="2" |7:6
|-
|18
|6L 12s
|-
|19
|[[19edo]]
|equal
|-
|20
|14L 6s
| rowspan="3" |6:5
|-
|21
|9L 12s
|-
|22
|4L 18s
|-
|23
|22L 1s
| rowspan="6" |5:4
|-
|24
|18L 6s
|-
|25
|14L 11s
|-
|26
|10L 16s
|-
|27
|6L 21s
|-
|28
|2L 26s
|-
|29
|27L 2s
| rowspan="9" |4:3
|-
|30
|24L 6s
|-
|31
|21L 10s
|-
|32
|18L 14s
|-
|33
|15L 18s
|-
|34
|12L 22s
|-
|35
|9L 26s
|-
|36
|6L 30s
|-
|37
|3L 34s
|-
|38
|[[38edo]]
|equal
|-
|39
|36L 3s
| rowspan="18" |3:2
|-
|40
|34L 6s
|-
|41
|32L 9s
|-
|42
|30L 12s
|-
|43
|28L 15s
|-
|44
|26L 18s
|-
|45
|24L 21s
|-
|46
|22L 24s
|-
|47
|20L 27s
|-
|48
|18L 30L
|-
|49
|16L 33s
|-
|50
|14L 36s
|-
|51
|12L 39s
|-
|52
|10L 42s
|-
|53
|8L 45s
|-
|54
|6L 48s
|-
|55
|4L 52s
|-
|56
|2L 54s
|-
|57
|[[57edo]]
|equal
|-
|58
|56L 2s
| rowspan="34" |2:1
|-
|59
|55L 4s
|-
|60
|54L 6s
|-
|61
|53L 8s
|-
|62
|52L 10s
|-
|63
|51L 12s
|-
|64
|50L 14s
|-
|65
|49L 16s
|-
|66
|48L 18s
|-
|67
|47L 20s
|-
|68
|46L 22s
|-
|69
|45L 24s
|-
|70
|44L 26s
|-
|71
|43L 28s
|-
|72
|42L 30s
|-
|73
|41L 32s
|-
|74
|40L 34s
|-
|75
|39L 36s
|-
|76
|38L 38s
|-
|77
|37L 40s
|-
|78
|36L 42s
|-
|79
|35L 44s
|-
|80
|34L 46s
|-
|81
|33L 48s
|-
|82
|32L 50s
|-
|83
|31L 52s
|-
|84
|30L 54s
|-
|85
|29L 56s
|-
|86
|28L 58s
|-
|87
|27L 60s
|-
|88
|26L 62s
|-
|89
|25L 64s
|-
|90
|24L 66s
|-
|91
|23L 68s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Shrutar]]</text>
      <sha1>bk3ez0o9x2x20kmdqftwtqna2ylczmm</sha1>
    </revision>
    <revision>
      <id>96193</id>
      <parentid>92924</parentid>
      <timestamp>2022-10-04T18:51:32Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96193</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3888" sha1="9s7ibjomskhjxzpnjed7ruz8wa7docn" xml:space="preserve">{{Infobox ET}}
'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

===Period of 19-limit Shrutar===

{| class="wikitable"
|-
! |Degree
! |Cents
!Difference from 68edo
|-
| |2
| |21.05263
|3.40557¢
|-
| |3
| |31.57895
| -3.71517¢
|-
| | 5
| | 52.63158
| -0.3096¢
|-
| |7
| |73.68421
|3.096¢
|-
| |8
| |84.21053
| -4.02477¢
|-
| |10
| |105.26316
|  -0.619195¢
|-
| |12
| | 126.31579
|2.78638¢
|-
| |13
| |136.842105
|  -4.334365¢
|-
| |15
| |157.89474
| -0.9288¢
|-
| | 17
| |178.94737
|2.47678¢
|-
| |18
| | 189.47369
| -4.644¢
|-
| |20
| |210.52632
| -1.23839¢
|-
| |22
| |231.57895
|2.16718¢
|-
| |23
| |242.10526
|  -4.953560372
|-
| |25
| |263.157895
| -1.548¢
|-
| |27
| |284.21053
|1.857585¢
|-
| |29
| |305.26316
|5.26316¢
|-
| |30
| |315.78947
| -1.857585¢
|-
| |32
| |336.842105
|1.548¢
|-
| |34
| |357.89474
|4.95356¢
|-
| |35
| | 368.42105
| -2.16718¢
|-
| |37
| |389.47368
|1.23839¢
|-
| |39
| |410.52632
|4.64396¢
|-
| |40
| |421.05263
| -2.47678¢
|-
| |42
| |442.10526
|0.92879¢
|-
| |44
| |463.157895
|4.334365¢
|-
| |45
| |473.68421
| -2.78638¢
|-
| |47
| |494.73684
|0.619195¢
|-
| |49
| |515.78947
|4.02477¢
|-
| |50
| |526.31579
| -3.095975¢
|-
| |52
| |547.36842
|0.3096¢
|-
| |54
| |568.42105
|3.71517¢
|-
| |55
| |578.94737
| -3.40557¢
|}
Since 114edo has a step of 10.52632 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 114edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|23:22
|-
|6
|[[6edo]]
|equal
|-
|7
|[[2L 5s]]
|17:16
|-
|8
|[[2L 6s]]
|15:14
|-
|9
|[[6L 3s]]
|13:12
|-
|10
|[[4L 6s]]
|12:11
|-
|11
|[[4L 7s]]
|11:10
|-
|12
|[[6L 6s]]
|10:9
|-
|13
|[[10L 3s]]
| rowspan="2" |9:8
|-
|14
|[[2L 12s]]
|-
|15
|[[9L 6s]]
| rowspan="2" |8:7
|-
|16
|2L 14s
|-
|17
|[[12L 5s]]
| rowspan="2" |7:6
|-
|18
|6L 12s
|-
|19
|[[19edo]]
|equal
|-
|20
|14L 6s
| rowspan="3" |6:5
|-
|21
|9L 12s
|-
|22
|4L 18s
|-
|23
|22L 1s
| rowspan="6" |5:4
|-
|24
|18L 6s
|-
|25
|14L 11s
|-
|26
|10L 16s
|-
|27
|6L 21s
|-
|28
|2L 26s
|-
|29
|27L 2s
| rowspan="9" |4:3
|-
|30
|24L 6s
|-
|31
|21L 10s
|-
|32
|18L 14s
|-
|33
|15L 18s
|-
|34
|12L 22s
|-
|35
|9L 26s
|-
|36
|6L 30s
|-
|37
|3L 34s
|-
|38
|[[38edo]]
|equal
|-
|39
|36L 3s
| rowspan="18" |3:2
|-
|40
|34L 6s
|-
|41
|32L 9s
|-
|42
|30L 12s
|-
|43
|28L 15s
|-
|44
|26L 18s
|-
|45
|24L 21s
|-
|46
|22L 24s
|-
|47
|20L 27s
|-
|48
|18L 30L
|-
|49
|16L 33s
|-
|50
|14L 36s
|-
|51
|12L 39s
|-
|52
|10L 42s
|-
|53
|8L 45s
|-
|54
|6L 48s
|-
|55
|4L 52s
|-
|56
|2L 54s
|-
|57
|[[57edo]]
|equal
|-
|58
|56L 2s
| rowspan="34" |2:1
|-
|59
|55L 4s
|-
|60
|54L 6s
|-
|61
|53L 8s
|-
|62
|52L 10s
|-
|63
|51L 12s
|-
|64
|50L 14s
|-
|65
|49L 16s
|-
|66
|48L 18s
|-
|67
|47L 20s
|-
|68
|46L 22s
|-
|69
|45L 24s
|-
|70
|44L 26s
|-
|71
|43L 28s
|-
|72
|42L 30s
|-
|73
|41L 32s
|-
|74
|40L 34s
|-
|75
|39L 36s
|-
|76
|38L 38s
|-
|77
|37L 40s
|-
|78
|36L 42s
|-
|79
|35L 44s
|-
|80
|34L 46s
|-
|81
|33L 48s
|-
|82
|32L 50s
|-
|83
|31L 52s
|-
|84
|30L 54s
|-
|85
|29L 56s
|-
|86
|28L 58s
|-
|87
|27L 60s
|-
|88
|26L 62s
|-
|89
|25L 64s
|-
|90
|24L 66s
|-
|91
|23L 68s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Shrutar]]</text>
      <sha1>9s7ibjomskhjxzpnjed7ruz8wa7docn</sha1>
    </revision>
    <revision>
      <id>111199</id>
      <parentid>96193</parentid>
      <timestamp>2023-05-02T11:11:22Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Cleaned up subheadings and added harmonics table</comment>
      <origin>111199</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3971" sha1="qmirtkcdop8ci660rhu4tlv4uf16fya" xml:space="preserve">{{Infobox ET}}
'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

== Prime harmonics ==
{{Harmonics in equal|114}}

== Period of 19-limit Shrutar ==

{| class="wikitable"
|-
! |Degree
! |Cents
!Difference from 68edo
|-
| |2
| |21.05263
|3.40557¢
|-
| |3
| |31.57895
| -3.71517¢
|-
| | 5
| | 52.63158
| -0.3096¢
|-
| |7
| |73.68421
|3.096¢
|-
| |8
| |84.21053
| -4.02477¢
|-
| |10
| |105.26316
|  -0.619195¢
|-
| |12
| | 126.31579
|2.78638¢
|-
| |13
| |136.842105
|  -4.334365¢
|-
| |15
| |157.89474
| -0.9288¢
|-
| | 17
| |178.94737
|2.47678¢
|-
| |18
| | 189.47369
| -4.644¢
|-
| |20
| |210.52632
| -1.23839¢
|-
| |22
| |231.57895
|2.16718¢
|-
| |23
| |242.10526
|  -4.953560372
|-
| |25
| |263.157895
| -1.548¢
|-
| |27
| |284.21053
|1.857585¢
|-
| |29
| |305.26316
|5.26316¢
|-
| |30
| |315.78947
| -1.857585¢
|-
| |32
| |336.842105
|1.548¢
|-
| |34
| |357.89474
|4.95356¢
|-
| |35
| | 368.42105
| -2.16718¢
|-
| |37
| |389.47368
|1.23839¢
|-
| |39
| |410.52632
|4.64396¢
|-
| |40
| |421.05263
| -2.47678¢
|-
| |42
| |442.10526
|0.92879¢
|-
| |44
| |463.157895
|4.334365¢
|-
| |45
| |473.68421
| -2.78638¢
|-
| |47
| |494.73684
|0.619195¢
|-
| |49
| |515.78947
|4.02477¢
|-
| |50
| |526.31579
| -3.095975¢
|-
| |52
| |547.36842
|0.3096¢
|-
| |54
| |568.42105
|3.71517¢
|-
| |55
| |578.94737
| -3.40557¢
|}

== Circulating temperaments ==

Since 114edo has a step of 10.52632 cents, it also allows one to use its MOS scales as circulating temperaments.
{| class="wikitable"
|+Circulating temperaments in 114edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|23:22
|-
|6
|[[6edo]]
|equal
|-
|7
|[[2L 5s]]
|17:16
|-
|8
|[[2L 6s]]
|15:14
|-
|9
|[[6L 3s]]
|13:12
|-
|10
|[[4L 6s]]
|12:11
|-
|11
|[[4L 7s]]
|11:10
|-
|12
|[[6L 6s]]
|10:9
|-
|13
|[[10L 3s]]
| rowspan="2" |9:8
|-
|14
|[[2L 12s]]
|-
|15
|[[9L 6s]]
| rowspan="2" |8:7
|-
|16
|2L 14s
|-
|17
|[[12L 5s]]
| rowspan="2" |7:6
|-
|18
|6L 12s
|-
|19
|[[19edo]]
|equal
|-
|20
|14L 6s
| rowspan="3" |6:5
|-
|21
|9L 12s
|-
|22
|4L 18s
|-
|23
|22L 1s
| rowspan="6" |5:4
|-
|24
|18L 6s
|-
|25
|14L 11s
|-
|26
|10L 16s
|-
|27
|6L 21s
|-
|28
|2L 26s
|-
|29
|27L 2s
| rowspan="9" |4:3
|-
|30
|24L 6s
|-
|31
|21L 10s
|-
|32
|18L 14s
|-
|33
|15L 18s
|-
|34
|12L 22s
|-
|35
|9L 26s
|-
|36
|6L 30s
|-
|37
|3L 34s
|-
|38
|[[38edo]]
|equal
|-
|39
|36L 3s
| rowspan="18" |3:2
|-
|40
|34L 6s
|-
|41
|32L 9s
|-
|42
|30L 12s
|-
|43
|28L 15s
|-
|44
|26L 18s
|-
|45
|24L 21s
|-
|46
|22L 24s
|-
|47
|20L 27s
|-
|48
|18L 30L
|-
|49
|16L 33s
|-
|50
|14L 36s
|-
|51
|12L 39s
|-
|52
|10L 42s
|-
|53
|8L 45s
|-
|54
|6L 48s
|-
|55
|4L 52s
|-
|56
|2L 54s
|-
|57
|[[57edo]]
|equal
|-
|58
|56L 2s
| rowspan="34" |2:1
|-
|59
|55L 4s
|-
|60
|54L 6s
|-
|61
|53L 8s
|-
|62
|52L 10s
|-
|63
|51L 12s
|-
|64
|50L 14s
|-
|65
|49L 16s
|-
|66
|48L 18s
|-
|67
|47L 20s
|-
|68
|46L 22s
|-
|69
|45L 24s
|-
|70
|44L 26s
|-
|71
|43L 28s
|-
|72
|42L 30s
|-
|73
|41L 32s
|-
|74
|40L 34s
|-
|75
|39L 36s
|-
|76
|38L 38s
|-
|77
|37L 40s
|-
|78
|36L 42s
|-
|79
|35L 44s
|-
|80
|34L 46s
|-
|81
|33L 48s
|-
|82
|32L 50s
|-
|83
|31L 52s
|-
|84
|30L 54s
|-
|85
|29L 56s
|-
|86
|28L 58s
|-
|87
|27L 60s
|-
|88
|26L 62s
|-
|89
|25L 64s
|-
|90
|24L 66s
|-
|91
|23L 68s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Shrutar]]</text>
      <sha1>qmirtkcdop8ci660rhu4tlv4uf16fya</sha1>
    </revision>
    <revision>
      <id>111200</id>
      <parentid>111199</parentid>
      <timestamp>2023-05-02T11:11:37Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Circulating temperaments */</comment>
      <origin>111200</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3975" sha1="70bdknhh73eg67nvkfq15hodeywvocs" xml:space="preserve">{{Infobox ET}}
'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

== Prime harmonics ==
{{Harmonics in equal|114}}

== Period of 19-limit Shrutar ==

{| class="wikitable"
|-
! |Degree
! |Cents
!Difference from 68edo
|-
| |2
| |21.05263
|3.40557¢
|-
| |3
| |31.57895
| -3.71517¢
|-
| | 5
| | 52.63158
| -0.3096¢
|-
| |7
| |73.68421
|3.096¢
|-
| |8
| |84.21053
| -4.02477¢
|-
| |10
| |105.26316
|  -0.619195¢
|-
| |12
| | 126.31579
|2.78638¢
|-
| |13
| |136.842105
|  -4.334365¢
|-
| |15
| |157.89474
| -0.9288¢
|-
| | 17
| |178.94737
|2.47678¢
|-
| |18
| | 189.47369
| -4.644¢
|-
| |20
| |210.52632
| -1.23839¢
|-
| |22
| |231.57895
|2.16718¢
|-
| |23
| |242.10526
|  -4.953560372
|-
| |25
| |263.157895
| -1.548¢
|-
| |27
| |284.21053
|1.857585¢
|-
| |29
| |305.26316
|5.26316¢
|-
| |30
| |315.78947
| -1.857585¢
|-
| |32
| |336.842105
|1.548¢
|-
| |34
| |357.89474
|4.95356¢
|-
| |35
| | 368.42105
| -2.16718¢
|-
| |37
| |389.47368
|1.23839¢
|-
| |39
| |410.52632
|4.64396¢
|-
| |40
| |421.05263
| -2.47678¢
|-
| |42
| |442.10526
|0.92879¢
|-
| |44
| |463.157895
|4.334365¢
|-
| |45
| |473.68421
| -2.78638¢
|-
| |47
| |494.73684
|0.619195¢
|-
| |49
| |515.78947
|4.02477¢
|-
| |50
| |526.31579
| -3.095975¢
|-
| |52
| |547.36842
|0.3096¢
|-
| |54
| |568.42105
|3.71517¢
|-
| |55
| |578.94737
| -3.40557¢
|}

== Circulating temperaments ==

Since 114edo has a step of 10.52632 cents, it also allows one to use its MOS scales as [[circulating temperament]]s.
{| class="wikitable"
|+Circulating temperaments in 114edo
!Tones
!Pattern
!L:s
|-
|5
|[[4L 1s]]
|23:22
|-
|6
|[[6edo]]
|equal
|-
|7
|[[2L 5s]]
|17:16
|-
|8
|[[2L 6s]]
|15:14
|-
|9
|[[6L 3s]]
|13:12
|-
|10
|[[4L 6s]]
|12:11
|-
|11
|[[4L 7s]]
|11:10
|-
|12
|[[6L 6s]]
|10:9
|-
|13
|[[10L 3s]]
| rowspan="2" |9:8
|-
|14
|[[2L 12s]]
|-
|15
|[[9L 6s]]
| rowspan="2" |8:7
|-
|16
|2L 14s
|-
|17
|[[12L 5s]]
| rowspan="2" |7:6
|-
|18
|6L 12s
|-
|19
|[[19edo]]
|equal
|-
|20
|14L 6s
| rowspan="3" |6:5
|-
|21
|9L 12s
|-
|22
|4L 18s
|-
|23
|22L 1s
| rowspan="6" |5:4
|-
|24
|18L 6s
|-
|25
|14L 11s
|-
|26
|10L 16s
|-
|27
|6L 21s
|-
|28
|2L 26s
|-
|29
|27L 2s
| rowspan="9" |4:3
|-
|30
|24L 6s
|-
|31
|21L 10s
|-
|32
|18L 14s
|-
|33
|15L 18s
|-
|34
|12L 22s
|-
|35
|9L 26s
|-
|36
|6L 30s
|-
|37
|3L 34s
|-
|38
|[[38edo]]
|equal
|-
|39
|36L 3s
| rowspan="18" |3:2
|-
|40
|34L 6s
|-
|41
|32L 9s
|-
|42
|30L 12s
|-
|43
|28L 15s
|-
|44
|26L 18s
|-
|45
|24L 21s
|-
|46
|22L 24s
|-
|47
|20L 27s
|-
|48
|18L 30L
|-
|49
|16L 33s
|-
|50
|14L 36s
|-
|51
|12L 39s
|-
|52
|10L 42s
|-
|53
|8L 45s
|-
|54
|6L 48s
|-
|55
|4L 52s
|-
|56
|2L 54s
|-
|57
|[[57edo]]
|equal
|-
|58
|56L 2s
| rowspan="34" |2:1
|-
|59
|55L 4s
|-
|60
|54L 6s
|-
|61
|53L 8s
|-
|62
|52L 10s
|-
|63
|51L 12s
|-
|64
|50L 14s
|-
|65
|49L 16s
|-
|66
|48L 18s
|-
|67
|47L 20s
|-
|68
|46L 22s
|-
|69
|45L 24s
|-
|70
|44L 26s
|-
|71
|43L 28s
|-
|72
|42L 30s
|-
|73
|41L 32s
|-
|74
|40L 34s
|-
|75
|39L 36s
|-
|76
|38L 38s
|-
|77
|37L 40s
|-
|78
|36L 42s
|-
|79
|35L 44s
|-
|80
|34L 46s
|-
|81
|33L 48s
|-
|82
|32L 50s
|-
|83
|31L 52s
|-
|84
|30L 54s
|-
|85
|29L 56s
|-
|86
|28L 58s
|-
|87
|27L 60s
|-
|88
|26L 62s
|-
|89
|25L 64s
|-
|90
|24L 66s
|-
|91
|23L 68s
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Shrutar]]</text>
      <sha1>70bdknhh73eg67nvkfq15hodeywvocs</sha1>
    </revision>
    <revision>
      <id>115200</id>
      <parentid>111200</parentid>
      <timestamp>2023-05-30T21:17:34Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Circulating temperaments */</comment>
      <origin>115200</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2119" sha1="lyk0kmmosnukfbtwgfluvdubikirnwx" xml:space="preserve">{{Infobox ET}}
'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

== Prime harmonics ==
{{Harmonics in equal|114}}

== Period of 19-limit Shrutar ==

{| class="wikitable"
|-
! |Degree
! |Cents
!Difference from 68edo
|-
| |2
| |21.05263
|3.40557¢
|-
| |3
| |31.57895
| -3.71517¢
|-
| | 5
| | 52.63158
| -0.3096¢
|-
| |7
| |73.68421
|3.096¢
|-
| |8
| |84.21053
| -4.02477¢
|-
| |10
| |105.26316
|  -0.619195¢
|-
| |12
| | 126.31579
|2.78638¢
|-
| |13
| |136.842105
|  -4.334365¢
|-
| |15
| |157.89474
| -0.9288¢
|-
| | 17
| |178.94737
|2.47678¢
|-
| |18
| | 189.47369
| -4.644¢
|-
| |20
| |210.52632
| -1.23839¢
|-
| |22
| |231.57895
|2.16718¢
|-
| |23
| |242.10526
|  -4.953560372
|-
| |25
| |263.157895
| -1.548¢
|-
| |27
| |284.21053
|1.857585¢
|-
| |29
| |305.26316
|5.26316¢
|-
| |30
| |315.78947
| -1.857585¢
|-
| |32
| |336.842105
|1.548¢
|-
| |34
| |357.89474
|4.95356¢
|-
| |35
| | 368.42105
| -2.16718¢
|-
| |37
| |389.47368
|1.23839¢
|-
| |39
| |410.52632
|4.64396¢
|-
| |40
| |421.05263
| -2.47678¢
|-
| |42
| |442.10526
|0.92879¢
|-
| |44
| |463.157895
|4.334365¢
|-
| |45
| |473.68421
| -2.78638¢
|-
| |47
| |494.73684
|0.619195¢
|-
| |49
| |515.78947
|4.02477¢
|-
| |50
| |526.31579
| -3.095975¢
|-
| |52
| |547.36842
|0.3096¢
|-
| |54
| |568.42105
|3.71517¢
|-
| |55
| |578.94737
| -3.40557¢
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Shrutar]]</text>
      <sha1>lyk0kmmosnukfbtwgfluvdubikirnwx</sha1>
    </revision>
    <revision>
      <id>132304</id>
      <parentid>115200</parentid>
      <timestamp>2024-01-21T07:17:12Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Changed subheading from prime harmonics to harmonics, added interval table</comment>
      <origin>132304</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2149" sha1="gpvih9hnhc48r1gymutb8wx6dffiwsp" xml:space="preserve">{{Infobox ET}}
'''114edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 114 parts, each of 10.52632 [[cent]]s. In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

== Harmonics ==
{{Harmonics in equal|114}}

== Intervals ==
{{Interval table}}

== Period of 19-limit Shrutar ==

{| class="wikitable"
|-
! |Degree
! |Cents
!Difference from 68edo
|-
| |2
| |21.05263
|3.40557¢
|-
| |3
| |31.57895
| -3.71517¢
|-
| | 5
| | 52.63158
| -0.3096¢
|-
| |7
| |73.68421
|3.096¢
|-
| |8
| |84.21053
| -4.02477¢
|-
| |10
| |105.26316
|  -0.619195¢
|-
| |12
| | 126.31579
|2.78638¢
|-
| |13
| |136.842105
|  -4.334365¢
|-
| |15
| |157.89474
| -0.9288¢
|-
| | 17
| |178.94737
|2.47678¢
|-
| |18
| | 189.47369
| -4.644¢
|-
| |20
| |210.52632
| -1.23839¢
|-
| |22
| |231.57895
|2.16718¢
|-
| |23
| |242.10526
|  -4.953560372
|-
| |25
| |263.157895
| -1.548¢
|-
| |27
| |284.21053
|1.857585¢
|-
| |29
| |305.26316
|5.26316¢
|-
| |30
| |315.78947
| -1.857585¢
|-
| |32
| |336.842105
|1.548¢
|-
| |34
| |357.89474
|4.95356¢
|-
| |35
| | 368.42105
| -2.16718¢
|-
| |37
| |389.47368
|1.23839¢
|-
| |39
| |410.52632
|4.64396¢
|-
| |40
| |421.05263
| -2.47678¢
|-
| |42
| |442.10526
|0.92879¢
|-
| |44
| |463.157895
|4.334365¢
|-
| |45
| |473.68421
| -2.78638¢
|-
| |47
| |494.73684
|0.619195¢
|-
| |49
| |515.78947
|4.02477¢
|-
| |50
| |526.31579
| -3.095975¢
|-
| |52
| |547.36842
|0.3096¢
|-
| |54
| |568.42105
|3.71517¢
|-
| |55
| |578.94737
| -3.40557¢
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Shrutar]]</text>
      <sha1>gpvih9hnhc48r1gymutb8wx6dffiwsp</sha1>
    </revision>
    <revision>
      <id>132897</id>
      <parentid>132304</parentid>
      <timestamp>2024-01-25T09:45:01Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Edo intro</comment>
      <origin>132897</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2043" sha1="ghr26q0wu21p1f3wmvw0dpwy4epjjlc" xml:space="preserve">{{Infobox ET}}
{{EDO intro|114}} 

In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

== Harmonics ==
{{Harmonics in equal|114}}

== Intervals ==
{{Interval table}}

== Period of 19-limit Shrutar ==

{| class="wikitable"
|-
! |Degree
! |Cents
!Difference from 68edo
|-
| |2
| |21.05263
|3.40557¢
|-
| |3
| |31.57895
| -3.71517¢
|-
| | 5
| | 52.63158
| -0.3096¢
|-
| |7
| |73.68421
|3.096¢
|-
| |8
| |84.21053
| -4.02477¢
|-
| |10
| |105.26316
|  -0.619195¢
|-
| |12
| | 126.31579
|2.78638¢
|-
| |13
| |136.842105
|  -4.334365¢
|-
| |15
| |157.89474
| -0.9288¢
|-
| | 17
| |178.94737
|2.47678¢
|-
| |18
| | 189.47369
| -4.644¢
|-
| |20
| |210.52632
| -1.23839¢
|-
| |22
| |231.57895
|2.16718¢
|-
| |23
| |242.10526
|  -4.953560372
|-
| |25
| |263.157895
| -1.548¢
|-
| |27
| |284.21053
|1.857585¢
|-
| |29
| |305.26316
|5.26316¢
|-
| |30
| |315.78947
| -1.857585¢
|-
| |32
| |336.842105
|1.548¢
|-
| |34
| |357.89474
|4.95356¢
|-
| |35
| | 368.42105
| -2.16718¢
|-
| |37
| |389.47368
|1.23839¢
|-
| |39
| |410.52632
|4.64396¢
|-
| |40
| |421.05263
| -2.47678¢
|-
| |42
| |442.10526
|0.92879¢
|-
| |44
| |463.157895
|4.334365¢
|-
| |45
| |473.68421
| -2.78638¢
|-
| |47
| |494.73684
|0.619195¢
|-
| |49
| |515.78947
|4.02477¢
|-
| |50
| |526.31579
| -3.095975¢
|-
| |52
| |547.36842
|0.3096¢
|-
| |54
| |568.42105
|3.71517¢
|-
| |55
| |578.94737
| -3.40557¢
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Shrutar]]</text>
      <sha1>ghr26q0wu21p1f3wmvw0dpwy4epjjlc</sha1>
    </revision>
    <revision>
      <id>145681</id>
      <parentid>132897</parentid>
      <timestamp>2024-06-04T06:23:19Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>-unparsable shit</comment>
      <origin>145681</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="746" sha1="sdvwy5h80om4eui52nq9eqqbdxnapsy" xml:space="preserve">{{Infobox ET}}
{{EDO intro|114}} 

In the [[5-limit|5-limit]] it [[tempering_out|tempers out]] 2048/2025, in the [[7-limit|7-limit]] 245/243, in the [[11-limit|11-limit]] 121/120, 176/175 and [[Quartisma|117440512/117406179]], in the [[13-limit|13-limit]] 196/195 and 325/324, in the [[17-limit|17-limit]] 136/135 and 154/153, in the [[19-limit|19-limit]] 286/285 and 343/342. These commas make for 114edo being an excellent tuning for [[Diaschismic_family|shrutar temperament]]; it is in fact the [[Optimal_patent_val|optimal patent val]] for [[Shrutar|shrutar]] in the 11- 13- 17- and 19-limit, as well as the rank three bisector temperament.

== Harmonics ==
{{Harmonics in equal|114}}

== Intervals ==
{{Interval table}}

[[Category:Shrutar]]</text>
      <sha1>sdvwy5h80om4eui52nq9eqqbdxnapsy</sha1>
    </revision>
    <revision>
      <id>145682</id>
      <parentid>145681</parentid>
      <timestamp>2024-06-04T06:27:50Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>145682</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="756" sha1="ostnggrbnvohovhtn66qmz0umvpyqgz" xml:space="preserve">{{Infobox ET}}
{{EDO intro|114}} 

== Theory ==
In the [[5-limit]] the equal temperament [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]), in the [[7-limit]] [[245/243]], in the [[11-limit]] [[121/120]], [[176/175]] and notably the [[quartisma]], in the [[13-limit]] [[196/195]] and [[325/324]], in the [[17-limit]] [[136/135]] and [[154/153]], in the [[19-limit]] [[286/285]] and [[343/342]]. These commas make for 114edo being an excellent tuning for the [[shrutar]] temperament; it is in fact the [[optimal patent val]] for shrutar in the 11-, 13-, 17-, and 19-limit, as well as the rank-3 [[bisector]] temperament.

=== Odd harmonics ===
{{Harmonics in equal|114}}

== Intervals ==
{{Interval table}}

[[Category:Shrutar]]
[[Category:Bisector]]</text>
      <sha1>ostnggrbnvohovhtn66qmz0umvpyqgz</sha1>
    </revision>
    <revision>
      <id>145683</id>
      <parentid>145682</parentid>
      <timestamp>2024-06-04T06:29:06Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+subsets and supersets</comment>
      <origin>145683</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="894" sha1="htlfve9es6qiqvxl7kou6mo90h8bmoz" xml:space="preserve">{{Infobox ET}}
{{EDO intro|114}} 

== Theory ==
In the [[5-limit]] the equal temperament [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]), in the [[7-limit]] [[245/243]], in the [[11-limit]] [[121/120]], [[176/175]] and notably the [[quartisma]], in the [[13-limit]] [[196/195]] and [[325/324]], in the [[17-limit]] [[136/135]] and [[154/153]], in the [[19-limit]] [[286/285]] and [[343/342]]. These commas make for 114edo being an excellent tuning for the [[shrutar]] temperament; it is in fact the [[optimal patent val]] for shrutar in the 11-, 13-, 17-, and 19-limit, as well as the rank-3 [[bisector]] temperament.

=== Odd harmonics ===
{{Harmonics in equal|114}}

=== Subsets and supersets ===
Since 114 factors into {{factorization|114}}, 114 edo has subset edos {{EDOs| 2, 3, 6, 19, 38, and 57 }}. 

== Intervals ==
{{Interval table}}

[[Category:Shrutar]]
[[Category:Bisector]]</text>
      <sha1>htlfve9es6qiqvxl7kou6mo90h8bmoz</sha1>
    </revision>
    <revision>
      <id>181319</id>
      <parentid>145683</parentid>
      <timestamp>2025-02-18T16:51:42Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181319</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="889" sha1="enu67829n3jl8u6dyov3v3116utpj3f" xml:space="preserve">{{Infobox ET}}
{{ED intro}} 

== Theory ==
In the [[5-limit]] the equal temperament [[tempering out|tempers out]] 2048/2025 ([[diaschisma]]), in the [[7-limit]] [[245/243]], in the [[11-limit]] [[121/120]], [[176/175]] and notably the [[quartisma]], in the [[13-limit]] [[196/195]] and [[325/324]], in the [[17-limit]] [[136/135]] and [[154/153]], in the [[19-limit]] [[286/285]] and [[343/342]]. These commas make for 114edo being an excellent tuning for the [[shrutar]] temperament; it is in fact the [[optimal patent val]] for shrutar in the 11-, 13-, 17-, and 19-limit, as well as the rank-3 [[bisector]] temperament.

=== Odd harmonics ===
{{Harmonics in equal|114}}

=== Subsets and supersets ===
Since 114 factors into {{factorization|114}}, 114 edo has subset edos {{EDOs| 2, 3, 6, 19, 38, and 57 }}. 

== Intervals ==
{{Interval table}}

[[Category:Shrutar]]
[[Category:Bisector]]</text>
      <sha1>enu67829n3jl8u6dyov3v3116utpj3f</sha1>
    </revision>
  </page>
  <page>
    <title>11664edo</title>
    <ns>0</ns>
    <id>61</id>
    <revision>
      <id>61</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>61</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="836" sha1="semz7zqo1gjcrqrw8vtlxn0vxl3psnx" xml:space="preserve">The 11664 division divides the octave into 11664 parts of 0.10288 cents each. It is a very strong 7-limit system, with a lower 7-limit  [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]]  than any division until [[18355edo|18355]]. It is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly consistent through the 27 limit and with a lower 23-limit relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also highly composite, since 11664 = 2^3 * 3^6. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]].</text>
      <sha1>semz7zqo1gjcrqrw8vtlxn0vxl3psnx</sha1>
    </revision>
    <revision>
      <id>4304</id>
      <timestamp>2015-08-22T18:14:55Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 557180667 - Original comment: **</comment>
      <origin>4304</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3369" sha1="kyi53ma7tt01i8k856ae9kun1s2cglw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-08-22 18:14:55 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;557180667&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 11664 division divides the octave into 11664 parts of 0.10288 cents each. It is a very strong 7-limit system, with a lower 7-limit  [[Tenney-Euclidean temperament measures#TE simple badness|relative error]]  than any division until [[18355edo|18355]]. It is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly consistent through the 27 limit and with a lower 23-limit relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also highly composite, since 11664 = 2^3 * 3^6. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]]. &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11664edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 11664 division divides the octave into 11664 parts of 0.10288 cents each. It is a very strong 7-limit system, with a lower 7-limit  &amp;lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&amp;gt;relative error&amp;lt;/a&amp;gt;  than any division until &amp;lt;a class="wiki_link" href="/18355edo"&amp;gt;18355&amp;lt;/a&amp;gt;. It is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt; unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly consistent through the 27 limit and with a lower 23-limit relative error than any division until &amp;lt;a class="wiki_link" href="/16808edo"&amp;gt;16808&amp;lt;/a&amp;gt;. Aside from this peculiar double threat property, it is also highly composite, since 11664 = 2^3 * 3^6. Among its divisiors are &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16edo"&amp;gt;16&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27edo"&amp;gt;27&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/243edo"&amp;gt;243&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>kyi53ma7tt01i8k856ae9kun1s2cglw</sha1>
    </revision>
    <revision>
      <id>68394</id>
      <parentid>61</parentid>
      <timestamp>2021-04-23T07:37:50Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>68394</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="868" sha1="3i1b99l2bmv3yap3a17eqem81xr00em" xml:space="preserve">The 11664 division divides the octave into 11664 parts of 0.10288 cents each. It is a very strong 7-limit system, with a lower 7-limit  [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]]  than any division until [[18355edo|18355]]. It is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly consistent through the 27 limit and with a lower 23-limit relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also highly composite, since 11664 = 2^3 * 3^6. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]].

{{Primes in edo|11664|prec=5}}</text>
      <sha1>3i1b99l2bmv3yap3a17eqem81xr00em</sha1>
    </revision>
    <revision>
      <id>81667</id>
      <parentid>68394</parentid>
      <timestamp>2021-12-10T02:21:53Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Categories</comment>
      <origin>81667</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="912" sha1="hhbnmjtz77a0ism4ow4ofpqmw2f1ybr" xml:space="preserve">The 11664 division divides the octave into 11664 parts of 0.10288 cents each. It is a very strong 7-limit system, with a lower 7-limit  [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]]  than any division until [[18355edo|18355]]. It is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly consistent through the 27 limit and with a lower 23-limit relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also highly composite, since 11664 = 2^3 * 3^6. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]].

{{Primes in edo|11664|prec=5}}

[[Category:Equal divisions of the octave]]</text>
      <sha1>hhbnmjtz77a0ism4ow4ofpqmw2f1ybr</sha1>
    </revision>
    <revision>
      <id>93252</id>
      <parentid>81667</parentid>
      <timestamp>2022-07-04T01:41:19Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>93252</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="948" sha1="a8sag3q129s59nd83f4s1sle7zjqi2h" xml:space="preserve">The '''11664 division''' divides the octave into 11664 parts of 0.10288 cents each. It is a very strong 7-limit system, with a lower 7-limit  [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]]  than any division until [[18355edo|18355]]. It is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly consistent through the 27 limit and with a lower 23-limit relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also highly composite, since 11664 = 2^3 * 3^6. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]].

{{Primes in edo|11664|prec=5}}

[[Category:Equal divisions of the octave|#####]] &lt;!-- 5-digit number --&gt;</text>
      <sha1>a8sag3q129s59nd83f4s1sle7zjqi2h</sha1>
    </revision>
    <revision>
      <id>96528</id>
      <parentid>93252</parentid>
      <timestamp>2022-10-04T22:21:36Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
      <origin>96528</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="963" sha1="fh638swqz4311t71ry6r3q6ftnmtfg2" xml:space="preserve">{{Infobox ET}}
The '''11664 division''' divides the octave into 11664 parts of 0.10288 cents each. It is a very strong 7-limit system, with a lower 7-limit  [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]]  than any division until [[18355edo|18355]]. It is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly consistent through the 27 limit and with a lower 23-limit relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also highly composite, since 11664 = 2^3 * 3^6. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]].

{{Primes in edo|11664|prec=5}}

[[Category:Equal divisions of the octave|#####]] &lt;!-- 5-digit number --&gt;</text>
      <sha1>fh638swqz4311t71ry6r3q6ftnmtfg2</sha1>
    </revision>
    <revision>
      <id>101054</id>
      <parentid>96528</parentid>
      <timestamp>2023-01-07T11:18:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Improve intro and update the error table</comment>
      <origin>101054</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="888" sha1="fzu5aau9l347v0altbtukdptmixuysh" xml:space="preserve">{{Infobox ET}}
{{EDO intro|11664}}

11664edo is a very strong 7-limit system, with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until [[18355edo|18355]]. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly [[consistent]] through the 27-odd-limit and with a lower 23-limit relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also very composite, since 11664 = 2&lt;sup&gt;3&lt;/sup&gt; × 3&lt;sup&gt;6&lt;/sup&gt;. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]].

=== Prime harmonics ===
{{Harmonics in equal|11664|prec=5}}</text>
      <sha1>fzu5aau9l347v0altbtukdptmixuysh</sha1>
    </revision>
    <revision>
      <id>119269</id>
      <parentid>101054</parentid>
      <timestamp>2023-07-09T04:16:41Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119269</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="907" sha1="gv71e9ikjzbaz907legnrt8z2t6mr3f" xml:space="preserve">{{novelty}}{{stub}}{{Infobox ET}}
{{EDO intro|11664}}

11664edo is a very strong 7-limit system, with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until [[18355edo|18355]]. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly [[consistent]] through the 27-odd-limit and with a lower 23-limit relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also very composite, since 11664 = 2&lt;sup&gt;3&lt;/sup&gt; × 3&lt;sup&gt;6&lt;/sup&gt;. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]].

=== Prime harmonics ===
{{Harmonics in equal|11664|prec=5}}</text>
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    </revision>
    <revision>
      <id>119717</id>
      <parentid>119269</parentid>
      <timestamp>2023-07-09T07:11:30Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
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      <minor/>
      <origin>119717</origin>
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      <text bytes="888" sha1="fzu5aau9l347v0altbtukdptmixuysh" xml:space="preserve">{{Infobox ET}}
{{EDO intro|11664}}

11664edo is a very strong 7-limit system, with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until [[18355edo|18355]]. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly [[consistent]] through the 27-odd-limit and with a lower 23-limit relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also very composite, since 11664 = 2&lt;sup&gt;3&lt;/sup&gt; × 3&lt;sup&gt;6&lt;/sup&gt;. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]].

=== Prime harmonics ===
{{Harmonics in equal|11664|prec=5}}</text>
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    </revision>
    <revision>
      <id>181320</id>
      <parentid>119717</parentid>
      <timestamp>2025-02-18T16:52:08Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181320</origin>
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      <format>text/x-wiki</format>
      <text bytes="896" sha1="hsowlbqtb9qmnk5tf4qxi2hxf115rgy" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

11664edo is a very strong 7-limit system, with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until [[18355edo|18355]]. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly [[consistent]] through the 27-odd-limit and with a lower 23-limit relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also very composite, since {{nowrap|11664 {{=}} 2&lt;sup&gt;3&lt;/sup&gt; × 3&lt;sup&gt;6&lt;/sup&gt;}}. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]].

=== Prime harmonics ===
{{Harmonics in equal|11664|prec=5}}</text>
      <sha1>hsowlbqtb9qmnk5tf4qxi2hxf115rgy</sha1>
    </revision>
    <revision>
      <id>206244</id>
      <parentid>181320</parentid>
      <timestamp>2025-08-03T13:20:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Expand a little</comment>
      <origin>206244</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1414" sha1="r2n2f3ptda87tsrn2lhnah98xn6p3k5" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

11664edo is a very strong [[7-limit]] system, with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until [[18355edo|18355]]. It is a [[zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly [[consistent]] through the [[27-odd-limit]] and with a lower [[23-limit]] relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also very composite, giving itself another edge over similar systems. 

Some of the simpler commas [[tempering out|tempered out]] include [[123201/123200]] and [[1990656/1990625]] in the [[13-limit]]; [[194481/194480]] and [[336141/336140]] in the [[17-limit]]; 23409/23408 and 89376/89375 in the [[19-limit]]; 43264/43263, 71875/71874, and 76545/76544 in the [[23-limit]]. 

=== Prime harmonics ===
{{Harmonics in equal|11664|intervals=prime|columns=9}}
{{Harmonics in equal|11664|intervals=prime|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 11664edo (continued)}}

=== Subsets and supersets ===
11664 factors into primes as {{nowrap| 2&lt;sup&gt;3&lt;/sup&gt; × 3&lt;sup&gt;6&lt;/sup&gt; }}. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]].</text>
      <sha1>r2n2f3ptda87tsrn2lhnah98xn6p3k5</sha1>
    </revision>
    <revision>
      <id>211771</id>
      <parentid>206244</parentid>
      <timestamp>2025-10-02T15:59:43Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Expand the harmonics table a little</comment>
      <origin>211771</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1416" sha1="s2eo0zfkjhbuxqign0hdrmt71lvfe3v" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

11664edo is a very strong [[7-limit]] system, with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until [[18355edo|18355]]. It is a [[zeta peak edo]] unlike other very strong 7-limit divisions in this size range, which has to do with the fact that it is also very strong in higher limits, being distinctly [[consistent]] through the [[27-odd-limit]] and with a lower [[23-limit]] relative error than any division until [[16808edo|16808]]. Aside from this peculiar double threat property, it is also very composite, giving itself another edge over similar systems. 

Some of the simpler commas [[tempering out|tempered out]] include [[123201/123200]] and [[1990656/1990625]] in the [[13-limit]]; [[194481/194480]] and [[336141/336140]] in the [[17-limit]]; 23409/23408 and 89376/89375 in the [[19-limit]]; 43264/43263, 71875/71874, and 76545/76544 in the [[23-limit]]. 

=== Prime harmonics ===
{{Harmonics in equal|11664|intervals=prime|columns=11}}
{{Harmonics in equal|11664|intervals=prime|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 11664edo (continued)}}

=== Subsets and supersets ===
11664 factors into primes as {{nowrap| 2&lt;sup&gt;3&lt;/sup&gt; × 3&lt;sup&gt;6&lt;/sup&gt; }}. Among its divisiors are [[12edo|12]], [[16edo|16]], [[24edo|24]], [[27edo|27]], [[72edo|72]], [[81edo|81]] and [[243edo|243]].</text>
      <sha1>s2eo0zfkjhbuxqign0hdrmt71lvfe3v</sha1>
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  </page>
  <page>
    <title>1171edo</title>
    <ns>0</ns>
    <id>62</id>
    <revision>
      <id>62</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>62</origin>
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      <text bytes="540" sha1="49f9u80oq5hr0kv97g6t50gg90fe8td" xml:space="preserve">The 1171 equal division divides the octave into 1171 parts of size 1.0248 cents each. It is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]]. It has a 5-limit comma basis consisting of the monzisma, |54 -37 2&amp;gt; and whoosh, |37 25 -33&amp;gt;. While not a strong higher-limit system, it is uniquely consistent through the 27-limit, and is very strong on the 2.3.5.11 subgroup. We might also note that 1171 is a prime number.</text>
      <sha1>49f9u80oq5hr0kv97g6t50gg90fe8td</sha1>
    </revision>
    <revision>
      <id>4305</id>
      <timestamp>2015-08-14T11:50:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 556692091 - Original comment: **</comment>
      <origin>4305</origin>
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      <text bytes="2404" sha1="7hhc6qm1257kc4zqiwdmreus215z2lw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-08-14 11:50:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;556692091&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 1171 equal division divides the octave into 1171 parts of size 1.0248 cents each. It is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]]. It has a 5-limit comma basis consisting of the monzisma, |54 -37 2&amp;gt; and whoosh, |37 25 -33&amp;gt;. While not a strong higher-limit system, it is uniquely consistent through the 27-limit, and is very strong on the 2.3.5.11 subgroup. We might also note that 1171 is a prime number.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1171edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 1171 equal division divides the octave into 1171 parts of size 1.0248 cents each. It is a very strong 5-limit division, being the first one past &amp;lt;a class="wiki_link" href="/612edo"&amp;gt;612&amp;lt;/a&amp;gt; with a lower 5-limit &amp;lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&amp;gt;relative error&amp;lt;/a&amp;gt;. It has a 5-limit comma basis consisting of the monzisma, |54 -37 2&amp;amp;gt; and whoosh, |37 25 -33&amp;amp;gt;. While not a strong higher-limit system, it is uniquely consistent through the 27-limit, and is very strong on the 2.3.5.11 subgroup. We might also note that 1171 is a prime number.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <timestamp>2015-08-14T11:50:07Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 556692051 - Original comment: **</comment>
      <origin>4306</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2308" sha1="6eyf88dazsvzyy0cojxi4rpu2yw7lmq" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-08-14 11:50:07 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;556692051&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 1171 equal division divides the octave into 1171 parts of size 1.0248 cents each. It is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]]. It has a 5-limit comma basis consisting of the monzisma, |54 -37 2&amp;gt; and whoosh, |37 25 -33&amp;gt;. While not a strong higher-limit system, it is uniquely consistent through the 27-limit, and is very strong on the 2.3.5.11 subgroup.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1171edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 1171 equal division divides the octave into 1171 parts of size 1.0248 cents each. It is a very strong 5-limit division, being the first one past &amp;lt;a class="wiki_link" href="/612edo"&amp;gt;612&amp;lt;/a&amp;gt; with a lower 5-limit &amp;lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&amp;gt;relative error&amp;lt;/a&amp;gt;. It has a 5-limit comma basis consisting of the monzisma, |54 -37 2&amp;amp;gt; and whoosh, |37 25 -33&amp;amp;gt;. While not a strong higher-limit system, it is uniquely consistent through the 27-limit, and is very strong on the 2.3.5.11 subgroup.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>6eyf88dazsvzyy0cojxi4rpu2yw7lmq</sha1>
    </revision>
    <revision>
      <id>80487</id>
      <parentid>62</parentid>
      <timestamp>2021-11-08T22:42:14Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>hyphenate and link comma-basis</comment>
      <origin>80487</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="544" sha1="gw3bylyg51j07qtrbfagkcxvt0d01ji" xml:space="preserve">The 1171 equal division divides the octave into 1171 parts of size 1.0248 cents each. It is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]]. It has a 5-limit [[comma-basis]] consisting of the monzisma, |54 -37 2&amp;gt; and whoosh, |37 25 -33&amp;gt;. While not a strong higher-limit system, it is uniquely consistent through the 27-limit, and is very strong on the 2.3.5.11 subgroup. We might also note that 1171 is a prime number.</text>
      <sha1>gw3bylyg51j07qtrbfagkcxvt0d01ji</sha1>
    </revision>
    <revision>
      <id>80632</id>
      <parentid>80487</parentid>
      <timestamp>2021-11-12T20:17:40Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>unhyphenate "comma basis"</comment>
      <origin>80632</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="544" sha1="dw9f78sm6q7x2ym0vtjwlm523thv2vr" xml:space="preserve">The 1171 equal division divides the octave into 1171 parts of size 1.0248 cents each. It is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the monzisma, |54 -37 2&amp;gt; and whoosh, |37 25 -33&amp;gt;. While not a strong higher-limit system, it is uniquely consistent through the 27-limit, and is very strong on the 2.3.5.11 subgroup. We might also note that 1171 is a prime number.</text>
      <sha1>dw9f78sm6q7x2ym0vtjwlm523thv2vr</sha1>
    </revision>
    <revision>
      <id>90811</id>
      <parentid>80632</parentid>
      <timestamp>2022-04-04T15:10:05Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+prime error table, style</comment>
      <origin>90811</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="717" sha1="ry0k4dqljm7r7e76z7iebqs3nnj3m2q" xml:space="preserve">The '''1171 equal divisions of the octave''' ('''1171edo''') divides the [[octave]] into 1171 parts of size 1.0248 [[cent]]s each. It is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is uniquely consistent through the [[27-odd-limit]], and is very strong on the 2.3.5.11 subgroup. We might also note that 1171 is a [[prime number]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

[[Category:Equal divisions of the octave]]</text>
      <sha1>ry0k4dqljm7r7e76z7iebqs3nnj3m2q</sha1>
    </revision>
    <revision>
      <id>93191</id>
      <parentid>90811</parentid>
      <timestamp>2022-07-04T01:12:10Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>93191</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="746" sha1="je0ypewy38o7cl59olka2r1ulu87wqr" xml:space="preserve">The '''1171 equal divisions of the octave''' ('''1171edo''') divides the [[octave]] into 1171 parts of size 1.0248 [[cent]]s each. It is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is uniquely consistent through the [[27-odd-limit]], and is very strong on the 2.3.5.11 subgroup. We might also note that 1171 is a [[prime number]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>je0ypewy38o7cl59olka2r1ulu87wqr</sha1>
    </revision>
    <revision>
      <id>96477</id>
      <parentid>93191</parentid>
      <timestamp>2022-10-04T22:08:12Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
      <origin>96477</origin>
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      <text bytes="761" sha1="5aryris6iaq99c4mak9vwljjrf4hrtd" xml:space="preserve">{{Infobox ET}}
The '''1171 equal divisions of the octave''' ('''1171edo''') divides the [[octave]] into 1171 parts of size 1.0248 [[cent]]s each. It is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is uniquely consistent through the [[27-odd-limit]], and is very strong on the 2.3.5.11 subgroup. We might also note that 1171 is a [[prime number]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>5aryris6iaq99c4mak9vwljjrf4hrtd</sha1>
    </revision>
    <revision>
      <id>119381</id>
      <parentid>96477</parentid>
      <timestamp>2023-07-09T05:19:25Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119381</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="780" sha1="700hmsnr9jnam7823d69tdax5qmmn5u" xml:space="preserve">{{novelty}}{{stub}}{{Infobox ET}}
The '''1171 equal divisions of the octave''' ('''1171edo''') divides the [[octave]] into 1171 parts of size 1.0248 [[cent]]s each. It is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is uniquely consistent through the [[27-odd-limit]], and is very strong on the 2.3.5.11 subgroup. We might also note that 1171 is a [[prime number]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>700hmsnr9jnam7823d69tdax5qmmn5u</sha1>
    </revision>
    <revision>
      <id>119607</id>
      <parentid>119381</parentid>
      <timestamp>2023-07-09T06:50:47Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119607</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="761" sha1="5aryris6iaq99c4mak9vwljjrf4hrtd" xml:space="preserve">{{Infobox ET}}
The '''1171 equal divisions of the octave''' ('''1171edo''') divides the [[octave]] into 1171 parts of size 1.0248 [[cent]]s each. It is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is uniquely consistent through the [[27-odd-limit]], and is very strong on the 2.3.5.11 subgroup. We might also note that 1171 is a [[prime number]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>5aryris6iaq99c4mak9vwljjrf4hrtd</sha1>
    </revision>
    <revision>
      <id>124857</id>
      <parentid>119607</parentid>
      <timestamp>2023-09-27T21:22:49Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
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      <text bytes="670" sha1="57g771fxaj4woaagwcn7xwg49qjzizc" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1171}}

1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is uniquely consistent through the [[27-odd-limit]], and is very strong on the 2.3.5.11 subgroup.

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>57g771fxaj4woaagwcn7xwg49qjzizc</sha1>
    </revision>
    <revision>
      <id>124858</id>
      <parentid>124857</parentid>
      <timestamp>2023-09-27T21:34:20Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>rank-2 temperaments</comment>
      <origin>124858</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1116" sha1="o2fahx25qlq86s8zasbuyo73ct7mzj2" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1171}}

1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is uniquely consistent through the [[27-odd-limit]], and is very strong on the 2.3.5.11 subgroup.

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

[[Category:Equal divisions of the octave|####]]

== Regular temperament properties ==

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
!Periods
per 8ve
!Generator
(Reduced)
!Cents
(Reduced)
!Associated
Ratio
!Temperaments
|-
|1
|129\1171
|132.195
|[-38 5 13⟩
|[[Astro]]
|-
|1
|243\1171
|249.018
|[-26 18 -1⟩
|[[Monzismic]]
|-
|1
|315\1171
|322.801
|[-6 23 -13⟩
|[[Senior]]
|-
|1
|335\1171
|343.296
|8000/6561
|[[Raider]]
|-
|1
|547\1171
|560.547
|864/625
|[[Whoosh]]
|}&lt;!-- 4-digit number --&gt;</text>
      <sha1>o2fahx25qlq86s8zasbuyo73ct7mzj2</sha1>
    </revision>
    <revision>
      <id>124859</id>
      <parentid>124858</parentid>
      <timestamp>2023-09-27T21:36:23Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>style</comment>
      <origin>124859</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1146" sha1="r8cxs2behvsgw480xapfvc39dv6oxg1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1171}}
== Theory ==
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is uniquely consistent through the [[27-odd-limit]], and is very strong on the 2.3.5.11 subgroup.

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

[[Category:Equal divisions of the octave|####]]

== Regular temperament properties ==

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
!Periods
per 8ve
!Generator
(Reduced)
!Cents
(Reduced)
!Associated
Ratio
!Temperaments
|-
|1
|129\1171
|132.195
|{{Monzo|-38 5 13}}
|[[Astro]]
|-
|1
|243\1171
|249.018
|{{Monzo|-26 18 -1}}
|[[Monzismic]]
|-
|1
|315\1171
|322.801
|{{Monzo|-6 23 -13}}
|[[Senior]]
|-
|1
|335\1171
|343.296
|8000/6561
|[[Raider]]
|-
|1
|547\1171
|560.547
|864/625
|[[Whoosh]]
|}&lt;!-- 4-digit number --&gt;</text>
      <sha1>r8cxs2behvsgw480xapfvc39dv6oxg1</sha1>
    </revision>
    <revision>
      <id>124886</id>
      <parentid>124859</parentid>
      <timestamp>2023-09-29T11:14:02Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>-redundant (and broken) category</comment>
      <origin>124886</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1120" sha1="f6r9rx28ryctyq7nstwshdtpqm0ffxo" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1171}}
== Theory ==
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is uniquely consistent through the [[27-odd-limit]], and is very strong on the 2.3.5.11 subgroup.

=== Prime harmonics ===
{{Harmonics in equal|1171}}

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]].

== Regular temperament properties ==

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 129\1171
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
|1
| 243\1171
| 249.018
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
|-
|1
| 315\1171
| 322.801
| {{Monzo| -6 23 -13 }}
| [[Senior]]
|-
| 1
| 335\1171
| 343.296
| 8000/6561
| [[Raider]]
|-
| 1
| 547\1171
| 560.547
| 864/625
| [[Whoosh]]
|}</text>
      <sha1>f6r9rx28ryctyq7nstwshdtpqm0ffxo</sha1>
    </revision>
    <revision>
      <id>126109</id>
      <parentid>124886</parentid>
      <timestamp>2023-10-17T13:10:21Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Clarify the title row of the rank-2 temp table</comment>
      <origin>126109</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1254" sha1="7zzxkozl8aiu7ii9a0m3d5fv1kmmlfs" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1171}}

== Theory ==
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is uniquely consistent through the [[27-odd-limit]], and is very strong on the 2.3.5.11 subgroup.

=== Prime harmonics ===
{{Harmonics in equal|1171}}

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]].

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 129\1171
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
|1
| 243\1171
| 249.018
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
|-
|1
| 315\1171
| 322.801
| {{Monzo| -6 23 -13 }}
| [[Senior]]
|-
| 1
| 335\1171
| 343.296
| 8000/6561
| [[Raider]]
|-
| 1
| 547\1171
| 560.547
| 864/625
| [[Whoosh]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct</text>
      <sha1>7zzxkozl8aiu7ii9a0m3d5fv1kmmlfs</sha1>
    </revision>
    <revision>
      <id>126663</id>
      <parentid>126109</parentid>
      <timestamp>2023-10-31T10:16:40Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Note 2342edo; misc. cleanup</comment>
      <origin>126663</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1414" sha1="a84ktf5o6felt2qhcqeh5zqiam7fqpr" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1171}}

== Theory ==
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is [[consistency|distinctly consistent]] through the [[27-odd-limit]], and is very strong on the 2.3.5.11 [[subgroup]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]]. [[2342edo]] which doubles it, corrects its [[harmonic]] [[7/1|7]] to a near-just quality. 

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 129\1171
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
|1
| 243\1171
| 249.018
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
|-
|1
| 315\1171
| 322.801
| {{Monzo| -6 23 -13 }}
| [[Senior]]
|-
| 1
| 335\1171
| 343.296
| 8000/6561
| [[Raider]]
|-
| 1
| 547\1171
| 560.547
| 864/625
| [[Whoosh]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct</text>
      <sha1>a84ktf5o6felt2qhcqeh5zqiam7fqpr</sha1>
    </revision>
    <revision>
      <id>165552</id>
      <parentid>126663</parentid>
      <timestamp>2024-11-16T00:48:30Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>165552</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1298" sha1="lpfsa4z6t5scxqwxogruq3yovchz5oz" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1171}}

== Theory ==
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is [[consistency|distinctly consistent]] through the [[27-odd-limit]], and is very strong on the 2.3.5.11 [[subgroup]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]]. [[2342edo]] which doubles it, corrects its [[harmonic]] [[7/1|7]] to a near-just quality. 

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;Ratio
! Temperaments
|-
| 1
| 129\1171
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
| 1
| 243\1171
| 249.018
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
|-
| 1
| 315\1171
| 322.801
| {{Monzo| -6 23 -13 }}
| [[Senior]]
|-
| 1
| 335\1171
| 343.296
| 8000/6561
| [[Raider]]
|-
| 1
| 547\1171
| 560.547
| 864/625
| [[Whoosh]]
|}
{{orf}}</text>
      <sha1>lpfsa4z6t5scxqwxogruq3yovchz5oz</sha1>
    </revision>
    <revision>
      <id>165554</id>
      <parentid>165552</parentid>
      <timestamp>2024-11-16T00:48:42Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Rank-2 temperaments */</comment>
      <origin>165554</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1299" sha1="n1k6fv7ymota73z0xysjjcswwxxz195" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1171}}

== Theory ==
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is [[consistency|distinctly consistent]] through the [[27-odd-limit]], and is very strong on the 2.3.5.11 [[subgroup]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]]. [[2342edo]] which doubles it, corrects its [[harmonic]] [[7/1|7]] to a near-just quality. 

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 129\1171
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
| 1
| 243\1171
| 249.018
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
|-
| 1
| 315\1171
| 322.801
| {{Monzo| -6 23 -13 }}
| [[Senior]]
|-
| 1
| 335\1171
| 343.296
| 8000/6561
| [[Raider]]
|-
| 1
| 547\1171
| 560.547
| 864/625
| [[Whoosh]]
|}
{{orf}}</text>
      <sha1>n1k6fv7ymota73z0xysjjcswwxxz195</sha1>
    </revision>
    <revision>
      <id>166341</id>
      <parentid>165554</parentid>
      <timestamp>2024-11-16T13:25:00Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Partial undo</comment>
      <origin>166341</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1442" sha1="tdeq30j8g7566q29qyzkc5j138n32v6" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1171}}

== Theory ==
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is [[consistency|distinctly consistent]] through the [[27-odd-limit]], and is very strong on the 2.3.5.11 [[subgroup]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]]. [[2342edo]] which doubles it, corrects its [[harmonic]] [[7/1|7]] to a near-just quality. 

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 129\1171
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
| 1
| 243\1171
| 249.018
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
|-
| 1
| 315\1171
| 322.801
| {{Monzo| -6 23 -13 }}
| [[Senior]]
|-
| 1
| 335\1171
| 343.296
| 8000/6561
| [[Raider]]
|-
| 1
| 547\1171
| 560.547
| 864/625
| [[Whoosh]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct</text>
      <sha1>tdeq30j8g7566q29qyzkc5j138n32v6</sha1>
    </revision>
    <revision>
      <id>168247</id>
      <parentid>166341</parentid>
      <timestamp>2024-12-02T12:04:24Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>+music</comment>
      <origin>168247</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1759" sha1="49h9a9to4ob185hgs3iqoc4pgqrgz41" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1171}}

== Theory ==
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is [[consistency|distinctly consistent]] through the [[27-odd-limit]], and is very strong on the 2.3.5.11 [[subgroup]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]]. [[2342edo]] which doubles it, corrects its [[harmonic]] [[7/1|7]] to a near-just quality. 

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 129\1171
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
| 1
| 243\1171
| 249.018
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
|-
| 1
| 315\1171
| 322.801
| {{Monzo| -6 23 -13 }}
| [[Senior]]
|-
| 1
| 335\1171
| 343.296
| 8000/6561
| [[Raider]]
|-
| 1
| 547\1171
| 560.547
| 864/625
| [[Whoosh]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Francium]]
* "Right On Friendly Lovers" from ''Naughty Girl Era'' (2024) – [https://open.spotify.com/track/208kkCqXksQWhT0zdxFXYm Spotify] | [https://francium223.bandcamp.com/track/right-on-friendly-lovers Bandcamp] | [https://www.youtube.com/watch?v=_qqisIAkVRc YouTube] − friesachic in 1171edo</text>
      <sha1>49h9a9to4ob185hgs3iqoc4pgqrgz41</sha1>
    </revision>
    <revision>
      <id>176123</id>
      <parentid>168247</parentid>
      <timestamp>2025-01-15T16:04:46Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176123</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1753" sha1="iizs1jge6itdcofd4kwz4odm3pqawli" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1171}}

== Theory ==
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is [[consistency|distinctly consistent]] through the [[27-odd-limit]], and is very strong on the 2.3.5.11 [[subgroup]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]]. [[2342edo]] which doubles it, corrects its [[harmonic]] [[7/1|7]] to a near-just quality. 

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 129\1171
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
| 1
| 243\1171
| 249.018
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
|-
| 1
| 315\1171
| 322.801
| {{Monzo| -6 23 -13 }}
| [[Senior]]
|-
| 1
| 335\1171
| 343.296
| 8000/6561
| [[Raider]]
|-
| 1
| 547\1171
| 560.547
| 864/625
| [[Whoosh]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
; [[Francium]]
* "Right On Friendly Lovers" from ''Naughty Girl Era'' (2024) – [https://open.spotify.com/track/208kkCqXksQWhT0zdxFXYm Spotify] | [https://francium223.bandcamp.com/track/right-on-friendly-lovers Bandcamp] | [https://www.youtube.com/watch?v=_qqisIAkVRc YouTube] − friesachic in 1171edo</text>
      <sha1>iizs1jge6itdcofd4kwz4odm3pqawli</sha1>
    </revision>
    <revision>
      <id>180416</id>
      <parentid>176123</parentid>
      <timestamp>2025-02-12T12:27:31Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Music */ +music</comment>
      <origin>180416</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2033" sha1="m6drgao1yvjxuzembdgszofofjvduvo" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1171}}

== Theory ==
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is [[consistency|distinctly consistent]] through the [[27-odd-limit]], and is very strong on the 2.3.5.11 [[subgroup]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]]. [[2342edo]] which doubles it, corrects its [[harmonic]] [[7/1|7]] to a near-just quality. 

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 129\1171
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
| 1
| 243\1171
| 249.018
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
|-
| 1
| 315\1171
| 322.801
| {{Monzo| -6 23 -13 }}
| [[Senior]]
|-
| 1
| 335\1171
| 343.296
| 8000/6561
| [[Raider]]
|-
| 1
| 547\1171
| 560.547
| 864/625
| [[Whoosh]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
; [[Francium]]
* "Right On Friendly Lovers" from ''Naughty Girl Era'' (2024) – [https://open.spotify.com/track/208kkCqXksQWhT0zdxFXYm Spotify] | [https://francium223.bandcamp.com/track/right-on-friendly-lovers Bandcamp] | [https://www.youtube.com/watch?v=_qqisIAkVRc YouTube] − in Friesachic, 1171edo tuning
* "Apply to a Print" from ''I Want To'' (2025) – [https://open.spotify.com/track/5botrdPX1YdFmk2mP6kbOr Spotify] | [https://francium223.bandcamp.com/track/apply-to-a-print Bandcamp] | [https://www.youtube.com/watch?v=ui-ORXPgqaE YouTube] – in Viljevic, 1171edo tuning</text>
      <sha1>m6drgao1yvjxuzembdgszofofjvduvo</sha1>
    </revision>
    <revision>
      <id>181321</id>
      <parentid>180416</parentid>
      <timestamp>2025-02-18T16:52:38Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181321</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2027" sha1="530is4seqgugde8txfel56t1xw4qxlh" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is [[consistency|distinctly consistent]] through the [[27-odd-limit]], and is very strong on the 2.3.5.11 [[subgroup]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]]. [[2342edo]] which doubles it, corrects its [[harmonic]] [[7/1|7]] to a near-just quality. 

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 129\1171
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
| 1
| 243\1171
| 249.018
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
|-
| 1
| 315\1171
| 322.801
| {{Monzo| -6 23 -13 }}
| [[Senior]]
|-
| 1
| 335\1171
| 343.296
| 8000/6561
| [[Raider]]
|-
| 1
| 547\1171
| 560.547
| 864/625
| [[Whoosh]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
; [[Francium]]
* "Right On Friendly Lovers" from ''Naughty Girl Era'' (2024) – [https://open.spotify.com/track/208kkCqXksQWhT0zdxFXYm Spotify] | [https://francium223.bandcamp.com/track/right-on-friendly-lovers Bandcamp] | [https://www.youtube.com/watch?v=_qqisIAkVRc YouTube] − in Friesachic, 1171edo tuning
* "Apply to a Print" from ''I Want To'' (2025) – [https://open.spotify.com/track/5botrdPX1YdFmk2mP6kbOr Spotify] | [https://francium223.bandcamp.com/track/apply-to-a-print Bandcamp] | [https://www.youtube.com/watch?v=ui-ORXPgqaE YouTube] – in Viljevic, 1171edo tuning</text>
      <sha1>530is4seqgugde8txfel56t1xw4qxlh</sha1>
    </revision>
    <revision>
      <id>225835</id>
      <parentid>181321</parentid>
      <timestamp>2026-03-13T13:32:21Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct" to "[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct"</comment>
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      <text bytes="2082" sha1="oxruj8g7sz3iwtwec9k4ecnq8y71hla" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1171edo is a very strong 5-limit division, being the first one past [[612edo|612]] with a lower 5-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It has a 5-limit [[comma basis]] consisting of the [[monzisma]], {{monzo| 54 -37 2 }} and whoosh, {{monzo| 37 25 -33 }}. While not a strong higher-limit system, it is [[consistency|distinctly consistent]] through the [[27-odd-limit]], and is very strong on the 2.3.5.11 [[subgroup]].

=== Prime harmonics ===
{{Harmonics in equal|1171}}

=== Subsets and supersets ===
1171edo is the 193rd [[prime edo]]. [[2342edo]] which doubles it, corrects its [[harmonic]] [[7/1|7]] to a near-just quality. 

== Regular temperament properties ==
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 129\1171
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
| 1
| 243\1171
| 249.018
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
|-
| 1
| 315\1171
| 322.801
| {{Monzo| -6 23 -13 }}
| [[Senior]]
|-
| 1
| 335\1171
| 343.296
| 8000/6561
| [[Raider]]
|-
| 1
| 547\1171
| 560.547
| 864/625
| [[Whoosh]]
|}
&lt;nowiki /&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Music ==
; [[Francium]]
* "Right On Friendly Lovers" from ''Naughty Girl Era'' (2024) – [https://open.spotify.com/track/208kkCqXksQWhT0zdxFXYm Spotify] | [https://francium223.bandcamp.com/track/right-on-friendly-lovers Bandcamp] | [https://www.youtube.com/watch?v=_qqisIAkVRc YouTube] − in Friesachic, 1171edo tuning
* "Apply to a Print" from ''I Want To'' (2025) – [https://open.spotify.com/track/5botrdPX1YdFmk2mP6kbOr Spotify] | [https://francium223.bandcamp.com/track/apply-to-a-print Bandcamp] | [https://www.youtube.com/watch?v=ui-ORXPgqaE YouTube] – in Viljevic, 1171edo tuning</text>
      <sha1>oxruj8g7sz3iwtwec9k4ecnq8y71hla</sha1>
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  </page>
  <page>
    <title>1178edo</title>
    <ns>0</ns>
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    <revision>
      <id>63</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
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        <id>0</id>
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      <text bytes="587" sha1="qw5it82duoqe9b7c5ibvp2zsjd9vhhv" xml:space="preserve">The 1178 equal tuning divides the octave into 1178 parts of 1.0187 cents each. It is a very strong 19-limit system, and is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak, integral and gap edo]]. It is also distinctly consistent through to the 21 odd limit, and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]]. It is also notable for being divisible by both 19 and 31. A basis for its 19-limit commas is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984.</text>
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        <id>0</id>
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      <comment>**Imported revision 556730503 - Original comment: **</comment>
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      <text bytes="2541" sha1="8xgy6t06hhbczf42g7racwyznwq8fj1" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-08-15 12:15:40 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;556730503&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 1178 equal tuning divides the octave into 1178 parts of 1.0187 cents each. It is a very strong 19-limit system, and is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak, integral and gap edo]]. It is also distinctly consistent through to the 21 odd limit, and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]]. It is also notable for being divisible by both 19 and 31. A basis for its 19-limit commas is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1178edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 1178 equal tuning divides the octave into 1178 parts of 1.0187 cents each. It is a very strong 19-limit system, and is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta peak, integral and gap edo&amp;lt;/a&amp;gt;. It is also distinctly consistent through to the 21 odd limit, and is the first edo past &amp;lt;a class="wiki_link" href="/742edo"&amp;gt;742&amp;lt;/a&amp;gt; with a lower 19-limit &amp;lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&amp;gt;relative error&amp;lt;/a&amp;gt;. It is also notable for being divisible by both 19 and 31. A basis for its 19-limit commas is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 556639225 - Original comment: **</comment>
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      <text bytes="2541" sha1="j8x24k82nms5eg0b6rbjgo774h5k5ru" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-08-13 13:15:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;556639225&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 1178 equal tuning divides the octave into 1178 parts of 1.0187 cents each. It is a very strong 19-limit system, and is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|Zeta peak, integral and gap edo]]. It is also distinctly consistent through to the 21 odd limit, and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]]. It is also notable for being divisible by both 19 and 31. A basis for its 19-limit commas is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984.&lt;/pre&gt;&lt;/div&gt;
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&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1178edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 1178 equal tuning divides the octave into 1178 parts of 1.0187 cents each. It is a very strong 19-limit system, and is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;Zeta peak, integral and gap edo&amp;lt;/a&amp;gt;. It is also distinctly consistent through to the 21 odd limit, and is the first edo past &amp;lt;a class="wiki_link" href="/742edo"&amp;gt;742&amp;lt;/a&amp;gt; with a lower 19-limit &amp;lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&amp;gt;relative error&amp;lt;/a&amp;gt;. It is also notable for being divisible by both 19 and 31. A basis for its 19-limit commas is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 556638869 - Original comment: **</comment>
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      <text bytes="2425" sha1="6mvseow6jwk6fs5n9glv1njyzg6afq4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-08-13 13:08:53 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;556638869&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 1178 equal tuning divides the octave into 1178 parts of 1.0187 cents each. It is a very strong 19-limit system, and is a [[The Riemann Zeta Function and Tuning#Zeta EDO lists|Zeta peak, integral and gap edo]]. It is also distinctly consistent through to the 21 odd limit, and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures#TE simple badness|relative error]]. A basis for its 19-limit commas is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1178edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 1178 equal tuning divides the octave into 1178 parts of 1.0187 cents each. It is a very strong 19-limit system, and is a &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;Zeta peak, integral and gap edo&amp;lt;/a&amp;gt;. It is also distinctly consistent through to the 21 odd limit, and is the first edo past &amp;lt;a class="wiki_link" href="/742edo"&amp;gt;742&amp;lt;/a&amp;gt; with a lower 19-limit &amp;lt;a class="wiki_link" href="/Tenney-Euclidean%20temperament%20measures#TE simple badness"&amp;gt;relative error&amp;lt;/a&amp;gt;. A basis for its 19-limit commas is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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        <id>342</id>
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      <text bytes="666" sha1="flpn9q44whum0x3veo5m6y2wrt1wu9o" xml:space="preserve">The '''1178 equal tuning''' divides the octave into 1178 parts of 1.0187 cents each. It is a very strong 19-limit system, and is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak, integral and gap edo]]. It is also distinctly consistent through to the 21 odd limit, and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]]. It is also notable for being divisible by both 19 and 31. A basis for its 19-limit commas is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984.

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
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    <revision>
      <id>95772</id>
      <parentid>93192</parentid>
      <timestamp>2022-09-27T12:04:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+prime error table</comment>
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      <text bytes="737" sha1="orbrqn951fmmo3d9qjeuap6jkq7hi1g" xml:space="preserve">The '''1178 equal tuning''' divides the octave into 1178 parts of 1.0187 cents each. It is a very strong 19-limit system, and is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]]. It is also distinctly [[consistent]] through to the 21-odd-limit, and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It is also notable for being divisible by both 19 and 31. A basis for its 19-limit commas is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
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      <timestamp>2022-09-29T05:41:47Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+semihemienneadecal</comment>
      <origin>95796</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1003" sha1="rj813twg99kpr3plkpqvumk40tajgja" xml:space="preserve">The '''1178 equal tuning''' divides the octave into 1178 parts of 1.0187 cents each. It is a very strong 19-limit system, and is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]]. It is also distinctly [[consistent]] through to the 21-odd-limit, and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A basis for its 19-limit commas is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984. It supports and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Miscellaneous properties ===
Since 1178 = 2 × 19 × 31, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}. 

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;
[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]</text>
      <sha1>rj813twg99kpr3plkpqvumk40tajgja</sha1>
    </revision>
    <revision>
      <id>96478</id>
      <parentid>95796</parentid>
      <timestamp>2022-10-04T22:08:33Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
      <origin>96478</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1018" sha1="9b4itjy3hdtys7ukaewntu17cmape7l" xml:space="preserve">{{Infobox ET}}
The '''1178 equal tuning''' divides the octave into 1178 parts of 1.0187 cents each. It is a very strong 19-limit system, and is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]]. It is also distinctly [[consistent]] through to the 21-odd-limit, and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A basis for its 19-limit commas is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984. It supports and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Miscellaneous properties ===
Since 1178 = 2 × 19 × 31, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}. 

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;
[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]</text>
      <sha1>9b4itjy3hdtys7ukaewntu17cmape7l</sha1>
    </revision>
    <revision>
      <id>101083</id>
      <parentid>96478</parentid>
      <timestamp>2023-01-08T03:51:36Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Improve intro and section titles</comment>
      <origin>101083</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="874" sha1="7kis8ulgxznbwfy9u1h5ndmb48gpdm1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

1178edo is a very strong 19-limit system, and is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]]. It is also distinctly [[consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A basis for its 19-limit commas is 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4375/4374, 4914/4913 and 5985/5984. It supports and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Divisors ===
Since 1178 = 2 × 19 × 31, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}. 

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]</text>
      <sha1>7kis8ulgxznbwfy9u1h5ndmb48gpdm1</sha1>
    </revision>
    <revision>
      <id>105042</id>
      <parentid>101083</parentid>
      <timestamp>2023-03-08T09:19:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Links; section title</comment>
      <origin>105042</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="899" sha1="a4d5g9hfm1wdgq1v49bcqlps369y88c" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

1178edo is a very strong 19-limit system, and is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]]. It is also distinctly [[consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A basis for its 19-limit commas is [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4375/4374]], 4914/4913 and 5985/5984. It supports and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = 2 × 19 × 31, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}. 

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]</text>
      <sha1>a4d5g9hfm1wdgq1v49bcqlps369y88c</sha1>
    </revision>
    <revision>
      <id>119380</id>
      <parentid>105042</parentid>
      <timestamp>2023-07-09T05:19:14Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119380</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="918" sha1="19cobv3y7gasr6omsj2ygzr1r4xmxur" xml:space="preserve">{{novelty}}{{stub}}{{Infobox ET}}
{{EDO intro|1178}}

1178edo is a very strong 19-limit system, and is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]]. It is also distinctly [[consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A basis for its 19-limit commas is [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4375/4374]], 4914/4913 and 5985/5984. It supports and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = 2 × 19 × 31, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}. 

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]</text>
      <sha1>19cobv3y7gasr6omsj2ygzr1r4xmxur</sha1>
    </revision>
    <revision>
      <id>119608</id>
      <parentid>119380</parentid>
      <timestamp>2023-07-09T06:50:58Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119608</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="899" sha1="a4d5g9hfm1wdgq1v49bcqlps369y88c" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

1178edo is a very strong 19-limit system, and is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak, integral and gap edo]]. It is also distinctly [[consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A basis for its 19-limit commas is [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4375/4374]], 4914/4913 and 5985/5984. It supports and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = 2 × 19 × 31, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}. 

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]</text>
      <sha1>a4d5g9hfm1wdgq1v49bcqlps369y88c</sha1>
    </revision>
    <revision>
      <id>126662</id>
      <parentid>119608</parentid>
      <timestamp>2023-10-31T10:13:08Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Adopt template: Factorization; misc. cleanup</comment>
      <origin>126662</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="888" sha1="imfhc84fcnljcf8au427xkgmlgv13x1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A basis for its 19-limit [[comma]]s is [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4375/4374]], [[4914/4913]] and 5985/5984. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}. 

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]</text>
      <sha1>imfhc84fcnljcf8au427xkgmlgv13x1</sha1>
    </revision>
    <revision>
      <id>126904</id>
      <parentid>126662</parentid>
      <timestamp>2023-11-04T18:14:59Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>126904</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="976" sha1="94lt7c7b6cl8dwx5wqgqh3n6dhv1wzw" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A basis for its 19-limit [[comma]]s is [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4375/4374]], [[4914/4913]] and 5985/5984. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}. 

== Music ==
[https://www.youtube.com/watch?v=c9e7MTsIDc4 Listening] (2023)
[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]</text>
      <sha1>94lt7c7b6cl8dwx5wqgqh3n6dhv1wzw</sha1>
    </revision>
    <revision>
      <id>126905</id>
      <parentid>126904</parentid>
      <timestamp>2023-11-04T18:15:29Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Music */ added music</comment>
      <origin>126905</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1043" sha1="1lznwg976pln59btwmkddpotf5ij5j4" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A basis for its 19-limit [[comma]]s is [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4375/4374]], [[4914/4913]] and 5985/5984. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}. 

== Music ==
; '''[[Eliora]]'''
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 Listening] (2023) - 217 &amp; 1178 and enneadecal in 1178edo tuning
[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]</text>
      <sha1>1lznwg976pln59btwmkddpotf5ij5j4</sha1>
    </revision>
    <revision>
      <id>126908</id>
      <parentid>126905</parentid>
      <timestamp>2023-11-04T19:11:18Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Theory */ +regular temperament properties</comment>
      <origin>126908</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2689" sha1="6oo01yuub640qxqf1itfltab1fa516x" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A basis for its 19-limit [[comma]]s is [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4375/4374]], [[4914/4913]] and 5985/5984. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" |Tuning Error
|-	
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo|-1867 1178}}
|{{val|1178 1867}}
| 0.0276
| 0.0276
| 2.71
|-
|2.3.5
|{{monzo|-14 -19-19}}, {{monzo|-99 61 1}}
|{{val|1178 1867 2735}}
| 0.0522
| 0.0415
| 4.07
|-
|2.3.5.7
|4375/4374, 703125/702464, 2460375/2458624
|{{val|1178 1867 2735 3307}}
| 0.0450
| 0.0380
| 3.73
|-
|2.3.5.7.11
|3025/3024, 4375/4374, 9801/9800, 43923/43904
|{{val|1178 1867 2735 3307 4075}}
| 0.0484
| 0.0347
| 3.41
|-
|2.3.5.7.11.13
|3025/3024, 4225/4224, 9801/9800, 91125/91091, 1664000/1663893
|{{val|1178 1867 2735 3307 4075 4359}}
| 0.0457
| 0.0322
| 3.16
|-
|2.3.5.7.11.13.17
|3025/3024, 4225/4224, 5832/5831, 9801/9800, 14875/14872, 31213/31212
|{{val|1178 1867 2735 3307 4075 4359 4815}}
| 0.0403
| 0.0327
| 3.21
|-
|2.3.5.7.11.13.17.19
|4200/4199, 5985/5984, 9801/9800, 10830/10829, 12636/12635, 14365/14364, 23409/23408
|{{val|1178 1867 2735 3307 4075 4359 4815 5004}}
| 0.0370
| 0.0318
| 3.12
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
|1
|337\1178
|343.29
|8000/6561
|[[Raider]]
|-
|19
|489\1178&lt;br&gt;(7\1178)
|498.13&lt;br&gt;(7.13)
|4/3&lt;br&gt;(225/224)
|[[Enneadecal]]
|-
|38
|489\1178&lt;br&gt;(7\1178)
|498.13&lt;br&gt;(7.13)
|4/3&lt;br&gt;(225/224)
|[[Hemienneadecal]]
|}

== Music ==
; '''[[Eliora]]'''
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 Listening] (2023) - 217 &amp; 1178 and enneadecal in 1178edo tuning
[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]</text>
      <sha1>6oo01yuub640qxqf1itfltab1fa516x</sha1>
    </revision>
    <revision>
      <id>126916</id>
      <parentid>126908</parentid>
      <timestamp>2023-11-04T19:51:33Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>if there is a music plz add to listen</comment>
      <origin>126916</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2709" sha1="2goqzjcjky9mnajxwc286w04bo6k0q8" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A basis for its 19-limit [[comma]]s is [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4375/4374]], [[4914/4913]] and 5985/5984. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" |Tuning Error
|-	
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo|-1867 1178}}
|{{val|1178 1867}}
| 0.0276
| 0.0276
| 2.71
|-
|2.3.5
|{{monzo|-14 -19-19}}, {{monzo|-99 61 1}}
|{{val|1178 1867 2735}}
| 0.0522
| 0.0415
| 4.07
|-
|2.3.5.7
|4375/4374, 703125/702464, 2460375/2458624
|{{val|1178 1867 2735 3307}}
| 0.0450
| 0.0380
| 3.73
|-
|2.3.5.7.11
|3025/3024, 4375/4374, 9801/9800, 43923/43904
|{{val|1178 1867 2735 3307 4075}}
| 0.0484
| 0.0347
| 3.41
|-
|2.3.5.7.11.13
|3025/3024, 4225/4224, 9801/9800, 91125/91091, 1664000/1663893
|{{val|1178 1867 2735 3307 4075 4359}}
| 0.0457
| 0.0322
| 3.16
|-
|2.3.5.7.11.13.17
|3025/3024, 4225/4224, 5832/5831, 9801/9800, 14875/14872, 31213/31212
|{{val|1178 1867 2735 3307 4075 4359 4815}}
| 0.0403
| 0.0327
| 3.21
|-
|2.3.5.7.11.13.17.19
|4200/4199, 5985/5984, 9801/9800, 10830/10829, 12636/12635, 14365/14364, 23409/23408
|{{val|1178 1867 2735 3307 4075 4359 4815 5004}}
| 0.0370
| 0.0318
| 3.12
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
|1
|337\1178
|343.29
|8000/6561
|[[Raider]]
|-
|19
|489\1178&lt;br&gt;(7\1178)
|498.13&lt;br&gt;(7.13)
|4/3&lt;br&gt;(225/224)
|[[Enneadecal]]
|-
|38
|489\1178&lt;br&gt;(7\1178)
|498.13&lt;br&gt;(7.13)
|4/3&lt;br&gt;(225/224)
|[[Hemienneadecal]]
|}

== Music ==
; '''[[Eliora]]'''
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 Listening] (2023) - 217 &amp; 1178 and enneadecal in 1178edo tuning
[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]
[[Category:Listen]]</text>
      <sha1>2goqzjcjky9mnajxwc286w04bo6k0q8</sha1>
    </revision>
    <revision>
      <id>126944</id>
      <parentid>126916</parentid>
      <timestamp>2023-11-05T13:38:46Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Review</comment>
      <origin>126944</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2989" sha1="ss23brfs7js1aolpxfgtf7jzqg3i59e" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-	
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -1867 1178 }}
| {{mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893
| {{mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| 0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| 0.0370
| 0.0318
| 3.12
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 489\1178&lt;br&gt;(7\1178)
| 498.13&lt;br&gt;(7.13)
| 4/3&lt;br&gt;(225/224)
| [[Enneadecal]]
|-
| 38
| 489\1178&lt;br&gt;(7\1178)
| 498.13&lt;br&gt;(7.13)
| 4/3&lt;br&gt;(225/224)
| [[Hemienneadecal]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – 217 &amp; 1178 and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]
[[Category:Listen]]</text>
      <sha1>ss23brfs7js1aolpxfgtf7jzqg3i59e</sha1>
    </revision>
    <revision>
      <id>126946</id>
      <parentid>126944</parentid>
      <timestamp>2023-11-05T13:50:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ +semihemienneadecal</comment>
      <origin>126946</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3089" sha1="o9y6otmm859qr33f8z891cijfa1ncj4" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-	
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -1867 1178 }}
| {{mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893
| {{mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| 0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| 0.0370
| 0.0318
| 3.12
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 489\1178&lt;br&gt;(7\1178)
| 498.13&lt;br&gt;(7.13)
| 4/3&lt;br&gt;(225/224)
| [[Enneadecal]]
|-
| 38
| 260\1178&lt;br&gt;(12\1178)
| 264.86&lt;br&gt;(12.22)
| 500/429&lt;br&gt;(144/143)
| [[Semihemienneadecal]]
|-
| 38
| 489\1178&lt;br&gt;(7\1178)
| 498.13&lt;br&gt;(7.13)
| 4/3&lt;br&gt;(225/224)
| [[Hemienneadecal]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – 217 &amp; 1178 and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]
[[Category:Listen]]</text>
      <sha1>o9y6otmm859qr33f8z891cijfa1ncj4</sha1>
    </revision>
    <revision>
      <id>127010</id>
      <parentid>126946</parentid>
      <timestamp>2023-11-06T22:07:50Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Rank-2 temperaments */ + 217 &amp; 1178 as music is composed in it</comment>
      <origin>127010</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3164" sha1="gewlo90whcod95xopf02z1a4ood65c5" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-	
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -1867 1178 }}
| {{mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893
| {{mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| 0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| 0.0370
| 0.0318
| 3.12
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 489\1178&lt;br&gt;(7\1178)
| 498.13&lt;br&gt;(7.13)
| 4/3&lt;br&gt;(225/224)
| [[Enneadecal]]
|-
|31
|581\1178
(11\1178)
|591.851
(11.205)
|936/665

(?)
|[[217 &amp; 1178]]
|-
| 38
| 260\1178&lt;br&gt;(12\1178)
| 264.86&lt;br&gt;(12.22)
| 500/429&lt;br&gt;(144/143)
| [[Semihemienneadecal]]
|-
| 38
| 489\1178&lt;br&gt;(7\1178)
| 498.13&lt;br&gt;(7.13)
| 4/3&lt;br&gt;(225/224)
| [[Hemienneadecal]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – 217 &amp; 1178 and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]
[[Category:Listen]]</text>
      <sha1>gewlo90whcod95xopf02z1a4ood65c5</sha1>
    </revision>
    <revision>
      <id>127011</id>
      <parentid>127010</parentid>
      <timestamp>2023-11-06T22:08:29Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Rank-2 temperaments */ style</comment>
      <origin>127011</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3213" sha1="8i6ukrl8qf9cdfymq1cwrawytnjld2z" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-	
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -1867 1178 }}
| {{mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893
| {{mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| 0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| 0.0370
| 0.0318
| 3.12
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 489\1178&lt;br&gt;(7\1178)
| 498.13&lt;br&gt;(7.13)
| 4/3&lt;br&gt;(225/224)
| [[Enneadecal]]
|-
| 31
| 581\1178&lt;br&gt;(11\1178)
| 591.851&lt;br&gt;(11.205)
| 936/665&lt;br&gt;(?)
| [[31st-octave temperaments#217 &amp; 1178|217 &amp; 1178]]
|-
| 38
| 260\1178&lt;br&gt;(12\1178)
| 264.86&lt;br&gt;(12.22)
| 500/429&lt;br&gt;(144/143)
| [[Semihemienneadecal]]
|-
| 38
| 489\1178&lt;br&gt;(7\1178)
| 498.13&lt;br&gt;(7.13)
| 4/3&lt;br&gt;(225/224)
| [[Hemienneadecal]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – 217 &amp; 1178 and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Semihemienneadecal]]
[[Category:Listen]]</text>
      <sha1>8i6ukrl8qf9cdfymq1cwrawytnjld2z</sha1>
    </revision>
    <revision>
      <id>165891</id>
      <parentid>127011</parentid>
      <timestamp>2024-11-16T04:21:04Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>165891</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2705" sha1="ghcvpr0485l8pcyv7h0vcv1h7oopyr7" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{{comma basis begin}}
|-
| 2.3
| {{monzo| -1867 1178 }}
| {{mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893
| {{mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| 0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| 0.0370
| 0.0318
| 3.12
{{comma basis end}}

=== Rank-2 temperaments ===
{{rank-2 begin}}
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 489\1178&lt;br /&gt;(7\1178)
| 498.13&lt;br /&gt;(7.13)
| 4/3&lt;br /&gt;(225/224)
| [[Enneadecal]]
|-
| 31
| 581\1178&lt;br /&gt;(11\1178)
| 591.851&lt;br /&gt;(11.205)
| 936/665&lt;br /&gt;(?)
| [[31st-octave temperaments#217 &amp; 1178|217 &amp; 1178]]
|-
| 38
| 260\1178&lt;br /&gt;(12\1178)
| 264.86&lt;br /&gt;(12.22)
| 500/429&lt;br /&gt;(144/143)
| [[Semihemienneadecal]]
|-
| 38
| 489\1178&lt;br /&gt;(7\1178)
| 498.13&lt;br /&gt;(7.13)
| 4/3&lt;br /&gt;(225/224)
| [[Hemienneadecal]]
{{rank-2 end}}
{{orf}}

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) &amp;ndash; 217 &amp; 1178 and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Listen]]
[[Category:Semihemienneadecal]]</text>
      <sha1>ghcvpr0485l8pcyv7h0vcv1h7oopyr7</sha1>
    </revision>
    <revision>
      <id>166186</id>
      <parentid>165891</parentid>
      <timestamp>2024-11-16T12:35:28Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Partial undo</comment>
      <origin>166186</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3262" sha1="ke9ibu1cs1wk5dbn8zikqn25h7sn8nk" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 = {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -1867 1178 }}
| {{mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893
| {{mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| 0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| 0.0370
| 0.0318
| 3.12
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 489\1178&lt;br /&gt;(7\1178)
| 498.13&lt;br /&gt;(7.13)
| 4/3&lt;br /&gt;(225/224)
| [[Enneadecal]]
|-
| 31
| 581\1178&lt;br /&gt;(11\1178)
| 591.851&lt;br /&gt;(11.205)
| 936/665&lt;br /&gt;(?)
| [[31st-octave temperaments#217 &amp; 1178|217 &amp; 1178]]
|-
| 38
| 260\1178&lt;br /&gt;(12\1178)
| 264.86&lt;br /&gt;(12.22)
| 500/429&lt;br /&gt;(144/143)
| [[Semihemienneadecal]]
|-
| 38
| 489\1178&lt;br /&gt;(7\1178)
| 498.13&lt;br /&gt;(7.13)
| 4/3&lt;br /&gt;(225/224)
| [[Hemienneadecal]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) &amp;ndash; 217 &amp; 1178 and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Listen]]
[[Category:Semihemienneadecal]]</text>
      <sha1>ke9ibu1cs1wk5dbn8zikqn25h7sn8nk</sha1>
    </revision>
    <revision>
      <id>176354</id>
      <parentid>166186</parentid>
      <timestamp>2025-01-15T19:43:13Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176354</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3280" sha1="3xo808c6wwee3kydzn1n4t5t2l8uilh" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1178}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 factors into {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -1867 1178 }}
| {{mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893
| {{mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| +0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| +0.0370
| 0.0318
| 3.12
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 489\1178&lt;br /&gt;(7\1178)
| 498.13&lt;br /&gt;(7.13)
| 4/3&lt;br /&gt;(225/224)
| [[Enneadecal]]
|-
| 31
| 581\1178&lt;br /&gt;(11\1178)
| 591.851&lt;br /&gt;(11.205)
| 936/665&lt;br /&gt;(?)
| [[31st-octave temperaments#217 &amp; 1178|217 &amp; 1178]]
|-
| 38
| 260\1178&lt;br /&gt;(12\1178)
| 264.86&lt;br /&gt;(12.22)
| 500/429&lt;br /&gt;(144/143)
| [[Semihemienneadecal]]
|-
| 38
| 489\1178&lt;br /&gt;(7\1178)
| 498.13&lt;br /&gt;(7.13)
| 4/3&lt;br /&gt;(225/224)
| [[Hemienneadecal]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – {{nowrap|217 &amp;amp; 1178}} and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Listen]]
[[Category:Semihemienneadecal]]</text>
      <sha1>3xo808c6wwee3kydzn1n4t5t2l8uilh</sha1>
    </revision>
    <revision>
      <id>181322</id>
      <parentid>176354</parentid>
      <timestamp>2025-02-18T16:53:05Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181322</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3274" sha1="n8vz23yboid293677livpp7q8j4n0kc" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 factors into {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -1867 1178 }}
| {{mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893
| {{mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| +0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| +0.0370
| 0.0318
| 3.12
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 489\1178&lt;br /&gt;(7\1178)
| 498.13&lt;br /&gt;(7.13)
| 4/3&lt;br /&gt;(225/224)
| [[Enneadecal]]
|-
| 31
| 581\1178&lt;br /&gt;(11\1178)
| 591.851&lt;br /&gt;(11.205)
| 936/665&lt;br /&gt;(?)
| [[31st-octave temperaments#217 &amp; 1178|217 &amp; 1178]]
|-
| 38
| 260\1178&lt;br /&gt;(12\1178)
| 264.86&lt;br /&gt;(12.22)
| 500/429&lt;br /&gt;(144/143)
| [[Semihemienneadecal]]
|-
| 38
| 489\1178&lt;br /&gt;(7\1178)
| 498.13&lt;br /&gt;(7.13)
| 4/3&lt;br /&gt;(225/224)
| [[Hemienneadecal]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – {{nowrap|217 &amp;amp; 1178}} and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Listen]]
[[Category:Semihemienneadecal]]</text>
      <sha1>n8vz23yboid293677livpp7q8j4n0kc</sha1>
    </revision>
    <revision>
      <id>225880</id>
      <parentid>181322</parentid>
      <timestamp>2026-03-13T13:32:43Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct" to "[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct"</comment>
      <origin>225880</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3329" sha1="r9j0axzvc6lbltdtqljp5smr4uio5f5" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 factors into {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -1867 1178 }}
| {{mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 1664000/1663893
| {{mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| +0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| +0.0370
| 0.0318
| 3.12
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 489\1178&lt;br /&gt;(7\1178)
| 498.13&lt;br /&gt;(7.13)
| 4/3&lt;br /&gt;(225/224)
| [[Enneadecal]]
|-
| 31
| 581\1178&lt;br /&gt;(11\1178)
| 591.851&lt;br /&gt;(11.205)
| 936/665&lt;br /&gt;(?)
| [[31st-octave temperaments#217 &amp; 1178|217 &amp; 1178]]
|-
| 38
| 260\1178&lt;br /&gt;(12\1178)
| 264.86&lt;br /&gt;(12.22)
| 500/429&lt;br /&gt;(144/143)
| [[Semihemienneadecal]]
|-
| 38
| 489\1178&lt;br /&gt;(7\1178)
| 498.13&lt;br /&gt;(7.13)
| 4/3&lt;br /&gt;(225/224)
| [[Hemienneadecal]]
|}
&lt;nowiki /&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – {{nowrap|217 &amp;amp; 1178}} and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Listen]]
[[Category:Semihemienneadecal]]</text>
      <sha1>r9j0axzvc6lbltdtqljp5smr4uio5f5</sha1>
    </revision>
    <revision>
      <id>233742</id>
      <parentid>225880</parentid>
      <timestamp>2026-07-13T09:51:25Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ comma basis</comment>
      <origin>233742</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3296" sha1="c1vniv6tc1zo4hrdsh0z49hw5eot08z" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 factors into {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -1867 1178 }}
| {{Mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{Monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{Mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{Mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{Mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 655473/655360
| {{Mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| +0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| +0.0370
| 0.0318
| 3.12
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 489\1178&lt;br&gt;(7\1178)
| 498.13&lt;br&gt;(7.13)
| 4/3&lt;br&gt;(225/224)
| [[Enneadecal]]
|-
| 31
| 581\1178&lt;br&gt;(11\1178)
| 591.851&lt;br&gt;(11.205)
| 936/665&lt;br&gt;(?)
| [[31st-octave temperaments#217 &amp; 1178|217 &amp; 1178]]
|-
| 38
| 260\1178&lt;br&gt;(12\1178)
| 264.86&lt;br&gt;(12.22)
| 500/429&lt;br&gt;(144/143)
| [[Semihemienneadecal]]
|-
| 38
| 489\1178&lt;br&gt;(7\1178)
| 498.13&lt;br&gt;(7.13)
| 4/3&lt;br&gt;(225/224)
| [[Hemienneadecal]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – {{nowrap|217 &amp;amp; 1178}} and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Listen]]
[[Category:Semihemienneadecal]]</text>
      <sha1>c1vniv6tc1zo4hrdsh0z49hw5eot08z</sha1>
    </revision>
    <revision>
      <id>233743</id>
      <parentid>233742</parentid>
      <timestamp>2026-07-13T09:53:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ convert the table of rank-2 temps to minimal-generator form</comment>
      <origin>233743</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2942" sha1="0kippkzwhusbzn73jbrx2v8xzpg2rmx" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1178edo is a very strong 19-limit system, and is a [[zeta edo|zeta peak, integral and gap edo]]. It is also [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. A [[comma basis|basis]] for its 19-limit [[comma]]s consists of [[2500/2499]], [[3025/3024]], 3250/3249, 4200/4199, [[4225/4224]], [[4375/4374]], and [[4914/4913]]. It [[support]]s and provides a great tuning for [[semihemienneadecal]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}

=== Subsets and supersets ===
Since 1178 factors into {{factorization|1178}}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -1867 1178 }}
| {{Mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{Monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{Mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{Mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{Mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 655473/655360
| {{Mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| +0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| +0.0370
| 0.0318
| 3.12
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 7\1178
| 7.13
| 225/224
| [[Enneadecal]]
|-
| 38
| 12\1178
| 12.22
| 144/143
| [[Semihemienneadecal]]
|-
| 38
| 7\1178
| 7.13
| 225/224
| [[Hemienneadecal]]
|}
&lt;nowiki/&gt;* In [[normal forms #Minimal-generator form|minimal-generator form]]

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – {{nowrap|217 &amp;amp; 1178}} and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Listen]]
[[Category:Semihemienneadecal]]</text>
      <sha1>0kippkzwhusbzn73jbrx2v8xzpg2rmx</sha1>
    </revision>
    <revision>
      <id>233744</id>
      <parentid>233743</parentid>
      <timestamp>2026-07-13T10:10:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Expand</comment>
      <origin>233744</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3512" sha1="93gpb0lz4hx6409r634x19ucxlcdgef" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1178edo is a very strong [[19-limit]] system, [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It is a [[zeta peak edo|zeta peak]], [[zeta integral edo|integral]] and [[zeta gap edo|gap edo]].

As an equal temperament, it tempers out {{monzo| -14 -19-19 }} ([[enneadeca]]) in the [[5-limit]], 4375/4374 ([[ragisma]]) and 703125/702464  ([[meter]]) in the [[7-limit]], so that it [[support]]s [[enneadecal]]. In the [[11-limit]] it tempers out [[3025/3024]], [[9801/9800]], and [[234375/234256]], supporting [[hemienneadecal]], and in the [[13-limit]] [[4225/4224]] and [[10648/10647]], supporting and providing a great tuning for [[semihemienneadecal]]. It further tempers out [[2500/2499]], [[4914/4913]] in the [[17-limit]]; [[3250/3249]], [[4200/4199]] in the 19-limit; and [[2025/2024]] among others in the [[23-limit]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}
{{Harmonics in equal|1178|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 1178edo (continued)}}

=== Subsets and supersets ===
Since 1178 factors into primes as {{nowrap| 2 × 19 × 31 }}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -1867 1178 }}
| {{Mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{Monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{Mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{Mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{Mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 655473/655360
| {{Mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| +0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| +0.0370
| 0.0318
| 3.12
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 7\1178
| 7.13
| 225/224
| [[Enneadecal]]
|-
| 38
| 12\1178
| 12.22
| 144/143
| [[Semihemienneadecal]]
|-
| 38
| 7\1178
| 7.13
| 225/224
| [[Hemienneadecal]]
|}
&lt;nowiki/&gt;* In [[normal forms #Minimal-generator form|minimal-generator form]]

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – {{nowrap|217 &amp;amp; 1178}} and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Listen]]
[[Category:Semihemienneadecal]]</text>
      <sha1>93gpb0lz4hx6409r634x19ucxlcdgef</sha1>
    </revision>
    <revision>
      <id>233745</id>
      <parentid>233744</parentid>
      <timestamp>2026-07-13T10:27:29Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ + 23-limit and records</comment>
      <origin>233745</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4120" sha1="m5dwlul85f44dspy22kgysvkp4r7isi" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
1178edo is a very strong [[19-limit]] system, [[consistency|distinctly consistent]] through to the [[21-odd-limit]], and is the first edo past [[742edo|742]] with a lower 19-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. It is a [[zeta peak edo|zeta peak]], [[zeta integral edo|integral]] and [[zeta gap edo|gap edo]].

As an equal temperament, it tempers out {{monzo| -14 -19-19 }} ([[enneadeca]]) in the [[5-limit]], 4375/4374 ([[ragisma]]) and 703125/702464  ([[meter]]) in the [[7-limit]], so that it [[support]]s [[enneadecal]]. In the [[11-limit]] it tempers out [[3025/3024]], [[9801/9800]], and [[234375/234256]], supporting [[hemienneadecal]], and in the [[13-limit]] [[4225/4224]] and [[10648/10647]], supporting and providing a great tuning for [[semihemienneadecal]]. It further tempers out [[2500/2499]], [[4914/4913]] in the [[17-limit]]; [[3250/3249]], [[4200/4199]] in the 19-limit; and [[2025/2024]] among others in the [[23-limit]]. 

=== Prime harmonics ===
{{Harmonics in equal|1178|columns=11}}
{{Harmonics in equal|1178|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 1178edo (continued)}}

=== Subsets and supersets ===
Since 1178 factors into primes as {{nowrap| 2 × 19 × 31 }}, 1178edo is notable for containing both 19 and 31. Its subset edos are {{EDOs| 2, 19, 31, 38, 62, and 589 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -1867 1178 }}
| {{Mapping| 1178 1867 }}
| +0.0276
| 0.0276
| 2.71
|-
| 2.3.5
| {{Monzo| -14 -19-19 }}, {{monzo| -99 61 1 }}
| {{Mapping| 1178 1867 2735 }}
| +0.0522
| 0.0415
| 4.07
|-
| 2.3.5.7
| 4375/4374, 703125/702464, {{monzo| -52 -5 -2 23 }}
| {{Mapping| 1178 1867 2735 3307 }}
| +0.0450
| 0.0380
| 3.73
|-
| 2.3.5.7.11
| 3025/3024, 4375/4374, 234375/234256, {{monzo| -27 3 -4 10 1 }}
| {{Mapping| 1178 1867 2735 3307 4075 }}
| +0.0484
| 0.0347
| 3.41
|-
| 2.3.5.7.11.13
| 3025/3024, 4225/4224, 4375/4374, 78125/78078, 655473/655360
| {{Mapping| 1178 1867 2735 3307 4075 4359 }}
| +0.0457
| 0.0322
| 3.16
|-
| 2.3.5.7.11.13.17
| 2500/2499, 3025/3024, 4225/4224, 4375/4374, 4914/4913, 14875/14872
| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 }}
| +0.0403
| 0.0327
| 3.21
|-
| 2.3.5.7.11.13.17.19
| 2500/2499, 3025/3024, 3250/3249, 4200/4199, 4225/4224, 4375/4374, 4914/4913
| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 5004 }}
| +0.0370
| 0.0318
| 3.12
|-
| 2.3.5.7.11.13.17.19.23
| 2025/2024, 2500/2499, 3025/3024, 3060/3059, 3250/3249, 3520/3519, 4200/4199, 4375/4374
| {{Mapping| 1178 1867 2735 3307 4075 4359 4815 5004 5329 }}
| +0.0292
| 0.0436
| 4.28
|}
* 1178et holds the record of lowest relative error in the 19-limit, being the first to beat [[742edo|742]] in relative error and [[954edo|954h]] in absolute error, before getting superseded by [[1578edo|1578]] in both. 
* It holds the record of lowest absolute errors in the 13-, 17-, and 23-limit, after 954, 954, [[1106edo|1106]] and before [[1236edo|1236]], 1236, and [[1308edo|1308]], respectively. 

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 337\1178
| 343.29
| 8000/6561
| [[Raider]]
|-
| 19
| 7\1178
| 7.13
| 225/224
| [[Enneadecal]]
|-
| 38
| 12\1178
| 12.22
| 144/143
| [[Semihemienneadecal]]
|-
| 38
| 7\1178
| 7.13
| 225/224
| [[Hemienneadecal]]
|}
&lt;nowiki/&gt;* In [[normal forms #Minimal-generator form|minimal-generator form]]

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=c9e7MTsIDc4 ''Listening''] (2023) – {{nowrap|217 &amp;amp; 1178}} and enneadecal in 1178edo tuning

[[Category:Enneadecal]]
[[Category:Hemienneadecal]]
[[Category:Listen]]
[[Category:Semihemienneadecal]]</text>
      <sha1>m5dwlul85f44dspy22kgysvkp4r7isi</sha1>
    </revision>
  </page>
  <page>
    <title>118edo</title>
    <ns>0</ns>
    <id>64</id>
    <revision>
      <id>64</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>64</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1114" sha1="afb8v4tj9vpx32tlvox9zq8ce0xrlwc" xml:space="preserve">''118edo'' divides the octave into 118 equal parts, each of 10.17 cents. It represents the intersection of the 5-limit schismatic and parakleismic temperaments, tempering out both the schisma, |-15 8 1&amp;gt; and the parakleisma, |8 14 -13&amp;gt;, as well as the vishnuzma, |23 6 -14&amp;gt;, the hemithirds comma, |38 -2 -15&amp;gt; and the kwazy, |-53 10 16&amp;gt;. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit it is particularly notable for tempering out the gamelisma, 1029/1024, and is an excellent tuning for the rank three gamelismic temperament, and for guiron, the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo|99edo]] does better with that.

In the 11-limit, it tempers out 385/384 and 441/440, and is an excellent tuning for portent, the temperament tempering out both, and for the 11-limit version of guiron, which does also.
[[Category:edo]]
[[Category:gamelisma]]
[[Category:guiron]]
[[Category:parakleismic]]
[[Category:portent]]
[[Category:schismic]]</text>
      <sha1>afb8v4tj9vpx32tlvox9zq8ce0xrlwc</sha1>
    </revision>
    <revision>
      <id>4310</id>
      <timestamp>2012-06-04T18:08:36Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;phylingual</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 342593762 - Original comment: **</comment>
      <origin>4310</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3313" sha1="lgol7n3c262qbggdvd2z8vfevhke6my" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:phylingual|phylingual]] and made on &lt;tt&gt;2012-06-04 18:08:36 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;342593762&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//118edo// divides the octave into 118 equal parts, each of 10.17 cents. It represents the intersection of the 5-limit schismatic and parakleismic temperaments, tempering out both the schisma, |-15 8 1&amp;gt; and the parakleisma, |8 14 -13&amp;gt;, as well as the vishnuzma, |23 6 -14&amp;gt;, the hemithirds comma, |38 -2 -15&amp;gt; and the kwazy, |-53 10 16&amp;gt;. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit it is particularly notable for tempering out the gamelisma, 1029/1024, and is an excellent tuning for the rank three gamelismic temperament, and for guiron, the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out 385/384 and 441/440, and is an excellent tuning for portent, the temperament tempering out both, and for the 11-limit version of guiron, which does also.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;118edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;em&amp;gt;118edo&amp;lt;/em&amp;gt; divides the octave into 118 equal parts, each of 10.17 cents. It represents the intersection of the 5-limit schismatic and parakleismic temperaments, tempering out both the schisma, |-15 8 1&amp;amp;gt; and the parakleisma, |8 14 -13&amp;amp;gt;, as well as the vishnuzma, |23 6 -14&amp;amp;gt;, the hemithirds comma, |38 -2 -15&amp;amp;gt; and the kwazy, |-53 10 16&amp;amp;gt;. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 7-limit it is particularly notable for tempering out the gamelisma, 1029/1024, and is an excellent tuning for the rank three gamelismic temperament, and for guiron, the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but &amp;lt;a class="wiki_link" href="/99edo"&amp;gt;99edo&amp;lt;/a&amp;gt; does better with that.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit, it tempers out 385/384 and 441/440, and is an excellent tuning for portent, the temperament tempering out both, and for the 11-limit version of guiron, which does also.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>lgol7n3c262qbggdvd2z8vfevhke6my</sha1>
    </revision>
    <revision>
      <id>4311</id>
      <timestamp>2010-08-08T20:03:50Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 155650931 - Original comment: **</comment>
      <origin>4311</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3321" sha1="85cfjott7f82ussq9239m6zgqfeigbj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2010-08-08 20:03:50 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;155650931&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//118edo// divides the octave into 118 equal parts, each of 10.17 cents. It represents the intersection of the 5-limit schismatic and parakleismic temperaments, tempering out both the schisma, |-15 8 1&amp;gt; and the parakleisma, |8 14 -13&amp;gt;, as well as the vishnuzma, |23 6 -14&amp;gt;, the hemithirds comma, |38 -2 -15&amp;gt; and the kwazy, |-53 10 16&amp;gt;. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit it is particularly notable for tempering out the gammelisma, 1029/1024, and is an excellent tuning for the rank three gamelismic temperament, and for guiron, the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out 385/384 and 441/440, and is an excellent tuning for portent, the temperament tempering out both, and for the 11-limit version of guiron, which does also.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;118edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;em&amp;gt;118edo&amp;lt;/em&amp;gt; divides the octave into 118 equal parts, each of 10.17 cents. It represents the intersection of the 5-limit schismatic and parakleismic temperaments, tempering out both the schisma, |-15 8 1&amp;amp;gt; and the parakleisma, |8 14 -13&amp;amp;gt;, as well as the vishnuzma, |23 6 -14&amp;amp;gt;, the hemithirds comma, |38 -2 -15&amp;amp;gt; and the kwazy, |-53 10 16&amp;amp;gt;. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 7-limit it is particularly notable for tempering out the gammelisma, 1029/1024, and is an excellent tuning for the rank three gamelismic temperament, and for guiron, the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but &amp;lt;a class="wiki_link" href="/99edo"&amp;gt;99edo&amp;lt;/a&amp;gt; does better with that.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit, it tempers out 385/384 and 441/440, and is an excellent tuning for portent, the temperament tempering out both, and for the 11-limit version of guiron, which does also.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>85cfjott7f82ussq9239m6zgqfeigbj</sha1>
    </revision>
    <revision>
      <id>40250</id>
      <parentid>64</parentid>
      <timestamp>2019-02-15T11:54:17Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>40250</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1209" sha1="j79odmzqjpc31jjbwpooi51tj9pnqog" xml:space="preserve">'''118edo''' is the [[EDO|equal division of the octave]] into 400 parts of 10.1695 cents each. It represents the intersection of the 5-limit schismatic and parakleismic temperaments, tempering out both the schisma, |-15 8 1&amp;gt; and the parakleisma, |8 14 -13&amp;gt;, as well as the vishnuzma, |23 6 -14&amp;gt;, the hemithirds comma, |38 -2 -15&amp;gt; and the kwazy, |-53 10 16&amp;gt;. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the gamelisma, 1029/1024, and is an excellent tuning for the rank three gamelismic temperament, and for guiron, the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out 385/384 and 441/440, and is an excellent tuning for portent, the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

[[Category:Edo]]
[[Category:gamelismic]]
[[Category:guiron]]
[[Category:parakleismic]]
[[Category:portent]]
[[Category:schismic]]</text>
      <sha1>j79odmzqjpc31jjbwpooi51tj9pnqog</sha1>
    </revision>
    <revision>
      <id>44723</id>
      <parentid>40250</parentid>
      <timestamp>2020-05-22T00:04:56Z</timestamp>
      <contributor>
        <username>Winspear</username>
        <id>181</id>
      </contributor>
      <comment>Fixed EDo error</comment>
      <origin>44723</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1209" sha1="qr1uxj2aq1uyy23rsjxi1rccqm7ysfa" xml:space="preserve">'''118edo''' is the [[EDO|equal division of the octave]] into 118 parts of 10.1695 cents each. It represents the intersection of the 5-limit schismatic and parakleismic temperaments, tempering out both the schisma, |-15 8 1&amp;gt; and the parakleisma, |8 14 -13&amp;gt;, as well as the vishnuzma, |23 6 -14&amp;gt;, the hemithirds comma, |38 -2 -15&amp;gt; and the kwazy, |-53 10 16&amp;gt;. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the gamelisma, 1029/1024, and is an excellent tuning for the rank three gamelismic temperament, and for guiron, the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out 385/384 and 441/440, and is an excellent tuning for portent, the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

[[Category:Edo]]
[[Category:gamelismic]]
[[Category:guiron]]
[[Category:parakleismic]]
[[Category:portent]]
[[Category:schismic]]</text>
      <sha1>qr1uxj2aq1uyy23rsjxi1rccqm7ysfa</sha1>
    </revision>
    <revision>
      <id>49174</id>
      <parentid>44723</parentid>
      <timestamp>2020-09-11T10:11:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Cleanup</comment>
      <origin>49174</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1352" sha1="lc6thl0us680166vhy8kgpwarvj000h" xml:space="preserve">'''118edo''' is the [[equal division of the octave]] into 118 parts of 10.1695 cents each. 

118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo|-15 8 1}} and the [[parakleisma]], {{monzo|8 14 -13}}, as well as the [[vishnuzma]], {{monzo|23 6 -14}}, the [[hemithirds comma]], {{monzo|38 -2 -15}}, and the [[kwazy]], {{monzo|-53 10 16}}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

[[Category:Edo]]
[[Category:118edo]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>lc6thl0us680166vhy8kgpwarvj000h</sha1>
    </revision>
    <revision>
      <id>49215</id>
      <parentid>49174</parentid>
      <timestamp>2020-09-11T18:27:22Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>removed singleton category</comment>
      <origin>49215</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1332" sha1="dc8hkefmymc8p4m3cnfah4oiol947wu" xml:space="preserve">'''118edo''' is the [[equal division of the octave]] into 118 parts of 10.1695 cents each. 

118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo|-15 8 1}} and the [[parakleisma]], {{monzo|8 14 -13}}, as well as the [[vishnuzma]], {{monzo|23 6 -14}}, the [[hemithirds comma]], {{monzo|38 -2 -15}}, and the [[kwazy]], {{monzo|-53 10 16}}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

[[Category:Edo]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>dc8hkefmymc8p4m3cnfah4oiol947wu</sha1>
    </revision>
    <revision>
      <id>50785</id>
      <parentid>49215</parentid>
      <timestamp>2020-10-05T08:17:57Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+error table</comment>
      <origin>50785</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1693" sha1="i37kqrthw16ontyrvoe6wo22nx4emd4" xml:space="preserve">'''118edo''' is the [[equal division of the octave]] into 118 parts of 10.1695 cents each. 

118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo|-15 8 1}} and the [[parakleisma]], {{monzo|8 14 -13}}, as well as the [[vishnuzma]], {{monzo|23 6 -14}}, the [[hemithirds comma]], {{monzo|38 -2 -15}}, and the [[kwazy]], {{monzo|-53 10 16}}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

== Just approximation ==
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" |Error
! absolute (¢)
| 0.00
| -0.26
| +0.13
| -2.72
| -2.17
| +3.54
| -3.26
| -2.60
|-
! [[Relative error|relative]] (%)
| 0.0
| -2.6
| +1.2
| -26.8
| -21.3
| +34.8
| -32.1
| -25.5
|}

[[Category:Edo]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>i37kqrthw16ontyrvoe6wo22nx4emd4</sha1>
    </revision>
    <revision>
      <id>54967</id>
      <parentid>50785</parentid>
      <timestamp>2020-12-04T23:13:06Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54967</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1719" sha1="hagnibbb0bj9h8ubch30z9ygol0155p" xml:space="preserve">'''118edo''' is the [[equal division of the octave]] into 118 parts of 10.1695 cents each. 

118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo|-15 8 1}} and the [[parakleisma]], {{monzo|8 14 -13}}, as well as the [[vishnuzma]], {{monzo|23 6 -14}}, the [[hemithirds comma]], {{monzo|38 -2 -15}}, and the [[kwazy]], {{monzo|-53 10 16}}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

== Just approximation ==
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" |Error
! absolute (¢)
| 0.00
| -0.26
| +0.13
| -2.72
| -2.17
| +3.54
| -3.26
| -2.60
|-
! [[Relative error|relative]] (%)
| 0.0
| -2.6
| +1.2
| -26.8
| -21.3
| +34.8
| -32.1
| -25.5
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>hagnibbb0bj9h8ubch30z9ygol0155p</sha1>
    </revision>
    <revision>
      <id>59168</id>
      <parentid>54967</parentid>
      <timestamp>2021-01-17T11:47:44Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>prime intervals via template, monzo formatting</comment>
      <origin>59168</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1423" sha1="c1t24ipeq4oxtd4ij48toop5lrbq0aj" xml:space="preserve">'''118edo''' is the [[equal division of the octave]] into 118 parts of 10.1695 cents each. 

118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

== Just approximation ==
{{primes in edo|118|prec=2}}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>c1t24ipeq4oxtd4ij48toop5lrbq0aj</sha1>
    </revision>
    <revision>
      <id>72247</id>
      <parentid>59168</parentid>
      <timestamp>2021-06-09T09:39:29Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+table of rank-2 temperaments</comment>
      <origin>72247</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2563" sha1="pxvm13u9j6n68x3s3lkyjvvih0wi4t0" xml:space="preserve">'''118edo''' is the [[equal division of the octave]] into 118 parts of 10.1695 cents each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].{{primes in edo|118|prec=2}}

== Rank-2 temperaments ==
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
|1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
|2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
|2
| 7\118
| 71.19
| 25/24
| [[Vishnuzmic]] / [[vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]]&lt;br&gt;Bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>pxvm13u9j6n68x3s3lkyjvvih0wi4t0</sha1>
    </revision>
    <revision>
      <id>73572</id>
      <parentid>72247</parentid>
      <timestamp>2021-07-02T09:25:22Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+RTT property table</comment>
      <origin>73572</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3301" sha1="3hcix2n1sgcrtwr2hpooitxlej4ktga" xml:space="preserve">'''118edo''' is the [[equal division of the octave]] into 118 parts of 10.1695 cents each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

=== Prime harmonics ===
{{primes in edo|118|prec=2}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, 1224440064/1220703125
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnuzmic]] / [[vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]]&lt;br&gt;Bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>3hcix2n1sgcrtwr2hpooitxlej4ktga</sha1>
    </revision>
    <revision>
      <id>73644</id>
      <parentid>73572</parentid>
      <timestamp>2021-07-04T10:58:08Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* Rank-2 temperaments */ the wide line created by the &lt;br&gt; between [[Bischismic]] and Bipont (118) attracted unnecessary attention in my opinion</comment>
      <origin>73644</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3300" sha1="bz0day88sj0wch1zf0tg0besd0gbv53" xml:space="preserve">'''118edo''' is the [[equal division of the octave]] into 118 parts of 10.1695 cents each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

=== Prime harmonics ===
{{primes in edo|118|prec=2}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, 1224440064/1220703125
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnuzmic]] / [[vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / Bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>bz0day88sj0wch1zf0tg0besd0gbv53</sha1>
    </revision>
    <revision>
      <id>74133</id>
      <parentid>73644</parentid>
      <timestamp>2021-07-11T12:08:22Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* Prime harmonics */ the same prec is now estimated by EDO magnitude</comment>
      <origin>74133</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3293" sha1="kj8se0gb98wgx5mqkvwc0k5qwvgy3lx" xml:space="preserve">'''118edo''' is the [[equal division of the octave]] into 118 parts of 10.1695 cents each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

=== Prime harmonics ===
{{primes in edo|118}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, 1224440064/1220703125
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnuzmic]] / [[vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / Bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>kj8se0gb98wgx5mqkvwc0k5qwvgy3lx</sha1>
    </revision>
    <revision>
      <id>75080</id>
      <parentid>74133</parentid>
      <timestamp>2021-07-26T10:21:07Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>Added a small factoid</comment>
      <origin>75080</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3449" sha1="8s9v1gx1s2l02lgjfsrvj0f09z6dvm3" xml:space="preserve">'''118edo''' is the [[equal division of the octave]] into 118 parts of 10.1695 cents each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

=== Prime harmonics ===
{{primes in edo|118}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, 1224440064/1220703125
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnuzmic]] / [[vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / Bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Trivia ==
118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names.
[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>8s9v1gx1s2l02lgjfsrvj0f09z6dvm3</sha1>
    </revision>
    <revision>
      <id>75215</id>
      <parentid>75080</parentid>
      <timestamp>2021-07-29T10:21:51Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Improve intro</comment>
      <origin>75215</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3626" sha1="1u31017ed79hmyy3y9mlva0yf3oqiel" xml:space="preserve">The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of 10.1695 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

=== Prime harmonics ===
{{primes in edo|118}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, 1224440064/1220703125
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnuzmic]] / [[vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / Bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Trivia ==
118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names.
[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>1u31017ed79hmyy3y9mlva0yf3oqiel</sha1>
    </revision>
    <revision>
      <id>77034</id>
      <parentid>75215</parentid>
      <timestamp>2021-08-28T13:08:46Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+infobox</comment>
      <origin>77034</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3821" sha1="e57tz58qtda2p7wj4rrjtn65klih63m" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.16949¢
| Fifth = 69\118 (701.69¢)
| Major 2nd = 20\118 (203¢)
| Minor 2nd = 9\118 (92¢)
| Augmented 1sn = 11\118  (112¢)
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

=== Prime harmonics ===
{{primes in edo|118}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, 1224440064/1220703125
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnuzmic]] / [[vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / Bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Trivia ==
118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names.
[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>e57tz58qtda2p7wj4rrjtn65klih63m</sha1>
    </revision>
    <revision>
      <id>78237</id>
      <parentid>77034</parentid>
      <timestamp>2021-09-16T20:24:13Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Expansion</comment>
      <origin>78237</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4231" sha1="2702rqpgxgnipth8ra884pbt47izny9" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.16949¢
| Fifth = 69\118 (701.69¢)
| Major 2nd = 20\118 (203¢)
| Minor 2nd = 9\118 (92¢)
| Augmented 1sn = 11\118  (112¢)
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

=== Prime harmonics ===
{{primes in edo|118}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, 1224440064/1220703125
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / Bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Trivia ==
118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names. 

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>2702rqpgxgnipth8ra884pbt47izny9</sha1>
    </revision>
    <revision>
      <id>78246</id>
      <parentid>78237</parentid>
      <timestamp>2021-09-16T21:40:56Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ expansion</comment>
      <origin>78246</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4776" sha1="0iki7lkq1ud3kmltizjexe92fczbeat" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.16949¢
| Fifth = 69\118 (701.69¢)
| Major 2nd = 20\118 (203¢)
| Minor 2nd = 9\118 (92¢)
| Augmented 1sn = 11\118  (112¢)
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

=== Prime harmonics ===
{{primes in edo|118}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Trivia ==
118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names. 

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>0iki7lkq1ud3kmltizjexe92fczbeat</sha1>
    </revision>
    <revision>
      <id>78247</id>
      <parentid>78246</parentid>
      <timestamp>2021-09-16T21:41:21Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ correction</comment>
      <origin>78247</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4780" sha1="sx0t1ucvmkaj947adpbg8zv1g4kdfyf" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.16949¢
| Fifth = 69\118 (701.69¢)
| Major 2nd = 20\118 (203¢)
| Minor 2nd = 9\118 (92¢)
| Augmented 1sn = 11\118  (112¢)
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

=== Prime harmonics ===
{{primes in edo|118}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Trivia ==
118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names. 

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>sx0t1ucvmkaj947adpbg8zv1g4kdfyf</sha1>
    </revision>
    <revision>
      <id>78481</id>
      <parentid>78247</parentid>
      <timestamp>2021-09-24T13:07:15Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ +squarschmidt</comment>
      <origin>78481</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4832" sha1="opiz1bc4oh16jig722mjb11iza84apu" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.16949¢
| Fifth = 69\118 (701.69¢)
| Major 2nd = 20\118 (203¢)
| Minor 2nd = 9\118 (92¢)
| Augmented 1sn = 11\118  (112¢)
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

=== Prime harmonics ===
{{primes in edo|118}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Trivia ==
118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names. 

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>opiz1bc4oh16jig722mjb11iza84apu</sha1>
    </revision>
    <revision>
      <id>79776</id>
      <parentid>78481</parentid>
      <timestamp>2021-10-22T10:42:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Update infobox</comment>
      <origin>79776</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4823" sha1="abi0ifprpf6vnxwsij2tk3kq4c9vxcu" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].

=== Prime harmonics ===
{{primes in edo|118}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Trivia ==
118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names. 

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>abi0ifprpf6vnxwsij2tk3kq4c9vxcu</sha1>
    </revision>
    <revision>
      <id>80380</id>
      <parentid>79776</parentid>
      <timestamp>2021-11-07T10:53:21Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>80380</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4966" sha1="miwhzupea5gq8jyrh439o18fawbohb0" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a stack of 10 [[12edo]]&lt;nowiki/&gt;s minus the said comma. 

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Trivia ==
118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names. 

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>miwhzupea5gq8jyrh439o18fawbohb0</sha1>
    </revision>
    <revision>
      <id>80381</id>
      <parentid>80380</parentid>
      <timestamp>2021-11-07T10:54:08Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>Note page in process of editing</comment>
      <origin>80381</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4988" sha1="j42hvv6r7j52lu40a9clp7ic24vuydh" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a stack of 10 [[12edo]]&lt;nowiki/&gt;s minus the said comma. 

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 
=== Prime harmonics ===
{{primes in edo|118}}
==Table of intervals==

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Trivia ==
118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names. 

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>j42hvv6r7j52lu40a9clp7ic24vuydh</sha1>
    </revision>
    <revision>
      <id>80382</id>
      <parentid>80381</parentid>
      <timestamp>2021-11-07T11:04:28Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Table of intervals */</comment>
      <origin>80382</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5719" sha1="db1ef2kb5plgkq8lm84dvumbaowthup" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a stack of 10 [[12edo]]&lt;nowiki/&gt;s minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 
=== Prime harmonics ===
{{primes in edo|118}}
==Table of intervals==
{| class="wikitable"
|+Selected 118 EDO intervals
!Step
!Name
!Associated ratio
|-
|0
|unison
|1/1 exact
|-
|2
|comma
|[[531441/524288]], [[81/80]]
|-
|9
|limma
|[[256/243]]
|-
|11
|apotome
|[[2187/2048]]
|-
|20
|whole tone
|[[9/8]]
|-
|23
|septimal second
|[[8/7]]
|-
|26
|septimal third
|[[7/6]]
|-
|29
|Pythagorean minor 3rd
|[[32/27]]
|-
|31
|Classical minor 3rd
|[[6/5]]
|-
|38
|Classical major 3rd
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|[[81/64]]
|-
|49
|perfect 4th
|[[4/3]]
|-
|59
|symmetric tritone
|
|-
|69
|perfect 5th
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|[[128/81]]
|-
|80
|Classical minor 6th
|[[8/5]]
|-
|87
|Classical major 6th
|[[5/3]]
|-
|89
|Pythagorean major 6th
|[[27/16]]
|-
|118
|perfect 8ve
|2/1 exact
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Trivia ==
118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names. 

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>db1ef2kb5plgkq8lm84dvumbaowthup</sha1>
    </revision>
    <revision>
      <id>80436</id>
      <parentid>80382</parentid>
      <timestamp>2021-11-08T10:08:51Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>proposed notation</comment>
      <origin>80436</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6806" sha1="lhglz92x4212lk1oky1elfetplm1q3k" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a stack of 10 [[12edo]]&lt;nowiki/&gt;s minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Proposed chemical notation ===
118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. 

Interval names can be made using the suffix "-ide", as to how a 2-atom molecule would be named in chemistry. For example, fluoride for limma, chloride for 17 steps, strontide for classical major third. These are appended to the initial note. For example, an interval of 17 steps on the 1st note would be called hydrogen chloride, a limma on the 11th note would be called sodium fluoride. Classical minor 6th on 29th note would be called copper mercuride, etc.

It should be noted that vast majority of these intervals do not make chemical sense, such as apotome on the 9th note, which would be chlorine sodiide. In real life, this is impossible, as flow of electrons is always to chlorine, not from it.

=== Prime harmonics ===
{{primes in edo|118}}
==Table of intervals==
{| class="wikitable"
|+Selected 118 EDO intervals
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma
|fluorine
|[[256/243]]
|-
|11
|apotome
|sodium
|[[2187/2048]]
|-
|20
|whole tone
|calcium
|[[9/8]]
|-
|23
|septimal second
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd
|gallium
|[[6/5]]
|-
|38
|Classical major 3rd
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|59
|symmetric tritone
|praseodymium
|
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>lhglz92x4212lk1oky1elfetplm1q3k</sha1>
    </revision>
    <revision>
      <id>80438</id>
      <parentid>80436</parentid>
      <timestamp>2021-11-08T10:15:34Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Table of intervals */ collapse by default</comment>
      <origin>80438</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6834" sha1="eb5ciami70nc6xmvhpq6af3vh5i57vl" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a stack of 10 [[12edo]]&lt;nowiki/&gt;s minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Proposed chemical notation ===
118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. 

Interval names can be made using the suffix "-ide", as to how a 2-atom molecule would be named in chemistry. For example, fluoride for limma, chloride for 17 steps, strontide for classical major third. These are appended to the initial note. For example, an interval of 17 steps on the 1st note would be called hydrogen chloride, a limma on the 11th note would be called sodium fluoride. Classical minor 6th on 29th note would be called copper mercuride, etc.

It should be noted that vast majority of these intervals do not make chemical sense, such as apotome on the 9th note, which would be chlorine sodiide. In real life, this is impossible, as flow of electrons is always to chlorine, not from it.

=== Prime harmonics ===
{{primes in edo|118}}
==Table of intervals==
{| class="wikitable collapsible mw-collapsed"
|+Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma
|fluorine
|[[256/243]]
|-
|11
|apotome
|sodium
|[[2187/2048]]
|-
|20
|whole tone
|calcium
|[[9/8]]
|-
|23
|septimal second
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd
|gallium
|[[6/5]]
|-
|38
|Classical major 3rd
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|59
|symmetric tritone
|praseodymium
|
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>eb5ciami70nc6xmvhpq6af3vh5i57vl</sha1>
    </revision>
    <revision>
      <id>80439</id>
      <parentid>80438</parentid>
      <timestamp>2021-11-08T10:31:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Fix the table; move chemical notation to a dedicated notation section</comment>
      <origin>80439</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6889" sha1="6t41r7bmd9yy393c2myzvfm1s6eyu46" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a stack of ten [[12edo]]s minus the said comma{{clarify}}.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma
|fluorine
|[[256/243]]
|-
|11
|apotome
|sodium
|[[2187/2048]]
|-
|20
|whole tone
|calcium
|[[9/8]]
|-
|23
|septimal second
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd
|gallium
|[[6/5]]
|-
|38
|Classical major 3rd
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|59
|symmetric tritone
|praseodymium
|
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Proposed chemical notation ===
118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. 

Interval names can be made using the suffix "-ide", as to how a 2-atom molecule would be named in chemistry. For example, fluoride for limma, chloride for 17 steps, strontide for classical major third. These are appended to the initial note. For example, an interval of 17 steps on the 1st note would be called hydrogen chloride, a limma on the 11th note would be called sodium fluoride. Classical minor 6th on 29th note would be called copper mercuride, etc.

It should be noted that vast majority of these intervals do not make chemical sense, such as apotome on the 9th note, which would be chlorine sodiide. In real life, this is impossible, as flow of electrons is always to chlorine, not from it.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>6t41r7bmd9yy393c2myzvfm1s6eyu46</sha1>
    </revision>
    <revision>
      <id>80440</id>
      <parentid>80439</parentid>
      <timestamp>2021-11-08T10:41:54Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Table of intervals */</comment>
      <origin>80440</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7693" sha1="c2bhbb90swo8sk6nzv7p3j0odx7yk7e" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a stack of ten [[12edo]]s minus the said comma{{clarify}}.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|1
|semicomma
|hydrogen
|[[243/242]], many others
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma
|fluorine
|[[256/243]]
|-
|10
|dodecaic semitone
|neon
|[[17/16]]
|-
|11
|apotome
|sodium
|[[2187/2048]]
|-
|18
|diminished tone
|argon
|[[10/9]]
|-
|19
|minor tone
|potassium
|[[19/17]]
|-
|20
|major tone
|calcium
|[[9/8]]
|-
|23
|septimal second, slendro gulu
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd
|gallium
|[[6/5]]
|-
|33
|Lesser tridecimal third
|germanium
|[[39/32]]
|-
|34
|Minor-neutral third
|selenium
|[[11/9]]
|-
|35
|Minor tridecimal neurtral third, "major-neutral" third
|bromine
|[[16/13]], 70/57
|-
|36
|Golden ratio 3rd, major-tridecimal neutral third
|krypton
|[[16/13]], [[26/21]], [[21/17]]
|-
|38
|Classical major 3rd
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|59
|symmetric tritone
|praseodymium
|[[99/70]], [[140/99]]
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|82
|Golden ratio sixth, minor-neutral tridecimal sixth
|lead
|13/8, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|83
|Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
|13/8, 57/35
|-
|84
|Major-neutral sixth
|polonium
|[[18/11]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|109
|Pythagorean major 7th
|meitnerium
|[[243/128]]
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Proposed chemical notation ===
118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. 

Interval names can be made using the suffix "-ide", as to how a 2-atom molecule would be named in chemistry. For example, fluoride for limma, chloride for 17 steps, strontide for classical major third. These are appended to the initial note. For example, an interval of 17 steps on the 1st note would be called hydrogen chloride, a limma on the 11th note would be called sodium fluoride. Classical minor 6th on 29th note would be called copper mercuride, etc.

It should be noted that vast majority of these intervals do not make chemical sense, such as apotome on the 9th note, which would be chlorine sodiide. In real life, this is impossible, as flow of electrons is always to chlorine, not from it.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>c2bhbb90swo8sk6nzv7p3j0odx7yk7e</sha1>
    </revision>
    <revision>
      <id>80527</id>
      <parentid>80440</parentid>
      <timestamp>2021-11-09T08:51:20Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Notation */ motivation as to why it is used</comment>
      <origin>80527</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7684" sha1="qqvczbnthsm00j3ztpsmqsobv4wp93n" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a stack of ten [[12edo]]s minus the said comma{{clarify}}.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|1
|semicomma
|hydrogen
|[[243/242]], many others
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma
|fluorine
|[[256/243]]
|-
|10
|dodecaic semitone
|neon
|[[17/16]]
|-
|11
|apotome
|sodium
|[[2187/2048]]
|-
|18
|diminished tone
|argon
|[[10/9]]
|-
|19
|minor tone
|potassium
|[[19/17]]
|-
|20
|major tone
|calcium
|[[9/8]]
|-
|23
|septimal second, slendro gulu
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd
|gallium
|[[6/5]]
|-
|33
|Lesser tridecimal third
|germanium
|[[39/32]]
|-
|34
|Minor-neutral third
|selenium
|[[11/9]]
|-
|35
|Minor tridecimal neurtral third, "major-neutral" third
|bromine
|[[16/13]], 70/57
|-
|36
|Golden ratio 3rd, major-tridecimal neutral third
|krypton
|[[16/13]], [[26/21]], [[21/17]]
|-
|38
|Classical major 3rd
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|59
|symmetric tritone
|praseodymium
|[[99/70]], [[140/99]]
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|82
|Golden ratio sixth, minor-neutral tridecimal sixth
|lead
|13/8, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|83
|Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
|13/8, 57/35
|-
|84
|Major-neutral sixth
|polonium
|[[18/11]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|109
|Pythagorean major 7th
|meitnerium
|[[243/128]]
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Possible chemical notation ===
118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct and provides a fine one-to-one correspondence between notes and their names, as opposed to extending small scales into large EDOs which create excessive labels. Some may argue that other notations. like [[Ups and Downs Notation|ups and downs]] favor [[12edo]] or [[5L 2s|the diatonic scale]], while the chemical notation system has no such issue. 

However, chemical notation's properties can also be a disadvantage - it requires a certain level of knowledge of chemistry, or at least memorizing the names of the elements of the periodic table. In addition, since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>qqvczbnthsm00j3ztpsmqsobv4wp93n</sha1>
    </revision>
    <revision>
      <id>80528</id>
      <parentid>80527</parentid>
      <timestamp>2021-11-09T08:52:37Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Possible chemical notation */</comment>
      <origin>80528</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7629" sha1="d80r3zp0nkreb4o4g38m6e2bdfjccx0" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a stack of ten [[12edo]]s minus the said comma{{clarify}}.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|1
|semicomma
|hydrogen
|[[243/242]], many others
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma
|fluorine
|[[256/243]]
|-
|10
|dodecaic semitone
|neon
|[[17/16]]
|-
|11
|apotome
|sodium
|[[2187/2048]]
|-
|18
|diminished tone
|argon
|[[10/9]]
|-
|19
|minor tone
|potassium
|[[19/17]]
|-
|20
|major tone
|calcium
|[[9/8]]
|-
|23
|septimal second, slendro gulu
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd
|gallium
|[[6/5]]
|-
|33
|Lesser tridecimal third
|germanium
|[[39/32]]
|-
|34
|Minor-neutral third
|selenium
|[[11/9]]
|-
|35
|Minor tridecimal neurtral third, "major-neutral" third
|bromine
|[[16/13]], 70/57
|-
|36
|Golden ratio 3rd, major-tridecimal neutral third
|krypton
|[[16/13]], [[26/21]], [[21/17]]
|-
|38
|Classical major 3rd
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|59
|symmetric tritone
|praseodymium
|[[99/70]], [[140/99]]
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|82
|Golden ratio sixth, minor-neutral tridecimal sixth
|lead
|13/8, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|83
|Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
|13/8, 57/35
|-
|84
|Major-neutral sixth
|polonium
|[[18/11]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|109
|Pythagorean major 7th
|meitnerium
|[[243/128]]
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Possible chemical notation ===
118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct and provides a fine one-to-one correspondence between notes and their names, as opposed to extending small scales into large EDOs which create excessive labels. Some may argue that other notations. like [[Ups and Downs Notation|ups and downs]] favor [[12edo]] or [[5L 2s|the diatonic scale]], while the chemical notation system has no such issue. 

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>d80r3zp0nkreb4o4g38m6e2bdfjccx0</sha1>
    </revision>
    <revision>
      <id>80540</id>
      <parentid>80528</parentid>
      <timestamp>2021-11-09T16:04:08Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>80540</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7641" sha1="he4zsx6fh1xrgmo6uwomgdid59lrip4" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a stack of ten sets of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|1
|semicomma
|hydrogen
|[[243/242]], many others
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma
|fluorine
|[[256/243]]
|-
|10
|dodecaic semitone
|neon
|[[17/16]]
|-
|11
|apotome
|sodium
|[[2187/2048]]
|-
|18
|diminished tone
|argon
|[[10/9]]
|-
|19
|minor tone
|potassium
|[[19/17]]
|-
|20
|major tone
|calcium
|[[9/8]]
|-
|23
|septimal second, slendro gulu
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd
|gallium
|[[6/5]]
|-
|33
|Lesser tridecimal third
|germanium
|[[39/32]]
|-
|34
|Minor-neutral third
|selenium
|[[11/9]]
|-
|35
|Minor tridecimal neurtral third, "major-neutral" third
|bromine
|[[16/13]], 70/57
|-
|36
|Golden ratio 3rd, major-tridecimal neutral third
|krypton
|[[16/13]], [[26/21]], [[21/17]]
|-
|38
|Classical major 3rd
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|59
|symmetric tritone
|praseodymium
|[[99/70]], [[140/99]]
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|82
|Golden ratio sixth, minor-neutral tridecimal sixth
|lead
|13/8, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|83
|Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
|13/8, 57/35
|-
|84
|Major-neutral sixth
|polonium
|[[18/11]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|109
|Pythagorean major 7th
|meitnerium
|[[243/128]]
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Possible chemical notation ===
118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct and provides a fine one-to-one correspondence between notes and their names, as opposed to extending small scales into large EDOs which create excessive labels. Some may argue that other notations. like [[Ups and Downs Notation|ups and downs]] favor [[12edo]] or [[5L 2s|the diatonic scale]], while the chemical notation system has no such issue. 

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>he4zsx6fh1xrgmo6uwomgdid59lrip4</sha1>
    </revision>
    <revision>
      <id>80541</id>
      <parentid>80540</parentid>
      <timestamp>2021-11-09T17:07:08Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Improve wording (maybe)</comment>
      <origin>80541</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7646" sha1="6xybhpo5oclbgxzmu6bsx3gxo77uv8c" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|1
|semicomma
|hydrogen
|[[243/242]], many others
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma
|fluorine
|[[256/243]]
|-
|10
|dodecaic semitone
|neon
|[[17/16]]
|-
|11
|apotome
|sodium
|[[2187/2048]]
|-
|18
|diminished tone
|argon
|[[10/9]]
|-
|19
|minor tone
|potassium
|[[19/17]]
|-
|20
|major tone
|calcium
|[[9/8]]
|-
|23
|septimal second, slendro gulu
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd
|gallium
|[[6/5]]
|-
|33
|Lesser tridecimal third
|germanium
|[[39/32]]
|-
|34
|Minor-neutral third
|selenium
|[[11/9]]
|-
|35
|Minor tridecimal neurtral third, "major-neutral" third
|bromine
|[[16/13]], 70/57
|-
|36
|Golden ratio 3rd, major-tridecimal neutral third
|krypton
|[[16/13]], [[26/21]], [[21/17]]
|-
|38
|Classical major 3rd
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|59
|symmetric tritone
|praseodymium
|[[99/70]], [[140/99]]
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|82
|Golden ratio sixth, minor-neutral tridecimal sixth
|lead
|13/8, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|83
|Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
|13/8, 57/35
|-
|84
|Major-neutral sixth
|polonium
|[[18/11]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|109
|Pythagorean major 7th
|meitnerium
|[[243/128]]
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Possible chemical notation ===
118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct and provides a fine one-to-one correspondence between notes and their names, as opposed to extending small scales into large EDOs which create excessive labels. Some may argue that other notations. like [[Ups and Downs Notation|ups and downs]] favor [[12edo]] or [[5L 2s|the diatonic scale]], while the chemical notation system has no such issue. 

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>6xybhpo5oclbgxzmu6bsx3gxo77uv8c</sha1>
    </revision>
    <revision>
      <id>80548</id>
      <parentid>80541</parentid>
      <timestamp>2021-11-09T22:57:32Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Possible chemical notation */</comment>
      <origin>80548</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7703" sha1="h38kiryjbven5lg21uuklcmzf3ftrjk" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|1
|semicomma
|hydrogen
|[[243/242]], many others
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma
|fluorine
|[[256/243]]
|-
|10
|dodecaic semitone
|neon
|[[17/16]]
|-
|11
|apotome
|sodium
|[[2187/2048]]
|-
|18
|diminished tone
|argon
|[[10/9]]
|-
|19
|minor tone
|potassium
|[[19/17]]
|-
|20
|major tone
|calcium
|[[9/8]]
|-
|23
|septimal second, slendro gulu
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd
|gallium
|[[6/5]]
|-
|33
|Lesser tridecimal third
|germanium
|[[39/32]]
|-
|34
|Minor-neutral third
|selenium
|[[11/9]]
|-
|35
|Minor tridecimal neurtral third, "major-neutral" third
|bromine
|[[16/13]], 70/57
|-
|36
|Golden ratio 3rd, major-tridecimal neutral third
|krypton
|[[16/13]], [[26/21]], [[21/17]]
|-
|38
|Classical major 3rd
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|59
|symmetric tritone
|praseodymium
|[[99/70]], [[140/99]]
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|82
|Golden ratio sixth, minor-neutral tridecimal sixth
|lead
|13/8, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|83
|Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
|13/8, 57/35
|-
|84
|Major-neutral sixth
|polonium
|[[18/11]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|109
|Pythagorean major 7th
|meitnerium
|[[243/128]]
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct and provides a fine one-to-one correspondence between notes and their names, as opposed to extending small scales into large EDOs which create excessive labels. Some may argue that other notations. like [[Ups and Downs Notation|ups and downs]] favor [[12edo]] or [[5L 2s|the diatonic scale]], while the chemical notation system has no such issue. 

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>h38kiryjbven5lg21uuklcmzf3ftrjk</sha1>
    </revision>
    <revision>
      <id>80770</id>
      <parentid>80548</parentid>
      <timestamp>2021-11-13T20:10:39Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Intervals */ Shruti</comment>
      <origin>80770</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8002" sha1="e39knint5ix1ewj3k00gedqliuijpgm" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently represents the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|1
|semicomma
|hydrogen
|[[243/242]], many others
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma, dayavati
|fluorine
|[[256/243]]
|-
|10
|dodecaic semitone
|neon
|[[17/16]]
|-
|11
|apotome, ranjani
|sodium
|[[16/15]], [[2187/2048]]
|-
|18
|diminished tone, ratika
|argon
|[[10/9]]
|-
|19
|minor tone
|potassium
|[[19/17]]
|-
|20
|major tone, raudri
|calcium
|[[9/8]]
|-
|23
|septimal second, slendro gulu
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd, krodha
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd, vajrika
|gallium
|[[6/5]]
|-
|33
|Lesser tridecimal third
|germanium
|[[39/32]]
|-
|34
|Minor-neutral third
|selenium
|[[11/9]]
|-
|35
|Minor tridecimal neurtral third, "major-neutral" third
|bromine
|[[16/13]], 70/57
|-
|36
|Golden ratio 3rd, major-tridecimal neutral third
|krypton
|[[16/13]], [[26/21]], [[21/17]]
|-
|38
|Classical major 3rd, prasarini
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|51
|Kshiti
|antimony
|[[27/20]]
|-
|58
|Rakta
|cerium
|[[45/32]]
|-
|59
|symmetric tritone
|praseodymium
|[[99/70]], [[140/99]]
|-
|60
|Literal tritone, sandipani
|neodymium
|[[729/512]]
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|82
|Golden ratio sixth, minor-neutral tridecimal sixth
|lead
|13/8, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|83
|Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
|13/8, 57/35
|-
|84
|Major-neutral sixth
|polonium
|[[18/11]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|100
|Tivra
|fermium
|[[9/5]]
|-
|109
|Pythagorean major 7th
|meitnerium
|[[243/128]]
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct and provides a fine one-to-one correspondence between notes and their names, as opposed to extending small scales into large EDOs which create excessive labels. Some may argue that other notations. like [[Ups and Downs Notation|ups and downs]] favor [[12edo]] or [[5L 2s|the diatonic scale]], while the chemical notation system has no such issue. 

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>e39knint5ix1ewj3k00gedqliuijpgm</sha1>
    </revision>
    <revision>
      <id>80771</id>
      <parentid>80770</parentid>
      <timestamp>2021-11-13T20:10:54Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>80771</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8004" sha1="jl9to5rzznoiu0tvpm0l168s0npmozn" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|1
|semicomma
|hydrogen
|[[243/242]], many others
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma, dayavati
|fluorine
|[[256/243]]
|-
|10
|dodecaic semitone
|neon
|[[17/16]]
|-
|11
|apotome, ranjani
|sodium
|[[16/15]], [[2187/2048]]
|-
|18
|diminished tone, ratika
|argon
|[[10/9]]
|-
|19
|minor tone
|potassium
|[[19/17]]
|-
|20
|major tone, raudri
|calcium
|[[9/8]]
|-
|23
|septimal second, slendro gulu
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd, krodha
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd, vajrika
|gallium
|[[6/5]]
|-
|33
|Lesser tridecimal third
|germanium
|[[39/32]]
|-
|34
|Minor-neutral third
|selenium
|[[11/9]]
|-
|35
|Minor tridecimal neurtral third, "major-neutral" third
|bromine
|[[16/13]], 70/57
|-
|36
|Golden ratio 3rd, major-tridecimal neutral third
|krypton
|[[16/13]], [[26/21]], [[21/17]]
|-
|38
|Classical major 3rd, prasarini
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|51
|Kshiti
|antimony
|[[27/20]]
|-
|58
|Rakta
|cerium
|[[45/32]]
|-
|59
|symmetric tritone
|praseodymium
|[[99/70]], [[140/99]]
|-
|60
|Literal tritone, sandipani
|neodymium
|[[729/512]]
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|82
|Golden ratio sixth, minor-neutral tridecimal sixth
|lead
|13/8, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|83
|Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
|13/8, 57/35
|-
|84
|Major-neutral sixth
|polonium
|[[18/11]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|100
|Tivra
|fermium
|[[9/5]]
|-
|109
|Pythagorean major 7th
|meitnerium
|[[243/128]]
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct and provides a fine one-to-one correspondence between notes and their names, as opposed to extending small scales into large EDOs which create excessive labels. Some may argue that other notations. like [[Ups and Downs Notation|ups and downs]] favor [[12edo]] or [[5L 2s|the diatonic scale]], while the chemical notation system has no such issue. 

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>jl9to5rzznoiu0tvpm0l168s0npmozn</sha1>
    </revision>
    <revision>
      <id>81151</id>
      <parentid>80771</parentid>
      <timestamp>2021-11-21T17:31:45Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Possible chemical notation */</comment>
      <origin>81151</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7817" sha1="mfksthgogyey3wqk1fi0v61r2zdbpkm" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|1
|semicomma
|hydrogen
|[[243/242]], many others
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma, dayavati
|fluorine
|[[256/243]]
|-
|10
|dodecaic semitone
|neon
|[[17/16]]
|-
|11
|apotome, ranjani
|sodium
|[[16/15]], [[2187/2048]]
|-
|18
|diminished tone, ratika
|argon
|[[10/9]]
|-
|19
|minor tone
|potassium
|[[19/17]]
|-
|20
|major tone, raudri
|calcium
|[[9/8]]
|-
|23
|septimal second, slendro gulu
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd, krodha
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd, vajrika
|gallium
|[[6/5]]
|-
|33
|Lesser tridecimal third
|germanium
|[[39/32]]
|-
|34
|Minor-neutral third
|selenium
|[[11/9]]
|-
|35
|Minor tridecimal neurtral third, "major-neutral" third
|bromine
|[[16/13]], 70/57
|-
|36
|Golden ratio 3rd, major-tridecimal neutral third
|krypton
|[[16/13]], [[26/21]], [[21/17]]
|-
|38
|Classical major 3rd, prasarini
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|51
|Kshiti
|antimony
|[[27/20]]
|-
|58
|Rakta
|cerium
|[[45/32]]
|-
|59
|symmetric tritone
|praseodymium
|[[99/70]], [[140/99]]
|-
|60
|Literal tritone, sandipani
|neodymium
|[[729/512]]
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|82
|Golden ratio sixth, minor-neutral tridecimal sixth
|lead
|13/8, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|83
|Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
|13/8, 57/35
|-
|84
|Major-neutral sixth
|polonium
|[[18/11]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|100
|Tivra
|fermium
|[[9/5]]
|-
|109
|Pythagorean major 7th
|meitnerium
|[[243/128]]
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>mfksthgogyey3wqk1fi0v61r2zdbpkm</sha1>
    </revision>
    <revision>
      <id>81226</id>
      <parentid>81151</parentid>
      <timestamp>2021-11-24T10:09:16Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>81226</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8106" sha1="69d4loea87reu3m54gacgzvvjutkp81" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|1
|semicomma
|hydrogen
|[[243/242]], many others
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma, dayavati
|fluorine
|[[256/243]]
|-
|10
|dodecaic semitone
|neon
|[[17/16]]
|-
|11
|apotome, ranjani
|sodium
|[[16/15]], [[2187/2048]]
|-
|18
|diminished tone, ratika
|argon
|[[10/9]]
|-
|19
|minor tone
|potassium
|[[19/17]]
|-
|20
|major tone, raudri
|calcium
|[[9/8]]
|-
|23
|septimal second, slendric 2
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd, krodha
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd, vajrika
|gallium
|[[6/5]]
|-
|33
|Lesser tridecimal third
|germanium
|[[39/32]]
|-
|34
|Minor-neutral third
|selenium
|[[11/9]]
|-
|35
|Minor tridecimal neurtral third, "major-neutral" third
|bromine
|[[16/13]], 70/57
|-
|36
|Golden ratio 3rd, major-tridecimal neutral third
|krypton
|[[16/13]], [[26/21]], [[21/17]]
|-
|38
|Classical major 3rd, prasarini
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|45
|Barbados 3rd
|rhodium
|[[13/10]],
|-
|46
|Slendric 3
|palladium
|[[21/16]],
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|51
|Kshiti
|antimony
|[[27/20]]
|-
|58
|Rakta
|cerium
|[[45/32]]
|-
|59
|symmetric tritone
|praseodymium
|[[99/70]], [[140/99]]
|-
|60
|Literal tritone, sandipani
|neodymium
|[[729/512]]
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|82
|Golden ratio sixth, minor-neutral tridecimal sixth
|lead
|13/8 I, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|83
|Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
|13/8 II, 57/35
|-
|84
|Major-neutral sixth
|polonium
|[[18/11]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|92
|Septimal supermajor 6th, slendro 5
|uranium
|[[12/7]]
|-
|95
|Harmonic 7th
|americium
|[[7/4]]
|-
|100
|Tivra
|fermium
|[[9/5]]
|-
|109
|Pythagorean major 7th
|meitnerium
|[[243/128]]
|-
|116
|Comma 7th
|livermorium
|[[160/81]]
|-
|117
|Semicomma supermajor 7th
|tenessine
|multiple
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>69d4loea87reu3m54gacgzvvjutkp81</sha1>
    </revision>
    <revision>
      <id>81470</id>
      <parentid>81226</parentid>
      <timestamp>2021-12-04T15:48:35Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ +remark on 5-limit relative error</comment>
      <origin>81470</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8305" sha1="gbpf2gz52hj9w24595b3qek7vn680cr" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|1
|semicomma
|hydrogen
|[[243/242]], many others
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma, dayavati
|fluorine
|[[256/243]]
|-
|10
|dodecaic semitone
|neon
|[[17/16]]
|-
|11
|apotome, ranjani
|sodium
|[[16/15]], [[2187/2048]]
|-
|18
|diminished tone, ratika
|argon
|[[10/9]]
|-
|19
|minor tone
|potassium
|[[19/17]]
|-
|20
|major tone, raudri
|calcium
|[[9/8]]
|-
|23
|septimal second, slendric 2
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd, krodha
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd, vajrika
|gallium
|[[6/5]]
|-
|33
|Lesser tridecimal third
|germanium
|[[39/32]]
|-
|34
|Minor-neutral third
|selenium
|[[11/9]]
|-
|35
|Minor tridecimal neurtral third, "major-neutral" third
|bromine
|[[16/13]], 70/57
|-
|36
|Golden ratio 3rd, major-tridecimal neutral third
|krypton
|[[16/13]], [[26/21]], [[21/17]]
|-
|38
|Classical major 3rd, prasarini
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|45
|Barbados 3rd
|rhodium
|[[13/10]],
|-
|46
|Slendric 3
|palladium
|[[21/16]],
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|51
|Kshiti
|antimony
|[[27/20]]
|-
|58
|Rakta
|cerium
|[[45/32]]
|-
|59
|symmetric tritone
|praseodymium
|[[99/70]], [[140/99]]
|-
|60
|Literal tritone, sandipani
|neodymium
|[[729/512]]
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|82
|Golden ratio sixth, minor-neutral tridecimal sixth
|lead
|13/8 I, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|83
|Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
|13/8 II, 57/35
|-
|84
|Major-neutral sixth
|polonium
|[[18/11]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|92
|Septimal supermajor 6th, slendro 5
|uranium
|[[12/7]]
|-
|95
|Harmonic 7th
|americium
|[[7/4]]
|-
|100
|Tivra
|fermium
|[[9/5]]
|-
|109
|Pythagorean major 7th
|meitnerium
|[[243/128]]
|-
|116
|Comma 7th
|livermorium
|[[160/81]]
|-
|117
|Semicomma supermajor 7th
|tenessine
|multiple
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until 171 do we find a better ET in terms of absolute error, and not until 441 do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>gbpf2gz52hj9w24595b3qek7vn680cr</sha1>
    </revision>
    <revision>
      <id>81475</id>
      <parentid>81470</parentid>
      <timestamp>2021-12-04T18:10:03Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ link EDOs mentioned</comment>
      <origin>81475</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8327" sha1="t48jqcq4njrza4uijyd3c3663bkewo2" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
!Step
!Name
!Chemical notation
&lt;small&gt;if base note = 0&lt;/small&gt;
!Associated ratio
|-
|0
|unison
|oganesson / neutronium
|1/1 exact
|-
|1
|semicomma
|hydrogen
|[[243/242]], many others
|-
|2
|comma
|helium
|[[531441/524288]], [[81/80]]
|-
|9
|limma, dayavati
|fluorine
|[[256/243]]
|-
|10
|dodecaic semitone
|neon
|[[17/16]]
|-
|11
|apotome, ranjani
|sodium
|[[16/15]], [[2187/2048]]
|-
|18
|diminished tone, ratika
|argon
|[[10/9]]
|-
|19
|minor tone
|potassium
|[[19/17]]
|-
|20
|major tone, raudri
|calcium
|[[9/8]]
|-
|23
|septimal second, slendric 2
|vanadium
|[[8/7]]
|-
|26
|septimal third
|iron
|[[7/6]]
|-
|29
|Pythagorean minor 3rd, krodha
|copper
|[[32/27]]
|-
|31
|Classical minor 3rd, vajrika
|gallium
|[[6/5]]
|-
|33
|Lesser tridecimal third
|germanium
|[[39/32]]
|-
|34
|Minor-neutral third
|selenium
|[[11/9]]
|-
|35
|Minor tridecimal neurtral third, "major-neutral" third
|bromine
|[[16/13]], 70/57
|-
|36
|Golden ratio 3rd, major-tridecimal neutral third
|krypton
|[[16/13]], [[26/21]], [[21/17]]
|-
|38
|Classical major 3rd, prasarini
|strontium
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|zirconium
|[[81/64]]
|-
|45
|Barbados 3rd
|rhodium
|[[13/10]],
|-
|46
|Slendric 3
|palladium
|[[21/16]],
|-
|49
|perfect 4th
|indium
|[[4/3]]
|-
|51
|Kshiti
|antimony
|[[27/20]]
|-
|58
|Rakta
|cerium
|[[45/32]]
|-
|59
|symmetric tritone
|praseodymium
|[[99/70]], [[140/99]]
|-
|60
|Literal tritone, sandipani
|neodymium
|[[729/512]]
|-
|69
|perfect 5th
|thulium
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|platinum
|[[128/81]]
|-
|80
|Classical minor 6th
|mercury
|[[8/5]]
|-
|82
|Golden ratio sixth, minor-neutral tridecimal sixth
|lead
|13/8 I, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|83
|Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
|13/8 II, 57/35
|-
|84
|Major-neutral sixth
|polonium
|[[18/11]]
|-
|87
|Classical major 6th
|francium
|[[5/3]]
|-
|89
|Pythagorean major 6th
|actinium
|[[27/16]]
|-
|92
|Septimal supermajor 6th, slendro 5
|uranium
|[[12/7]]
|-
|95
|Harmonic 7th
|americium
|[[7/4]]
|-
|100
|Tivra
|fermium
|[[9/5]]
|-
|109
|Pythagorean major 7th
|meitnerium
|[[243/128]]
|-
|116
|Comma 7th
|livermorium
|[[160/81]]
|-
|117
|Semicomma supermajor 7th
|tenessine
|multiple
|-
|118
|perfect 8ve
|oganesson / neutronium
|2/1 exact
|}

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>t48jqcq4njrza4uijyd3c3663bkewo2</sha1>
    </revision>
    <revision>
      <id>81476</id>
      <parentid>81475</parentid>
      <timestamp>2021-12-04T18:21:29Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ cleanup</comment>
      <origin>81476</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8513" sha1="l7sixmuhbq7t241gsf6135ivkomt8gf" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Eliora's Naming System
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
! Approximate Ratios
|-
| 0
| unison
| oganesson / neutronium
| 1/1 exact
|-
| 1
| semicomma
| hydrogen
| [[243/242]], many others
|-
| 2
| comma
| helium
| [[531441/524288]], [[81/80]]
|-
| 9
| limma, dayavati
| fluorine
| [[256/243]]
|-
| 10
| dodecaic semitone
| neon
| [[17/16]]
|-
| 11
| apotome, ranjani
| sodium
| [[16/15]], [[2187/2048]]
|-
| 18
| diminished tone, ratika
| argon
| [[10/9]]
|-
| 19
| minor tone
| potassium
| [[19/17]]
|-
| 20
| major tone, raudri
| calcium
| [[9/8]]
|-
| 23
| septimal second, slendric 2
| vanadium
| [[8/7]]
|-
| 26
| septimal third
| iron
| [[7/6]]
|-
| 29
| Pythagorean minor 3rd, krodha
| copper
| [[32/27]]
|-
| 31
| Classical minor 3rd, vajrika
| gallium
| [[6/5]]
|-
| 33
| Lesser tridecimal third
| germanium
| [[39/32]]
|-
| 34
| Minor-neutral third
| selenium
| [[11/9]]
|-
| 35
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| [[16/13]], 70/57
|-
| 36
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| [[16/13]], [[26/21]], [[21/17]]
|-
| 38
| Classical major 3rd, prasarini
| strontium
| [[5/4]]
|-
| 40
| Pythagorean major 3rd
| zirconium
| [[81/64]]
|-
| 45
| Barbados 3rd
| rhodium
| [[13/10]],
|-
| 46
| Slendric 3
| palladium
| [[21/16]],
|-
| 49
| perfect 4th
| indium
| [[4/3]]
|-
| 51
| Kshiti
| antimony
| [[27/20]]
|-
| 58
| Rakta
| cerium
| [[45/32]]
|-
| 59
| symmetric tritone
| praseodymium
| [[99/70]], [[140/99]]
|-
| 60
| Literal tritone, sandipani
| neodymium
| [[729/512]]
|-
| 69
| perfect 5th
| thulium
| [[3/2]]
|-
| 78
| Pythagorean minor 6th
| platinum
| [[128/81]]
|-
| 80
| Classical minor 6th
| mercury
| [[8/5]]
|-
| 82
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| 13/8 I, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
| 83
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| 13/8 II, 57/35
|-
| 84
| Major-neutral sixth
| polonium
| [[18/11]]
|-
| 87
| Classical major 6th
| francium
| [[5/3]]
|-
| 89
| Pythagorean major 6th
| actinium
| [[27/16]]
|-
| 92
| Septimal supermajor 6th, slendro 5
| uranium
| [[12/7]]
|-
| 95
| Harmonic 7th
| americium
| [[7/4]]
|-
| 100
| Tivra
| fermium
| [[9/5]]
|-
| 109
| Pythagorean major 7th
| meitnerium
| [[243/128]]
|-
| 116
| Comma 7th
| livermorium
| [[160/81]]
|-
| 117
| Semicomma supermajor 7th
| tenessine
| multiple
|-
| 118
| perfect 8ve
| oganesson / neutronium
| 2/1 exact
|}

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>l7sixmuhbq7t241gsf6135ivkomt8gf</sha1>
    </revision>
    <revision>
      <id>81478</id>
      <parentid>81476</parentid>
      <timestamp>2021-12-04T19:17:35Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ complete the table</comment>
      <origin>81478</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10883" sha1="3g81tqtj0k95ntrvkt38i0sx0y99m6v" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Eliora's Naming System
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
! Approximate Ratios
|-
| 0
| unison
| oganesson / neutronium
| [[1/1]]
|-
| 1
| semicomma
| hydrogen
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
|-
| 2
| comma
| helium
| [[81/80]]
|-
| 3
| 
| 
| [[64/63]], [[49/48]]
|-
| 4
| 
| 
| [[50/49]]
|-
| 5
| 
| 
| [[36/35]]
|-
| 6
| 
| 
| [[28/27]]
|-
| 7
| 
| 
| [[25/24]]
|-
| 8
| 
| 
| [[21/20]], [[22/21]]
|-
| 9
| limma, dayavati
| fluorine
| [[19/18]], [[20/19]]
|-
| 10
| dodecaic semitone
| neon
| [[17/16]], [[18/17]]
|-
| 11
| apotome, ranjani
| sodium
| [[16/15]]
|-
| 12
| 
| 
| [[15/14]]
|-
| 13
| 
| 
| [[27/25]]
|-
| 14
| 
| 
| [[88/81]]
|-
| 15
| 
| 
| [[12/11]]
|-
| 16
| 
| 
| [[11/10]]
|-
| 17
| 
| 
| [[21/19]]

|-
| 18
| diminished tone, ratika
| argon
| [[10/9]]
|-
| 19
| minor tone
| potassium
| [[28/25]], [[19/17]]
|-
| 20
| major tone, raudri
| calcium
| [[9/8]]
|-
| 21
| 
| 
| [[17/15]]
|-
| 22
| 
| 
| [[256/225]]
|-
| 23
| septimal second, slendric 2
| vanadium
| [[8/7]]
|-
| 24
| 
| 
| [[144/125]], [[121/105]]
|-
| 25
| 
| 
| [[125/108]], [[81/70]], [[22/19]]
|-
| 26
| septimal third
| iron
| [[7/6]]
|-
| 27
| 
| 
| [[75/64]]
|-
| 28
| 
| 
| [[33/28]]
|-
| 29
| Pythagorean minor 3rd, krodha
| copper
| [[32/27]], [[19/16]]
|-
| 30
| 
| 
| [[25/21]]
|-
| 31
| Classical minor 3rd, vajrika
| gallium
| [[6/5]]
|-
| 32
| 
| 
| [[135/112]]
|-
| 33
| Lesser tridecimal third
| germanium
| [[40/33]], [[17/14]]
|-
| 34
| Minor-neutral third
| selenium
| [[11/9]]
|-
| 35
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| [[27/22]]
|-
| 36
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| [[99/80]], [[21/17]]
|-
| 37
| 
| 
| [[56/45]]
|-
| 38
| Classical major 3rd, prasarini
| strontium
| [[5/4]]
|-
| 39
| 
| 
| [[63/50]]
|-
| 40
| Pythagorean major 3rd
| zirconium
| [[24/19]], [[19/15]]
|-
| 41
| 
| 
| [[14/11]]
|-
| 42
| 
| 
| [[77/60]]
|-
| 43
| 
| 
| [[9/7]]
|-
| 44
| 
| 
| [[35/27]], [[22/17]]
|-
| 45
| Barbados 3rd
| rhodium
| [[98/75]]
|-
| 46
| Slendric 3
| palladium
| [[21/16]]
|-
| 47
| 
| 
| [[320/243]]
|-
| 48
| 
| 
| [[160/121]], [[85/64]]
|-
| 49
| perfect 4th
| indium
| [[4/3]]
|-
| 50
| 
| 
| [[75/56]], [[51/38]]
|-
| 51
| Kshiti
| antimony
| [[27/20]]
|-
| 52
| 
| 
| [[49/36]]
|-
| 53
| 
| 
| [[15/11]]
|-
| 54
| 
| 
| [[48/35]], [[11/8]]
|-
| 55
| 
| 
| [[112/81]]
|-
| 56
| 
| 
| [[25/18]]
|-
| 57
| 
| 
| [[7/5]]
|-
| 58
| Rakta
| cerium
| [[45/32]]
|-
| 59
| symmetric tritone
| praseodymium
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
|-
| 60
| Literal tritone, sandipani
| neodymium
| [[64/45]]
|-
| 61
| 
| 
| [[10/7]]
|-
| 62
| 
| 
| [[36/25]]
|-
| 63
| 
| 
| [[81/56]]
|-
| 64
| 
| 
| [[35/24]], [[16/11]]
|-
| 65
| 
| 
| [[22/15]]
|-
| 66
| 
| 
| [[72/49]]
|-
| 67
| 
| 
| [[40/27]]
|-
| 68
| 
| 
| [[112/75]], [[76/51]]
|-
| 69
| perfect 5th
| thulium
| [[3/2]]
|-
| 70
| 
| 
| [[121/80]], [[128/85]]
|-
| 71
| 
| 
| [[243/160]]
|-
| 72
| 
| 
| [[32/21]]
|-
| 73
| 
| 
| [[75/49]]
|-
| 74
| 
| 
| [[54/35]], [[17/11]]
|-
| 75
| 
| 
| [[14/9]]
|-
| 76
| 
| 
| [[120/77]]
|-
| 77
| 
| 
| [[11/7]]
|-
| 78
| Pythagorean minor 6th
| platinum
| [[19/12]], [[30/19]]
|-
| 79
| 
| 
| [[100/63]]
|-
| 80
| Classical minor 6th
| mercury
| [[8/5]]
|-
| 81
| 
| 
| [[45/28]]
|-
| 82
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| [[160/99]], [[34/21]]
|-
| 83
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| [[44/27]]
|-
| 84
| Major-neutral sixth
| polonium
| [[18/11]]
|-
| 85
| 
| 
| [[28/17]]
|-
| 86
| 
| 
| [[224/135]]
|-
| 87
| Classical major 6th
| francium
| [[5/3]]
|-
| 88
| 
| 
| [[42/25]]
|-
| 89
| Pythagorean major 6th
| actinium
| [[27/16]], [[32/19]]
|-
| 90
| 
| 
| [[56/33]]
|-
| 91
| 
| 
| [[128/75]]
|-
| 92
| Septimal supermajor 6th, slendro 5
| uranium
| [[12/7]]
|-
| 93
| 
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
|-
| 94
| 
| 
| [[125/72]]
|-
| 95
| Harmonic 7th
| americium
| [[7/4]]
|-
| 96
| 
| 
| [[225/128]]
|-
| 97
| 
| 
| [[30/17]]
|-
| 98
| 
| 
| [[16/9]]
|-
| 99
| 
| 
| [[25/14]]
|-
| 100
| Tivra
| fermium
| [[9/5]]
|-
| 101
| 
| 
| [[38/21]]
|-
| 102
| 
| 
| [[20/11]]
|-
| 103
| 
| 
| [[11/6]]
|-
| 104
| 
| 
| [[81/44]]
|-
| 105
| 
| 
| [[50/27]]
|-
| 106
| 
| 
| [[28/15]]
|-
| 107
| 
| 
| [[15/8]]
|-
| 108
| 
| 
| [[32/17]], [[17/9]]
|-
| 109
| Pythagorean major 7th
| meitnerium
| [[36/19]], [[19/10]]
|-
| 110
| 
| 
| [[40/21]], [[21/11]]
|-
| 111
| 
| 
| [[48/25]]
|-
| 112
| 
| 
| [[27/14]]
|-
| 113
| 
| 
| [[35/18]], [[64/33]]
|-
| 114
| 
| 
| [[49/25]]
|-
| 115
| 
| 
| [[63/32]], [[96/49]]
|-
| 116
| Comma 7th
| livermorium
| [[160/81]]
|-
| 117
| Semicomma supermajor 7th
| tenessine
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
|-
| 118
| perfect 8ve
| oganesson / neutronium
| [[2/1]]
|}

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>3g81tqtj0k95ntrvkt38i0sx0y99m6v</sha1>
    </revision>
    <revision>
      <id>81479</id>
      <parentid>81478</parentid>
      <timestamp>2021-12-04T19:45:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ +cents and diatonic marks</comment>
      <origin>81479</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12447" sha1="lzfr3dt06ppktytyr9kuxze05utcy4u" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Eliora's Naming System
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
! Approximate Ratios *
|-
| 0
| 0.00
| P1
| unison
| oganesson / neutronium
| [[1/1]]
|-
| 1
| 10.17
| 
| semicomma
| hydrogen
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
|-
| 2
| 20.34
| 
| comma
| helium
| [[81/80]]
|-
| 3
| 30.51
| 
| 
| 
| [[64/63]], [[49/48]]
|-
| 4
| 40.68
| 
| 
| 
| [[50/49]]
|-
| 5
| 50.85
| 
| 
| 
| [[36/35]]
|-
| 6
| 61.02
| 
| 
| 
| [[28/27]]
|-
| 7
| 71.19
| 
| 
| 
| [[25/24]]
|-
| 8
| 81.36
| 
| 
| 
| [[21/20]], [[22/21]]
|-
| 9
| 91.53
| m2
| limma, dayavati
| fluorine
| [[19/18]], [[20/19]]
|-
| 10
| 101.69
| 
| dodecaic semitone
| neon
| [[17/16]], [[18/17]]
|-
| 11
| 111.86
| 
| apotome, ranjani
| sodium
| [[16/15]]
|-
| 12
| 122.03
| 
| 
| 
| [[15/14]]
|-
| 13
| 132.20
| 
| 
| 
| [[27/25]]
|-
| 14
| 142.37
| 
| 
| 
| [[88/81]]
|-
| 15
| 152.54
| 
| 
| 
| [[12/11]]
|-
| 16
| 162.71
| 
| 
| 
| [[11/10]]
|-
| 17
| 172.88
| 
| 
| 
| [[21/19]]
|-
| 18
| 183.05
| 
| diminished tone, ratika
| argon
| [[10/9]]
|-
| 19
| 193.22
| 
| minor tone
| potassium
| [[28/25]], [[19/17]]
|-
| 20
| 203.39
| M2
| major tone, raudri
| calcium
| [[9/8]]
|-
| 21
| 213.56
| 
| 
| 
| [[17/15]]
|-
| 22
| 223.73
| 
| 
| 
| [[256/225]]
|-
| 23
| 233.90
| 
| septimal second, slendric 2
| vanadium
| [[8/7]]
|-
| 24
| 244.07
| 
| 
| 
| [[144/125]], [[121/105]]
|-
| 25
| 254.24
| 
| 
| 
| [[125/108]], [[81/70]], [[22/19]]
|-
| 26
| 260.41
| 
| septimal third
| iron
| [[7/6]]
|-
| 27
| 274.58
| 
| 
| 
| [[75/64]]
|-
| 28
| 284.75
| 
| 
| 
| [[33/28]]
|-
| 29
| 294.92
| m3
| Pythagorean minor 3rd, krodha
| copper
| [[32/27]], [[19/16]]
|-
| 30
| 305.08
| 
| 
| 
| [[25/21]]
|-
| 31
| 315.25
| 
| Classical minor 3rd, vajrika
| gallium
| [[6/5]]
|-
| 32
| 325.42
| 
| 
| 
| [[135/112]]
|-
| 33
| 335.59
| 
| Lesser tridecimal third
| germanium
| [[40/33]], [[17/14]]
|-
| 34
| 345.76
| 
| Minor-neutral third
| selenium
| [[11/9]]
|-
| 35
| 355.93
| 
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| [[27/22]]
|-
| 36
| 366.10
| 
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| [[99/80]], [[21/17]]
|-
| 37
| 376.27
| 
| 
| 
| [[56/45]]
|-
| 38
| 386.44
| 
| Classical major 3rd, prasarini
| strontium
| [[5/4]]
|-
| 39
| 396.61
| 
| 
| 
| [[63/50]]
|-
| 40
| 406.78
| M3
| Pythagorean major 3rd
| zirconium
| [[24/19]], [[19/15]]
|-
| 41
| 416.95
| 
| 
| 
| [[14/11]]
|-
| 42
| 427.12
| 
| 
| 
| [[77/60]]
|-
| 43
| 437.29
| 
| 
| 
| [[9/7]]
|-
| 44
| 447.46
| 
| 
| 
| [[35/27]], [[22/17]]
|-
| 45
| 457.63
| 
| Barbados 3rd
| rhodium
| [[98/75]]
|-
| 46
| 467.80
| 
| Slendric 3
| palladium
| [[21/16]]
|-
| 47
| 477.97
| 
| 
| 
| [[320/243]]
|-
| 48
| 488.14
| 
| 
| 
| [[160/121]], [[85/64]]
|-
| 49
| 498.31
| P4
| perfect 4th
| indium
| [[4/3]]
|-
| 50
| 508.47
| 
| 
| 
| [[75/56]], [[51/38]]
|-
| 51
| 518.64
| 
| Kshiti
| antimony
| [[27/20]]
|-
| 52
| 528.81
| 
| 
| 
| [[49/36]]
|-
| 53
| 538.98
| 
| 
| 
| [[15/11]]
|-
| 54
| 549.15
| 
| 
| 
| [[48/35]], [[11/8]]
|-
| 55
| 559.32
| 
| 
| 
| [[112/81]]
|-
| 56
| 569.49
| 
| 
| 
| [[25/18]]
|-
| 57
| 579.66
| 
| 
| 
| [[7/5]]
|-
| 58
| 589.83
| d5
| Rakta
| cerium
| [[45/32]]
|-
| 59
| 600.00
| 
| symmetric tritone
| praseodymium
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
|-
| 60
| 610.17
| A4
| Literal tritone, sandipani
| neodymium
| [[64/45]]
|-
| 61
| 620.34
| 
| 
| 
| [[10/7]]
|-
| 62
| 630.51
| 
| 
| 
| [[36/25]]
|-
| 63
| 640.68
| 
| 
| 
| [[81/56]]
|-
| 64
| 650.85
| 
| 
| 
| [[35/24]], [[16/11]]
|-
| 65
| 661.02
| 
| 
| 
| [[22/15]]
|-
| 66
| 671.19
| 
| 
| 
| [[72/49]]
|-
| 67
| 681.36
| 
| 
| 
| [[40/27]]
|-
| 68
| 691.53
| 
| 
| 
| [[112/75]], [[76/51]]
|-
| 69
| 701.69
| P5
| perfect 5th
| thulium
| [[3/2]]
|-
| 70
| 711.86
| 
| 
| 
| [[121/80]], [[128/85]]
|-
| 71
| 722.03
| 
| 
| 
| [[243/160]]
|-
| 72
| 732.20
| 
| 
| 
| [[32/21]]
|-
| 73
| 742.37
| 
| 
| 
| [[75/49]]
|-
| 74
| 752.54
| 
| 
| 
| [[54/35]], [[17/11]]
|-
| 75
| 762.71
| 
| 
| 
| [[14/9]]
|-
| 76
| 772.88
| 
| 
| 
| [[120/77]]
|-
| 77
| 783.05
| 
| 
| 
| [[11/7]]
|-
| 78
| 793.22
| m6
| Pythagorean minor 6th
| platinum
| [[19/12]], [[30/19]]
|-
| 79
| 803.39
| 
| 
| 
| [[100/63]]
|-
| 80
| 813.56
| 
| Classical minor 6th
| mercury
| [[8/5]]
|-
| 81
| 823.73
| 
| 
| 
| [[45/28]]
|-
| 82
| 833.90
| 
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| [[160/99]], [[34/21]]
|-
| 83
| 844.07
| 
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| [[44/27]]
|-
| 84
| 854.24
| 
| Major-neutral sixth
| polonium
| [[18/11]]
|-
| 85
| 864.41
| 
| 
| 
| [[28/17]]
|-
| 86
| 874.58
| 
| 
| 
| [[224/135]]
|-
| 87
| 884.75
| 
| Classical major 6th
| francium
| [[5/3]]
|-
| 88
| 894.92
| 
| 
| 
| [[42/25]]
|-
| 89
| 905.08
| M6
| Pythagorean major 6th
| actinium
| [[27/16]], [[32/19]]
|-
| 90
| 915.25
| 
| 
| 
| [[56/33]]
|-
| 91
| 925.42
| 
| 
| 
| [[128/75]]
|-
| 92
| 935.59
| 
| Septimal supermajor 6th, slendro 5
| uranium
| [[12/7]]
|-
| 93
| 945.76
| 
| 
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
|-
| 94
| 955.93
| 
| 
| 
| [[125/72]]
|-
| 95
| 966.10
| 
| Harmonic 7th
| americium
| [[7/4]]
|-
| 96
| 976.27
| 
| 
| 
| [[225/128]]
|-
| 97
| 986.44
| 
| 
| 
| [[30/17]]
|-
| 98
| 996.61
| m7
| 
| 
| [[16/9]]
|-
| 99
| 1006.78
| 
| 
| 
| [[25/14]]
|-
| 100
| 1016.95
| 
| Tivra
| fermium
| [[9/5]]
|-
| 101
| 1027.12
| 
| 
| 
| [[38/21]]
|-
| 102
| 1037.29
| 
| 
| 
| [[20/11]]
|-
| 103
| 1047.46
| 
| 
| 
| [[11/6]]
|-
| 104
| 1057.63
| 
| 
| 
| [[81/44]]
|-
| 105
| 1067.80
| 
| 
| 
| [[50/27]]
|-
| 106
| 1077.97
| 
| 
| 
| [[28/15]]
|-
| 107
| 1088.14
| 
| 
| 
| [[15/8]]
|-
| 108
| 1098.31
| 
| 
| 
| [[32/17]], [[17/9]]
|-
| 109
| 1108.47
| M7
| Pythagorean major 7th
| meitnerium
| [[36/19]], [[19/10]]
|-
| 110
| 1118.64
| 
| 
| 
| [[40/21]], [[21/11]]
|-
| 111
| 1128.81
| 
| 
| 
| [[48/25]]
|-
| 112
| 1138.98
| 
| 
| 
| [[27/14]]
|-
| 113
| 1149.15
| 
| 
| 
| [[35/18]], [[64/33]]
|-
| 114
| 1159.32
| 
| 
| 
| [[49/25]]
|-
| 115
| 1169.49
| 
| 
| 
| [[63/32]], [[96/49]]
|-
| 116
| 1179.66
| 
| Comma 7th
| livermorium
| [[160/81]]
|-
| 117
| 1189.83
| 
| Semicomma supermajor 7th
| tenessine
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
|-
| 118
| 1200.00
| P8
| perfect 8ve
| oganesson / neutronium
| [[2/1]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>lzfr3dt06ppktytyr9kuxze05utcy4u</sha1>
    </revision>
    <revision>
      <id>81480</id>
      <parentid>81479</parentid>
      <timestamp>2021-12-04T19:52:52Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ ratios</comment>
      <origin>81480</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12465" sha1="l48kazzp0rksf84ml6vpjziauavrz3m" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Eliora's Naming System
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
! Approximate Ratios *
|-
| 0
| 0.00
| P1
| unison
| oganesson / neutronium
| [[1/1]]
|-
| 1
| 10.17
| 
| semicomma
| hydrogen
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
|-
| 2
| 20.34
| 
| comma
| helium
| [[81/80]]
|-
| 3
| 30.51
| 
| 
| 
| [[64/63]], [[49/48]]
|-
| 4
| 40.68
| 
| 
| 
| [[50/49]]
|-
| 5
| 50.85
| 
| 
| 
| [[36/35]]
|-
| 6
| 61.02
| 
| 
| 
| [[28/27]]
|-
| 7
| 71.19
| 
| 
| 
| [[25/24]]
|-
| 8
| 81.36
| 
| 
| 
| [[21/20]], [[22/21]]
|-
| 9
| 91.53
| m2
| limma, dayavati
| fluorine
| [[19/18]], [[20/19]]
|-
| 10
| 101.69
| 
| dodecaic semitone
| neon
| [[17/16]], [[18/17]]
|-
| 11
| 111.86
| 
| apotome, ranjani
| sodium
| [[16/15]]
|-
| 12
| 122.03
| 
| 
| 
| [[15/14]]
|-
| 13
| 132.20
| 
| 
| 
| [[27/25]]
|-
| 14
| 142.37
| 
| 
| 
| [[88/81]]
|-
| 15
| 152.54
| 
| 
| 
| [[12/11]]
|-
| 16
| 162.71
| 
| 
| 
| [[11/10]]
|-
| 17
| 172.88
| 
| 
| 
| [[21/19]]
|-
| 18
| 183.05
| 
| diminished tone, ratika
| argon
| [[10/9]]
|-
| 19
| 193.22
| 
| minor tone
| potassium
| [[28/25]], [[19/17]]
|-
| 20
| 203.39
| M2
| major tone, raudri
| calcium
| [[9/8]]
|-
| 21
| 213.56
| 
| 
| 
| [[17/15]]
|-
| 22
| 223.73
| 
| 
| 
| [[256/225]]
|-
| 23
| 233.90
| 
| septimal second, slendric 2
| vanadium
| [[8/7]]
|-
| 24
| 244.07
| 
| 
| 
| [[144/125]], [[121/105]]
|-
| 25
| 254.24
| 
| 
| 
| [[125/108]], [[81/70]], [[22/19]]
|-
| 26
| 260.41
| 
| septimal third
| iron
| [[7/6]]
|-
| 27
| 274.58
| 
| 
| 
| [[75/64]]
|-
| 28
| 284.75
| 
| 
| 
| [[33/28]]
|-
| 29
| 294.92
| m3
| Pythagorean minor 3rd, krodha
| copper
| [[32/27]], [[19/16]]
|-
| 30
| 305.08
| 
| 
| 
| [[25/21]]
|-
| 31
| 315.25
| 
| Classical minor 3rd, vajrika
| gallium
| [[6/5]]
|-
| 32
| 325.42
| 
| 
| 
| [[98/81]]
|-
| 33
| 335.59
| 
| Lesser tridecimal third
| germanium
| [[40/33]], [[17/14]]
|-
| 34
| 345.76
| 
| Minor-neutral third
| selenium
| [[11/9]]
|-
| 35
| 355.93
| 
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| [[27/22]]
|-
| 36
| 366.10
| 
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| [[99/80]], [[21/17]]
|-
| 37
| 376.27
| 
| 
| 
| [[56/45]]
|-
| 38
| 386.44
| 
| Classical major 3rd, prasarini
| strontium
| [[5/4]]
|-
| 39
| 396.61
| 
| 
| 
| [[63/50]]
|-
| 40
| 406.78
| M3
| Pythagorean major 3rd
| zirconium
| [[24/19]], [[19/15]]
|-
| 41
| 416.95
| 
| 
| 
| [[14/11]]
|-
| 42
| 427.12
| 
| 
| 
| [[77/60]]
|-
| 43
| 437.29
| 
| 
| 
| [[9/7]]
|-
| 44
| 447.46
| 
| 
| 
| [[35/27]], [[22/17]]
|-
| 45
| 457.63
| 
| Barbados 3rd
| rhodium
| [[98/75]]
|-
| 46
| 467.80
| 
| Slendric 3
| palladium
| [[21/16]]
|-
| 47
| 477.97
| 
| 
| 
| [[320/243]]
|-
| 48
| 488.14
| 
| 
| 
| [[160/121]], [[85/64]]
|-
| 49
| 498.31
| P4
| perfect 4th
| indium
| [[4/3]]
|-
| 50
| 508.47
| 
| 
| 
| [[75/56]], [[51/38]]
|-
| 51
| 518.64
| 
| Kshiti
| antimony
| [[27/20]]
|-
| 52
| 528.81
| 
| 
| 
| [[49/36]], [[19/14]]
|-
| 53
| 538.98
| 
| 
| 
| [[15/11]]
|-
| 54
| 549.15
| 
| 
| 
| [[48/35]], [[11/8]]
|-
| 55
| 559.32
| 
| 
| 
| [[112/81]]
|-
| 56
| 569.49
| 
| 
| 
| [[25/18]]
|-
| 57
| 579.66
| 
| 
| 
| [[7/5]]
|-
| 58
| 589.83
| d5
| Rakta
| cerium
| [[45/32]]
|-
| 59
| 600.00
| 
| symmetric tritone
| praseodymium
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
|-
| 60
| 610.17
| A4
| Literal tritone, sandipani
| neodymium
| [[64/45]]
|-
| 61
| 620.34
| 
| 
| 
| [[10/7]]
|-
| 62
| 630.51
| 
| 
| 
| [[36/25]]
|-
| 63
| 640.68
| 
| 
| 
| [[81/56]]
|-
| 64
| 650.85
| 
| 
| 
| [[35/24]], [[16/11]]
|-
| 65
| 661.02
| 
| 
| 
| [[22/15]]
|-
| 66
| 671.19
| 
| 
| 
| [[72/49]], [[28/19]]
|-
| 67
| 681.36
| 
| 
| 
| [[40/27]]
|-
| 68
| 691.53
| 
| 
| 
| [[112/75]], [[76/51]]
|-
| 69
| 701.69
| P5
| perfect 5th
| thulium
| [[3/2]]
|-
| 70
| 711.86
| 
| 
| 
| [[121/80]], [[128/85]]
|-
| 71
| 722.03
| 
| 
| 
| [[243/160]]
|-
| 72
| 732.20
| 
| 
| 
| [[32/21]]
|-
| 73
| 742.37
| 
| 
| 
| [[75/49]]
|-
| 74
| 752.54
| 
| 
| 
| [[54/35]], [[17/11]]
|-
| 75
| 762.71
| 
| 
| 
| [[14/9]]
|-
| 76
| 772.88
| 
| 
| 
| [[120/77]]
|-
| 77
| 783.05
| 
| 
| 
| [[11/7]]
|-
| 78
| 793.22
| m6
| Pythagorean minor 6th
| platinum
| [[19/12]], [[30/19]]
|-
| 79
| 803.39
| 
| 
| 
| [[100/63]]
|-
| 80
| 813.56
| 
| Classical minor 6th
| mercury
| [[8/5]]
|-
| 81
| 823.73
| 
| 
| 
| [[45/28]]
|-
| 82
| 833.90
| 
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| [[160/99]], [[34/21]]
|-
| 83
| 844.07
| 
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| [[44/27]]
|-
| 84
| 854.24
| 
| Major-neutral sixth
| polonium
| [[18/11]]
|-
| 85
| 864.41
| 
| 
| 
| [[28/17]]
|-
| 86
| 874.58
| 
| 
| 
| [[81/49]]
|-
| 87
| 884.75
| 
| Classical major 6th
| francium
| [[5/3]]
|-
| 88
| 894.92
| 
| 
| 
| [[42/25]]
|-
| 89
| 905.08
| M6
| Pythagorean major 6th
| actinium
| [[27/16]], [[32/19]]
|-
| 90
| 915.25
| 
| 
| 
| [[56/33]]
|-
| 91
| 925.42
| 
| 
| 
| [[128/75]]
|-
| 92
| 935.59
| 
| Septimal supermajor 6th, slendro 5
| uranium
| [[12/7]]
|-
| 93
| 945.76
| 
| 
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
|-
| 94
| 955.93
| 
| 
| 
| [[125/72]]
|-
| 95
| 966.10
| 
| Harmonic 7th
| americium
| [[7/4]]
|-
| 96
| 976.27
| 
| 
| 
| [[225/128]]
|-
| 97
| 986.44
| 
| 
| 
| [[30/17]]
|-
| 98
| 996.61
| m7
| 
| 
| [[16/9]]
|-
| 99
| 1006.78
| 
| 
| 
| [[25/14]]
|-
| 100
| 1016.95
| 
| Tivra
| fermium
| [[9/5]]
|-
| 101
| 1027.12
| 
| 
| 
| [[38/21]]
|-
| 102
| 1037.29
| 
| 
| 
| [[20/11]]
|-
| 103
| 1047.46
| 
| 
| 
| [[11/6]]
|-
| 104
| 1057.63
| 
| 
| 
| [[81/44]]
|-
| 105
| 1067.80
| 
| 
| 
| [[50/27]]
|-
| 106
| 1077.97
| 
| 
| 
| [[28/15]]
|-
| 107
| 1088.14
| 
| 
| 
| [[15/8]]
|-
| 108
| 1098.31
| 
| 
| 
| [[32/17]], [[17/9]]
|-
| 109
| 1108.47
| M7
| Pythagorean major 7th
| meitnerium
| [[36/19]], [[19/10]]
|-
| 110
| 1118.64
| 
| 
| 
| [[40/21]], [[21/11]]
|-
| 111
| 1128.81
| 
| 
| 
| [[48/25]]
|-
| 112
| 1138.98
| 
| 
| 
| [[27/14]]
|-
| 113
| 1149.15
| 
| 
| 
| [[35/18]], [[64/33]]
|-
| 114
| 1159.32
| 
| 
| 
| [[49/25]]
|-
| 115
| 1169.49
| 
| 
| 
| [[63/32]], [[96/49]]
|-
| 116
| 1179.66
| 
| Comma 7th
| livermorium
| [[160/81]]
|-
| 117
| 1189.83
| 
| Semicomma supermajor 7th
| tenessine
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
|-
| 118
| 1200.00
| P8
| perfect 8ve
| oganesson / neutronium
| [[2/1]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>l48kazzp0rksf84ml6vpjziauavrz3m</sha1>
    </revision>
    <revision>
      <id>83191</id>
      <parentid>81480</parentid>
      <timestamp>2022-01-03T18:41:11Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Intervals */ I didn't invent shruti, attributing naming system to me makes it sound like I did.</comment>
      <origin>83191</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12659" sha1="fombjiszv7bbwq8iymsle8xwwsud8vb" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! FJS
Name
! Eliora's Naming System 
&lt;small&gt;(+Shruti correspondence)&lt;/small&gt;
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
! Approximate Ratios *
|-
| 0
| 0.00
| P1
| unison
| oganesson / neutronium
| [[1/1]]
|-
| 1
| 10.17
| 
| semicomma
| hydrogen
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
|-
| 2
| 20.34
| 
| comma
| helium
| [[81/80]], [[531441/524288]]
|-
| 3
| 30.51
| 
| 
| 
| [[64/63]], [[49/48]]
|-
| 4
| 40.68
| 
| 
| 
| [[50/49]]
|-
| 5
| 50.85
| 
| 
| 
| [[36/35]]
|-
| 6
| 61.02
| 
| 
| 
| [[28/27]]
|-
| 7
| 71.19
| 
| 
| 
| [[25/24]]
|-
| 8
| 81.36
| 
| 
| 
| [[21/20]], [[22/21]]
|-
| 9
| 91.53
| m2
| limma, dayavati
| fluorine
| [[19/18]], [[20/19]]
|-
| 10
| 101.69
| 
| dodecaic semitone
| neon
| [[17/16]], [[18/17]]
|-
| 11
| 111.86
| 
| apotome, ranjani
| sodium
| [[16/15]]
|-
| 12
| 122.03
| 
| 
| 
| [[15/14]]
|-
| 13
| 132.20
| 
| 
| 
| [[27/25]]
|-
| 14
| 142.37
| 
| 
| 
| [[88/81]]
|-
| 15
| 152.54
| 
| 
| 
| [[12/11]]
|-
| 16
| 162.71
| 
| 
| 
| [[11/10]]
|-
| 17
| 172.88
| 
| 
| 
| [[21/19]]
|-
| 18
| 183.05
| 
| diminished tone, ratika
| argon
| [[10/9]]
|-
| 19
| 193.22
| 
| minor tone
| potassium
| [[28/25]], [[19/17]]
|-
| 20
| 203.39
| M2
| major tone, raudri
| calcium
| [[9/8]]
|-
| 21
| 213.56
| 
| 
| 
| [[17/15]]
|-
| 22
| 223.73
| 
| 
| 
| [[256/225]]
|-
| 23
| 233.90
| 
| septimal second, slendric 2
| vanadium
| [[8/7]]
|-
| 24
| 244.07
| 
| 
| 
| [[144/125]], [[121/105]]
|-
| 25
| 254.24
| 
| 
| 
| [[125/108]], [[81/70]], [[22/19]]
|-
| 26
| 260.41
| 
| septimal third
| iron
| [[7/6]]
|-
| 27
| 274.58
| 
| 
| 
| [[75/64]]
|-
| 28
| 284.75
| 
| 
| 
| [[33/28]]
|-
| 29
| 294.92
| m3
| Pythagorean minor 3rd, krodha
| copper
| [[32/27]], [[19/16]]
|-
| 30
| 305.08
| 
| 
| 
| [[25/21]]
|-
| 31
| 315.25
| 
| Classical minor 3rd, vajrika
| gallium
| [[6/5]]
|-
| 32
| 325.42
| 
| 
| 
| [[98/81]]
|-
| 33
| 335.59
| 
| Lesser tridecimal third
| germanium
| [[40/33]], [[17/14]]
|-
| 34
| 345.76
| 
| Minor-neutral third
| selenium
| [[11/9]]
|-
| 35
| 355.93
| 
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| [[27/22]], 16/13 I**
|-
| 36
| 366.10
| 
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| [[99/80]], [[21/17]], 16/13 II**
|-
| 37
| 376.27
| 
| 
| 
| [[56/45]]
|-
| 38
| 386.44
| 
| Classical major 3rd, prasarini
| strontium
| [[5/4]]
|-
| 39
| 396.61
| 
| 
| 
| [[63/50]]
|-
| 40
| 406.78
| M3
| Pythagorean major 3rd
| zirconium
| [[24/19]], [[19/15]]
|-
| 41
| 416.95
| 
| 
| 
| [[14/11]]
|-
| 42
| 427.12
| 
| 
| 
| [[77/60]]
|-
| 43
| 437.29
| 
| 
| 
| [[9/7]]
|-
| 44
| 447.46
| 
| 
| 
| [[35/27]], [[22/17]]
|-
| 45
| 457.63
| 
| Barbados 3rd
| rhodium
| [[98/75]]
|-
| 46
| 467.80
| 
| Slendric 3
| palladium
| [[21/16]]
|-
| 47
| 477.97
| 
| 
| 
| [[320/243]]
|-
| 48
| 488.14
| 
| 
| 
| [[160/121]], [[85/64]]
|-
| 49
| 498.31
| P4
| perfect 4th
| indium
| [[4/3]]
|-
| 50
| 508.47
| 
| 
| 
| [[75/56]], [[51/38]]
|-
| 51
| 518.64
| 
| Kshiti
| antimony
| [[27/20]]
|-
| 52
| 528.81
| 
| 
| 
| [[49/36]], [[19/14]]
|-
| 53
| 538.98
| 
| 
| 
| [[15/11]]
|-
| 54
| 549.15
| 
| 
| 
| [[48/35]], [[11/8]]
|-
| 55
| 559.32
| 
| 
| 
| [[112/81]]
|-
| 56
| 569.49
| 
| 
| 
| [[25/18]]
|-
| 57
| 579.66
| 
| 
| 
| [[7/5]]
|-
| 58
| 589.83
| d5
| Rakta
| cerium
| [[45/32]]
|-
| 59
| 600.00
| 
| symmetric tritone
| praseodymium
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
|-
| 60
| 610.17
| A4
| Literal tritone, sandipani
| neodymium
| [[64/45]]
|-
| 61
| 620.34
| 
| 
| 
| [[10/7]]
|-
| 62
| 630.51
| 
| 
| 
| [[36/25]]
|-
| 63
| 640.68
| 
| 
| 
| [[81/56]]
|-
| 64
| 650.85
| 
| 
| 
| [[35/24]], [[16/11]]
|-
| 65
| 661.02
| 
| 
| 
| [[22/15]]
|-
| 66
| 671.19
| 
| 
| 
| [[72/49]], [[28/19]]
|-
| 67
| 681.36
| 
| 
| 
| [[40/27]]
|-
| 68
| 691.53
| 
| 
| 
| [[112/75]], [[76/51]]
|-
| 69
| 701.69
| P5
| perfect 5th
| thulium
| [[3/2]]
|-
| 70
| 711.86
| 
| 
| 
| [[121/80]], [[128/85]]
|-
| 71
| 722.03
| 
| 
| 
| [[243/160]]
|-
| 72
| 732.20
| 
| 
| 
| [[32/21]]
|-
| 73
| 742.37
| 
| 
| 
| [[75/49]]
|-
| 74
| 752.54
| 
| 
| 
| [[54/35]], [[17/11]]
|-
| 75
| 762.71
| 
| 
| 
| [[14/9]]
|-
| 76
| 772.88
| 
| 
| 
| [[120/77]]
|-
| 77
| 783.05
| 
| 
| 
| [[11/7]]
|-
| 78
| 793.22
| m6
| Pythagorean minor 6th
| platinum
| [[19/12]], [[30/19]]
|-
| 79
| 803.39
| 
| 
| 
| [[100/63]]
|-
| 80
| 813.56
| 
| Classical minor 6th
| mercury
| [[8/5]]
|-
| 81
| 823.73
| 
| 
| 
| [[45/28]]
|-
| 82
| 833.90
| 
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| [[160/99]], [[34/21]], [[13/8]] I**
|-
| 83
| 844.07
| 
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| [[44/27]], 13/8 II**
|-
| 84
| 854.24
| 
| Major-neutral sixth
| polonium
| [[18/11]]
|-
| 85
| 864.41
| 
| 
| 
| [[28/17]]
|-
| 86
| 874.58
| 
| 
| 
| [[81/49]]
|-
| 87
| 884.75
| 
| Classical major 6th
| francium
| [[5/3]]
|-
| 88
| 894.92
| 
| 
| 
| [[42/25]]
|-
| 89
| 905.08
| M6
| Pythagorean major 6th
| actinium
| [[27/16]], [[32/19]]
|-
| 90
| 915.25
| 
| 
| 
| [[56/33]]
|-
| 91
| 925.42
| 
| 
| 
| [[128/75]]
|-
| 92
| 935.59
| 
| Septimal supermajor 6th, slendro 5
| uranium
| [[12/7]]
|-
| 93
| 945.76
| 
| 
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
|-
| 94
| 955.93
| 
| 
| 
| [[125/72]]
|-
| 95
| 966.10
| 
| Harmonic 7th
| americium
| [[7/4]]
|-
| 96
| 976.27
| 
| 
| 
| [[225/128]]
|-
| 97
| 986.44
| 
| 
| 
| [[30/17]]
|-
| 98
| 996.61
| m7
| 
| 
| [[16/9]]
|-
| 99
| 1006.78
| 
| 
| 
| [[25/14]]
|-
| 100
| 1016.95
| 
| Tivra
| fermium
| [[9/5]]
|-
| 101
| 1027.12
| 
| 
| 
| [[38/21]]
|-
| 102
| 1037.29
| 
| 
| 
| [[20/11]]
|-
| 103
| 1047.46
| 
| 
| 
| [[11/6]]
|-
| 104
| 1057.63
| 
| 
| 
| [[81/44]]
|-
| 105
| 1067.80
| 
| 
| 
| [[50/27]]
|-
| 106
| 1077.97
| 
| 
| 
| [[28/15]]
|-
| 107
| 1088.14
| 
| 
| 
| [[15/8]]
|-
| 108
| 1098.31
| 
| 
| 
| [[32/17]], [[17/9]]
|-
| 109
| 1108.47
| M7
| Pythagorean major 7th
| meitnerium
| [[36/19]], [[19/10]]
|-
| 110
| 1118.64
| 
| 
| 
| [[40/21]], [[21/11]]
|-
| 111
| 1128.81
| 
| 
| 
| [[48/25]]
|-
| 112
| 1138.98
| 
| 
| 
| [[27/14]]
|-
| 113
| 1149.15
| 
| 
| 
| [[35/18]], [[64/33]]
|-
| 114
| 1159.32
| 
| 
| 
| [[49/25]]
|-
| 115
| 1169.49
| 
| 
| 
| [[63/32]], [[96/49]]
|-
| 116
| 1179.66
| 
| Comma 7th
| livermorium
| [[160/81]]
|-
| 117
| 1189.83
| 
| Semicomma supermajor 7th
| tenessine
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
|-
| 118
| 1200.00
| P8
| perfect 8ve
| oganesson / neutronium
| [[2/1]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>fombjiszv7bbwq8iymsle8xwwsud8vb</sha1>
    </revision>
    <revision>
      <id>83192</id>
      <parentid>83191</parentid>
      <timestamp>2022-01-03T18:46:16Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Notation */</comment>
      <origin>83192</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13131" sha1="dty7lxywgyye6hzkj71h4ia27g4jqs6" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! FJS
Name
! Eliora's Naming System 
&lt;small&gt;(+Shruti correspondence)&lt;/small&gt;
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
! Approximate Ratios *
|-
| 0
| 0.00
| P1
| unison
| oganesson / neutronium
| [[1/1]]
|-
| 1
| 10.17
| 
| semicomma
| hydrogen
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
|-
| 2
| 20.34
| 
| comma
| helium
| [[81/80]], [[531441/524288]]
|-
| 3
| 30.51
| 
| 
| 
| [[64/63]], [[49/48]]
|-
| 4
| 40.68
| 
| 
| 
| [[50/49]]
|-
| 5
| 50.85
| 
| 
| 
| [[36/35]]
|-
| 6
| 61.02
| 
| 
| 
| [[28/27]]
|-
| 7
| 71.19
| 
| 
| 
| [[25/24]]
|-
| 8
| 81.36
| 
| 
| 
| [[21/20]], [[22/21]]
|-
| 9
| 91.53
| m2
| limma, dayavati
| fluorine
| [[19/18]], [[20/19]]
|-
| 10
| 101.69
| 
| dodecaic semitone
| neon
| [[17/16]], [[18/17]]
|-
| 11
| 111.86
| 
| apotome, ranjani
| sodium
| [[16/15]]
|-
| 12
| 122.03
| 
| 
| 
| [[15/14]]
|-
| 13
| 132.20
| 
| 
| 
| [[27/25]]
|-
| 14
| 142.37
| 
| 
| 
| [[88/81]]
|-
| 15
| 152.54
| 
| 
| 
| [[12/11]]
|-
| 16
| 162.71
| 
| 
| 
| [[11/10]]
|-
| 17
| 172.88
| 
| 
| 
| [[21/19]]
|-
| 18
| 183.05
| 
| diminished tone, ratika
| argon
| [[10/9]]
|-
| 19
| 193.22
| 
| minor tone
| potassium
| [[28/25]], [[19/17]]
|-
| 20
| 203.39
| M2
| major tone, raudri
| calcium
| [[9/8]]
|-
| 21
| 213.56
| 
| 
| 
| [[17/15]]
|-
| 22
| 223.73
| 
| 
| 
| [[256/225]]
|-
| 23
| 233.90
| 
| septimal second, slendric 2
| vanadium
| [[8/7]]
|-
| 24
| 244.07
| 
| 
| 
| [[144/125]], [[121/105]]
|-
| 25
| 254.24
| 
| 
| 
| [[125/108]], [[81/70]], [[22/19]]
|-
| 26
| 260.41
| 
| septimal third
| iron
| [[7/6]]
|-
| 27
| 274.58
| 
| 
| 
| [[75/64]]
|-
| 28
| 284.75
| 
| 
| 
| [[33/28]]
|-
| 29
| 294.92
| m3
| Pythagorean minor 3rd, krodha
| copper
| [[32/27]], [[19/16]]
|-
| 30
| 305.08
| 
| 
| 
| [[25/21]]
|-
| 31
| 315.25
| 
| Classical minor 3rd, vajrika
| gallium
| [[6/5]]
|-
| 32
| 325.42
| 
| 
| 
| [[98/81]]
|-
| 33
| 335.59
| 
| Lesser tridecimal third
| germanium
| [[40/33]], [[17/14]]
|-
| 34
| 345.76
| 
| Minor-neutral third
| selenium
| [[11/9]]
|-
| 35
| 355.93
| 
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| [[27/22]], 16/13 I**
|-
| 36
| 366.10
| 
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| [[99/80]], [[21/17]], 16/13 II**
|-
| 37
| 376.27
| 
| 
| 
| [[56/45]]
|-
| 38
| 386.44
| 
| Classical major 3rd, prasarini
| strontium
| [[5/4]]
|-
| 39
| 396.61
| 
| 
| 
| [[63/50]]
|-
| 40
| 406.78
| M3
| Pythagorean major 3rd
| zirconium
| [[24/19]], [[19/15]]
|-
| 41
| 416.95
| 
| 
| 
| [[14/11]]
|-
| 42
| 427.12
| 
| 
| 
| [[77/60]]
|-
| 43
| 437.29
| 
| 
| 
| [[9/7]]
|-
| 44
| 447.46
| 
| 
| 
| [[35/27]], [[22/17]]
|-
| 45
| 457.63
| 
| Barbados 3rd
| rhodium
| [[98/75]]
|-
| 46
| 467.80
| 
| Slendric 3
| palladium
| [[21/16]]
|-
| 47
| 477.97
| 
| 
| 
| [[320/243]]
|-
| 48
| 488.14
| 
| 
| 
| [[160/121]], [[85/64]]
|-
| 49
| 498.31
| P4
| perfect 4th
| indium
| [[4/3]]
|-
| 50
| 508.47
| 
| 
| 
| [[75/56]], [[51/38]]
|-
| 51
| 518.64
| 
| Kshiti
| antimony
| [[27/20]]
|-
| 52
| 528.81
| 
| 
| 
| [[49/36]], [[19/14]]
|-
| 53
| 538.98
| 
| 
| 
| [[15/11]]
|-
| 54
| 549.15
| 
| 
| 
| [[48/35]], [[11/8]]
|-
| 55
| 559.32
| 
| 
| 
| [[112/81]]
|-
| 56
| 569.49
| 
| 
| 
| [[25/18]]
|-
| 57
| 579.66
| 
| 
| 
| [[7/5]]
|-
| 58
| 589.83
| d5
| Rakta
| cerium
| [[45/32]]
|-
| 59
| 600.00
| 
| symmetric tritone
| praseodymium
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
|-
| 60
| 610.17
| A4
| Literal tritone, sandipani
| neodymium
| [[64/45]]
|-
| 61
| 620.34
| 
| 
| 
| [[10/7]]
|-
| 62
| 630.51
| 
| 
| 
| [[36/25]]
|-
| 63
| 640.68
| 
| 
| 
| [[81/56]]
|-
| 64
| 650.85
| 
| 
| 
| [[35/24]], [[16/11]]
|-
| 65
| 661.02
| 
| 
| 
| [[22/15]]
|-
| 66
| 671.19
| 
| 
| 
| [[72/49]], [[28/19]]
|-
| 67
| 681.36
| 
| 
| 
| [[40/27]]
|-
| 68
| 691.53
| 
| 
| 
| [[112/75]], [[76/51]]
|-
| 69
| 701.69
| P5
| perfect 5th
| thulium
| [[3/2]]
|-
| 70
| 711.86
| 
| 
| 
| [[121/80]], [[128/85]]
|-
| 71
| 722.03
| 
| 
| 
| [[243/160]]
|-
| 72
| 732.20
| 
| 
| 
| [[32/21]]
|-
| 73
| 742.37
| 
| 
| 
| [[75/49]]
|-
| 74
| 752.54
| 
| 
| 
| [[54/35]], [[17/11]]
|-
| 75
| 762.71
| 
| 
| 
| [[14/9]]
|-
| 76
| 772.88
| 
| 
| 
| [[120/77]]
|-
| 77
| 783.05
| 
| 
| 
| [[11/7]]
|-
| 78
| 793.22
| m6
| Pythagorean minor 6th
| platinum
| [[19/12]], [[30/19]]
|-
| 79
| 803.39
| 
| 
| 
| [[100/63]]
|-
| 80
| 813.56
| 
| Classical minor 6th
| mercury
| [[8/5]]
|-
| 81
| 823.73
| 
| 
| 
| [[45/28]]
|-
| 82
| 833.90
| 
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| [[160/99]], [[34/21]], [[13/8]] I**
|-
| 83
| 844.07
| 
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| [[44/27]], 13/8 II**
|-
| 84
| 854.24
| 
| Major-neutral sixth
| polonium
| [[18/11]]
|-
| 85
| 864.41
| 
| 
| 
| [[28/17]]
|-
| 86
| 874.58
| 
| 
| 
| [[81/49]]
|-
| 87
| 884.75
| 
| Classical major 6th
| francium
| [[5/3]]
|-
| 88
| 894.92
| 
| 
| 
| [[42/25]]
|-
| 89
| 905.08
| M6
| Pythagorean major 6th
| actinium
| [[27/16]], [[32/19]]
|-
| 90
| 915.25
| 
| 
| 
| [[56/33]]
|-
| 91
| 925.42
| 
| 
| 
| [[128/75]]
|-
| 92
| 935.59
| 
| Septimal supermajor 6th, slendro 5
| uranium
| [[12/7]]
|-
| 93
| 945.76
| 
| 
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
|-
| 94
| 955.93
| 
| 
| 
| [[125/72]]
|-
| 95
| 966.10
| 
| Harmonic 7th
| americium
| [[7/4]]
|-
| 96
| 976.27
| 
| 
| 
| [[225/128]]
|-
| 97
| 986.44
| 
| 
| 
| [[30/17]]
|-
| 98
| 996.61
| m7
| 
| 
| [[16/9]]
|-
| 99
| 1006.78
| 
| 
| 
| [[25/14]]
|-
| 100
| 1016.95
| 
| Tivra
| fermium
| [[9/5]]
|-
| 101
| 1027.12
| 
| 
| 
| [[38/21]]
|-
| 102
| 1037.29
| 
| 
| 
| [[20/11]]
|-
| 103
| 1047.46
| 
| 
| 
| [[11/6]]
|-
| 104
| 1057.63
| 
| 
| 
| [[81/44]]
|-
| 105
| 1067.80
| 
| 
| 
| [[50/27]]
|-
| 106
| 1077.97
| 
| 
| 
| [[28/15]]
|-
| 107
| 1088.14
| 
| 
| 
| [[15/8]]
|-
| 108
| 1098.31
| 
| 
| 
| [[32/17]], [[17/9]]
|-
| 109
| 1108.47
| M7
| Pythagorean major 7th
| meitnerium
| [[36/19]], [[19/10]]
|-
| 110
| 1118.64
| 
| 
| 
| [[40/21]], [[21/11]]
|-
| 111
| 1128.81
| 
| 
| 
| [[48/25]]
|-
| 112
| 1138.98
| 
| 
| 
| [[27/14]]
|-
| 113
| 1149.15
| 
| 
| 
| [[35/18]], [[64/33]]
|-
| 114
| 1159.32
| 
| 
| 
| [[49/25]]
|-
| 115
| 1169.49
| 
| 
| 
| [[63/32]], [[96/49]]
|-
| 116
| 1179.66
| 
| Comma 7th
| livermorium
| [[160/81]]
|-
| 117
| 1189.83
| 
| Semicomma supermajor 7th
| tenessine
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
|-
| 118
| 1200.00
| P8
| perfect 8ve
| oganesson / neutronium
| [[2/1]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

This being said, there are a few correspondences - 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. In addition, 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively. Mercury, ending the 6d-block, corresponds to 8/5. The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (magnesium), the 2s metal.  

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>dty7lxywgyye6hzkj71h4ia27g4jqs6</sha1>
    </revision>
    <revision>
      <id>83194</id>
      <parentid>83192</parentid>
      <timestamp>2022-01-03T18:55:42Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>83194</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13583" sha1="lu9cqjnqbtlwojbz4gdsa73amz7l2fk" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! FJS
Name
! Eliora's Naming System 
&lt;small&gt;(+Shruti correspondence)&lt;/small&gt;
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
! Approximate Ratios *
|-
| 0
| 0.00
| P1
| unison
| oganesson / neutronium
| [[1/1]]
|-
| 1
| 10.17
| 
| semicomma
| hydrogen
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
|-
| 2
| 20.34
| 
| comma
| helium
| [[81/80]], [[531441/524288]]
|-
| 3
| 30.51
| 
| 
| lithium
| [[64/63]], [[49/48]]
|-
| 4
| 40.68
| 
| 
| beryllium
| [[50/49]]
|-
| 5
| 50.85
| 
| 
| boron
| [[36/35]]
|-
| 6
| 61.02
| 
| 
| carbon
| [[28/27]]
|-
| 7
| 71.19
| 
| 
| nitrogen
| [[25/24]]
|-
| 8
| 81.36
| 
| 
| oxygen
| [[21/20]], [[22/21]]
|-
| 9
| 91.53
| m2
| limma, dayavati
| fluorine
| [[19/18]], [[20/19]]
|-
| 10
| 101.69
| 
| dodecaic semitone
| neon
| [[17/16]], [[18/17]]
|-
| 11
| 111.86
| 
| apotome, ranjani
| sodium
| [[16/15]]
|-
| 12
| 122.03
| 
| 
| magnesium
| [[15/14]]
|-
| 13
| 132.20
| 
| 
| aluminium
| [[27/25]]
|-
| 14
| 142.37
| 
| 
| silicon
| [[88/81]]
|-
| 15
| 152.54
| 
| 
| phosphorus
| [[12/11]]
|-
| 16
| 162.71
| 
| 
| sulphur
| [[11/10]]
|-
| 17
| 172.88
| 
| 
| chlorine
| [[21/19]]
|-
| 18
| 183.05
| 
| diminished tone, ratika
| argon
| [[10/9]]
|-
| 19
| 193.22
| 
| minor tone
| potassium
| [[28/25]], [[19/17]]
|-
| 20
| 203.39
| M2
| major tone, raudri
| calcium
| [[9/8]]
|-
| 21
| 213.56
| 
| 
| scandium
| [[17/15]]
|-
| 22
| 223.73
| 
| 
| titanium
| [[256/225]]
|-
| 23
| 233.90
| 
| septimal second, slendric 2
| vanadium
| [[8/7]]
|-
| 24
| 244.07
| 
| 
| chromium
| [[144/125]], [[121/105]]
|-
| 25
| 254.24
| 
| 
| manganese
| [[125/108]], [[81/70]], [[22/19]]
|-
| 26
| 260.41
| 
| septimal third
| iron
| [[7/6]]
|-
| 27
| 274.58
| 
| 
| cobalt
| [[75/64]]
|-
| 28
| 284.75
| 
| 
| nickel
| [[33/28]]
|-
| 29
| 294.92
| m3
| Pythagorean minor 3rd, krodha
| copper
| [[32/27]], [[19/16]]
|-
| 30
| 305.08
| 
| 
| zinc
| [[25/21]]
|-
| 31
| 315.25
| 
| Classical minor 3rd, vajrika
| gallium
| [[6/5]]
|-
| 32
| 325.42
| 
| 
| germanium
| [[98/81]]
|-
| 33
| 335.59
| 
| Lesser tridecimal third
| arsenic
| [[40/33]], [[17/14]]
|-
| 34
| 345.76
| 
| Minor-neutral third
| selenium
| [[11/9]]
|-
| 35
| 355.93
| 
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| [[27/22]], 16/13 I**
|-
| 36
| 366.10
| 
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| [[99/80]], [[21/17]], 16/13 II**
|-
| 37
| 376.27
| 
| 
| rubidium
| [[56/45]]
|-
| 38
| 386.44
| 
| Classical major 3rd, prasarini
| strontium
| [[5/4]]
|-
| 39
| 396.61
| 
| 
| yttrium
| [[63/50]]
|-
| 40
| 406.78
| M3
| Pythagorean major 3rd
| zirconium
| [[24/19]], [[19/15]]
|-
| 41
| 416.95
| 
| 
| niobium
| [[14/11]]
|-
| 42
| 427.12
| 
| 
| molybdenum
| [[77/60]]
|-
| 43
| 437.29
| 
| 
| technetium
| [[9/7]]
|-
| 44
| 447.46
| 
| 
| ruthenium
| [[35/27]], [[22/17]]
|-
| 45
| 457.63
| 
| Barbados 3rd
| rhodium
| [[98/75]]
|-
| 46
| 467.80
| 
| Slendric 3
| palladium
| [[21/16]]
|-
| 47
| 477.97
| 
| 
| silver
| [[320/243]]
|-
| 48
| 488.14
| 
| 
| cadmium
| [[160/121]], [[85/64]]
|-
| 49
| 498.31
| P4
| perfect 4th
| indium
| [[4/3]]
|-
| 50
| 508.47
| 
| 
| tin
| [[75/56]], [[51/38]]
|-
| 51
| 518.64
| 
| Kshiti
| antimony
| [[27/20]]
|-
| 52
| 528.81
| 
| 
| tellurium
| [[49/36]], [[19/14]]
|-
| 53
| 538.98
| 
| 
| iodine
| [[15/11]]
|-
| 54
| 549.15
| 
| 
| xenon
| [[48/35]], [[11/8]]
|-
| 55
| 559.32
| 
| 
| caesium
| [[112/81]]
|-
| 56
| 569.49
| 
| 
| barium
| [[25/18]]
|-
| 57
| 579.66
| 
| 
| lanthanum
| [[7/5]]
|-
| 58
| 589.83
| d5
| Rakta
| cerium
| [[45/32]]
|-
| 59
| 600.00
| 
| symmetric tritone
| praseodymium
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
|-
| 60
| 610.17
| A4
| Literal tritone, sandipani
| neodymium
| [[64/45]]
|-
| 61
| 620.34
| 
| 
| 
| [[10/7]]
|-
| 62
| 630.51
| 
| 
| 
| [[36/25]]
|-
| 63
| 640.68
| 
| 
| 
| [[81/56]]
|-
| 64
| 650.85
| 
| 
| 
| [[35/24]], [[16/11]]
|-
| 65
| 661.02
| 
| 
| 
| [[22/15]]
|-
| 66
| 671.19
| 
| 
| 
| [[72/49]], [[28/19]]
|-
| 67
| 681.36
| 
| wolf 5th
| holmium
| [[40/27]]
|-
| 68
| 691.53
| 
| wolf cub 5th
| erbium
| [[112/75]], [[76/51]]
|-
| 69
| 701.69
| P5
| perfect 5th
| thulium
| [[3/2]]
|-
| 70
| 711.86
| 
| 
| ytterbium
| [[121/80]], [[128/85]]
|-
| 71
| 722.03
| 
| 
| lutetium
| [[243/160]]
|-
| 72
| 732.20
| 
| 
| hafnium
| [[32/21]]
|-
| 73
| 742.37
| 
| 
| tantalum
| [[75/49]]
|-
| 74
| 752.54
| 
| 
| tungsten
| [[54/35]], [[17/11]]
|-
| 75
| 762.71
| 
| 
| rhenium
| [[14/9]]
|-
| 76
| 772.88
| 
| 
| osmium
| [[120/77]]
|-
| 77
| 783.05
| 
| 
| iridium
| [[11/7]]
|-
| 78
| 793.22
| m6
| Pythagorean minor 6th
| platinum
| [[19/12]], [[30/19]]
|-
| 79
| 803.39
| 
| 
| gold
| [[100/63]]
|-
| 80
| 813.56
| 
| Classical minor 6th
| mercury
| [[8/5]]
|-
| 81
| 823.73
| 
| 
| thallium
| [[45/28]]
|-
| 82
| 833.90
| 
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| [[160/99]], [[34/21]], [[13/8]] I**
|-
| 83
| 844.07
| 
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| [[44/27]], 13/8 II**
|-
| 84
| 854.24
| 
| Major-neutral sixth
| polonium
| [[18/11]]
|-
| 85
| 864.41
| 
| 
| 
| [[28/17]]
|-
| 86
| 874.58
| 
| 
| 
| [[81/49]]
|-
| 87
| 884.75
| 
| Classical major 6th
| francium
| [[5/3]]
|-
| 88
| 894.92
| 
| 
| 
| [[42/25]]
|-
| 89
| 905.08
| M6
| Pythagorean major 6th
| actinium
| [[27/16]], [[32/19]]
|-
| 90
| 915.25
| 
| 
| 
| [[56/33]]
|-
| 91
| 925.42
| 
| 
| 
| [[128/75]]
|-
| 92
| 935.59
| 
| Septimal supermajor 6th, slendro 5
| uranium
| [[12/7]]
|-
| 93
| 945.76
| 
| 
| neptunium
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
|-
| 94
| 955.93
| 
| 
| plutonium
| [[125/72]]
|-
| 95
| 966.10
| 
| Harmonic 7th
| americium
| [[7/4]]
|-
| 96
| 976.27
| 
| 
| curium
| [[225/128]]
|-
| 97
| 986.44
| 
| 
| berkelium
| [[30/17]]
|-
| 98
| 996.61
| m7
| 
| 
| [[16/9]]
|-
| 99
| 1006.78
| 
| 
| 
| [[25/14]]
|-
| 100
| 1016.95
| 
| Tivra
| fermium
| [[9/5]]
|-
| 101
| 1027.12
| 
| 
| 
| [[38/21]]
|-
| 102
| 1037.29
| 
| 
| 
| [[20/11]]
|-
| 103
| 1047.46
| 
| 
| 
| [[11/6]]
|-
| 104
| 1057.63
| 
| 
| 
| [[81/44]]
|-
| 105
| 1067.80
| 
| 
| 
| [[50/27]]
|-
| 106
| 1077.97
| 
| 
| 
| [[28/15]]
|-
| 107
| 1088.14
| 
| 
| 
| [[15/8]]
|-
| 108
| 1098.31
| 
| 
| 
| [[32/17]], [[17/9]]
|-
| 109
| 1108.47
| M7
| Pythagorean major 7th
| meitnerium
| [[36/19]], [[19/10]]
|-
| 110
| 1118.64
| 
| 
| darmstadtium
| [[40/21]], [[21/11]]
|-
| 111
| 1128.81
| 
| 
| roentgenium
| [[48/25]]
|-
| 112
| 1138.98
| 
| 
| copernicium
| [[27/14]]
|-
| 113
| 1149.15
| 
| 
| nihonium
| [[35/18]], [[64/33]]
|-
| 114
| 1159.32
| 
| 
| flerovium
| [[49/25]]
|-
| 115
| 1169.49
| 
| 
| moscovium
| [[63/32]], [[96/49]]
|-
| 116
| 1179.66
| 
| Comma 7th
| livermorium
| [[160/81]]
|-
| 117
| 1189.83
| 
| Semicomma supermajor 7th
| tenessine
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
|-
| 118
| 1200.00
| P8
| perfect 8ve
| oganesson / neutronium
| [[2/1]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

This being said, there are a few correspondences - 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. In addition, 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively. Mercury, ending the 6d-block, corresponds to 8/5. The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>lu9cqjnqbtlwojbz4gdsa73amz7l2fk</sha1>
    </revision>
    <revision>
      <id>83195</id>
      <parentid>83194</parentid>
      <timestamp>2022-01-03T18:59:17Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>83195</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13586" sha1="0gtgex6litq10yll5ge7rfeqzrwgkgk" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! FJS
Name
! Eliora's Naming System 
&lt;small&gt;(+Shruti 22 correspondence)&lt;/small&gt;
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
! Approximate Ratios *
|-
| 0
| 0.00
| P1
| unison
| oganesson / neutronium
| [[1/1]]
|-
| 1
| 10.17
| 
| semicomma
| hydrogen
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
|-
| 2
| 20.34
| 
| comma
| helium
| [[81/80]], [[531441/524288]]
|-
| 3
| 30.51
| 
| 
| lithium
| [[64/63]], [[49/48]]
|-
| 4
| 40.68
| 
| 
| beryllium
| [[50/49]]
|-
| 5
| 50.85
| 
| 
| boron
| [[36/35]]
|-
| 6
| 61.02
| 
| 
| carbon
| [[28/27]]
|-
| 7
| 71.19
| 
| 
| nitrogen
| [[25/24]]
|-
| 8
| 81.36
| 
| 
| oxygen
| [[21/20]], [[22/21]]
|-
| 9
| 91.53
| m2
| limma, dayavati
| fluorine
| [[19/18]], [[20/19]]
|-
| 10
| 101.69
| 
| dodecaic semitone
| neon
| [[17/16]], [[18/17]]
|-
| 11
| 111.86
| 
| apotome, ranjani
| sodium
| [[16/15]]
|-
| 12
| 122.03
| 
| 
| magnesium
| [[15/14]]
|-
| 13
| 132.20
| 
| 
| aluminium
| [[27/25]]
|-
| 14
| 142.37
| 
| 
| silicon
| [[88/81]]
|-
| 15
| 152.54
| 
| 
| phosphorus
| [[12/11]]
|-
| 16
| 162.71
| 
| 
| sulphur
| [[11/10]]
|-
| 17
| 172.88
| 
| 
| chlorine
| [[21/19]]
|-
| 18
| 183.05
| 
| diminished tone, ratika
| argon
| [[10/9]]
|-
| 19
| 193.22
| 
| minor tone
| potassium
| [[28/25]], [[19/17]]
|-
| 20
| 203.39
| M2
| major tone, raudri
| calcium
| [[9/8]]
|-
| 21
| 213.56
| 
| 
| scandium
| [[17/15]]
|-
| 22
| 223.73
| 
| 
| titanium
| [[256/225]]
|-
| 23
| 233.90
| 
| septimal second, slendric 2
| vanadium
| [[8/7]]
|-
| 24
| 244.07
| 
| 
| chromium
| [[144/125]], [[121/105]]
|-
| 25
| 254.24
| 
| 
| manganese
| [[125/108]], [[81/70]], [[22/19]]
|-
| 26
| 260.41
| 
| septimal third
| iron
| [[7/6]]
|-
| 27
| 274.58
| 
| 
| cobalt
| [[75/64]]
|-
| 28
| 284.75
| 
| 
| nickel
| [[33/28]]
|-
| 29
| 294.92
| m3
| Pythagorean minor 3rd, krodha
| copper
| [[32/27]], [[19/16]]
|-
| 30
| 305.08
| 
| 
| zinc
| [[25/21]]
|-
| 31
| 315.25
| 
| Classical minor 3rd, vajrika
| gallium
| [[6/5]]
|-
| 32
| 325.42
| 
| 
| germanium
| [[98/81]]
|-
| 33
| 335.59
| 
| Lesser tridecimal third
| arsenic
| [[40/33]], [[17/14]]
|-
| 34
| 345.76
| 
| Minor-neutral third
| selenium
| [[11/9]]
|-
| 35
| 355.93
| 
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| [[27/22]], 16/13 I**
|-
| 36
| 366.10
| 
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| [[99/80]], [[21/17]], 16/13 II**
|-
| 37
| 376.27
| 
| 
| rubidium
| [[56/45]]
|-
| 38
| 386.44
| 
| Classical major 3rd, prasarini
| strontium
| [[5/4]]
|-
| 39
| 396.61
| 
| 
| yttrium
| [[63/50]]
|-
| 40
| 406.78
| M3
| Pythagorean major 3rd
| zirconium
| [[24/19]], [[19/15]]
|-
| 41
| 416.95
| 
| 
| niobium
| [[14/11]]
|-
| 42
| 427.12
| 
| 
| molybdenum
| [[77/60]]
|-
| 43
| 437.29
| 
| 
| technetium
| [[9/7]]
|-
| 44
| 447.46
| 
| 
| ruthenium
| [[35/27]], [[22/17]]
|-
| 45
| 457.63
| 
| Barbados 3rd
| rhodium
| [[98/75]]
|-
| 46
| 467.80
| 
| Slendric 3
| palladium
| [[21/16]]
|-
| 47
| 477.97
| 
| 
| silver
| [[320/243]]
|-
| 48
| 488.14
| 
| 
| cadmium
| [[160/121]], [[85/64]]
|-
| 49
| 498.31
| P4
| perfect 4th
| indium
| [[4/3]]
|-
| 50
| 508.47
| 
| 
| tin
| [[75/56]], [[51/38]]
|-
| 51
| 518.64
| 
| Kshiti
| antimony
| [[27/20]]
|-
| 52
| 528.81
| 
| 
| tellurium
| [[49/36]], [[19/14]]
|-
| 53
| 538.98
| 
| 
| iodine
| [[15/11]]
|-
| 54
| 549.15
| 
| 
| xenon
| [[48/35]], [[11/8]]
|-
| 55
| 559.32
| 
| 
| caesium
| [[112/81]]
|-
| 56
| 569.49
| 
| 
| barium
| [[25/18]]
|-
| 57
| 579.66
| 
| 
| lanthanum
| [[7/5]]
|-
| 58
| 589.83
| d5
| Rakta
| cerium
| [[45/32]]
|-
| 59
| 600.00
| 
| symmetric tritone
| praseodymium
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
|-
| 60
| 610.17
| A4
| Literal tritone, sandipani
| neodymium
| [[64/45]]
|-
| 61
| 620.34
| 
| 
| 
| [[10/7]]
|-
| 62
| 630.51
| 
| 
| 
| [[36/25]]
|-
| 63
| 640.68
| 
| 
| 
| [[81/56]]
|-
| 64
| 650.85
| 
| 
| 
| [[35/24]], [[16/11]]
|-
| 65
| 661.02
| 
| 
| 
| [[22/15]]
|-
| 66
| 671.19
| 
| 
| 
| [[72/49]], [[28/19]]
|-
| 67
| 681.36
| 
| wolf 5th
| holmium
| [[40/27]]
|-
| 68
| 691.53
| 
| wolf cub 5th
| erbium
| [[112/75]], [[76/51]]
|-
| 69
| 701.69
| P5
| perfect 5th
| thulium
| [[3/2]]
|-
| 70
| 711.86
| 
| 
| ytterbium
| [[121/80]], [[128/85]]
|-
| 71
| 722.03
| 
| 
| lutetium
| [[243/160]]
|-
| 72
| 732.20
| 
| 
| hafnium
| [[32/21]]
|-
| 73
| 742.37
| 
| 
| tantalum
| [[75/49]]
|-
| 74
| 752.54
| 
| 
| tungsten
| [[54/35]], [[17/11]]
|-
| 75
| 762.71
| 
| 
| rhenium
| [[14/9]]
|-
| 76
| 772.88
| 
| 
| osmium
| [[120/77]]
|-
| 77
| 783.05
| 
| 
| iridium
| [[11/7]]
|-
| 78
| 793.22
| m6
| Pythagorean minor 6th
| platinum
| [[19/12]], [[30/19]]
|-
| 79
| 803.39
| 
| 
| gold
| [[100/63]]
|-
| 80
| 813.56
| 
| Classical minor 6th
| mercury
| [[8/5]]
|-
| 81
| 823.73
| 
| 
| thallium
| [[45/28]]
|-
| 82
| 833.90
| 
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| [[160/99]], [[34/21]], [[13/8]] I**
|-
| 83
| 844.07
| 
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| [[44/27]], 13/8 II**
|-
| 84
| 854.24
| 
| Major-neutral sixth
| polonium
| [[18/11]]
|-
| 85
| 864.41
| 
| 
| 
| [[28/17]]
|-
| 86
| 874.58
| 
| 
| 
| [[81/49]]
|-
| 87
| 884.75
| 
| Classical major 6th
| francium
| [[5/3]]
|-
| 88
| 894.92
| 
| 
| 
| [[42/25]]
|-
| 89
| 905.08
| M6
| Pythagorean major 6th
| actinium
| [[27/16]], [[32/19]]
|-
| 90
| 915.25
| 
| 
| 
| [[56/33]]
|-
| 91
| 925.42
| 
| 
| 
| [[128/75]]
|-
| 92
| 935.59
| 
| Septimal supermajor 6th, slendro 5
| uranium
| [[12/7]]
|-
| 93
| 945.76
| 
| 
| neptunium
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
|-
| 94
| 955.93
| 
| 
| plutonium
| [[125/72]]
|-
| 95
| 966.10
| 
| Harmonic 7th
| americium
| [[7/4]]
|-
| 96
| 976.27
| 
| 
| curium
| [[225/128]]
|-
| 97
| 986.44
| 
| 
| berkelium
| [[30/17]]
|-
| 98
| 996.61
| m7
| 
| 
| [[16/9]]
|-
| 99
| 1006.78
| 
| 
| 
| [[25/14]]
|-
| 100
| 1016.95
| 
| Tivra
| fermium
| [[9/5]]
|-
| 101
| 1027.12
| 
| 
| 
| [[38/21]]
|-
| 102
| 1037.29
| 
| 
| 
| [[20/11]]
|-
| 103
| 1047.46
| 
| 
| 
| [[11/6]]
|-
| 104
| 1057.63
| 
| 
| 
| [[81/44]]
|-
| 105
| 1067.80
| 
| 
| 
| [[50/27]]
|-
| 106
| 1077.97
| 
| 
| 
| [[28/15]]
|-
| 107
| 1088.14
| 
| 
| 
| [[15/8]]
|-
| 108
| 1098.31
| 
| 
| 
| [[32/17]], [[17/9]]
|-
| 109
| 1108.47
| M7
| Pythagorean major 7th
| meitnerium
| [[36/19]], [[19/10]]
|-
| 110
| 1118.64
| 
| 
| darmstadtium
| [[40/21]], [[21/11]]
|-
| 111
| 1128.81
| 
| 
| roentgenium
| [[48/25]]
|-
| 112
| 1138.98
| 
| 
| copernicium
| [[27/14]]
|-
| 113
| 1149.15
| 
| 
| nihonium
| [[35/18]], [[64/33]]
|-
| 114
| 1159.32
| 
| 
| flerovium
| [[49/25]]
|-
| 115
| 1169.49
| 
| 
| moscovium
| [[63/32]], [[96/49]]
|-
| 116
| 1179.66
| 
| Comma 7th
| livermorium
| [[160/81]]
|-
| 117
| 1189.83
| 
| Semicomma supermajor 7th
| tenessine
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
|-
| 118
| 1200.00
| P8
| perfect 8ve
| oganesson / neutronium
| [[2/1]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

This being said, there are a few correspondences - 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. In addition, 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively. Mercury, ending the 6d-block, corresponds to 8/5. The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>0gtgex6litq10yll5ge7rfeqzrwgkgk</sha1>
    </revision>
    <revision>
      <id>83207</id>
      <parentid>83195</parentid>
      <timestamp>2022-01-03T21:50:20Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ the "marks" column is just diatonic degrees and doesn't really involve FJS. And plz save from text sizes</comment>
      <origin>83207</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13570" sha1="9wale24wqb1wxm3268pnl98s9v1utry" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
! Approximate Ratios *
|-
| 0
| 0.00
| P1
| unison
| oganesson / neutronium
| [[1/1]]
|-
| 1
| 10.17
| 
| semicomma
| hydrogen
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
|-
| 2
| 20.34
| 
| comma
| helium
| [[81/80]], [[531441/524288]]
|-
| 3
| 30.51
| 
| 
| lithium
| [[64/63]], [[49/48]]
|-
| 4
| 40.68
| 
| 
| beryllium
| [[50/49]]
|-
| 5
| 50.85
| 
| 
| boron
| [[36/35]]
|-
| 6
| 61.02
| 
| 
| carbon
| [[28/27]]
|-
| 7
| 71.19
| 
| 
| nitrogen
| [[25/24]]
|-
| 8
| 81.36
| 
| 
| oxygen
| [[21/20]], [[22/21]]
|-
| 9
| 91.53
| m2
| limma, dayavati
| fluorine
| [[19/18]], [[20/19]]
|-
| 10
| 101.69
| 
| dodecaic semitone
| neon
| [[17/16]], [[18/17]]
|-
| 11
| 111.86
| 
| apotome, ranjani
| sodium
| [[16/15]]
|-
| 12
| 122.03
| 
| 
| magnesium
| [[15/14]]
|-
| 13
| 132.20
| 
| 
| aluminium
| [[27/25]]
|-
| 14
| 142.37
| 
| 
| silicon
| [[88/81]]
|-
| 15
| 152.54
| 
| 
| phosphorus
| [[12/11]]
|-
| 16
| 162.71
| 
| 
| sulphur
| [[11/10]]
|-
| 17
| 172.88
| 
| 
| chlorine
| [[21/19]]
|-
| 18
| 183.05
| 
| diminished tone, ratika
| argon
| [[10/9]]
|-
| 19
| 193.22
| 
| minor tone
| potassium
| [[28/25]], [[19/17]]
|-
| 20
| 203.39
| M2
| major tone, raudri
| calcium
| [[9/8]]
|-
| 21
| 213.56
| 
| 
| scandium
| [[17/15]]
|-
| 22
| 223.73
| 
| 
| titanium
| [[256/225]]
|-
| 23
| 233.90
| 
| septimal second, slendric 2
| vanadium
| [[8/7]]
|-
| 24
| 244.07
| 
| 
| chromium
| [[144/125]], [[121/105]]
|-
| 25
| 254.24
| 
| 
| manganese
| [[125/108]], [[81/70]], [[22/19]]
|-
| 26
| 260.41
| 
| septimal third
| iron
| [[7/6]]
|-
| 27
| 274.58
| 
| 
| cobalt
| [[75/64]]
|-
| 28
| 284.75
| 
| 
| nickel
| [[33/28]]
|-
| 29
| 294.92
| m3
| Pythagorean minor 3rd, krodha
| copper
| [[32/27]], [[19/16]]
|-
| 30
| 305.08
| 
| 
| zinc
| [[25/21]]
|-
| 31
| 315.25
| 
| Classical minor 3rd, vajrika
| gallium
| [[6/5]]
|-
| 32
| 325.42
| 
| 
| germanium
| [[98/81]]
|-
| 33
| 335.59
| 
| Lesser tridecimal third
| arsenic
| [[40/33]], [[17/14]]
|-
| 34
| 345.76
| 
| Minor-neutral third
| selenium
| [[11/9]]
|-
| 35
| 355.93
| 
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| [[27/22]], 16/13 I**
|-
| 36
| 366.10
| 
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| [[99/80]], [[21/17]], 16/13 II**
|-
| 37
| 376.27
| 
| 
| rubidium
| [[56/45]]
|-
| 38
| 386.44
| 
| Classical major 3rd, prasarini
| strontium
| [[5/4]]
|-
| 39
| 396.61
| 
| 
| yttrium
| [[63/50]]
|-
| 40
| 406.78
| M3
| Pythagorean major 3rd
| zirconium
| [[24/19]], [[19/15]]
|-
| 41
| 416.95
| 
| 
| niobium
| [[14/11]]
|-
| 42
| 427.12
| 
| 
| molybdenum
| [[77/60]]
|-
| 43
| 437.29
| 
| 
| technetium
| [[9/7]]
|-
| 44
| 447.46
| 
| 
| ruthenium
| [[35/27]], [[22/17]]
|-
| 45
| 457.63
| 
| Barbados 3rd
| rhodium
| [[98/75]]
|-
| 46
| 467.80
| 
| Slendric 3
| palladium
| [[21/16]]
|-
| 47
| 477.97
| 
| 
| silver
| [[320/243]]
|-
| 48
| 488.14
| 
| 
| cadmium
| [[160/121]], [[85/64]]
|-
| 49
| 498.31
| P4
| perfect 4th
| indium
| [[4/3]]
|-
| 50
| 508.47
| 
| 
| tin
| [[75/56]], [[51/38]]
|-
| 51
| 518.64
| 
| Kshiti
| antimony
| [[27/20]]
|-
| 52
| 528.81
| 
| 
| tellurium
| [[49/36]], [[19/14]]
|-
| 53
| 538.98
| 
| 
| iodine
| [[15/11]]
|-
| 54
| 549.15
| 
| 
| xenon
| [[48/35]], [[11/8]]
|-
| 55
| 559.32
| 
| 
| caesium
| [[112/81]]
|-
| 56
| 569.49
| 
| 
| barium
| [[25/18]]
|-
| 57
| 579.66
| 
| 
| lanthanum
| [[7/5]]
|-
| 58
| 589.83
| d5
| Rakta
| cerium
| [[45/32]]
|-
| 59
| 600.00
| 
| symmetric tritone
| praseodymium
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
|-
| 60
| 610.17
| A4
| Literal tritone, sandipani
| neodymium
| [[64/45]]
|-
| 61
| 620.34
| 
| 
| 
| [[10/7]]
|-
| 62
| 630.51
| 
| 
| 
| [[36/25]]
|-
| 63
| 640.68
| 
| 
| 
| [[81/56]]
|-
| 64
| 650.85
| 
| 
| 
| [[35/24]], [[16/11]]
|-
| 65
| 661.02
| 
| 
| 
| [[22/15]]
|-
| 66
| 671.19
| 
| 
| 
| [[72/49]], [[28/19]]
|-
| 67
| 681.36
| 
| wolf 5th
| holmium
| [[40/27]]
|-
| 68
| 691.53
| 
| wolf cub 5th
| erbium
| [[112/75]], [[76/51]]
|-
| 69
| 701.69
| P5
| perfect 5th
| thulium
| [[3/2]]
|-
| 70
| 711.86
| 
| 
| ytterbium
| [[121/80]], [[128/85]]
|-
| 71
| 722.03
| 
| 
| lutetium
| [[243/160]]
|-
| 72
| 732.20
| 
| 
| hafnium
| [[32/21]]
|-
| 73
| 742.37
| 
| 
| tantalum
| [[75/49]]
|-
| 74
| 752.54
| 
| 
| tungsten
| [[54/35]], [[17/11]]
|-
| 75
| 762.71
| 
| 
| rhenium
| [[14/9]]
|-
| 76
| 772.88
| 
| 
| osmium
| [[120/77]]
|-
| 77
| 783.05
| 
| 
| iridium
| [[11/7]]
|-
| 78
| 793.22
| m6
| Pythagorean minor 6th
| platinum
| [[19/12]], [[30/19]]
|-
| 79
| 803.39
| 
| 
| gold
| [[100/63]]
|-
| 80
| 813.56
| 
| Classical minor 6th
| mercury
| [[8/5]]
|-
| 81
| 823.73
| 
| 
| thallium
| [[45/28]]
|-
| 82
| 833.90
| 
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| [[160/99]], [[34/21]], [[13/8]] I**
|-
| 83
| 844.07
| 
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| [[44/27]], 13/8 II**
|-
| 84
| 854.24
| 
| Major-neutral sixth
| polonium
| [[18/11]]
|-
| 85
| 864.41
| 
| 
| 
| [[28/17]]
|-
| 86
| 874.58
| 
| 
| 
| [[81/49]]
|-
| 87
| 884.75
| 
| Classical major 6th
| francium
| [[5/3]]
|-
| 88
| 894.92
| 
| 
| 
| [[42/25]]
|-
| 89
| 905.08
| M6
| Pythagorean major 6th
| actinium
| [[27/16]], [[32/19]]
|-
| 90
| 915.25
| 
| 
| 
| [[56/33]]
|-
| 91
| 925.42
| 
| 
| 
| [[128/75]]
|-
| 92
| 935.59
| 
| Septimal supermajor 6th, slendro 5
| uranium
| [[12/7]]
|-
| 93
| 945.76
| 
| 
| neptunium
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
|-
| 94
| 955.93
| 
| 
| plutonium
| [[125/72]]
|-
| 95
| 966.10
| 
| Harmonic 7th
| americium
| [[7/4]]
|-
| 96
| 976.27
| 
| 
| curium
| [[225/128]]
|-
| 97
| 986.44
| 
| 
| berkelium
| [[30/17]]
|-
| 98
| 996.61
| m7
| 
| 
| [[16/9]]
|-
| 99
| 1006.78
| 
| 
| 
| [[25/14]]
|-
| 100
| 1016.95
| 
| Tivra
| fermium
| [[9/5]]
|-
| 101
| 1027.12
| 
| 
| 
| [[38/21]]
|-
| 102
| 1037.29
| 
| 
| 
| [[20/11]]
|-
| 103
| 1047.46
| 
| 
| 
| [[11/6]]
|-
| 104
| 1057.63
| 
| 
| 
| [[81/44]]
|-
| 105
| 1067.80
| 
| 
| 
| [[50/27]]
|-
| 106
| 1077.97
| 
| 
| 
| [[28/15]]
|-
| 107
| 1088.14
| 
| 
| 
| [[15/8]]
|-
| 108
| 1098.31
| 
| 
| 
| [[32/17]], [[17/9]]
|-
| 109
| 1108.47
| M7
| Pythagorean major 7th
| meitnerium
| [[36/19]], [[19/10]]
|-
| 110
| 1118.64
| 
| 
| darmstadtium
| [[40/21]], [[21/11]]
|-
| 111
| 1128.81
| 
| 
| roentgenium
| [[48/25]]
|-
| 112
| 1138.98
| 
| 
| copernicium
| [[27/14]]
|-
| 113
| 1149.15
| 
| 
| nihonium
| [[35/18]], [[64/33]]
|-
| 114
| 1159.32
| 
| 
| flerovium
| [[49/25]]
|-
| 115
| 1169.49
| 
| 
| moscovium
| [[63/32]], [[96/49]]
|-
| 116
| 1179.66
| 
| Comma 7th
| livermorium
| [[160/81]]
|-
| 117
| 1189.83
| 
| Semicomma supermajor 7th
| tenessine
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
|-
| 118
| 1200.00
| P8
| perfect 8ve
| oganesson / neutronium
| [[2/1]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

This being said, there are a few correspondences - 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. In addition, 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively. Mercury, ending the 6d-block, corresponds to 8/5. The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>9wale24wqb1wxm3268pnl98s9v1utry</sha1>
    </revision>
    <revision>
      <id>83208</id>
      <parentid>83207</parentid>
      <timestamp>2022-01-03T21:52:36Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ move the ratios column before naming attempts</comment>
      <origin>83208</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13451" sha1="ldec8tt66et1bbgbbtrs70hy1hs9xor" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| 
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| 
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| diminished tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| minor tone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| 
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| 
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| 
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| 
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| 
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| 
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| 
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| 
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| 
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| 
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| 
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| 
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| 
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| 
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| 
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| 
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| 
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| 
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendro 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| 
| 
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| 
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| 
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| 
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| 
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| 
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| 
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| 
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| 
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| 
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

This being said, there are a few correspondences - 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. In addition, 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively. Mercury, ending the 6d-block, corresponds to 8/5. The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>ldec8tt66et1bbgbbtrs70hy1hs9xor</sha1>
    </revision>
    <revision>
      <id>83257</id>
      <parentid>83208</parentid>
      <timestamp>2022-01-04T23:17:57Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>83257</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13891" sha1="52fz8y67rleblt8vj7fvvib76pjlmwp" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

This being said, there are a few correspondences - 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. In addition, 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively. Mercury, ending the 6d-block, corresponds to 8/5. The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>52fz8y67rleblt8vj7fvvib76pjlmwp</sha1>
    </revision>
    <revision>
      <id>86237</id>
      <parentid>83257</parentid>
      <timestamp>2022-02-22T12:42:23Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Possible chemical notation */</comment>
      <origin>86237</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14059" sha1="jz8xv6z2bni27ta9xdik8roklzm5vo9" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo. 

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one step of the maximally even parakleismic scale, created by stacking 6/5.  

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>jz8xv6z2bni27ta9xdik8roklzm5vo9</sha1>
    </revision>
    <revision>
      <id>86238</id>
      <parentid>86237</parentid>
      <timestamp>2022-02-22T12:42:45Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Possible chemical notation */</comment>
      <origin>86238</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13896" sha1="16sk6y0wp6f70q8gylfojr4url0rqf5" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. 

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one step of the maximally even parakleismic scale, created by stacking 6/5.  

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>16sk6y0wp6f70q8gylfojr4url0rqf5</sha1>
    </revision>
    <revision>
      <id>86239</id>
      <parentid>86238</parentid>
      <timestamp>2022-02-22T12:54:13Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Possible chemical notation */</comment>
      <origin>86239</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13900" sha1="5yh4wzmta8e12neyz8grlx333fiybjs" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. 

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>5yh4wzmta8e12neyz8grlx333fiybjs</sha1>
    </revision>
    <revision>
      <id>86274</id>
      <parentid>86239</parentid>
      <timestamp>2022-02-22T22:25:12Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>86274</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13929" sha1="cnh1297a5hsyin50dqp0fipq05xzzev" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each, a size close to [[169/168]]. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. 

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>cnh1297a5hsyin50dqp0fipq05xzzev</sha1>
    </revision>
    <revision>
      <id>87442</id>
      <parentid>86274</parentid>
      <timestamp>2022-02-26T16:48:49Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87442</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13909" sha1="nij17f2p245sfm1wdy53cuvyml6nwlb" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each, a size close to [[169/168]]. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. 

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>nij17f2p245sfm1wdy53cuvyml6nwlb</sha1>
    </revision>
    <revision>
      <id>91685</id>
      <parentid>87442</parentid>
      <timestamp>2022-05-04T11:58:12Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>91685</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14020" sha1="08wzedhsfxs31evl06hmn6jjk49la64" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each, a size close to [[169/168]]. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. 

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==

* [https://www.youtube.com/watch?v=uBQc_1M9Sa4 Pops (Orchestral Demo)] by [[User:Eliora|Eliora]]

[[Category:Equal divisions of the octave]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>08wzedhsfxs31evl06hmn6jjk49la64</sha1>
    </revision>
    <revision>
      <id>92928</id>
      <parentid>91685</parentid>
      <timestamp>2022-07-02T16:23:04Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92928</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14047" sha1="acvhtgvr72o1dbp0ir7myp8lmmybhy7" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each, a size close to [[169/168]]. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. 

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=uBQc_1M9Sa4 Pops (Orchestral Demo)] by [[User:Eliora|Eliora]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>acvhtgvr72o1dbp0ir7myp8lmmybhy7</sha1>
    </revision>
    <revision>
      <id>95590</id>
      <parentid>92928</parentid>
      <timestamp>2022-09-20T17:58:24Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Music */ edited to include the full song and not just demo</comment>
      <origin>95590</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14026" sha1="4oifohagbghkk0ay9xhdz6mitdlm92m" xml:space="preserve">{{Infobox ET
| Prime factorization = 2 × 59
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 11
}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each, a size close to [[169/168]]. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. 

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>4oifohagbghkk0ay9xhdz6mitdlm92m</sha1>
    </revision>
    <revision>
      <id>96084</id>
      <parentid>95590</parentid>
      <timestamp>2022-10-04T16:02:51Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET now computes most parameters automatically</comment>
      <origin>96084</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13859" sha1="q7njisx5arklww1k95b8n2ekjiz6y9v" xml:space="preserve">{{Infobox ET}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each, a size close to [[169/168]]. 

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. 

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>q7njisx5arklww1k95b8n2ekjiz6y9v</sha1>
    </revision>
    <revision>
      <id>99324</id>
      <parentid>96084</parentid>
      <timestamp>2022-11-27T14:40:43Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>added info about commas.</comment>
      <origin>99324</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13939" sha1="f0n49diygtiezdv6ewbykwh0g384r4x" xml:space="preserve">{{Infobox ET}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each.

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the [[bronzisma]], 2097152/2083725, [[169/168]], and [[170/169]].

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. 

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>f0n49diygtiezdv6ewbykwh0g384r4x</sha1>
    </revision>
    <revision>
      <id>99325</id>
      <parentid>99324</parentid>
      <timestamp>2022-11-27T14:42:31Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>99325</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13944" sha1="l3yuiaxwfm4klzimish2qbydkbm7saa" xml:space="preserve">{{Infobox ET}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each.

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. 

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>l3yuiaxwfm4klzimish2qbydkbm7saa</sha1>
    </revision>
    <revision>
      <id>113913</id>
      <parentid>99325</parentid>
      <timestamp>2023-05-20T19:33:19Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Rank-2 temperaments */</comment>
      <origin>113913</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13941" sha1="cnxqzkrls0owqpxbha9jkbbji8shp8f" xml:space="preserve">{{Infobox ET}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each.

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.  

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. 

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>cnxqzkrls0owqpxbha9jkbbji8shp8f</sha1>
    </revision>
    <revision>
      <id>116242</id>
      <parentid>113913</parentid>
      <timestamp>2023-06-10T13:24:28Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Possible chemical notation */ reword</comment>
      <origin>116242</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14001" sha1="f4zud8jy2x3ywm2ev1y2nv1hp2wj7fn" xml:space="preserve">{{Infobox ET}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each.

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Eliora's Chemical Notation&lt;br&gt;(if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>f4zud8jy2x3ywm2ev1y2nv1hp2wj7fn</sha1>
    </revision>
    <revision>
      <id>116243</id>
      <parentid>116242</parentid>
      <timestamp>2023-06-10T13:25:56Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Intervals */ I appreciate the credit but it's tautological in my opiniion, there is a section devoted to chemical notation explaining it's made by me</comment>
      <origin>116243</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14003" sha1="hgf5pe2q0rpu45nd2dzdwlbl3d9gbba" xml:space="preserve">{{Infobox ET}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each.

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br&gt;(see below, if base note = 0)
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
|-
| 53
| 538.98
| 
|[[15/11]]
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosiu
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>hgf5pe2q0rpu45nd2dzdwlbl3d9gbba</sha1>
    </revision>
    <revision>
      <id>116902</id>
      <parentid>116243</parentid>
      <timestamp>2023-06-16T08:19:26Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>116902</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16562" sha1="4fmb580pcg886zfxjqmtl86nfi5ixil" xml:space="preserve">{{Infobox ET}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each.

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
| {{UDnote|step=5}}
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
| {{UDnote|step=6}}
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
|-
| 10
| 101.69
| {{UDnote|step=10}}
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
| {{UDnote|step=14}}
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
| {{UDnote|step=28}}
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
| {{UDnote|step=30}}
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
| {{UDnote|step=32}}
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
| {{UDnote|step=41}}
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
| {{UDnote|step=43}}
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
| {{UDnote|step=47}}
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
|-
| 53
| 538.98
| 
|[[15/11]]
| {{UDnote|step=53}}
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
| {{UDnote|step=55}}
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
| {{UDnote|step=56}}
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
| {{UDnote|step=61}}
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
| {{UDnote|step=62}}
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
| {{UDnote|step=63}}
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
| {{UDnote|step=65}}
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
| {{UDnote|step=76}}
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
| {{UDnote|step=77}}
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
| {{UDnote|step=79}}
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
| {{UDnote|step=81}}
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
| {{UDnote|step=85}}
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
| {{UDnote|step=86}}
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
| {{UDnote|step=88}}
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
| {{UDnote|step=90}}
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
| {{UDnote|step=96}}
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>4fmb580pcg886zfxjqmtl86nfi5ixil</sha1>
    </revision>
    <revision>
      <id>116903</id>
      <parentid>116902</parentid>
      <timestamp>2023-06-16T08:20:35Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>116903</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16541" sha1="pljc3u4k4o2te27ge37ho87gr75r819" xml:space="preserve">{{Infobox ET}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each.

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{primes in edo|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
| {{UDnote|step=5}}
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
| {{UDnote|step=6}}
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
| {{UDnote|step=14}}
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
| {{UDnote|step=28}}
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
| {{UDnote|step=30}}
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
| {{UDnote|step=32}}
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
| {{UDnote|step=41}}
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
| {{UDnote|step=43}}
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
| {{UDnote|step=47}}
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
|-
| 53
| 538.98
| 
|[[15/11]]
| {{UDnote|step=53}}
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
| {{UDnote|step=55}}
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
| {{UDnote|step=56}}
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
| {{UDnote|step=61}}
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
| {{UDnote|step=62}}
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
| {{UDnote|step=63}}
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
| {{UDnote|step=65}}
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
| {{UDnote|step=76}}
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
| {{UDnote|step=77}}
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
| {{UDnote|step=79}}
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
| {{UDnote|step=81}}
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
| {{UDnote|step=85}}
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
| {{UDnote|step=86}}
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
| {{UDnote|step=88}}
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
| {{UDnote|step=90}}
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
| {{UDnote|step=96}}
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>pljc3u4k4o2te27ge37ho87gr75r819</sha1>
    </revision>
    <revision>
      <id>121123</id>
      <parentid>116903</parentid>
      <timestamp>2023-07-27T11:11:03Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <comment>/* Prime harmonics */</comment>
      <origin>121123</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16542" sha1="rmeua9qjoqmx8f3kk9zt24wv3y5p0p0" xml:space="preserve">{{Infobox ET}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each.

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{harmonics in equal}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
| {{UDnote|step=5}}
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
| {{UDnote|step=6}}
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
| {{UDnote|step=14}}
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
| {{UDnote|step=28}}
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
| {{UDnote|step=30}}
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
| {{UDnote|step=32}}
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
| {{UDnote|step=41}}
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
| {{UDnote|step=43}}
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
| {{UDnote|step=47}}
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
|-
| 53
| 538.98
| 
|[[15/11]]
| {{UDnote|step=53}}
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
| {{UDnote|step=55}}
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
| {{UDnote|step=56}}
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
| {{UDnote|step=61}}
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
| {{UDnote|step=62}}
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
| {{UDnote|step=63}}
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
| {{UDnote|step=65}}
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
| {{UDnote|step=76}}
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
| {{UDnote|step=77}}
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
| {{UDnote|step=79}}
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
| {{UDnote|step=81}}
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
| {{UDnote|step=85}}
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
| {{UDnote|step=86}}
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
| {{UDnote|step=88}}
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
| {{UDnote|step=90}}
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
| {{UDnote|step=96}}
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>rmeua9qjoqmx8f3kk9zt24wv3y5p0p0</sha1>
    </revision>
    <revision>
      <id>121124</id>
      <parentid>121123</parentid>
      <timestamp>2023-07-27T11:11:19Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <comment>/* Prime harmonics */</comment>
      <origin>121124</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16546" sha1="kglaj29ivcg4pwv3prp5ufoa9nyt1ng" xml:space="preserve">{{Infobox ET}}

The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each.

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{harmonics in equal|118}}

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
| {{UDnote|step=5}}
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
| {{UDnote|step=6}}
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
| {{UDnote|step=14}}
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
| {{UDnote|step=28}}
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
| {{UDnote|step=30}}
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
| {{UDnote|step=32}}
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
| {{UDnote|step=41}}
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
| {{UDnote|step=43}}
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
| {{UDnote|step=47}}
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
|-
| 53
| 538.98
| 
|[[15/11]]
| {{UDnote|step=53}}
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
| {{UDnote|step=55}}
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
| {{UDnote|step=56}}
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
| {{UDnote|step=61}}
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
| {{UDnote|step=62}}
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
| {{UDnote|step=63}}
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
| {{UDnote|step=65}}
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
| {{UDnote|step=76}}
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
| {{UDnote|step=77}}
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
| {{UDnote|step=79}}
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
| {{UDnote|step=81}}
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
| {{UDnote|step=85}}
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
| {{UDnote|step=86}}
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
| {{UDnote|step=88}}
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
| {{UDnote|step=90}}
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
| {{UDnote|step=96}}
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| +0.125
| 0.604
| 5.93
|-
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| +0.583
| 0.650
| 6.39
|-
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>kglaj29ivcg4pwv3prp5ufoa9nyt1ng</sha1>
    </revision>
    <revision>
      <id>121217</id>
      <parentid>121124</parentid>
      <timestamp>2023-07-29T06:31:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup; +subsets and supersets</comment>
      <origin>121217</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17021" sha1="jejof2ol355tfe6k8q4za4tv46995gq" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|gamelan]] temperament, and for [[guiron]], the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]]. 

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. 

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
| {{UDnote|step=5}}
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
| {{UDnote|step=6}}
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
| {{UDnote|step=14}}
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
| {{UDnote|step=28}}
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
| {{UDnote|step=30}}
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
| {{UDnote|step=32}}
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
| {{UDnote|step=41}}
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
| {{UDnote|step=43}}
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
| {{UDnote|step=47}}
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
|-
| 53
| 538.98
| 
|[[15/11]]
| {{UDnote|step=53}}
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
| {{UDnote|step=55}}
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
| {{UDnote|step=56}}
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
| {{UDnote|step=61}}
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
| {{UDnote|step=62}}
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
| {{UDnote|step=63}}
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
| {{UDnote|step=65}}
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
| {{UDnote|step=76}}
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
| {{UDnote|step=77}}
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
| {{UDnote|step=79}}
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
| {{UDnote|step=81}}
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
| {{UDnote|step=85}}
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
| {{UDnote|step=86}}
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
| {{UDnote|step=88}}
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
| {{UDnote|step=90}}
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
| {{UDnote|step=96}}
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 196/195, 352/351, 384/384, 625/624, 729/728
| style="border-top: double;" | [{{val| 118 187 274 331 408 437 }}] (118)
| style="border-top: double;" | +0.125
| style="border-top: double;" | 0.604
| style="border-top: double;" | 5.93
|-
| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 169/168, 325/324, 364/363, 385/384, 3136/3125
| style="border-top: double;" | [{{val| 118 187 274 331 408 436 }}] (118f)
| style="border-top: double;" | +0.583
| style="border-top: double;" | 0.650
| style="border-top: double;" | 6.39
|-
| style="border-top: double;" | 2.3.5.7.11.17
| style="border-top: double;" | 289/288, 385/384, 441/440, 561/560, 3136/3125
| style="border-top: double;" | [{{val| 118 187 274 331 408 482 }}]
| style="border-top: double;" | +0.417
| style="border-top: double;" | 0.399
| style="border-top: double;" | 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>jejof2ol355tfe6k8q4za4tv46995gq</sha1>
    </revision>
    <revision>
      <id>121218</id>
      <parentid>121217</parentid>
      <timestamp>2023-07-29T06:37:14Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ 5-limit JI and 22 shruti are connected. Zeta peak possibly has to do with its higher limit so we should mention it. Gamelan has been renamed to gamelismic</comment>
      <origin>121218</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17124" sha1="r13i9c51xt7dq8r3fw4stm3n2s95by1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11.

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
| {{UDnote|step=5}}
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
| {{UDnote|step=6}}
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
| {{UDnote|step=14}}
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
| {{UDnote|step=28}}
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
| {{UDnote|step=30}}
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
| {{UDnote|step=32}}
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
| {{UDnote|step=41}}
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
| {{UDnote|step=43}}
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
| {{UDnote|step=47}}
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
|-
| 53
| 538.98
| 
|[[15/11]]
| {{UDnote|step=53}}
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
| {{UDnote|step=55}}
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
| {{UDnote|step=56}}
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
| {{UDnote|step=61}}
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
| {{UDnote|step=62}}
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
| {{UDnote|step=63}}
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
| {{UDnote|step=65}}
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
| {{UDnote|step=76}}
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
| {{UDnote|step=77}}
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
| {{UDnote|step=79}}
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
| {{UDnote|step=81}}
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
| {{UDnote|step=85}}
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
| {{UDnote|step=86}}
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
| {{UDnote|step=88}}
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
| {{UDnote|step=90}}
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
| {{UDnote|step=96}}
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 196/195, 352/351, 384/384, 625/624, 729/728
| style="border-top: double;" | [{{val| 118 187 274 331 408 437 }}] (118)
| style="border-top: double;" | +0.125
| style="border-top: double;" | 0.604
| style="border-top: double;" | 5.93
|-
| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 169/168, 325/324, 364/363, 385/384, 3136/3125
| style="border-top: double;" | [{{val| 118 187 274 331 408 436 }}] (118f)
| style="border-top: double;" | +0.583
| style="border-top: double;" | 0.650
| style="border-top: double;" | 6.39
|-
| style="border-top: double;" | 2.3.5.7.11.17
| style="border-top: double;" | 289/288, 385/384, 441/440, 561/560, 3136/3125
| style="border-top: double;" | [{{val| 118 187 274 331 408 482 }}]
| style="border-top: double;" | +0.417
| style="border-top: double;" | 0.399
| style="border-top: double;" | 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>r13i9c51xt7dq8r3fw4stm3n2s95by1</sha1>
    </revision>
    <revision>
      <id>121234</id>
      <parentid>121218</parentid>
      <timestamp>2023-07-30T23:47:33Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Subsets and supersets */ fractional octave temps</comment>
      <origin>121234</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17164" sha1="380tv8w2exo4gwmu31ocun781n9n1in" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
| {{UDnote|step=5}}
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
| {{UDnote|step=6}}
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
| {{UDnote|step=14}}
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
| {{UDnote|step=28}}
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
| {{UDnote|step=30}}
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
| {{UDnote|step=32}}
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
| {{UDnote|step=41}}
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
| {{UDnote|step=43}}
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
| {{UDnote|step=47}}
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
|-
| 53
| 538.98
| 
|[[15/11]]
| {{UDnote|step=53}}
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
| {{UDnote|step=55}}
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
| {{UDnote|step=56}}
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
| {{UDnote|step=61}}
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
| {{UDnote|step=62}}
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
| {{UDnote|step=63}}
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
| {{UDnote|step=65}}
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
| {{UDnote|step=76}}
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
| {{UDnote|step=77}}
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
| {{UDnote|step=79}}
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
| {{UDnote|step=81}}
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
| {{UDnote|step=85}}
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
| {{UDnote|step=86}}
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
| {{UDnote|step=88}}
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
| {{UDnote|step=90}}
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
| {{UDnote|step=96}}
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 196/195, 352/351, 384/384, 625/624, 729/728
| style="border-top: double;" | [{{val| 118 187 274 331 408 437 }}] (118)
| style="border-top: double;" | +0.125
| style="border-top: double;" | 0.604
| style="border-top: double;" | 5.93
|-
| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 169/168, 325/324, 364/363, 385/384, 3136/3125
| style="border-top: double;" | [{{val| 118 187 274 331 408 436 }}] (118f)
| style="border-top: double;" | +0.583
| style="border-top: double;" | 0.650
| style="border-top: double;" | 6.39
|-
| style="border-top: double;" | 2.3.5.7.11.17
| style="border-top: double;" | 289/288, 385/384, 441/440, 561/560, 3136/3125
| style="border-top: double;" | [{{val| 118 187 274 331 408 482 }}]
| style="border-top: double;" | +0.417
| style="border-top: double;" | 0.399
| style="border-top: double;" | 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>380tv8w2exo4gwmu31ocun781n9n1in</sha1>
    </revision>
    <revision>
      <id>135444</id>
      <parentid>121234</parentid>
      <timestamp>2024-02-13T17:44:58Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <minor/>
      <comment>+categories</comment>
      <origin>135444</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17184" sha1="8pyrxkh9i91mhzonkzkdldy01nvzcgm" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
| {{UDnote|step=5}}
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
| {{UDnote|step=6}}
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
| {{UDnote|step=14}}
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
| {{UDnote|step=28}}
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
| {{UDnote|step=30}}
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
| {{UDnote|step=32}}
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
| {{UDnote|step=41}}
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
| {{UDnote|step=43}}
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
| {{UDnote|step=47}}
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
|-
| 53
| 538.98
| 
|[[15/11]]
| {{UDnote|step=53}}
| 
| iodine
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
| {{UDnote|step=55}}
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
| {{UDnote|step=56}}
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
| {{UDnote|step=61}}
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
| {{UDnote|step=62}}
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
| {{UDnote|step=63}}
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
| {{UDnote|step=65}}
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
| {{UDnote|step=76}}
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
| {{UDnote|step=77}}
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
| {{UDnote|step=79}}
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
| {{UDnote|step=81}}
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
| {{UDnote|step=85}}
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
| {{UDnote|step=86}}
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
| {{UDnote|step=88}}
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
| {{UDnote|step=90}}
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
| {{UDnote|step=96}}
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 196/195, 352/351, 384/384, 625/624, 729/728
| style="border-top: double;" | [{{val| 118 187 274 331 408 437 }}] (118)
| style="border-top: double;" | +0.125
| style="border-top: double;" | 0.604
| style="border-top: double;" | 5.93
|-
| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 169/168, 325/324, 364/363, 385/384, 3136/3125
| style="border-top: double;" | [{{val| 118 187 274 331 408 436 }}] (118f)
| style="border-top: double;" | +0.583
| style="border-top: double;" | 0.650
| style="border-top: double;" | 6.39
|-
| style="border-top: double;" | 2.3.5.7.11.17
| style="border-top: double;" | 289/288, 385/384, 441/440, 561/560, 3136/3125
| style="border-top: double;" | [{{val| 118 187 274 331 408 482 }}]
| style="border-top: double;" | +0.417
| style="border-top: double;" | 0.399
| style="border-top: double;" | 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]
[[Category:Listen]]</text>
      <sha1>8pyrxkh9i91mhzonkzkdldy01nvzcgm</sha1>
    </revision>
    <revision>
      <id>139614</id>
      <parentid>135444</parentid>
      <timestamp>2024-03-19T05:44:18Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <comment>/* Intervals */ added SKULO notation</comment>
      <origin>139614</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17963" sha1="smm2run61g2itr0jc2diz4zdisi3unh" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
![[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
|D
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
|LD
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
|KD
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
|SD
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
|OD, uEb
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
| {{UDnote|step=5}}
|UD, oEb
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
| {{UDnote|step=6}}
|sEb, uD#
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
|kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
|lEb, sD#
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
|Eb, kD#
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
|LEb, lD#
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
|KEb, D#
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
|SEb
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
|OEb
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
| {{UDnote|step=14}}
|UEb
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
|uE
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
|oE
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
|sE
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
|kE
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
|lE
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
|E
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
|LE
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
|KE
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
|SE
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
|OE, uF
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
|UE, oF
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
|sF
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
|kF
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
| {{UDnote|step=28}}
|lF
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
|F
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
| {{UDnote|step=30}}
|LF
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
|KF
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
| {{UDnote|step=32}}
|SF
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
|OF
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
|UF
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
|uF#
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
|oF#
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
|sF#
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
|kF#
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
|lF#
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
|F#
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
| {{UDnote|step=41}}
|LF#
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
|KF#
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
| {{UDnote|step=43}}
|SF#
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
|OF#, uG
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
|UF#, oG
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
|sG
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
| {{UDnote|step=47}}
|kG
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
|lG
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
|G
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
|LG
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
|KG
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
|SG
|-
| 53
| 538.98
| 
|[[15/11]]
| {{UDnote|step=53}}
| 
| iodine
|OG, uGb
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
|UG, oAb
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
| {{UDnote|step=55}}
|uG#, sAb
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
| {{UDnote|step=56}}
|oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
|sG#, lAb
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
|kG#, Ab
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
|lG#, LAb
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
|G#, KAb
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
| {{UDnote|step=61}}
|LG#, SAb
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
| {{UDnote|step=62}}
|KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
| {{UDnote|step=63}}
|SG#, UAb
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
|OG#, uA
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
| {{UDnote|step=65}}
|oA
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
|sA
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
|kA
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
|lA
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
|A
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
|lA
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
|KA
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
|SA
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
|OA, uBb
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
|UA, oBb
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
|sBb
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
| {{UDnote|step=76}}
|kBb
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
| {{UDnote|step=77}}
|lBb
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
|Bb
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
| {{UDnote|step=79}}
|LBb
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
|KBb
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
| {{UDnote|step=81}}
|SBb
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
|OBb
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
|UBb
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
|uB
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
| {{UDnote|step=85}}
|oB
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
| {{UDnote|step=86}}
|sB
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
|kB
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
| {{UDnote|step=88}}
|lB
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
|B
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
| {{UDnote|step=90}}
|LB
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
|KB
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
|SB
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
|OB, uC
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
|UB, oC
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
|sC
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
| {{UDnote|step=96}}
|kC
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
|lC
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
|C
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
|LC
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
|KC
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
|SC
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
|OC, uDb
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
|UC, oDb
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
|uC#, sDb
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
|oC#, kDb
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
|sC#, lDb
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
|kC#, Db
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
|lC#, LDb
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
|C#, KDb
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
|LC#, SDb
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
|KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
|SC#, UDb
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
|OC#, uD
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
|UC#, oD
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
|sD
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
|kD
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
|lD
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
|D
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Possible chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| +0.341
| 0.370
| 3.89
|-
| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 196/195, 352/351, 384/384, 625/624, 729/728
| style="border-top: double;" | [{{val| 118 187 274 331 408 437 }}] (118)
| style="border-top: double;" | +0.125
| style="border-top: double;" | 0.604
| style="border-top: double;" | 5.93
|-
| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 169/168, 325/324, 364/363, 385/384, 3136/3125
| style="border-top: double;" | [{{val| 118 187 274 331 408 436 }}] (118f)
| style="border-top: double;" | +0.583
| style="border-top: double;" | 0.650
| style="border-top: double;" | 6.39
|-
| style="border-top: double;" | 2.3.5.7.11.17
| style="border-top: double;" | 289/288, 385/384, 441/440, 561/560, 3136/3125
| style="border-top: double;" | [{{val| 118 187 274 331 408 482 }}]
| style="border-top: double;" | +0.417
| style="border-top: double;" | 0.399
| style="border-top: double;" | 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}

== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]
[[Category:Listen]]</text>
      <sha1>smm2run61g2itr0jc2diz4zdisi3unh</sha1>
    </revision>
    <revision>
      <id>145235</id>
      <parentid>139614</parentid>
      <timestamp>2024-05-29T09:27:09Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>145235</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17656" sha1="md9tpxdvxysygro6hlxtjjfvko0ap0a" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
![[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
|D
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
|LD
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
|KD
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
|SD
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
|OD, uEb
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
| {{UDnote|step=5}}
|UD, oEb
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
| {{UDnote|step=6}}
|sEb, uD#
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
|kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
|lEb, sD#
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
|Eb, kD#
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
|LEb, lD#
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
|KEb, D#
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
|SEb
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
|OEb
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
| {{UDnote|step=14}}
|UEb
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
|uE
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
|oE
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
|sE
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
|kE
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
|lE
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
|E
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
|LE
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
|KE
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
|SE
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
|OE, uF
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
|UE, oF
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
|sF
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
|kF
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
| {{UDnote|step=28}}
|lF
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
|F
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
| {{UDnote|step=30}}
|LF
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
|KF
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
| {{UDnote|step=32}}
|SF
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
|OF
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
|UF
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
|uF#
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
|oF#
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
|sF#
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
|kF#
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
|lF#
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
|F#
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
| {{UDnote|step=41}}
|LF#
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
|KF#
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
| {{UDnote|step=43}}
|SF#
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
|OF#, uG
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
|UF#, oG
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
|sG
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
| {{UDnote|step=47}}
|kG
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
|lG
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
|G
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
|LG
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
|KG
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
|SG
|-
| 53
| 538.98
| 
|[[15/11]]
| {{UDnote|step=53}}
| 
| iodine
|OG, uGb
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
|UG, oAb
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
| {{UDnote|step=55}}
|uG#, sAb
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
| {{UDnote|step=56}}
|oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
|sG#, lAb
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
|kG#, Ab
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
|lG#, LAb
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
|G#, KAb
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
| {{UDnote|step=61}}
|LG#, SAb
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
| {{UDnote|step=62}}
|KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
| {{UDnote|step=63}}
|SG#, UAb
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
|OG#, uA
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
| {{UDnote|step=65}}
|oA
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
|sA
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
|kA
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
|lA
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
|A
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
|lA
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
|KA
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
|SA
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
|OA, uBb
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
|UA, oBb
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
|sBb
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
| {{UDnote|step=76}}
|kBb
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
| {{UDnote|step=77}}
|lBb
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
|Bb
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
| {{UDnote|step=79}}
|LBb
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
|KBb
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
| {{UDnote|step=81}}
|SBb
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
|OBb
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
|UBb
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
|uB
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
| {{UDnote|step=85}}
|oB
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
| {{UDnote|step=86}}
|sB
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
|kB
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
| {{UDnote|step=88}}
|lB
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
|B
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
| {{UDnote|step=90}}
|LB
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
|KB
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
|SB
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
|OB, uC
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
|UB, oC
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
|sC
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
| {{UDnote|step=96}}
|kC
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
|lC
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
|C
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
|LC
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
|KC
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
|SC
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
|OC, uDb
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
|UC, oDb
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
|uC#, sDb
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
|oC#, kDb
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
|sC#, lDb
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
|kC#, Db
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
|lC#, LDb
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
|C#, KDb
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
|LC#, SDb
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
|KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
|SC#, UDb
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
|OC#, uD
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
|UC#, oD
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
|sD
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
|kD
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
|lD
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
|D
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|-style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|-style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|-style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]
[[Category:Listen]]</text>
      <sha1>md9tpxdvxysygro6hlxtjjfvko0ap0a</sha1>
    </revision>
    <revision>
      <id>162252</id>
      <parentid>145235</parentid>
      <timestamp>2024-10-26T10:20:39Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Regular temperament properties */ Insert Instruments section after, with link to Lumatone mapping for 118edo</comment>
      <origin>162252</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17708" sha1="a7f70ime66p682ycxrawv88uctzbzcz" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
![[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
|[[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
|D
|-
| 1
| 10.17
| 
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
|LD
|-
| 2
| 20.34
| 
|[[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
|KD
|-
| 3
| 30.51
| 
|[[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
|SD
|-
| 4
| 40.68
| 
|[[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
|OD, uEb
|-
| 5
| 50.85
| 
|[[36/35]]
| 
| boron
| {{UDnote|step=5}}
|UD, oEb
|-
| 6
| 61.02
| 
|[[28/27]]
| 
| carbon
| {{UDnote|step=6}}
|sEb, uD#
|-
| 7
| 71.19
| 
|[[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
|kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
|[[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
|lEb, sD#
|-
| 9
| 91.53
| m2
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
|Eb, kD#
|-
| 10
| 101.69
| 
|[[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
|LEb, lD#
|-
| 11
| 111.86
| 
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
|KEb, D#
|-
| 12
| 122.03
| 
|[[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
|SEb
|-
| 13
| 132.20
| 
|[[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
|OEb
|-
| 14
| 142.37
| 
|[[88/81]]
| 
| silicon
| {{UDnote|step=14}}
|UEb
|-
| 15
| 152.54
| 
|[[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
|uE
|-
| 16
| 162.71
| 
|[[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
|oE
|-
| 17
| 172.88
| 
|[[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
|sE
|-
| 18
| 183.05
| 
|[[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
|kE
|-
| 19
| 193.22
| 
|[[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
|lE
|-
| 20
| 203.39
| M2
|[[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
|E
|-
| 21
| 213.56
| 
|[[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
|LE
|-
| 22
| 223.73
| 
|[[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
|KE
|-
| 23
| 233.90
| 
|[[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
|SE
|-
| 24
| 244.07
| 
|[[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
|OE, uF
|-
| 25
| 254.24
| 
|[[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
|UE, oF
|-
| 26
| 260.41
| 
|[[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
|sF
|-
| 27
| 274.58
| 
|[[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
|kF
|-
| 28
| 284.75
| 
|[[33/28]]
| 
| nickel
| {{UDnote|step=28}}
|lF
|-
| 29
| 294.92
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
|F
|-
| 30
| 305.08
| 
|[[25/21]]
| 
| zinc
| {{UDnote|step=30}}
|LF
|-
| 31
| 315.25
| 
|[[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
|KF
|-
| 32
| 325.42
| 
|[[98/81]]
| 
| germanium
| {{UDnote|step=32}}
|SF
|-
| 33
| 335.59
| 
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
|OF
|-
| 34
| 345.76
| 
|[[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
|UF
|-
| 35
| 355.93
| 
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
|uF#
|-
| 36
| 366.10
| 
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
|oF#
|-
| 37
| 376.27
| 
|[[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
|sF#
|-
| 38
| 386.44
| 
|[[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
|kF#
|-
| 39
| 396.61
| 
|[[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
|lF#
|-
| 40
| 406.78
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
|F#
|-
| 41
| 416.95
| 
|[[14/11]]
| 
| niobium
| {{UDnote|step=41}}
|LF#
|-
| 42
| 427.12
| 
|[[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
|KF#
|-
| 43
| 437.29
| 
|[[9/7]]
| 
| technetium
| {{UDnote|step=43}}
|SF#
|-
| 44
| 447.46
| 
|[[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
|OF#, uG
|-
| 45
| 457.63
| 
|[[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
|UF#, oG
|-
| 46
| 467.80
| 
|[[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
|sG
|-
| 47
| 477.97
| 
|[[320/243]]
| 
| silver
| {{UDnote|step=47}}
|kG
|-
| 48
| 488.14
| 
|[[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
|lG
|-
| 49
| 498.31
| P4
|[[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
|G
|-
| 50
| 508.47
| 
|[[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
|LG
|-
| 51
| 518.64
| 
|[[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
|KG
|-
| 52
| 528.81
| 
|[[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
|SG
|-
| 53
| 538.98
| 
|[[15/11]]
| {{UDnote|step=53}}
| 
| iodine
|OG, uGb
|-
| 54
| 549.15
| 
|[[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
|UG, oAb
|-
| 55
| 559.32
| 
|[[112/81]]
| 
| caesium
| {{UDnote|step=55}}
|uG#, sAb
|-
| 56
| 569.49
| 
|[[25/18]]
| 
| barium
| {{UDnote|step=56}}
|oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
|[[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
|sG#, lAb
|-
| 58
| 589.83
| d5
|[[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
|kG#, Ab
|-
| 59
| 600.00
| 
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
|lG#, LAb
|-
| 60
| 610.17
| A4
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
|G#, KAb
|-
| 61
| 620.34
| 
|[[10/7]]
| 
| promethium
| {{UDnote|step=61}}
|LG#, SAb
|-
| 62
| 630.51
| 
|[[36/25]]
| 
| samarium
| {{UDnote|step=62}}
|KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
|[[81/56]]
| 
| europium
| {{UDnote|step=63}}
|SG#, UAb
|-
| 64
| 650.85
| 
|[[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
|OG#, uA
|-
| 65
| 661.02
| 
|[[22/15]]
| 
| terbium
| {{UDnote|step=65}}
|oA
|-
| 66
| 671.19
| 
|[[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
|sA
|-
| 67
| 681.36
| 
|[[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
|kA
|-
| 68
| 691.53
| 
|[[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
|lA
|-
| 69
| 701.69
| P5
|[[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
|A
|-
| 70
| 711.86
| 
|[[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
|lA
|-
| 71
| 722.03
| 
|[[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
|KA
|-
| 72
| 732.20
| 
|[[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
|SA
|-
| 73
| 742.37
| 
|[[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
|OA, uBb
|-
| 74
| 752.54
| 
|[[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
|UA, oBb
|-
| 75
| 762.71
| 
|[[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
|sBb
|-
| 76
| 772.88
| 
|[[120/77]]
| 
| osmium
| {{UDnote|step=76}}
|kBb
|-
| 77
| 783.05
| 
|[[11/7]]
| 
| iridium
| {{UDnote|step=77}}
|lBb
|-
| 78
| 793.22
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
|Bb
|-
| 79
| 803.39
| 
|[[100/63]]
| 
| gold
| {{UDnote|step=79}}
|LBb
|-
| 80
| 813.56
| 
|[[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
|KBb
|-
| 81
| 823.73
| 
|[[45/28]]
| 
| thallium
| {{UDnote|step=81}}
|SBb
|-
| 82
| 833.90
| 
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
|OBb
|-
| 83
| 844.07
| 
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
|UBb
|-
| 84
| 854.24
| 
|[[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
|uB
|-
| 85
| 864.41
| 
|[[28/17]]
| 
| astatine
| {{UDnote|step=85}}
|oB
|-
| 86
| 874.58
| 
|[[81/49]]
| 
| radon
| {{UDnote|step=86}}
|sB
|-
| 87
| 884.75
| 
|[[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
|kB
|-
| 88
| 894.92
| 
|[[42/25]]
| 
| radium
| {{UDnote|step=88}}
|lB
|-
| 89
| 905.08
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
|B
|-
| 90
| 915.25
| 
|[[56/33]]
| 
| thorium
| {{UDnote|step=90}}
|LB
|-
| 91
| 925.42
| 
|[[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
|KB
|-
| 92
| 935.59
| 
|[[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
|SB
|-
| 93
| 945.76
| 
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
|OB, uC
|-
| 94
| 955.93
| 
|[[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
|UB, oC
|-
| 95
| 966.10
| 
|[[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
|sC
|-
| 96
| 976.27
| 
|[[225/128]]
| 
| curium
| {{UDnote|step=96}}
|kC
|-
| 97
| 986.44
| 
|[[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
|lC
|-
| 98
| 996.61
| m7
|[[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
|C
|-
| 99
| 1006.78
| 
|[[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
|LC
|-
| 100
| 1016.95
| 
|[[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
|KC
|-
| 101
| 1027.12
| 
|[[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
|SC
|-
| 102
| 1037.29
| 
|[[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
|OC, uDb
|-
| 103
| 1047.46
| 
|[[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
|UC, oDb
|-
| 104
| 1057.63
| 
|[[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
|uC#, sDb
|-
| 105
| 1067.80
| 
|[[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
|oC#, kDb
|-
| 106
| 1077.97
| 
|[[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
|sC#, lDb
|-
| 107
| 1088.14
| 
|[[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
|kC#, Db
|-
| 108
| 1098.31
| 
|[[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
|lC#, LDb
|-
| 109
| 1108.47
| M7
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
|C#, KDb
|-
| 110
| 1118.64
| 
|[[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
|LC#, SDb
|-
| 111
| 1128.81
| 
|[[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
|KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
|[[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
|SC#, UDb
|-
| 113
| 1149.15
| 
|[[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
|OC#, uD
|-
| 114
| 1159.32
| 
|[[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
|UC#, oD
|-
| 115
| 1169.49
| 
|[[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
|sD
|-
| 116
| 1179.66
| 
|[[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
|kD
|-
| 117
| 1189.83
| 
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
|lD
|-
| 118
| 1200.00
| P8
|[[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
|D
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; treated as a 2.3.5.7.11.17.19 system

&lt;nowiki&gt;**&lt;/nowiki&gt; based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| -0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|-style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|-style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|-style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Instruments ==

[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]
[[Category:Listen]]</text>
      <sha1>a7f70ime66p682ycxrawv88uctzbzcz</sha1>
    </revision>
    <revision>
      <id>164501</id>
      <parentid>162252</parentid>
      <timestamp>2024-11-06T14:18:53Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>164501</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18026" sha1="jd0w1qs9dtfvwtdchjc202xglk3f966" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* Treated as a 2.3.5.7.11.17.19 system

&lt;nowiki /&gt;** based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|-style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|-style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|-style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%; white-space: nowrap;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;Ratio*
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br /&gt;(28\118)
| 315.25&lt;br /&gt;(284.75)
| 6/5&lt;br /&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]
[[Category:Listen]]</text>
      <sha1>jd0w1qs9dtfvwtdchjc202xglk3f966</sha1>
    </revision>
    <revision>
      <id>164506</id>
      <parentid>164501</parentid>
      <timestamp>2024-11-06T15:20:30Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */</comment>
      <origin>164506</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18026" sha1="stck8hegz91fjyel6srfhgfymow0ne7" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* Treated as a 2.3.5.7.11.17.19 system

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|-style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|-style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|-style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%; white-space: nowrap;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;Ratio*
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br /&gt;(28\118)
| 315.25&lt;br /&gt;(284.75)
| 6/5&lt;br /&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]
[[Category:Listen]]</text>
      <sha1>stck8hegz91fjyel6srfhgfymow0ne7</sha1>
    </revision>
    <revision>
      <id>166006</id>
      <parentid>164506</parentid>
      <timestamp>2024-11-16T06:47:59Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>166006</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17446" sha1="eztgso3q1xv81cli9e68xs2vogal20o" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{{comma basis begin}}
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
{{comma basis end}}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{{rank-2 begin}}
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br /&gt;(28\118)
| 315.25&lt;br /&gt;(284.75)
| 6/5&lt;br /&gt;(33/28)
| [[Semiparakleismic]]
{{rank-2 end}}
{{orf}}

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>eztgso3q1xv81cli9e68xs2vogal20o</sha1>
    </revision>
    <revision>
      <id>166062</id>
      <parentid>166006</parentid>
      <timestamp>2024-11-16T11:20:42Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Undo revision 166006 by [[Special:Contributions/ArrowHead294|ArrowHead294]] ([[User talk:ArrowHead294|talk]])</comment>
      <origin>166062</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18026" sha1="stck8hegz91fjyel6srfhgfymow0ne7" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios *
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if base note = 0)
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* Treated as a 2.3.5.7.11.17.19 system

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I  
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|-style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|-style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|-style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%; white-space: nowrap;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;Ratio*
! Temperaments
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br /&gt;(28\118)
| 315.25&lt;br /&gt;(284.75)
| 6/5&lt;br /&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]
[[Category:Listen]]</text>
      <sha1>stck8hegz91fjyel6srfhgfymow0ne7</sha1>
    </revision>
    <revision>
      <id>166110</id>
      <parentid>166062</parentid>
      <timestamp>2024-11-16T11:55:24Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>166110</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18002" sha1="b8i08ttcfe6bdalez044tegs35u6poa" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br /&gt;(28\118)
| 315.25&lt;br /&gt;(284.75)
| 6/5&lt;br /&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>b8i08ttcfe6bdalez044tegs35u6poa</sha1>
    </revision>
    <revision>
      <id>174149</id>
      <parentid>166110</parentid>
      <timestamp>2025-01-01T20:50:33Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ Fix introductory list of commas tempered out to point to kwazy comma instead of kwazy temperament</comment>
      <origin>174149</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18008" sha1="92j5lv59bx82xv6kpkylcemgjmeyif1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Commatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br /&gt;(28\118)
| 315.25&lt;br /&gt;(284.75)
| 6/5&lt;br /&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>92j5lv59bx82xv6kpkylcemgjmeyif1</sha1>
    </revision>
    <revision>
      <id>175872</id>
      <parentid>174149</parentid>
      <timestamp>2025-01-13T13:25:39Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ cleanup and update</comment>
      <origin>175872</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17991" sha1="k0az2nkuehkployjvpewp7hicq7ojwz" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>k0az2nkuehkployjvpewp7hicq7ojwz</sha1>
    </revision>
    <revision>
      <id>176977</id>
      <parentid>175872</parentid>
      <timestamp>2025-01-18T23:12:35Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <origin>176977</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18388" sha1="qqxa7xnnottgqplk4gsivku9dl16f05" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Zeta properties ==
===Zeta peak index===
{| class="wikitable"
! colspan="3" |Tuning
! colspan="3" |Strength
! colspan="2" |Closest EDO
! colspan="2" |Integer limit
|-
!ZPI
!Steps per octave
!Step size (cents)
!Height
!Integral
!Gap
!EDO
!Octave (cents)
!Consistent
!Distinct
|-
|[[706zpi]]
|117.969513574257
|10.1721195895637
|9.850823
|1.544280
|18.861062
|118edo
|1200.31011156852
|12
|12
|}
== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>qqxa7xnnottgqplk4gsivku9dl16f05</sha1>
    </revision>
    <revision>
      <id>177060</id>
      <parentid>176977</parentid>
      <timestamp>2025-01-19T02:44:37Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Formatting</comment>
      <origin>177060</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18418" sha1="ewlehwd7g10d2b4xkaop8fppbxrja6i" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Zeta properties ==
=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[706zpi]]
| 117.969513574257
| 10.1721195895637
| 9.850823
| 1.544280
| 18.861062
| 118edo
| 1200.31011156852
| 12
| 12
|}

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>ewlehwd7g10d2b4xkaop8fppbxrja6i</sha1>
    </revision>
    <revision>
      <id>177136</id>
      <parentid>177060</parentid>
      <timestamp>2025-01-19T10:13:23Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Consolidate sections</comment>
      <origin>177136</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18433" sha1="scihdsovrvpb2gy8herzs7n0qul2uku" xml:space="preserve">{{Infobox ET}}
{{EDO intro|118}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[706zpi]]
| 117.969513574257
| 10.1721195895637
| 9.850823
| 1.544280
| 18.861062
| 118edo
| 1200.31011156852
| 12
| 12
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>scihdsovrvpb2gy8herzs7n0qul2uku</sha1>
    </revision>
    <revision>
      <id>181323</id>
      <parentid>177136</parentid>
      <timestamp>2025-02-18T16:54:04Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181323</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18428" sha1="e3np38fizxlzjclco6jj7k92y6lov9y" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[706zpi]]
| 117.969513574257
| 10.1721195895637
| 9.850823
| 1.544280
| 18.861062
| 118edo
| 1200.31011156852
| 12
| 12
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>e3np38fizxlzjclco6jj7k92y6lov9y</sha1>
    </revision>
    <revision>
      <id>187375</id>
      <parentid>181323</parentid>
      <timestamp>2025-03-20T15:53:33Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Zeta peak index */</comment>
      <origin>187375</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18272" sha1="48a4g51zaayd2gu0mja486anqyqwool" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>48a4g51zaayd2gu0mja486anqyqwool</sha1>
    </revision>
    <revision>
      <id>190558</id>
      <parentid>187375</parentid>
      <timestamp>2025-04-10T13:01:31Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>190558</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18272" sha1="gi89nro7tye8smltm8bnh77k66jcecu" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>gi89nro7tye8smltm8bnh77k66jcecu</sha1>
    </revision>
    <revision>
      <id>191054</id>
      <parentid>190558</parentid>
      <timestamp>2025-04-11T07:49:39Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Rank-2 temperaments */ (''See [[regular temperament]] for more about what all this means and how to use it.'')</comment>
      <origin>191054</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18359" sha1="kubrqr8wsl02rqbf8a21zjuv0rxrz8z" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
''See [[regular temperament]] for more about what all this means and how to use it.''

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>kubrqr8wsl02rqbf8a21zjuv0rxrz8z</sha1>
    </revision>
    <revision>
      <id>191161</id>
      <parentid>191054</parentid>
      <timestamp>2025-04-11T11:19:48Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>No need to remind readers of what a regular temperament is everywhere</comment>
      <origin>191161</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18272" sha1="gi89nro7tye8smltm8bnh77k66jcecu" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Subsets and supersets ===
118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>gi89nro7tye8smltm8bnh77k66jcecu</sha1>
    </revision>
    <revision>
      <id>192239</id>
      <parentid>191161</parentid>
      <timestamp>2025-04-15T11:51:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ +octave stretch; note prime factorization</comment>
      <origin>192239</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18601" sha1="h4kalsqpew5e12w5i7reiwj1uatzm1p" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. As a result, 118edo also excellently approximates the 22 Shruti scale.

118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Octave stretch ===
118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part. 

=== Subsets and supersets ===
Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>h4kalsqpew5e12w5i7reiwj1uatzm1p</sha1>
    </revision>
    <revision>
      <id>192240</id>
      <parentid>192239</parentid>
      <timestamp>2025-04-15T11:54:24Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ rework on the opening paragraph. -shruti (5-limit accuracy isn't a focus of it)</comment>
      <origin>192240</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18563" sha1="i87m3fbbnrcup5cpllwgfnotdpst3ni" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament|microtempering]], with [[error]]s well under half a cent. It represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It 

118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Octave stretch ===
118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part. 

=== Subsets and supersets ===
Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>i87m3fbbnrcup5cpllwgfnotdpst3ni</sha1>
    </revision>
    <revision>
      <id>192276</id>
      <parentid>192240</parentid>
      <timestamp>2025-04-15T17:39:04Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ Remove dangling word left over from 1 of the previous 2 edits</comment>
      <origin>192276</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18559" sha1="4hzuwki1fkf49qfk48zqoc9mderdil8" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament|microtempering]], with [[error]]s well under half a cent. It represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}.

118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Octave stretch ===
118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part. 

=== Subsets and supersets ===
Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>4hzuwki1fkf49qfk48zqoc9mderdil8</sha1>
    </revision>
    <revision>
      <id>192416</id>
      <parentid>192276</parentid>
      <timestamp>2025-04-16T15:33:40Z</timestamp>
      <contributor>
        <username>Nick Vuci</username>
        <id>286</id>
      </contributor>
      <minor/>
      <comment>Edited Helmholtz redirect</comment>
      <origin>192416</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18581" sha1="l34edjtjssfumre07zw04swzwunf3iy" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament|microtempering]], with [[error]]s well under half a cent. It represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}.

118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Octave stretch ===
118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part. 

=== Subsets and supersets ===
Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz temperament|Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>l34edjtjssfumre07zw04swzwunf3iy</sha1>
    </revision>
    <revision>
      <id>192610</id>
      <parentid>192416</parentid>
      <timestamp>2025-04-17T02:30:52Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Helmholtz temperament|" to "[[Helmholtz (temperament)|"</comment>
      <origin>192610</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18583" sha1="bf3yisss3z9vstrbwr2txklhvmv0lzz" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament|microtempering]], with [[error]]s well under half a cent. It represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}.

118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Octave stretch ===
118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part. 

=== Subsets and supersets ===
Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>bf3yisss3z9vstrbwr2txklhvmv0lzz</sha1>
    </revision>
    <revision>
      <id>200010</id>
      <parentid>192610</parentid>
      <timestamp>2025-06-19T13:43:23Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ Add link to 16/13</comment>
      <origin>200010</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18587" sha1="22pyl4myhx0t865z9qc6xb3uz8918qc" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament|microtempering]], with [[error]]s well under half a cent. It represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}.

118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Octave stretch ===
118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part. 

=== Subsets and supersets ===
Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], [[16/13]] I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
| {{UDnote|step=53}}
| 
| iodine
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| 
| xenon
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>22pyl4myhx0t865z9qc6xb3uz8918qc</sha1>
    </revision>
    <revision>
      <id>218164</id>
      <parentid>200010</parentid>
      <timestamp>2025-11-29T11:00:37Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Intervals */ Repair disordered columns at 53 (iodine) and 54 (xenon)</comment>
      <origin>218164</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18585" sha1="kaqssrfo8cyj8p16e582orcn5j6i7f8" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament|microtempering]], with [[error]]s well under half a cent. It represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}.

118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Octave stretch ===
118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part. 

=== Subsets and supersets ===
Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], [[16/13]] I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
|
| iodine
| {{UDnote|step=53}}
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
|
| xenon
| {{UDnote|step=54}}
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>kaqssrfo8cyj8p16e582orcn5j6i7f8</sha1>
    </revision>
    <revision>
      <id>225937</id>
      <parentid>218164</parentid>
      <timestamp>2026-03-13T13:33:11Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct" to "[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct"</comment>
      <origin>225937</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18640" sha1="jx5jfg2b519hnwp7ztnxlw3t56z1z7c" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament|microtempering]], with [[error]]s well under half a cent. It represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}.

118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Octave stretch ===
118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part. 

=== Subsets and supersets ===
Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], [[16/13]] I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
|
| iodine
| {{UDnote|step=53}}
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
|
| xenon
| {{UDnote|step=54}}
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>jx5jfg2b519hnwp7ztnxlw3t56z1z7c</sha1>
    </revision>
    <revision>
      <id>233019</id>
      <parentid>225937</parentid>
      <timestamp>2026-06-30T14:52:49Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ Fix typo; in the future; should probably split out columns for 13♯ and 13♭ if more ratios including 13 are wanted</comment>
      <origin>233019</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18639" sha1="3imql5eyg6fxqi1e1fe80fi188p3x5g" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament|microtempering]], with [[error]]s well under half a cent. It represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}.

118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Octave stretch ===
118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part. 

=== Subsets and supersets ===
Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], [[16/13]] I**
| Minor tridecimal neutral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
|
| iodine
| {{UDnote|step=53}}
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
|
| xenon
| {{UDnote|step=54}}
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>3imql5eyg6fxqi1e1fe80fi188p3x5g</sha1>
    </revision>
    <revision>
      <id>234145</id>
      <parentid>233019</parentid>
      <timestamp>2026-07-16T18:41:09Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[The Riemann zeta function and tuning|zeta peak edo]]" to "[[zeta peak edo]]"</comment>
      <origin>234145</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18602" sha1="qp5zuc8t7y2et8zanm08l3izmgp8hz5" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
118edo is the first [[5-limit]] equal division which clearly gives [[microtemperament|microtempering]], with [[error]]s well under half a cent. It represents the intersection of the [[5-limit]] [[schismatic]] and [[parakleismic]] temperaments, [[tempering out]] both the [[schisma]], {{monzo| -15 8 1 }} and the [[parakleisma]], {{monzo| 8 14 -13 }}, as well as the [[vishnuzma]], {{monzo| 23 6 -14 }}, the [[hemithirds comma]], {{monzo| 38 -2 -15 }}, and the [[kwazy comma]], {{monzo| -53 10 16 }}.

118edo is the 17th [[zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the [[hemimean comma]], but [[99edo]] does better with that.

In the 11-limit, it tempers out [[385/384]] and [[441/440]], and is an excellent tuning for [[portent]], the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]]. 

Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].

=== Prime harmonics ===
{{Harmonics in equal|118}}

=== Octave stretch ===
118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part. 

=== Subsets and supersets ===
Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].

== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Cents
! Marks
! Approximate Ratios*
! Eliora's Naming System&lt;br /&gt;(+Shruti 22 correspondence)
! Chemical Notation&lt;br /&gt;(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
| 
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
| 
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
| 
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
| 
| [[50/49]]
| 
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
| 
| [[36/35]]
| 
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
| 
| [[28/27]]
| 
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
| 
| [[25/24]]
| 
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
| 
| [[21/20]], [[22/21]]
| 
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
| 
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
| 
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
| 
| [[15/14]]
| 
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
| 
| [[27/25]]
| 
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
| 
| [[88/81]]
| 
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
| 
| [[12/11]]
| 
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
| 
| [[11/10]]
| 
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
| 
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
| 
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
| 
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
| 
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
| 
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
| 
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
| 
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
| 
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
| 
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
| 
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
| 
| [[33/28]]
| 
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
| 
| [[25/21]]
| 
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
| 
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
| 
| [[98/81]]
| 
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
| 
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
| 
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
| 
| [[27/22]], [[16/13]] I**
| Minor tridecimal neutral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
| 
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
| 
| [[56/45]]
| 
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
| 
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
| 
| [[63/50]]
| 
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
| 
| [[14/11]]
| 
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
| 
| [[77/60]]
| 
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
| 
| [[9/7]]
| 
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
| 
| [[35/27]], [[22/17]]
| 
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
| 
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
| 
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
| 
| [[320/243]]
| 
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
| 
| [[160/121]], [[85/64]]
| 
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
| 
| [[75/56]], [[51/38]]
| 
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
| 
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
| 
| [[49/36]], [[19/14]]
| 
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
| 
| [[15/11]]
|
| iodine
| {{UDnote|step=53}}
| OG, uGb
|-
| 54
| 549.15
| 
| [[48/35]], [[11/8]]
|
| xenon
| {{UDnote|step=54}}
| UG, oAb
|-
| 55
| 559.32
| 
| [[112/81]]
| 
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
| 
| [[25/18]]
| 
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
| 
| [[7/5]]
| 
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
| 
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
| 
| [[10/7]]
| 
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
| 
| [[36/25]]
| 
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
| 
| [[81/56]]
| 
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
| 
| [[35/24]], [[16/11]]
| 
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
| 
| [[22/15]]
| 
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
| 
| [[72/49]], [[28/19]]
| 
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
| 
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
| 
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
| 
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
| 
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
| 
| [[32/21]]
| 
| hafnium
| {{UDnote|step=72}}
| SA
|-
| 73
| 742.37
| 
| [[75/49]]
| 
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
| 74
| 752.54
| 
| [[54/35]], [[17/11]]
| 
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
| 75
| 762.71
| 
| [[14/9]]
| 
| rhenium
| {{UDnote|step=75}}
| sBb
|-
| 76
| 772.88
| 
| [[120/77]]
| 
| osmium
| {{UDnote|step=76}}
| kBb
|-
| 77
| 783.05
| 
| [[11/7]]
| 
| iridium
| {{UDnote|step=77}}
| lBb
|-
| 78
| 793.22
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
| 79
| 803.39
| 
| [[100/63]]
| 
| gold
| {{UDnote|step=79}}
| LBb
|-
| 80
| 813.56
| 
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
| 81
| 823.73
| 
| [[45/28]]
| 
| thallium
| {{UDnote|step=81}}
| SBb
|-
| 82
| 833.90
| 
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
| 83
| 844.07
| 
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
| 84
| 854.24
| 
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
| 85
| 864.41
| 
| [[28/17]]
| 
| astatine
| {{UDnote|step=85}}
| oB
|-
| 86
| 874.58
| 
| [[81/49]]
| 
| radon
| {{UDnote|step=86}}
| sB
|-
| 87
| 884.75
| 
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
| 88
| 894.92
| 
| [[42/25]]
| 
| radium
| {{UDnote|step=88}}
| lB
|-
| 89
| 905.08
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
| 90
| 915.25
| 
| [[56/33]]
| 
| thorium
| {{UDnote|step=90}}
| LB
|-
| 91
| 925.42
| 
| [[128/75]]
| 
| protactinium
| {{UDnote|step=91}}
| KB
|-
| 92
| 935.59
| 
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
| 93
| 945.76
| 
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| 
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
| 94
| 955.93
| 
| [[125/72]]
| 
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
| 95
| 966.10
| 
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
| 96
| 976.27
| 
| [[225/128]]
| 
| curium
| {{UDnote|step=96}}
| kC
|-
| 97
| 986.44
| 
| [[30/17]]
| 
| berkelium
| {{UDnote|step=97}}
| lC
|-
| 98
| 996.61
| m7
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
| 99
| 1006.78
| 
| [[25/14]]
| 
| einsteinium
| {{UDnote|step=99}}
| LC
|-
| 100
| 1016.95
| 
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
| 101
| 1027.12
| 
| [[38/21]]
| 
| mendelevium
| {{UDnote|step=101}}
| SC
|-
| 102
| 1037.29
| 
| [[20/11]]
| 
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
| 103
| 1047.46
| 
| [[11/6]]
| 
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
| 104
| 1057.63
| 
| [[81/44]]
| 
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
| 105
| 1067.80
| 
| [[50/27]]
| 
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
| 106
| 1077.97
| 
| [[28/15]]
| 
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
| 107
| 1088.14
| 
| [[15/8]]
| 
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
| 
| [[32/17]], [[17/9]]
| 
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
| 
| [[40/21]], [[21/11]]
| 
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
| 
| [[48/25]]
| 
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
| 
| [[27/14]]
| 
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
| 
| [[35/18]], [[64/33]]
| 
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
| 
| [[49/25]]
| 
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
| 
| [[63/32]], [[96/49]]
| 
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
| 
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
| 
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
&lt;nowiki /&gt;* {{sg|2.3.5.7.11.17.19 subgroup}}

&lt;nowiki /&gt;** Based on a dual-interval interpretation for the 13th harmonic

== Notation ==
=== Chemical notation ===
This notation was proposed by Eliora in November 2021. 

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.

The following are the correspondences of the periodic table structure with 118edo:  

* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.   
* Mercury, ending the 6d-block, corresponds to 8/5.   
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.  
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.

== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -187 118 }}
| {{mapping| 118 187 }}
| &amp;minus;0.119
| 0.082
| 0.81
|-
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| {{mapping| 118 187 274 }}
| +0.036
| 0.093
| 0.91
|-
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 }}
| +0.270
| 0.412
| 4.05
|-
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| {{mapping| 118 187 274 331 408 }}
| +0.341
| 0.370
| 3.89
|- style="border-top: double;"
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| 0.604
| 5.93
|- style="border-top: double;"
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| 0.650
| 6.39
|- style="border-top: double;"
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| 0.399
| 3.92
|-
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| 0.376
| 3.69
|}
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 11\118
| 111.86
| 16/15
| [[Vavoom]]
|-
| 1
| 19\118
| 193.22
| 28/25
| [[Luna]] / [[hemithirds]] / [[lunatic]]
|-
| 1
| 23\118
| 233.90
| 8/7
| [[Slendric]] / [[guiron]]
|-
| 1
| 31\118
| 315.25
| 6/5
| [[Parakleismic]] / paralytic
|-
| 1
| 39\118
| 396.61
| 44/35
| [[Squarschmidt]]
|-
| 1
| 49\118
| 498.31
| 4/3
| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
| 1
| 55\118
| 559.32
| 242/175
| [[Tritriple]]
|-
| 2
| 2\118
| 20.34
| 81/80
| [[Bicommatic]]
|-
| 2
| 5\118
| 50.85
| 33/32~36/35
| [[Kleischismic]]
|-
| 2
| 7\118
| 71.19
| 25/24
| [[Vishnu]] / ananta (118) / acyuta (118f)
|-
| 2
| 10\118
| 101.69
| 35/33
| [[Bischismic]] / bipont (118) / counterbipont (118f)
|-
| 2
| 16\118
| 162.71
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 18\118
| 183.05
| 10/9
| [[Unidec]] / ekadash (118) / hendec (118f)
|-
| 2
| 19\118
| 193.22
| 121/108
| [[Semiluna]]
|-
| 2
| 31\118&lt;br&gt;(28\118)
| 315.25&lt;br&gt;(284.75)
| 6/5&lt;br&gt;(33/28)
| [[Semiparakleismic]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Instruments ==
[[Lumatone mapping for 118edo]]

== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)

[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Schismic]]</text>
      <sha1>qp5zuc8t7y2et8zanm08l3izmgp8hz5</sha1>
    </revision>
  </page>
  <page>
    <title>119edo</title>
    <ns>0</ns>
    <id>65</id>
    <revision>
      <id>65</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>65</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="184" sha1="akbf264yi07h58f8byqb27mdb15rlnt" xml:space="preserve">Offers four fifths:

68 steps: 7edo superflat fifth

69 steps: Approximation of 2/7 comma meantone

70 steps: 17edo sharp fifth (patent)

71 steps: Approximation of half comma eventone</text>
      <sha1>akbf264yi07h58f8byqb27mdb15rlnt</sha1>
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    <revision>
      <id>4312</id>
      <timestamp>2016-11-28T14:09:58Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 600646744 - Original comment: **</comment>
      <origin>4312</origin>
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      <text bytes="1631" sha1="bxqfo56ngcstaalvvaiezyhu983ie8h" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-11-28 14:09:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;600646744&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;Offers four fifths:
68 steps: 7edo superflat fifth
69 steps: Approximation of 2/7 comma meantone
70 steps: 17edo sharp fifth (patent)
71 steps: Approximation of half comma eventone&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;119edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;Offers four fifths:&amp;lt;br /&amp;gt;
68 steps: 7edo superflat fifth&amp;lt;br /&amp;gt;
69 steps: Approximation of 2/7 comma meantone&amp;lt;br /&amp;gt;
70 steps: 17edo sharp fifth (patent)&amp;lt;br /&amp;gt;
71 steps: Approximation of half comma eventone&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4313</id>
      <timestamp>2016-10-09T07:04:49Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 594760378 - Original comment: **</comment>
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      <text bytes="1537" sha1="dn15e5yc9fus3jw30k9urfvledcb4yk" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-10-09 07:04:49 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;594760378&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;Offers three fifths:
68 steps: 7edo superflat fifth
69 steps: Approximation of 2/7 comma meantone
70 steps: 17edo sharp fifth (patent)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;119edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;Offers three fifths:&amp;lt;br /&amp;gt;
68 steps: 7edo superflat fifth&amp;lt;br /&amp;gt;
69 steps: Approximation of 2/7 comma meantone&amp;lt;br /&amp;gt;
70 steps: 17edo sharp fifth (patent)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>59497</id>
      <parentid>65</parentid>
      <timestamp>2021-01-19T05:01:40Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>added a proper opening sentence to this article</comment>
      <origin>59497</origin>
      <model>wikitext</model>
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      <text bytes="328" sha1="fr3fc1rax7oq9hkrblt8q0h4p7kla2f" xml:space="preserve">'''119edo''' is the [[equal division of the octave]] into 119 parts of 10.084 cents each.  It is notable for offering as many as four different options for fifths:

68 steps: 7edo superflat fifth

69 steps: Approximation of 2/7 comma meantone

70 steps: 17edo sharp fifth (patent)

71 steps: Approximation of half comma eventone</text>
      <sha1>fr3fc1rax7oq9hkrblt8q0h4p7kla2f</sha1>
    </revision>
    <revision>
      <id>67316</id>
      <parentid>59497</parentid>
      <timestamp>2021-04-14T20:41:28Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>67316</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="612" sha1="bpnq1ocgj0k50b35w4xukegq5by9c3b" xml:space="preserve">'''119edo''' is the [[equal division of the octave]] into 119 parts of 10.084 cents each.  It is notable for offering as many as four different options for fifths:

68 steps: 7edo superflat fifth: 17 17 17 17 17 17 17

69 steps: Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12

70 steps: 17edo sharp fifth (patent): 21 21 7 21 21 21 7, 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1

71 steps: Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2</text>
      <sha1>bpnq1ocgj0k50b35w4xukegq5by9c3b</sha1>
    </revision>
    <revision>
      <id>92929</id>
      <parentid>67316</parentid>
      <timestamp>2022-07-02T16:23:59Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Categories</comment>
      <origin>92929</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="683" sha1="3gzvstuyq9cca7w473u0o3c0epgja7c" xml:space="preserve">'''119edo''' is the [[equal division of the octave]] into 119 parts of 10.084 cents each. It is notable for offering as many as four different options for fifths:

68 steps: 7edo superflat fifth: 17 17 17 17 17 17 17

69 steps: Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12

70 steps: 17edo sharp fifth (patent): 21 21 7 21 21 21 7, 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1

71 steps: Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>3gzvstuyq9cca7w473u0o3c0epgja7c</sha1>
    </revision>
    <revision>
      <id>96197</id>
      <parentid>92929</parentid>
      <timestamp>2022-10-04T18:52:43Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96197</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="698" sha1="7t0sq0rkhkehl8fbsayuq64nznwu3ol" xml:space="preserve">{{Infobox ET}}
'''119edo''' is the [[equal division of the octave]] into 119 parts of 10.084 cents each. It is notable for offering as many as four different options for fifths:

68 steps: 7edo superflat fifth: 17 17 17 17 17 17 17

69 steps: Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12

70 steps: 17edo sharp fifth (patent): 21 21 7 21 21 21 7, 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1

71 steps: Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>7t0sq0rkhkehl8fbsayuq64nznwu3ol</sha1>
    </revision>
    <revision>
      <id>99695</id>
      <parentid>96197</parentid>
      <timestamp>2022-12-07T20:08:08Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>99695</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="667" sha1="trjlyfj624ec1srq2u51eo1hnnfh79m" xml:space="preserve">{{Infobox ET}}

== Theory ==
It is notable for offering as many as four different options for fifths:

68 steps: 7edo superflat fifth: 17 17 17 17 17 17 17

69 steps: Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12

70 steps: 17edo sharp fifth (patent): 21 21 7 21 21 21 7, 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1

71 steps: Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

=== Harmonics ===
{{Harmonics in equal|119}}
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>trjlyfj624ec1srq2u51eo1hnnfh79m</sha1>
    </revision>
    <revision>
      <id>99696</id>
      <parentid>99695</parentid>
      <timestamp>2022-12-07T20:08:25Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>real intro</comment>
      <origin>99696</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="680" sha1="pyfp77rx0kgf9yqx6s53wo7kp2wuyjt" xml:space="preserve">{{Infobox ET}}
{{EDO intro}}
== Theory ==
It is notable for offering as many as four different options for fifths:

68 steps: 7edo superflat fifth: 17 17 17 17 17 17 17

69 steps: Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12

70 steps: 17edo sharp fifth (patent): 21 21 7 21 21 21 7, 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1

71 steps: Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

=== Harmonics ===
{{Harmonics in equal|119}}
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>pyfp77rx0kgf9yqx6s53wo7kp2wuyjt</sha1>
    </revision>
    <revision>
      <id>99697</id>
      <parentid>99696</parentid>
      <timestamp>2022-12-07T20:08:36Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <minor/>
      <comment>oops</comment>
      <origin>99697</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="684" sha1="tvh7ms19y8l6yx964vm0xmt2ry3t3oy" xml:space="preserve">{{Infobox ET}}
{{EDO intro|119}}
== Theory ==
It is notable for offering as many as four different options for fifths:

68 steps: 7edo superflat fifth: 17 17 17 17 17 17 17

69 steps: Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12

70 steps: 17edo sharp fifth (patent): 21 21 7 21 21 21 7, 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1

71 steps: Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

=== Harmonics ===
{{Harmonics in equal|119}}
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>tvh7ms19y8l6yx964vm0xmt2ry3t3oy</sha1>
    </revision>
    <revision>
      <id>127744</id>
      <parentid>99697</parentid>
      <timestamp>2023-11-21T10:47:42Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>Rework</comment>
      <origin>127744</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1171" sha1="dkzgbepv9bacia386og02vp73kdq6vg" xml:space="preserve">{{Infobox ET}}
{{EDO intro|119}}
== Theory ==
119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit.

=== Odd harmonics ===
{{Harmonics in equal|119|columns=20}}
=== Subsets and supersets ===
Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

== Scales ==

* Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12
* Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>dkzgbepv9bacia386og02vp73kdq6vg</sha1>
    </revision>
    <revision>
      <id>132898</id>
      <parentid>127744</parentid>
      <timestamp>2024-01-25T09:48:18Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Broke single long harmonics table into 2 smaller harmonics tables (1 collapsed)</comment>
      <origin>132898</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1289" sha1="3n2mdwb602ph13ms0eehuxr6bsivrxf" xml:space="preserve">{{Infobox ET}}
{{EDO intro|119}}
== Theory ==
119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit.

=== Odd harmonics ===
{{Harmonics in equal|119|columns=12}}
{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 84edo (continued)}}

=== Subsets and supersets ===
Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

== Scales ==

* Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12
* Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>3n2mdwb602ph13ms0eehuxr6bsivrxf</sha1>
    </revision>
    <revision>
      <id>136286</id>
      <parentid>132898</parentid>
      <timestamp>2024-02-19T18:58:00Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <origin>136286</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1290" sha1="8tixmloufaaj3874gtguusyhili8reb" xml:space="preserve">{{Infobox ET}}
{{EDO intro|119}}
== Theory ==
119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit.

=== Odd harmonics ===
{{Harmonics in equal|119|columns=12}}
{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 119edo (continued)}}

=== Subsets and supersets ===
Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

== Scales ==

* Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12
* Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>8tixmloufaaj3874gtguusyhili8reb</sha1>
    </revision>
    <revision>
      <id>157509</id>
      <parentid>136286</parentid>
      <timestamp>2024-10-02T12:44:22Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>+collapsed interval table</comment>
      <origin>157509</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1326" sha1="cwnlz1ddon8qwznu5pfj57f8d2v85s7" xml:space="preserve">{{Infobox ET}}
{{EDO intro|119}}
== Theory ==
119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit.

=== Odd harmonics ===
{{Harmonics in equal|119|columns=12}}
{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 119edo (continued)}}

=== Subsets and supersets ===
Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

== Intervals ==
{{Interval table}}

== Scales ==

* Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12
* Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>cwnlz1ddon8qwznu5pfj57f8d2v85s7</sha1>
    </revision>
    <revision>
      <id>160101</id>
      <parentid>157509</parentid>
      <timestamp>2024-10-15T05:59:25Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <comment>added meantone mention</comment>
      <origin>160101</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1407" sha1="5t8e16krwagw9rcwzdu29hf7qhw0sgg" xml:space="preserve">{{Infobox ET}}
{{EDO intro|119}}
== Theory ==
119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit, while the 119b val is an extremely good approximation to [[2/7-comma meantone]].

=== Odd harmonics ===
{{Harmonics in equal|119|columns=12}}
{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 119edo (continued)}}

=== Subsets and supersets ===
Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

== Intervals ==
{{Interval table}}

== Scales ==

* Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12
* Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>5t8e16krwagw9rcwzdu29hf7qhw0sgg</sha1>
    </revision>
    <revision>
      <id>160104</id>
      <parentid>160101</parentid>
      <timestamp>2024-10-15T06:00:40Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <origin>160104</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1507" sha1="9v027x0o1fnokpppj451lbhhc3gofj5" xml:space="preserve">{{Infobox ET}}
{{EDO intro|119}}
== Theory ==
119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit, while the 119b val is an extremely good approximation to [[2/7-comma meantone]] in addition to supporting [[chlorine]] (by equating [[25/24]] very accurately to one step of 17edo).

=== Odd harmonics ===
{{Harmonics in equal|119|columns=12}}
{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 119edo (continued)}}

=== Subsets and supersets ===
Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

== Intervals ==
{{Interval table}}

== Scales ==

* Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12
* Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>9v027x0o1fnokpppj451lbhhc3gofj5</sha1>
    </revision>
    <revision>
      <id>181324</id>
      <parentid>160104</parentid>
      <timestamp>2025-02-18T16:54:17Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181324</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1502" sha1="buuyu9uamtxdf0olu7vkcgthczafd1z" xml:space="preserve">{{Infobox ET}}
{{ED intro}}
== Theory ==
119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit, while the 119b val is an extremely good approximation to [[2/7-comma meantone]] in addition to supporting [[chlorine]] (by equating [[25/24]] very accurately to one step of 17edo).

=== Odd harmonics ===
{{Harmonics in equal|119|columns=12}}
{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 119edo (continued)}}

=== Subsets and supersets ===
Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

== Intervals ==
{{Interval table}}

== Scales ==

* Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12
* Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>buuyu9uamtxdf0olu7vkcgthczafd1z</sha1>
    </revision>
    <revision>
      <id>204919</id>
      <parentid>181324</parentid>
      <timestamp>2025-07-20T22:21:15Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
      <origin>204919</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1517" sha1="i6650z1t5p0npldqg3oh0hyr0fdzyd5" xml:space="preserve">{{Infobox ET}}
{{ED intro}}
== Theory ==
119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit, while the 119b val is an extremely good approximation to [[2/7-comma meantone]] in addition to supporting [[chlorine]] (by equating [[25/24]] very accurately to one step of 17edo) and [[mothra]].

=== Odd harmonics ===
{{Harmonics in equal|119|columns=12}}
{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 119edo (continued)}}

=== Subsets and supersets ===
Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

== Intervals ==
{{Interval table}}

== Scales ==

* Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12
* Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>i6650z1t5p0npldqg3oh0hyr0fdzyd5</sha1>
    </revision>
    <revision>
      <id>204920</id>
      <parentid>204919</parentid>
      <timestamp>2025-07-20T22:21:41Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <origin>204920</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1525" sha1="ioyrr8ivz0ef6vpx82x73nvjo6ebmfq" xml:space="preserve">{{Infobox ET}}
{{ED intro}}
== Theory ==
119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit, while the 119b val is an extremely good approximation to [[2/7-comma meantone]] in addition to supporting [[chlorine]] (by equating [[25/24]] very accurately to one step of 17edo) and 7-limit [[mothra]].

=== Odd harmonics ===
{{Harmonics in equal|119|columns=12}}
{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 119edo (continued)}}

=== Subsets and supersets ===
Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

== Intervals ==
{{Interval table}}

== Scales ==

* Approximation of 2/7 comma meantone: 19 19 19 12 19 19 19 19 12
* Approximation of half comma eventone: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>ioyrr8ivz0ef6vpx82x73nvjo6ebmfq</sha1>
    </revision>
    <revision>
      <id>206607</id>
      <parentid>204920</parentid>
      <timestamp>2025-08-09T22:15:12Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>"eventone" is idiosyncratic</comment>
      <origin>206607</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1529" sha1="9gokflixv8k3azvh5a2l5g2aao2464b" xml:space="preserve">{{Infobox ET}}
{{ED intro}}
== Theory ==
119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit, while the 119b val is an extremely good approximation to [[2/7-comma meantone]] in addition to supporting [[chlorine]] (by equating [[25/24]] very accurately to one step of 17edo) and 7-limit [[mothra]].

=== Odd harmonics ===
{{Harmonics in equal|119|columns=12}}
{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 119edo (continued)}}

=== Subsets and supersets ===
Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

== Intervals ==
{{Interval table}}

== Scales ==
* Approximation of 2/7 comma [[meantone]]: 19 19 19 12 19 19 19 19 12
* Approximation of half comma [[archy]]: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>9gokflixv8k3azvh5a2l5g2aao2464b</sha1>
    </revision>
    <revision>
      <id>221971</id>
      <parentid>206607</parentid>
      <timestamp>2026-01-19T07:22:16Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Scales */ Approximation of Fix a/7 comma &amp;mdash; had 2 major seconds too many, as if for a superdiatonic tuning of 157edo</comment>
      <origin>221971</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1523" sha1="hb3sbu1bg1bbo931b1vgeyyo93xhp03" xml:space="preserve">{{Infobox ET}}
{{ED intro}}
== Theory ==
119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit, while the 119b val is an extremely good approximation to [[2/7-comma meantone]] in addition to supporting [[chlorine]] (by equating [[25/24]] very accurately to one step of 17edo) and 7-limit [[mothra]].

=== Odd harmonics ===
{{Harmonics in equal|119|columns=12}}
{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 119edo (continued)}}

=== Subsets and supersets ===
Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

== Intervals ==
{{Interval table}}

== Scales ==
* Approximation of 2/7 comma [[meantone]]: 19 19 12 19 19 19 12
* Approximation of half comma [[archy]]: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>hb3sbu1bg1bbo931b1vgeyyo93xhp03</sha1>
    </revision>
    <revision>
      <id>232252</id>
      <parentid>221971</parentid>
      <timestamp>2026-06-14T01:57:27Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Theory */ 119be</comment>
      <origin>232252</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1591" sha1="jf6f7gq24qnlpjggocybp7ius2w3mmi" xml:space="preserve">{{Infobox ET}}
{{ED intro}}
== Theory ==
119edo is in[[consistent]] in the 5-odd-limit, with both harmonics 3 and 5 falling halfway between steps. It does have potential as a 2.7.9.15 subgroup system. In higher limits, 2.7.15.29.37 is a strong interpretation.

Nonetheless, there is a number of mappings to be considered. In the 11-limit, 119edo's provides the [[optimal patent val]] for the 11-limit [[androboh]] and [[quasitemp]] temperaments. The patent val also tunes the 11-limit [[quadrawell]] temperament. 119c val tunes [[treecreeper]], [[sensus]], and [[senator]] as high as the 17-limit, while the 119b val is an extremely good approximation to [[2/7-comma meantone]] in addition to supporting [[chlorine]] (by equating [[25/24]] very accurately to one step of 17edo) and 7-limit [[mothra]], with the 119be val supporting 11-limit mothra with a flat tendency.

=== Odd harmonics ===
{{Harmonics in equal|119|columns=12}}
{{Harmonics in equal|119|columns=12|start=13|collapsed=1|title=Approximation of odd harmonics in 119edo (continued)}}

=== Subsets and supersets ===
Since 119edo factors as {{Factorization|119}}, it contains [[7edo]] and [[17edo]] as a subset. Hence it supports circles of fifths of those respective equal temperaments.

== Intervals ==
{{Interval table}}

== Scales ==
* Approximation of 2/7 comma [[meantone]]: 19 19 12 19 19 19 12
* Approximation of half comma [[archy]]: 23 23 2 23 23 23 2, 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 7 2 2 2 2 2 2 2 2 2

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
      <sha1>jf6f7gq24qnlpjggocybp7ius2w3mmi</sha1>
    </revision>
  </page>
  <page>
    <title>11/1</title>
    <ns>0</ns>
    <id>66</id>
    <revision>
      <id>66</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>66</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="186" sha1="dtkq10mcc42v5qwbd46xsto4h9ldu94" xml:space="preserve">'''11/1'''
|0 0 0 0 1&amp;gt;

4151.3179 cents

[[File:jid_11_1_pluck_adu_dr55.mp3]] [[:File:jid_11_1_pluck_adu_dr55.mp3|sound sample]]

The 11th harmonic; three octaves above [[11/8|11/8]].</text>
      <sha1>dtkq10mcc42v5qwbd46xsto4h9ldu94</sha1>
    </revision>
    <revision>
      <id>4314</id>
      <timestamp>2014-06-08T14:03:59Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513249370 - Original comment: **</comment>
      <origin>4314</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2919" sha1="66wblnnrm29qinonj5guyayxb2163je" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-08 14:03:59 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513249370&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/1**
|0 0 0 0 1&amp;gt;
4151.3179 cents
[[media type="file" key="jid_11_1_pluck_adu_dr55.mp3"]] [[file:xenharmonic/jid_11_1_pluck_adu_dr55.mp3|sound sample]]

The 11th harmonic; three octaves above [[11_8|11/8]].
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_1&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/1&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|0 0 0 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
4151.3179 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_1_pluck_adu_dr55.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_1_pluck_adu_dr55.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_1_pluck_adu_dr55.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_1_pluck_adu_dr55.mp3/513246290/jid_11_1_pluck_adu_dr55.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_1_pluck_adu_dr55.mp3/513246290/jid_11_1_pluck_adu_dr55.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 11th harmonic; three octaves above &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>66wblnnrm29qinonj5guyayxb2163je</sha1>
    </revision>
    <revision>
      <id>64437</id>
      <parentid>66</parentid>
      <timestamp>2021-03-27T04:02:03Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64437</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="235" sha1="hbo8z2qfkgfwrb3gafjbnuvsl38ia5h" xml:space="preserve">'''11/1'''
|0 0 0 0 1&amp;gt;

4151.3179 cents

[[File:jid_11_1_pluck_adu_dr55.mp3]] [[:File:jid_11_1_pluck_adu_dr55.mp3|sound sample]]

The 11th harmonic; three octaves above [[11/8|11/8]].

[[Category:Pages with internal sound examples]]</text>
      <sha1>hbo8z2qfkgfwrb3gafjbnuvsl38ia5h</sha1>
    </revision>
    <revision>
      <id>82505</id>
      <parentid>64437</parentid>
      <timestamp>2021-12-23T12:17:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+infobox</comment>
      <origin>82505</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="419" sha1="57fh9sr1exjr8wvpvhor8qiyqqyp1of" xml:space="preserve">{{Infobox Interval
| Ratio = 11/1
| Monzo = 0 0 0 0 1
| Cents = 4151.31794
| Name = 11th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;1o4, tricolo 4th
| FJS name = P25&lt;sup&gt;13&lt;/sup&gt;
| Sound = jid_11_1_pluck_adu_dr55.mp3
}}

'''11/1''' is the 11th harmonic; three octaves above [[11/8]].

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Harmonics]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>57fh9sr1exjr8wvpvhor8qiyqqyp1of</sha1>
    </revision>
    <revision>
      <id>86282</id>
      <parentid>82505</parentid>
      <timestamp>2022-02-22T23:46:38Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86282</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="371" sha1="fahwaq989jqr3sfg9oefiqyxthxc2d6" xml:space="preserve">{{Infobox Interval
| Ratio = 11/1
| Monzo = 0 0 0 0 1
| Cents = 4151.31794
| Name = 11th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;1o4, tricolo 4th
| FJS name = P25&lt;sup&gt;13&lt;/sup&gt;
| Sound = jid_11_1_pluck_adu_dr55.mp3
}}

'''11/1''' is the 11th harmonic; three octaves above [[11/8]].

== See also ==
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Harmonics]]</text>
      <sha1>fahwaq989jqr3sfg9oefiqyxthxc2d6</sha1>
    </revision>
    <revision>
      <id>97990</id>
      <parentid>86282</parentid>
      <timestamp>2022-10-25T11:58:34Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97990</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="239" sha1="q2blvrewuaui5uq16jro6zkxok4wlo4" xml:space="preserve">{{Infobox Interval
| Name = 11th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;1o4, tricolo 4th
| Sound = jid_11_1_pluck_adu_dr55.mp3
}}

'''11/1''' is the 11th harmonic; three octaves above [[11/8]].

== See also ==
* [[Gallery of just intervals]]</text>
      <sha1>q2blvrewuaui5uq16jro6zkxok4wlo4</sha1>
    </revision>
    <revision>
      <id>118786</id>
      <parentid>97990</parentid>
      <timestamp>2023-07-06T02:26:37Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added to category “equivalence interval”</comment>
      <origin>118786</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="273" sha1="pa5au0auvk484y9p0v369r394e4qkpm" xml:space="preserve">{{Infobox Interval
| Name = 11th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;1o4, tricolo 4th
| Sound = jid_11_1_pluck_adu_dr55.mp3
}}

'''11/1''' is the 11th harmonic; three octaves above [[11/8]].

== See also ==
* [[Gallery of just intervals]]
[[Category:Equivalence interval]]</text>
      <sha1>pa5au0auvk484y9p0v369r394e4qkpm</sha1>
    </revision>
    <revision>
      <id>118816</id>
      <parentid>118786</parentid>
      <timestamp>2023-07-06T02:56:13Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Equivalence interval]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>118816</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="239" sha1="q2blvrewuaui5uq16jro6zkxok4wlo4" xml:space="preserve">{{Infobox Interval
| Name = 11th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;1o4, tricolo 4th
| Sound = jid_11_1_pluck_adu_dr55.mp3
}}

'''11/1''' is the 11th harmonic; three octaves above [[11/8]].

== See also ==
* [[Gallery of just intervals]]</text>
      <sha1>q2blvrewuaui5uq16jro6zkxok4wlo4</sha1>
    </revision>
    <revision>
      <id>129170</id>
      <parentid>118816</parentid>
      <timestamp>2023-12-16T06:41:44Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Expansion</comment>
      <origin>129170</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="443" sha1="0mrnuptuudwmlhsyf2b1z4ot9zq59nk" xml:space="preserve">{{Infobox Interval
| Name = 11th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;1o4, tricolo 4th
| Sound = jid_11_1_pluck_adu_dr55.mp3
}}
'''11/1''', the '''11th harmonic''', is the [[harmonic]] past [[10/1]] and before [[12/1]]. It is three octaves above [[11/8]], and is the basis of [[11-limit]] harmony, as many 11-limit intervals can be expressed as the difference between this and another harmonic. 

== See also ==
* [[Gallery of just intervals]]</text>
      <sha1>0mrnuptuudwmlhsyf2b1z4ot9zq59nk</sha1>
    </revision>
    <revision>
      <id>129176</id>
      <parentid>129170</parentid>
      <timestamp>2023-12-16T06:45:25Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+link</comment>
      <origin>129176</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="447" sha1="qtoil4cuepdcdv7hoiyqbi2okiwjmxn" xml:space="preserve">{{Infobox Interval
| Name = 11th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;1o4, tricolo 4th
| Sound = jid_11_1_pluck_adu_dr55.mp3
}}
'''11/1''', the '''11th harmonic''', is the [[harmonic]] past [[10/1]] and before [[12/1]]. It is three [[octave]]s above [[11/8]], and is the basis of [[11-limit]] harmony, as many 11-limit intervals can be expressed as the difference between this and another harmonic. 

== See also ==
* [[Gallery of just intervals]]</text>
      <sha1>qtoil4cuepdcdv7hoiyqbi2okiwjmxn</sha1>
    </revision>
    <revision>
      <id>162434</id>
      <parentid>129176</parentid>
      <timestamp>2024-10-27T00:07:39Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* See also */ link</comment>
      <origin>162434</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="458" sha1="a5s5hn1jgicx5wdf4ev2tb29cvuohx9" xml:space="preserve">{{Infobox Interval
| Name = 11th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;1o4, tricolo 4th
| Sound = jid_11_1_pluck_adu_dr55.mp3
}}
'''11/1''', the '''11th harmonic''', is the [[harmonic]] past [[10/1]] and before [[12/1]]. It is three [[octave]]s above [[11/8]], and is the basis of [[11-limit]] harmony, as many 11-limit intervals can be expressed as the difference between this and another harmonic. 

== See also ==
* [[Gallery of just intervals]]
* [[Ed11]]</text>
      <sha1>a5s5hn1jgicx5wdf4ev2tb29cvuohx9</sha1>
    </revision>
    <revision>
      <id>189101</id>
      <parentid>162434</parentid>
      <timestamp>2025-03-31T10:33:00Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+"undecuple"</comment>
      <origin>189101</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="488" sha1="tczxxzex3b9lopdgcxsq1rztp19089x" xml:space="preserve">{{Infobox Interval
| Name = 11th harmonic, undecuple
| Color name = c&lt;sup&gt;3&lt;/sup&gt;1o4, tricolo 4th
| Sound = jid_11_1_pluck_adu_dr55.mp3
}}
'''11/1''', the '''11th harmonic''' or '''undecuple''', is the [[harmonic]] past [[10/1]] and before [[12/1]]. It is three [[octave]]s above [[11/8]], and is the basis of [[11-limit]] harmony, as many 11-limit intervals can be expressed as the difference between this and another harmonic. 

== See also ==
* [[Gallery of just intervals]]
* [[Ed11]]</text>
      <sha1>tczxxzex3b9lopdgcxsq1rztp19089x</sha1>
    </revision>
  </page>
  <page>
    <title>11 1</title>
    <ns>0</ns>
    <id>67</id>
    <redirect title="11/1" />
    <revision>
      <id>67</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>67</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18" sha1="62e3rmgouolckzwkkvn7itzm1weklt0" xml:space="preserve">#REDIRECT [[11/1]]</text>
      <sha1>62e3rmgouolckzwkkvn7itzm1weklt0</sha1>
    </revision>
  </page>
  <page>
    <title>11/10</title>
    <ns>0</ns>
    <id>68</id>
    <revision>
      <id>68</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>68</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="611" sha1="sgn8qw60gd8vtfgxi8n3pib9cl96fju" xml:space="preserve">'''11/10'''
|-1 0 -1 0 1&amp;gt;

165.00423 cents

[[File:jid_11_10_pluck_adu_dr220.mp3]] [[:File:jid_11_10_pluck_adu_dr220.mp3|sound sample]]

11/10, the large neutral second, is an interval favored by Ptolemy. Three of them are less than 4/3 by the wizardharry comma, 4000/3993.

Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.

See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]      [[Category:11-limit]]
[[Category:ji]]
[[Category:ratio]]
[[Category:todo:expand]]</text>
      <sha1>sgn8qw60gd8vtfgxi8n3pib9cl96fju</sha1>
    </revision>
    <revision>
      <id>4315</id>
      <timestamp>2016-12-28T13:57:26Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602876510 - Original comment: **</comment>
      <origin>4315</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3733" sha1="3ai5of3za0pten52cmg4kfz65w4g7ow" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2016-12-28 13:57:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602876510&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/10**
|-1 0 -1 0 1&amp;gt;
165.00423 cents
[[media type="file" key="jid_11_10_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_11_10_pluck_adu_dr220.mp3|sound sample]]

11/10, the large neutral second, is an interval favored by Ptolemy. Three of them are less than 4/3 by the wizardharry comma, 4000/3993.

Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_10&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/10&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|-1 0 -1 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
165.00423 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_10_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_10_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_10_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_10_pluck_adu_dr220.mp3/513194554/jid_11_10_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_10_pluck_adu_dr220.mp3/513194554/jid_11_10_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11/10, the large neutral second, is an interval favored by Ptolemy. Three of them are less than 4/3 by the wizardharry comma, 4000/3993.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>3ai5of3za0pten52cmg4kfz65w4g7ow</sha1>
    </revision>
    <revision>
      <id>4316</id>
      <timestamp>2016-09-25T09:05:43Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Chartrekhan</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 593235392 - Original comment: **</comment>
      <origin>4316</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3572" sha1="qjp8bfrpzmjv2ru29vmspwgnw4nzjk5" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Chartrekhan|Chartrekhan]] and made on &lt;tt&gt;2016-09-25 09:05:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;593235392&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/10**
|-1 0 -1 0 1&amp;gt;
165.00423 cents
[[media type="file" key="jid_11_10_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_11_10_pluck_adu_dr220.mp3|sound sample]]

11/10, the large neutral second, is an interval favored by Ptolemy. Three of them are less than 4/3 by the wizardharry comma, 4000/3993.

Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_10&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/10&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|-1 0 -1 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
165.00423 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_10_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_10_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_10_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_10_pluck_adu_dr220.mp3/513194554/jid_11_10_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_10_pluck_adu_dr220.mp3/513194554/jid_11_10_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11/10, the large neutral second, is an interval favored by Ptolemy. Three of them are less than 4/3 by the wizardharry comma, 4000/3993.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>qjp8bfrpzmjv2ru29vmspwgnw4nzjk5</sha1>
    </revision>
    <revision>
      <id>4317</id>
      <timestamp>2014-06-07T15:39:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513194580 - Original comment: **</comment>
      <origin>4317</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3068" sha1="t7kfgltpevjyqb689ut454o3ozsz1sn" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-07 15:39:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513194580&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/10**
|-1 0 -1 0 1&amp;gt;
165.00423 cents
[[media type="file" key="jid_11_10_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_11_10_pluck_adu_dr220.mp3|sound sample]]

11/10, the large neutral second, is an interval favored by Ptolemy. Three of them are less than 4/3 by the wizardharry comma, 4000/3993.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_10&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/10&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|-1 0 -1 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
165.00423 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_10_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_10_pluck_adu_dr220.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_10_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_10_pluck_adu_dr220.mp3/513194554/jid_11_10_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_10_pluck_adu_dr220.mp3/513194554/jid_11_10_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11/10, the large neutral second, is an interval favored by Ptolemy. Three of them are less than 4/3 by the wizardharry comma, 4000/3993.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>t7kfgltpevjyqb689ut454o3ozsz1sn</sha1>
    </revision>
    <revision>
      <id>4318</id>
      <timestamp>2011-08-25T16:45:43Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 248503783 - Original comment: **</comment>
      <origin>4318</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1500" sha1="gcteu33he37it7iv6n37l2rifln5eut" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-25 16:45:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;248503783&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;11/10, the large neutral second, is an interval favored by Ptolemy. Three of them are less than 4/3 by the wizardharry comma, 4000/3993.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_10&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;11/10, the large neutral second, is an interval favored by Ptolemy. Three of them are less than 4/3 by the wizardharry comma, 4000/3993.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>gcteu33he37it7iv6n37l2rifln5eut</sha1>
    </revision>
    <revision>
      <id>37705</id>
      <parentid>68</parentid>
      <timestamp>2018-10-23T22:17:36Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>added interval infobox and categories</comment>
      <origin>37705</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="728" sha1="3ga80wfd5bk0d7tbmui743u42xhnr04" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/10
| Monzo = -1 0 -1 0 1
| Cents = 165.00423
| Name = large neutral second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large neutral second''', is an interval favored by Ptolemy. Three of them are less than [[4/3]] by the wizardharry comma, 4000/3993.

Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.

: ''See also [[Gallery of Just Intervals]]''

[[Category:11-limit]]
[[Category:Interval]]
[[Category:Ji]]
[[Category:Ratio]]
[[Category:todo:expand]]
[[Category:Superparticular]]
[[Category:Second]]</text>
      <sha1>3ga80wfd5bk0d7tbmui743u42xhnr04</sha1>
    </revision>
    <revision>
      <id>38183</id>
      <parentid>37705</parentid>
      <timestamp>2018-10-29T18:00:29Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>replaced cat: Ji -&gt; Just interval; slightly reworked</comment>
      <origin>38183</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="781" sha1="rnrug2nc9tf43so19473dud44chiikc" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/10
| Monzo = -1 0 -1 0 1
| Cents = 165.00423
| Name = large neutral second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large neutral second''', is an interval favored by Ptolemy. Three of them are less than [[4/3]] by the wizardharry comma, 4000/3993.

Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[Gallery of Just Intervals]]
* [[20/11]] - its [[inverse interval]]

[[Category:11-limit]]
[[Category:Interval]]
[[Category:Ratio]]
[[Category:Just interval]]
[[Category:Superparticular]]
[[Category:Second]]

[[Category:todo:expand]]</text>
      <sha1>rnrug2nc9tf43so19473dud44chiikc</sha1>
    </revision>
    <revision>
      <id>46317</id>
      <parentid>38183</parentid>
      <timestamp>2020-06-10T07:12:14Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cats</comment>
      <origin>46317</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="768" sha1="8llrc5mhxpuyhctrrvrb0xj9sr5jx3r" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/10
| Monzo = -1 0 -1 0 1
| Cents = 165.00423
| Name = large neutral second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large neutral second''', is an interval favored by Ptolemy. Three of them are less than [[4/3]] by the wizardharry comma, 4000/3993.

Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[Gallery of Just Intervals]]
* [[20/11]] - its [[inverse interval]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Superparticular]]
[[Category:Second]]

[[Category:todo:expand]]</text>
      <sha1>8llrc5mhxpuyhctrrvrb0xj9sr5jx3r</sha1>
    </revision>
    <revision>
      <id>48555</id>
      <parentid>46317</parentid>
      <timestamp>2020-09-05T03:44:43Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>48555</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="787" sha1="p1qkig1b8ycgi2q2hagpedvssxj8mqh" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/10
| Monzo = -1 0 -1 0 1
| Cents = 165.00423
| Name = large neutral second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large neutral second''', is an interval favored by Ptolemy. Three of them are less than [[4/3]] by the wizardharry comma, 4000/3993.

Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[Gallery of Just Intervals]]
* [[20/11]] - its [[inverse interval]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Superparticular]]
[[Category:Second]]
[[Category:Over-5]]
[[Category:todo:expand]]</text>
      <sha1>p1qkig1b8ycgi2q2hagpedvssxj8mqh</sha1>
    </revision>
    <revision>
      <id>49884</id>
      <parentid>48555</parentid>
      <timestamp>2020-09-20T07:21:36Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+FJS name; cleanup; match the name with its octave complement</comment>
      <origin>49884</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1045" sha1="6xmi7rny5ak85z0p52pxd9yp7uhfk1o" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/10
| Monzo = -1 0 -1 0 1
| Cents = 165.00423
| Name = large undecimal neutral second, &lt;br&gt;undecimal major second
| Color name = 1og2, logu 2nd
| FJS name = m2&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;5&lt;/sub&gt;
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal major second''', is an interval favored by Ptolemy. Three of them are less than [[4/3]] by the wizardharry comma, 4000/3993.

Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Superparticular]]
[[Category:Second]]
[[Category:Neutral second]]
[[Category:Whole tone]]
[[Category:Over-5]]

[[Category:todo:expand]]</text>
      <sha1>6xmi7rny5ak85z0p52pxd9yp7uhfk1o</sha1>
    </revision>
    <revision>
      <id>51910</id>
      <parentid>49884</parentid>
      <timestamp>2020-10-25T13:43:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Name: major &gt; submajor</comment>
      <origin>51910</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1056" sha1="5bxmpr5rzrcxxb7ak8i7rgjevqw9jw9" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/10
| Monzo = -1 0 -1 0 1
| Cents = 165.00423
| Name = large undecimal neutral second, &lt;br&gt;undecimal submajor second
| Color name = 1og2, logu 2nd
| FJS name = m2&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;5&lt;/sub&gt;
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy. Three of them are less than [[4/3]] by the wizardharry comma, 4000/3993.

Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Superparticular]]
[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]

[[Category:todo:expand]]</text>
      <sha1>5bxmpr5rzrcxxb7ak8i7rgjevqw9jw9</sha1>
    </revision>
    <revision>
      <id>57408</id>
      <parentid>51910</parentid>
      <timestamp>2020-12-23T19:06:50Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>Moving some info to the new page on the Wizardharry comma, and combining paragraghs.</comment>
      <origin>57408</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="983" sha1="0q5po4tm89sgdsiy0tamwc6b862xn86" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/10
| Monzo = -1 0 -1 0 1
| Cents = 165.00423
| Name = large undecimal neutral second, &lt;br&gt;undecimal submajor second
| Color name = 1og2, logu 2nd
| FJS name = m2&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;5&lt;/sub&gt;
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy.  Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Superparticular]]
[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]

[[Category:todo:expand]]</text>
      <sha1>0q5po4tm89sgdsiy0tamwc6b862xn86</sha1>
    </revision>
    <revision>
      <id>64439</id>
      <parentid>57408</parentid>
      <timestamp>2021-03-27T04:02:22Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64439</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1031" sha1="iix8e6leffmm62dokf96l4got50j4s5" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/10
| Monzo = -1 0 -1 0 1
| Cents = 165.00423
| Name = large undecimal neutral second, &lt;br&gt;undecimal submajor second
| Color name = 1og2, logu 2nd
| FJS name = m2&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;5&lt;/sub&gt;
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy.  Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Superparticular]]
[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
[[Category:Pages with internal sound examples]]

[[Category:todo:expand]]</text>
      <sha1>iix8e6leffmm62dokf96l4got50j4s5</sha1>
    </revision>
    <revision>
      <id>81187</id>
      <parentid>64439</parentid>
      <timestamp>2021-11-22T16:20:09Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>81187</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1073" sha1="2045hz5bio0d7ror9h1yfbsx0avtvrh" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/10
| Monzo = -1 0 -1 0 1
| Cents = 165.00423
| Name = large undecimal neutral second, &lt;br&gt;undecimal submajor second
| Color name = 1og2, logu 2nd
| FJS name = m2&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;5&lt;/sub&gt;
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy.  Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Superparticular]]
[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
[[Category:Pages with internal sound examples]]

[[Category:todo:expand]]</text>
      <sha1>2045hz5bio0d7ror9h1yfbsx0avtvrh</sha1>
    </revision>
    <revision>
      <id>86102</id>
      <parentid>81187</parentid>
      <timestamp>2022-02-17T21:04:07Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Added Wikipedia box, misc. edits, categories</comment>
      <origin>86102</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1051" sha1="0ak689rq4z7nxa192gyxhkh739bwsy0" xml:space="preserve">{{Infobox Interval
| Ratio = 11/10
| Monzo = -1 0 -1 0 1
| Cents = 165.00423
| Name = large undecimal neutral second, &lt;br&gt;undecimal submajor second
| Color name = 1og2, logu 2nd
| FJS name = m2&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;5&lt;/sub&gt;
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral interval#Second}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy. Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Superparticular]]
[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
[[Category:Pages with internal sound examples]]

[[Category:todo:expand]]</text>
      <sha1>0ak689rq4z7nxa192gyxhkh739bwsy0</sha1>
    </revision>
    <revision>
      <id>86103</id>
      <parentid>86102</parentid>
      <timestamp>2022-02-17T21:05:00Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Categories (todo); introduction could be laid out better than this</comment>
      <origin>86103</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1059" sha1="sjzuil7pmoyo7odgba9lz05wjjtqr90" xml:space="preserve">{{Infobox Interval
| Ratio = 11/10
| Monzo = -1 0 -1 0 1
| Cents = 165.00423
| Name = large undecimal neutral second, &lt;br&gt;undecimal submajor second
| Color name = 1og2, logu 2nd
| FJS name = m2&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;5&lt;/sub&gt;
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral interval#Second}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy. Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Superparticular]]
[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
[[Category:Pages with internal sound examples]]

{{todo|expand|improve synopsis}}</text>
      <sha1>sjzuil7pmoyo7odgba9lz05wjjtqr90</sha1>
    </revision>
    <revision>
      <id>86292</id>
      <parentid>86103</parentid>
      <timestamp>2022-02-22T23:46:38Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86292</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1010" sha1="ls5x3uq6gqtvwr4jnkwziu5d3q7kdhe" xml:space="preserve">{{Infobox Interval
| Ratio = 11/10
| Monzo = -1 0 -1 0 1
| Cents = 165.00423
| Name = large undecimal neutral second, &lt;br&gt;undecimal submajor second
| Color name = 1og2, logu 2nd
| FJS name = m2&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;5&lt;/sub&gt;
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral interval#Second}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy. Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Superparticular]]
[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
{{todo|expand|improve synopsis}}</text>
      <sha1>ls5x3uq6gqtvwr4jnkwziu5d3q7kdhe</sha1>
    </revision>
    <revision>
      <id>97948</id>
      <parentid>86292</parentid>
      <timestamp>2022-10-25T11:33:30Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97948</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="856" sha1="89errr1kqdtpgq80wj28vdopcstsjc4" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral interval#Second}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy. Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
{{todo|expand|improve synopsis}}</text>
      <sha1>89errr1kqdtpgq80wj28vdopcstsjc4</sha1>
    </revision>
    <revision>
      <id>120819</id>
      <parentid>97948</parentid>
      <timestamp>2023-07-22T04:39:07Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <minor/>
      <origin>120819</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="855" sha1="q5agwz3kclwupiqegewsam9hcafe30c" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral interval#Second}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy. Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
[[Category:Equable heptatonic]]</text>
      <sha1>q5agwz3kclwupiqegewsam9hcafe30c</sha1>
    </revision>
    <revision>
      <id>124921</id>
      <parentid>120819</parentid>
      <timestamp>2023-10-01T08:57:18Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <comment>General expansion.</comment>
      <origin>124921</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1475" sha1="3imqmanjx3h8nwzeg1h4ha5fc04ke26" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral interval#Second}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy. Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. It is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 is treated as a comma in edos 1, 2, 3, 5, and some very low accuracy temperaments such as [[Very_low_accuracy_temperaments#Antietam|Antietam]]. If it is used as a generator instead, it produces [[porcupine]], although it is slightly sharper than the optimal tuning for porcupine and does not fit the 80edo patent val mapping.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
[[Category:Equable heptatonic]]</text>
      <sha1>3imqmanjx3h8nwzeg1h4ha5fc04ke26</sha1>
    </revision>
    <revision>
      <id>124995</id>
      <parentid>124921</parentid>
      <timestamp>2023-10-02T14:31:42Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Sectioning</comment>
      <origin>124995</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1519" sha1="mcg3j4yqwoipka6gbuetq92iofj6eqg" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy. 

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 is treated as a comma in edos 1, 2, 3, 5, and some very low accuracy temperaments such as [[Very low accuracy temperaments #Antietam|antietam]]. If it is used as a generator instead, it produces [[porcupine]], although it is slightly sharper than the optimal tuning for porcupine and does not fit the 80edo patent val mapping.

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
[[Category:Equable heptatonic]]</text>
      <sha1>mcg3j4yqwoipka6gbuetq92iofj6eqg</sha1>
    </revision>
    <revision>
      <id>147771</id>
      <parentid>124995</parentid>
      <timestamp>2024-07-13T12:52:29Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>147771</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1882" sha1="5gn7z52t5idhieizxr5ez4m8j69kz8h" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy.  Despite its being tempered together with [[12/11]] in systems like [[24edo]], this interval is considerably less exotic than 12/11 in terms of sound- in fact, one could make the argument that it functions a bit more like a narrowed [[10/9]] owing to its usage in such a capacity in systems like [[22edo]] and [[41edo]] where 11/10 and 10/9 are tempered together.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 is treated as a comma in edos 1, 2, 3, 5, and some very low accuracy temperaments such as [[Very low accuracy temperaments #Antietam|antietam]]. If it is used as a generator instead, it produces [[porcupine]], although it is slightly sharper than the optimal tuning for porcupine and does not fit the 80edo patent val mapping.

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
[[Category:Equable heptatonic]]</text>
      <sha1>5gn7z52t5idhieizxr5ez4m8j69kz8h</sha1>
    </revision>
    <revision>
      <id>147772</id>
      <parentid>147771</parentid>
      <timestamp>2024-07-13T12:53:28Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>147772</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1943" sha1="pc32xy9k16ztit61q2spyssxy0r1zub" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy.  Despite its being tempered together with [[12/11]] in systems like [[24edo]], this interval is considerably less exotic than 12/11 in terms of sound when compared to a number of other [[11-limit]] intervals- in fact, one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[22edo]] and [[41edo]] where 11/10 and 10/9 are tempered together.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 is treated as a comma in edos 1, 2, 3, 5, and some very low accuracy temperaments such as [[Very low accuracy temperaments #Antietam|antietam]]. If it is used as a generator instead, it produces [[porcupine]], although it is slightly sharper than the optimal tuning for porcupine and does not fit the 80edo patent val mapping.

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
[[Category:Equable heptatonic]]</text>
      <sha1>pc32xy9k16ztit61q2spyssxy0r1zub</sha1>
    </revision>
    <revision>
      <id>147773</id>
      <parentid>147772</parentid>
      <timestamp>2024-07-13T12:56:24Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>147773</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1956" sha1="n4lddgzkg4wwwjecxowns5tgtpdagu3" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy.  Despite its being tempered together with [[12/11]] in systems like [[24edo]], this interval, on its own, is considerably less exotic than 12/11 in terms of sound when compared to a number of other [[11-limit]] intervals- in fact, one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[22edo]] and [[41edo]] where 11/10 and 10/9 are tempered together.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 is treated as a comma in edos 1, 2, 3, 5, and some very low accuracy temperaments such as [[Very low accuracy temperaments #Antietam|antietam]]. If it is used as a generator instead, it produces [[porcupine]], although it is slightly sharper than the optimal tuning for porcupine and does not fit the 80edo patent val mapping.

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
[[Category:Equable heptatonic]]</text>
      <sha1>n4lddgzkg4wwwjecxowns5tgtpdagu3</sha1>
    </revision>
    <revision>
      <id>147774</id>
      <parentid>147773</parentid>
      <timestamp>2024-07-13T13:05:29Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>147774</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1963" sha1="il8hzb2aq7jlmd03g2msjxf5ub03ooo" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy.  Despite its being tempered together with [[12/11]] in systems like [[24edo]], this interval, on its own, is considerably less exotic than 12/11 in terms of sound when compared to a number of other simple [[11-limit]] intervals- in fact, one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[22edo]] and [[41edo]] where 11/10 and 10/9 are tempered together.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 is treated as a comma in edos 1, 2, 3, 5, and some very low accuracy temperaments such as [[Very low accuracy temperaments #Antietam|antietam]]. If it is used as a generator instead, it produces [[porcupine]], although it is slightly sharper than the optimal tuning for porcupine and does not fit the 80edo patent val mapping.

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5]]
[[Category:Equable heptatonic]]</text>
      <sha1>il8hzb2aq7jlmd03g2msjxf5ub03ooo</sha1>
    </revision>
    <revision>
      <id>157027</id>
      <parentid>147774</parentid>
      <timestamp>2024-09-30T08:39:47Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Recategorize</comment>
      <origin>157027</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1973" sha1="ow8ad1mai1h0j2n9kdlkexgnopib6yu" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy.  Despite its being tempered together with [[12/11]] in systems like [[24edo]], this interval, on its own, is considerably less exotic than 12/11 in terms of sound when compared to a number of other simple [[11-limit]] intervals- in fact, one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[22edo]] and [[41edo]] where 11/10 and 10/9 are tempered together.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 is treated as a comma in edos 1, 2, 3, 5, and some very low accuracy temperaments such as [[Very low accuracy temperaments #Antietam|antietam]]. If it is used as a generator instead, it produces [[porcupine]], although it is slightly sharper than the optimal tuning for porcupine and does not fit the 80edo patent val mapping.

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>ow8ad1mai1h0j2n9kdlkexgnopib6yu</sha1>
    </revision>
    <revision>
      <id>169706</id>
      <parentid>157027</parentid>
      <timestamp>2024-12-10T23:28:46Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>more balanced perspective</comment>
      <origin>169706</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1971" sha1="b1aizcmg9mena32ud0k4mryjmczvof7" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy.  Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals. If tempered sharp one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together (so that [[100/99|S10]] is tempered out).

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 is treated as a comma in edos 1, 2, 3, 5, and some very low accuracy temperaments such as [[Very low accuracy temperaments #Antietam|antietam]]. If it is used as a generator instead, it produces [[porcupine]], although it is slightly sharper than the optimal tuning for porcupine and does not fit the 80edo patent val mapping.

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>b1aizcmg9mena32ud0k4mryjmczvof7</sha1>
    </revision>
    <revision>
      <id>169707</id>
      <parentid>169706</parentid>
      <timestamp>2024-12-10T23:31:49Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <origin>169707</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2210" sha1="a60dsp6mdxzu0582zt21dae6zegd40x" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy.  Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals. If tempered sharp one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together (so that [[100/99|S10]] is tempered out). Meanwhile, when tuned just or near-just, it functions as almost exactly a third of [[4/3]], a very xenmelodic role corresponding to tempering out ([[100/99]])/([[121/120]]) = [[4000/3993|S10/S11]] = ([[4/3|12/9]])/([[11/10]])&lt;sup&gt;3&lt;/sup&gt;.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 is treated as a comma in edos 1, 2, 3, 5, and some very low accuracy temperaments such as [[Very low accuracy temperaments #Antietam|antietam]]. If it is used as a generator instead, it produces [[porcupine]], although it is slightly sharper than the optimal tuning for porcupine and does not fit the 80edo patent val mapping.

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>a60dsp6mdxzu0582zt21dae6zegd40x</sha1>
    </revision>
    <revision>
      <id>169708</id>
      <parentid>169707</parentid>
      <timestamp>2024-12-10T23:36:40Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>/* Temperaments */ better discussion of temperaments</comment>
      <origin>169708</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2523" sha1="9t49e0yge26euc4g4csusy8xhjrxjch" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy.  Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals. If tempered sharp one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together (so that [[100/99|S10]] is tempered out). Meanwhile, when tuned just or near-just, it functions as almost exactly a third of [[4/3]], a very xenmelodic role corresponding to tempering out ([[100/99]])/([[121/120]]) = [[4000/3993|S10/S11]] = ([[4/3|12/9]])/([[11/10]])&lt;sup&gt;3&lt;/sup&gt;.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 may be treated implicitly as a comma in JI scales that do not find [[11/8]] and [[5/4]] above the same degree, but usually it makes much more sense to use it as a generator, such as the aforementioned very accurate strategy of making it a third of [[4/3]], leading to scales that look like [[porcupine]] but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get [[echidna]], a [[hedgehog]] lookalike that detempers [[100/99]] and [[121/120]] and which efficiently and accurately finds [[11-limit]] and (no-13's) [[17-limit]] harmonies.

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>9t49e0yge26euc4g4csusy8xhjrxjch</sha1>
    </revision>
    <revision>
      <id>169709</id>
      <parentid>169708</parentid>
      <timestamp>2024-12-10T23:37:02Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>/* Temperaments */</comment>
      <origin>169709</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2537" sha1="fxtxqdnw0vbkag191ph1uqqkgoon5i5" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by Ptolemy.  Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals. If tempered sharp one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together (so that [[100/99|S10]] is tempered out). Meanwhile, when tuned just or near-just, it functions as almost exactly a third of [[4/3]], a very xenmelodic role corresponding to tempering out ([[100/99]])/([[121/120]]) = [[4000/3993|S10/S11]] = ([[4/3|12/9]])/([[11/10]])&lt;sup&gt;3&lt;/sup&gt;.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 may be treated implicitly as a comma in JI scales that (for example) do not find [[11/8]] and [[5/4]] above the same degree, but usually it makes much more sense to use it as a generator, such as the aforementioned very accurate strategy of making it a third of [[4/3]], leading to scales that look like [[porcupine]] but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get [[echidna]], a [[hedgehog]] lookalike that detempers [[100/99]] and [[121/120]] and which efficiently and accurately finds [[11-limit]] and (no-13's) [[17-limit]] harmonies.

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>fxtxqdnw0vbkag191ph1uqqkgoon5i5</sha1>
    </revision>
    <revision>
      <id>169848</id>
      <parentid>169709</parentid>
      <timestamp>2024-12-12T09:03:51Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Restore some old discussions. -s-expressions in the opening section since learning about the basics of a JI interval shouldn't rely on that knowledge</comment>
      <origin>169848</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2610" sha1="il5c7xmb54y67t0yzcl75lc2kv896rt" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by {{w|Ptolemy}}. Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals. If tempered sharp one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together (so that [[100/99]] is tempered out). Meanwhile, when tuned just or near-just, it functions as almost exactly a third of [[4/3]], a very xenmelodic role corresponding to tempering out [[4000/3993]].

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 is treated as a comma in edos 1, 2, 3, 5, and some very low accuracy temperaments such as [[Very low accuracy temperaments #Antietam|antietam]]. 

It may be treated implicitly as a comma in JI scales that (for example) do not find [[11/8]] and [[5/4]] above the same degree, but usually it makes much more sense to use it as a generator, such as the aforementioned very accurate strategy of making it a third of [[4/3]], leading to scales that look like [[porcupine]] but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get [[echidna]], a [[hedgehog]] lookalike that detempers [[100/99]] and [[121/120]] and which efficiently and accurately finds [[11-limit]] and (no-13's) [[17-limit]] harmonies.

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>il5c7xmb54y67t0yzcl75lc2kv896rt</sha1>
    </revision>
    <revision>
      <id>169971</id>
      <parentid>169848</parentid>
      <timestamp>2024-12-12T18:24:05Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>/* Temperaments */ please do not prioritize tempering out 11/10 over more useful information.</comment>
      <origin>169971</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2710" sha1="be4igvz6r740siectow0iswkk8z204u" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by {{w|Ptolemy}}. Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals. If tempered sharp one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together (so that [[100/99]] is tempered out). Meanwhile, when tuned just or near-just, it functions as almost exactly a third of [[4/3]], a very xenmelodic role corresponding to tempering out [[4000/3993]].

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
It may be treated implicitly as a comma in JI scales that (for example) do not find [[11/8]] and [[5/4]] above the same degree, but usually it makes much more sense to use it as a generator, such as the aforementioned very accurate strategy of making it a third of [[4/3]], leading to scales that look like [[porcupine]] but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get [[echidna]], a [[hedgehog]] lookalike that detempers [[100/99]] and [[121/120]] and which efficiently and accurately finds [[11-limit]] and (no-13's) [[17-limit]] harmonies.

=== [[Exotemperament]]s ===
Some [[val]]s that temper it out are the [[patent val]]s of [[edo]]s 1, 2, 3 and 5. An example rank 2 [[exotemperament]] that treats it as a comma to be tempered out is [[Very low accuracy temperaments #Antietam|antietam]].

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>be4igvz6r740siectow0iswkk8z204u</sha1>
    </revision>
    <revision>
      <id>170095</id>
      <parentid>169971</parentid>
      <timestamp>2024-12-13T10:39:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Temperaments */ there's lots of temps in the stearnsmic clan and echidna is but one of them</comment>
      <origin>170095</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2743" sha1="8hjj8ke7swzag014i8judiqgbuvjpou" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by {{w|Ptolemy}}. Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals. If tempered sharp one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together (so that [[100/99]] is tempered out). Meanwhile, when tuned just or near-just, it functions as almost exactly a third of [[4/3]], a very xenmelodic role corresponding to tempering out [[4000/3993]].

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 may be treated implicitly as a comma in JI scales that for example do not find [[11/8]] and [[5/4]] above the same degree, but usually it makes much more sense to use it as a generator, such as the aforementioned very accurate strategy of making it a third of [[4/3]], leading to scales that look like [[porcupine]] but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

=== Exotemperaments ===
11/10 is tempered out in the [[patent val]]s of [[edo]]s 1, 2, 3 and 5. An example rank-2 [[exotemperament]] that treats it as a comma to be tempered out is [[very low accuracy temperaments #Antietam|antietam]].

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>8hjj8ke7swzag014i8judiqgbuvjpou</sha1>
    </revision>
    <revision>
      <id>170139</id>
      <parentid>170095</parentid>
      <timestamp>2024-12-13T18:18:11Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>Added some more information from a seemingly overlooked comment on Discord</comment>
      <origin>170139</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3223" sha1="eh3bc7qnjv0qapqcjy4we3etqd56j3i" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by {{w|Ptolemy}}. Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals.  If tempered sharp, however, one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together due to [[100/99]] being tempered out.  Meanwhile, when tuned just or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to the tempering out of [[4000/3993]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the tempering out of the [[schisma]].  Assuming you go with either of the aforementioned options, keeping 11/10 distinct from 12/11 ensures that 11/10 has a way of bridging quartertone-based chords with more typical [[5-limit]] and Pythagorean chords as a sort of step between notes, however, if you temper out [[121/120]], expect this ability to vanish.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 may be treated implicitly as a comma in JI scales that for example do not find [[11/8]] and [[5/4]] above the same degree, but usually it makes much more sense to use it as a generator, such as the aforementioned very accurate strategy of making it a third of [[4/3]], leading to scales that look like [[porcupine]] but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

=== Exotemperaments ===
11/10 is tempered out in the [[patent val]]s of [[edo]]s 1, 2, 3 and 5. An example rank-2 [[exotemperament]] that treats it as a comma to be tempered out is [[very low accuracy temperaments #Antietam|antietam]].

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>eh3bc7qnjv0qapqcjy4we3etqd56j3i</sha1>
    </revision>
    <revision>
      <id>170140</id>
      <parentid>170139</parentid>
      <timestamp>2024-12-13T18:19:11Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>170140</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3219" sha1="t8lkiltpyzlyp1ib4hwu6cb7u83p6eh" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by {{w|Ptolemy}}. Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals.  If tempered sharp, however, one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together due to [[100/99]] being tempered out.  Meanwhile, when tuned just or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being tempered out, but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being tempered out.  Assuming you go with either of the aforementioned options, keeping 11/10 distinct from 12/11 ensures that 11/10 has a way of bridging quartertone-based chords with more typical [[5-limit]] and Pythagorean chords as a sort of step between notes, however, if you temper out [[121/120]], expect this ability to vanish.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 may be treated implicitly as a comma in JI scales that for example do not find [[11/8]] and [[5/4]] above the same degree, but usually it makes much more sense to use it as a generator, such as the aforementioned very accurate strategy of making it a third of [[4/3]], leading to scales that look like [[porcupine]] but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

=== Exotemperaments ===
11/10 is tempered out in the [[patent val]]s of [[edo]]s 1, 2, 3 and 5. An example rank-2 [[exotemperament]] that treats it as a comma to be tempered out is [[very low accuracy temperaments #Antietam|antietam]].

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>t8lkiltpyzlyp1ib4hwu6cb7u83p6eh</sha1>
    </revision>
    <revision>
      <id>170141</id>
      <parentid>170140</parentid>
      <timestamp>2024-12-13T18:21:50Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>170141</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3239" sha1="p9l9rk3cgah2rxowebge3v9x1jeb6p4" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by {{w|Ptolemy}}. Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals.  If tempered sharp, however, one could make the argument that it functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together due to [[100/99]] being tempered out.  Meanwhile, when tuned just or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being fudged or tempered out, but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged or tempered out.  Assuming you go with either of the aforementioned options, keeping 11/10 distinct from 12/11 ensures that 11/10 has a way of bridging quartertone-based chords with more typical [[5-limit]] and Pythagorean chords as a sort of step between notes, however, if you temper out [[121/120]], expect this ability to vanish.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 may be treated implicitly as a comma in JI scales that for example do not find [[11/8]] and [[5/4]] above the same degree, but usually it makes much more sense to use it as a generator, such as the aforementioned very accurate strategy of making it a third of [[4/3]], leading to scales that look like [[porcupine]] but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

=== Exotemperaments ===
11/10 is tempered out in the [[patent val]]s of [[edo]]s 1, 2, 3 and 5. An example rank-2 [[exotemperament]] that treats it as a comma to be tempered out is [[very low accuracy temperaments #Antietam|antietam]].

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>p9l9rk3cgah2rxowebge3v9x1jeb6p4</sha1>
    </revision>
    <revision>
      <id>170154</id>
      <parentid>170141</parentid>
      <timestamp>2024-12-13T22:50:48Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Added two sentences about it implying 11/5 in octave-repeating tunings</comment>
      <origin>170154</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3651" sha1="ef94foaz0jknb6rm7fwlzfxwmzf6a69" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by {{w|Ptolemy}}. Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals.  

If tempered sharp, however, one could make the argument that 11/10 functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together due to [[100/99]] being tempered out.  Meanwhile, when tuned just or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being fudged or tempered out, but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged or tempered out.  

Assuming you go with either of the aforementioned options, keeping 11/10 distinct from 12/11 ensures that 11/10 has a way of bridging quartertone-based chords with more typical [[5-limit]] and Pythagorean chords as a sort of step between notes, however, if you temper out [[121/120]], expect this ability to vanish.

11/10 is the [[octave-reduced]] form of [[11/5]], one of the three most [[concordant]] 11-limit intervals within the entire [[4/1|first two octaves]] along with [[11/4]] and [[11/3]]. In any [[Octave equivalence|octave-repeating]] tuning, a good approximation of 11/10 indicates a good approximation of 11/5. So, it could be argued that 11/10 is a high priority for any octave-repeating 11-limit tuning.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a cent, close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 may be treated implicitly as a comma in JI scales that for example do not find [[11/8]] and [[5/4]] above the same degree, but usually it makes much more sense to use it as a generator, such as the aforementioned very accurate strategy of making it a third of [[4/3]], leading to scales that look like [[porcupine]] but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

=== Exotemperaments ===
11/10 is tempered out in the [[patent val]]s of [[edo]]s 1, 2, 3 and 5. An example rank-2 [[exotemperament]] that treats it as a comma to be tempered out is [[very low accuracy temperaments #Antietam|antietam]].

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>ef94foaz0jknb6rm7fwlzfxwmzf6a69</sha1>
    </revision>
    <revision>
      <id>170155</id>
      <parentid>170154</parentid>
      <timestamp>2024-12-13T22:55:22Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Add links</comment>
      <origin>170155</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3710" sha1="8icexqj6r3j3an9zoc5be1lpzi7rp5f" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by {{w|Ptolemy}}. Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals.  

If tempered sharp, however, one could make the argument that 11/10 functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together due to [[100/99]] being tempered out.  Meanwhile, when tuned [[just]] or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being [[Fudging|fudged]] or [[tempered out]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged or tempered out.  

Assuming you go with either of the aforementioned options, keeping 11/10 distinct from 12/11 ensures that 11/10 has a way of bridging quartertone-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a sort of step between notes, however, if you temper out [[121/120]], expect this ability to vanish.

11/10 is the [[octave-reduced]] form of [[11/5]], one of the three most [[concordant]] 11-limit intervals within the entire [[4/1|first two octaves]] along with [[11/4]] and [[11/3]]. In any [[Octave equivalence|octave-repeating]] tuning, a good approximation of 11/10 indicates a good approximation of 11/5. So, it could be argued that 11/10 is a high priority for any octave-repeating 11-limit tuning.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a [[cent]], close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 may be treated implicitly as a [[comma]] in [[JI]] scales that for example do not find [[11/8]] and [[5/4]] above the same degree, but usually it makes much more sense to use it as a [[generator]], such as the aforementioned very accurate strategy of making it a third of [[4/3]], leading to scales that look like [[porcupine]] but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

=== Exotemperaments ===
11/10 is tempered out in the [[patent val]]s of [[edo]]s 1, 2, 3 and 5. An example rank-2 [[exotemperament]] that treats it as a comma to be tempered out is [[very low accuracy temperaments #Antietam|antietam]].

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>8icexqj6r3j3an9zoc5be1lpzi7rp5f</sha1>
    </revision>
    <revision>
      <id>170156</id>
      <parentid>170155</parentid>
      <timestamp>2024-12-13T22:55:47Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>170156</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3712" sha1="etxashy17hmlqc5jy8wtudwko95cdz3" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by {{w|Ptolemy}}. Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals.  

If tempered sharp, however, one could make the argument that 11/10 functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are tempered together due to [[100/99]] being tempered out.  

Meanwhile, when tuned [[just]] or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being [[Fudging|fudged]] or [[tempered out]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged or tempered out.  

Assuming you go with either of the aforementioned options, keeping 11/10 distinct from 12/11 ensures that 11/10 has a way of bridging quartertone-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a sort of step between notes, however, if you temper out [[121/120]], expect this ability to vanish.

11/10 is the [[octave-reduced]] form of [[11/5]], one of the three most [[concordant]] 11-limit intervals within the entire [[4/1|first two octaves]] along with [[11/4]] and [[11/3]]. In any [[Octave equivalence|octave-repeating]] tuning, a good approximation of 11/10 indicates a good approximation of 11/5. So, it could be argued that 11/10 is a high priority for any octave-repeating 11-limit tuning.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a [[cent]], close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 may be treated implicitly as a [[comma]] in [[JI]] scales that for example do not find [[11/8]] and [[5/4]] above the same degree, but usually it makes much more sense to use it as a [[generator]], such as the aforementioned very accurate strategy of making it a third of [[4/3]], leading to scales that look like [[porcupine]] but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

=== Exotemperaments ===
11/10 is tempered out in the [[patent val]]s of [[edo]]s 1, 2, 3 and 5. An example rank-2 [[exotemperament]] that treats it as a comma to be tempered out is [[very low accuracy temperaments #Antietam|antietam]].

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>etxashy17hmlqc5jy8wtudwko95cdz3</sha1>
    </revision>
    <revision>
      <id>172821</id>
      <parentid>170156</parentid>
      <timestamp>2024-12-24T06:27:33Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Links</comment>
      <origin>172821</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3720" sha1="2vj0lxl2z7slhdpah4amwzhrcnmabh5" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
{{Wikipedia| Neutral interval #Second }}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is an interval favored by {{w|Ptolemy}}. Depending on who you ask, this interval, on its own, is either considerably more or considerably less exotic than [[12/11]] or a number of other simple [[11-limit]] intervals.  

If tempered sharp, however, one could make the argument that 11/10 functions a bit more like a narrowed [[10/9]] in light of its usage in such a capacity in systems like [[41edo]] and [[63edo]] where 11/10 and 10/9 are [[tempered]] together due to [[100/99]] being tempered out.  

Meanwhile, when tuned [[just]] or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being [[Fudging|fudged]] or [[tempered out]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged or tempered out.  

Assuming you go with either of the aforementioned options, keeping 11/10 distinct from 12/11 ensures that 11/10 has a way of bridging [[quartertone]]-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a sort of step between notes, however, if you temper out [[121/120]], expect this ability to vanish.

11/10 is the [[octave-reduced]] form of [[11/5]], one of the three most [[concordant]] 11-limit intervals within the entire [[4/1|first two octaves]] along with [[11/4]] and [[11/3]]. In any [[Octave equivalence|octave-repeating]] tuning, a good approximation of 11/10 indicates a good approximation of 11/5. So, it could be argued that 11/10 is a high priority for any octave-repeating 11-limit tuning.

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a [[cent]], close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 

== Temperaments ==
11/10 may be treated implicitly as a [[comma]] in [[JI]] scales that for example do not find [[11/8]] and [[5/4]] above the same degree, but usually it makes much more sense to use it as a [[generator]], such as the aforementioned very accurate strategy of making it a third of [[4/3]], leading to scales that look like [[porcupine]] but whose harmonies can more accurately be explained in a number of ways depending partially on the exact tempering used. If you use a half-octave period you get temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

=== Exotemperaments ===
11/10 is tempered out in the [[patent val]]s of [[edo]]s 1, 2, 3 and 5. An example rank-2 [[exotemperament]] that treats it as a comma to be tempered out is [[very low accuracy temperaments #Antietam|antietam]].

== Trivia ==
Coincidentally, the interval between the most common tuning frequency (A440) and the second most common AC electrical frequency (50&amp;nbsp;Hz) is exactly 44/5, or three octaves above an 11/10. 

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>2vj0lxl2z7slhdpah4amwzhrcnmabh5</sha1>
    </revision>
    <revision>
      <id>189628</id>
      <parentid>172821</parentid>
      <timestamp>2025-04-02T18:49:17Z</timestamp>
      <contributor>
        <username>Hkm</username>
        <id>551</id>
      </contributor>
      <comment>cleanup</comment>
      <origin>189628</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2482" sha1="b3ne270j7iygunun0ded7c3u3quw6ke" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is the simplest submajor second. It is 15 cents sharp of 12/11 and 17 cents flat of 10/9. When tuned [[just]] or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being [[Fudging|fudged]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged. Keeping 11/10 distinct from 12/11 ensures that 11/10 bridges [[quartertone]]-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a step between notes.  

11/10 is the [[octave-reduced]] form of [[11/5]], one of the three most [[concordant]] 11-limit intervals within the entire [[4/1|first two octaves]] along with [[11/4]] and [[11/3]].

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a [[cent]], close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 11/10 is also approximated within 2 cents by [[22edo]], and is about 4c sharp of the [[chroma]] of [[BPS]]. 

== Temperaments ==
Using 11/10 as a generator tempering out 4000/3993 (as previously mentioned) leads to scales that look like [[porcupine]] but whose harmonies can more accurately be explained. With a half-octave period, a generator of 11/10 leads to temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

Using sqrt(11/10) as a generator leads to low-complexity [[Nautilus]] with one period to the octave and high-accuracy [[Harry]] with two periods; using cbrt(11/10) as a generator leads to [[Escapade]] with one period to the octave.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>b3ne270j7iygunun0ded7c3u3quw6ke</sha1>
    </revision>
    <revision>
      <id>189629</id>
      <parentid>189628</parentid>
      <timestamp>2025-04-02T18:51:47Z</timestamp>
      <contributor>
        <username>Hkm</username>
        <id>551</id>
      </contributor>
      <comment>/* Approximation */</comment>
      <origin>189629</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2501" sha1="kh118h8044092drsl85b87xs0u8tv2d" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is the simplest submajor second. It is 15 cents sharp of 12/11 and 17 cents flat of 10/9. When tuned [[just]] or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being [[Fudging|fudged]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged. Keeping 11/10 distinct from 12/11 ensures that 11/10 bridges [[quartertone]]-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a step between notes.  

11/10 is the [[octave-reduced]] form of [[11/5]], one of the three most [[concordant]] 11-limit intervals within the entire [[4/1|first two octaves]] along with [[11/4]] and [[11/3]].

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a [[cent]], close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 11/10 is also approximated within 2 cents by [[22edo]], and is 4c sharp of an octave-reduced stack of 9 generators in [[BPS]]. 

== Temperaments ==
Using 11/10 as a generator tempering out 4000/3993 (as previously mentioned) leads to scales that look like [[porcupine]] but whose harmonies can more accurately be explained. With a half-octave period, a generator of 11/10 leads to temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

Using sqrt(11/10) as a generator leads to low-complexity [[Nautilus]] with one period to the octave and high-accuracy [[Harry]] with two periods; using cbrt(11/10) as a generator leads to [[Escapade]] with one period to the octave.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>kh118h8044092drsl85b87xs0u8tv2d</sha1>
    </revision>
    <revision>
      <id>190406</id>
      <parentid>189629</parentid>
      <timestamp>2025-04-09T17:07:02Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>Add links to 12/11 and 10/9</comment>
      <origin>190406</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2509" sha1="5lnhe86x3pqz983cyu5n5vaqiw20ruy" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is the simplest submajor second. It is 15 cents sharp of [[12/11]] and 17 cents flat of [[10/9]]. When tuned [[just]] or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being [[Fudging|fudged]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged. Keeping 11/10 distinct from 12/11 ensures that 11/10 bridges [[quartertone]]-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a step between notes.  

11/10 is the [[octave-reduced]] form of [[11/5]], one of the three most [[concordant]] 11-limit intervals within the entire [[4/1|first two octaves]] along with [[11/4]] and [[11/3]].

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a [[cent]], close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 11/10 is also approximated within 2 cents by [[22edo]], and is 4c sharp of an octave-reduced stack of 9 generators in [[BPS]]. 

== Temperaments ==
Using 11/10 as a generator tempering out 4000/3993 (as previously mentioned) leads to scales that look like [[porcupine]] but whose harmonies can more accurately be explained. With a half-octave period, a generator of 11/10 leads to temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and efficiently and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies.

Using sqrt(11/10) as a generator leads to low-complexity [[Nautilus]] with one period to the octave and high-accuracy [[Harry]] with two periods; using cbrt(11/10) as a generator leads to [[Escapade]] with one period to the octave.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>5lnhe86x3pqz983cyu5n5vaqiw20ruy</sha1>
    </revision>
    <revision>
      <id>190449</id>
      <parentid>190406</parentid>
      <timestamp>2025-04-10T00:32:48Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>im not sure why i previously wrote that they are all find the (no-13's) 17-limit "efficiently"; this isnt true; echidna's 5 and 11 are much simpler than pogo's and supers'</comment>
      <origin>190449</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3109" sha1="5vg3uapusbd1fo41xyhavk3x5hmqoia" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is the simplest submajor second. It is 15 cents sharp of [[12/11]] and 17 cents flat of [[10/9]]. When tuned [[just]] or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being [[Fudging|fudged]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged. Keeping 11/10 distinct from 12/11 ensures that 11/10 bridges [[quartertone]]-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a step between notes.  

11/10 is the [[octave-reduced]] form of [[11/5]], one of the three most [[concordant]] 11-limit intervals within the entire [[4/1|first two octaves]] along with [[11/4]] and [[11/3]].

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a [[cent]], close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 11/10 is also approximated within 2 cents by [[22edo]], and is 4c sharp of an octave-reduced stack of 9 generators in [[BPS]]. 

== Temperaments ==
Using 11/10 as a generator tempering out 4000/3993 (as previously mentioned) leads to scales that look like [[porcupine]] but whose harmonies can more accurately be explained. With a half-octave period, a generator of 11/10 leads to temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies. Of these, echidna's mapping of the no-13's 17-limit is the simplest, though all three have the same mapping of the 2.3.7.11/10.17 subgroup so that they only differ on the mapping of 5 and 11. The complexity of 5 and 11 in pogo are used to increase accuracy, being a weak schismic extension. That leaves supers as the odd one out; if you are using an edo tuning for it, 58edo supports echidna while 94edo supports pogo, so it seems to exist as a portable alternate way of finding primes 5 and 11 across systems, unless you use the 152edo tuning, which requires using the second-best mapping of 13 (the 152f [[val]]).

Using sqrt(11/10) as a generator leads to low-complexity [[Nautilus]] with one period to the octave and high-accuracy [[Harry]] with two periods; using cbrt(11/10) as a generator leads to [[Escapade]] with one period to the octave.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>5vg3uapusbd1fo41xyhavk3x5hmqoia</sha1>
    </revision>
    <revision>
      <id>190450</id>
      <parentid>190449</parentid>
      <timestamp>2025-04-10T00:37:57Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>it's harry which is high accuracy; also clarify the gen</comment>
      <origin>190450</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3196" sha1="icqeho8i0n4i570hpi89t6n07wikhe7" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is the simplest submajor second. It is 15 cents sharp of [[12/11]] and 17 cents flat of [[10/9]]. When tuned [[just]] or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being [[Fudging|fudged]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged. Keeping 11/10 distinct from 12/11 ensures that 11/10 bridges [[quartertone]]-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a step between notes.  

11/10 is the [[octave-reduced]] form of [[11/5]], one of the three most [[concordant]] 11-limit intervals within the entire [[4/1|first two octaves]] along with [[11/4]] and [[11/3]].

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a [[cent]], close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 11/10 is also approximated within 2 cents by [[22edo]], and is 4c sharp of an octave-reduced stack of 9 generators in [[BPS]]. 

== Temperaments ==
Using 11/10 as a generator tempering out 4000/3993 (as previously mentioned) leads to scales that look like [[porcupine]] but whose harmonies can more accurately be explained. With a half-octave period, a generator of 11/10 leads to temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] and [[121/120]] and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies. Of these, echidna's mapping of the no-13's 17-limit is the simplest, though all three have the same mapping of the 2.3.7.11/10.17 subgroup so that they only differ on the mapping of 5 and 11. The complexity of 5 and 11 in pogo are used to increase accuracy, being a weak schismic extension. That leaves supers as the odd one out; if you are using an edo tuning for it, 58edo supports echidna while 94edo supports pogo, so it seems to exist as a portable alternate way of finding primes 5 and 11 across systems, unless you use the 152edo tuning, which requires using the second-best mapping of 13 (the 152f [[val]]).

Using sqrt(11/10) (22/21[[~]]21/20) as a generator leads to the low-complexity [[Nautilus]] with one period to the octave, and if you use two periods to the octave with this generator you get the high-accuracy temperament [[Harry]]; using cbrt(11/10) as a generator leads to [[Escapade]] with one period to the octave.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>icqeho8i0n4i570hpi89t6n07wikhe7</sha1>
    </revision>
    <revision>
      <id>190477</id>
      <parentid>190450</parentid>
      <timestamp>2025-04-10T04:53:07Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <comment>meow</comment>
      <origin>190477</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3406" sha1="5o5n4okeoi23k6w6lg0ulyu3467q8fj" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is the simplest submajor second. It is 15 cents sharp of [[12/11]] and 17 cents flat of [[10/9]]. When tuned [[just]] or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being [[Fudging|fudged]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged. Keeping 11/10 distinct from 12/11 ensures that 11/10 bridges [[quartertone]]-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a step between notes.  

11/10 is the [[octave-reduced]] form of [[11/5]], one of the three most [[concordant]] 11-limit intervals within the entire [[4/1|first two octaves]] along with [[11/4]] and [[11/3]].

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a [[cent]], close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 11/10 is also approximated within 2 cents by [[22edo]], and is 4c sharp of an octave-reduced stack of 9 generators in [[BPS]]. 

== Temperaments ==
Using 11/10 as a generator tempering out 4000/3993 (as previously mentioned) leads to scales that look like [[porcupine]] but whose harmonies can more accurately be explained. A half-octave period is exceptionally natural when 11/10 is a generator, because by virtue of making the (extremely accurate) approximation of the half-octave by [[99/70]], [[9/7]] is found as the period-complement of the generator. Taking this approach, this gives us temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] ~ [[121/120]] and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies. Of these, echidna's mapping of the no-13's 17-limit is the simplest, though all three have the same mapping of the 2.3.7.11/10.17 subgroup so that they only differ on the mapping of 5 and 11. The complexity of 5 and 11 in pogo are used to increase accuracy, being a weak schismic extension. That leaves supers as the odd one out; if you are using an edo tuning for it, 58edo supports echidna while 94edo supports pogo, so it seems to exist as a portable alternate way of finding primes 5 and 11 across systems, unless you use the 152edo tuning, which requires using the second-best mapping of 13 (the 152f [[val]]).

Using sqrt(11/10) (22/21[[~]]21/20) as a generator leads to the low-complexity [[Nautilus]] with one period to the octave, and if you use two periods to the octave with this generator you get the high-accuracy temperament [[Harry]]; using cbrt(11/10) as a generator leads to [[Escapade]] with one period to the octave.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>5o5n4okeoi23k6w6lg0ulyu3467q8fj</sha1>
    </revision>
    <revision>
      <id>215777</id>
      <parentid>190477</parentid>
      <timestamp>2025-11-03T06:51:04Z</timestamp>
      <contributor>
        <username>Pailiaq</username>
        <id>507</id>
      </contributor>
      <minor/>
      <origin>215777</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3445" sha1="gw5s19tvopaob4n23eqbv68vxvz95pr" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is the simplest submajor second. It is 15 cents sharp of [[12/11]] and 17 cents flat of [[10/9]]. When tuned [[just]] or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being [[Fudging|fudged]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged. Keeping 11/10 distinct from 12/11 ensures that 11/10 bridges [[quartertone]]-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a step between notes.  

11/10 is the [[octave-reduced]] form of [[11/5]], one of the three most [[concordant]] 11-limit intervals within the entire [[4/1|first two octaves]] along with [[11/4]] and [[11/3]].

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a [[cent]], close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 11/10 is also approximated within 2 cents by [[22edo]], and is 4c sharp of an octave-reduced stack of 9 generators in [[BPS]]. 
 {{Interval_Edo_Approximation | 11/10}}
== Temperaments ==
Using 11/10 as a generator tempering out 4000/3993 (as previously mentioned) leads to scales that look like [[porcupine]] but whose harmonies can more accurately be explained. A half-octave period is exceptionally natural when 11/10 is a generator, because by virtue of making the (extremely accurate) approximation of the half-octave by [[99/70]], [[9/7]] is found as the period-complement of the generator. Taking this approach, this gives us temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] ~ [[121/120]] and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies. Of these, echidna's mapping of the no-13's 17-limit is the simplest, though all three have the same mapping of the 2.3.7.11/10.17 subgroup so that they only differ on the mapping of 5 and 11. The complexity of 5 and 11 in pogo are used to increase accuracy, being a weak schismic extension. That leaves supers as the odd one out; if you are using an edo tuning for it, 58edo supports echidna while 94edo supports pogo, so it seems to exist as a portable alternate way of finding primes 5 and 11 across systems, unless you use the 152edo tuning, which requires using the second-best mapping of 13 (the 152f [[val]]).

Using sqrt(11/10) (22/21[[~]]21/20) as a generator leads to the low-complexity [[Nautilus]] with one period to the octave, and if you use two periods to the octave with this generator you get the high-accuracy temperament [[Harry]]; using cbrt(11/10) as a generator leads to [[Escapade]] with one period to the octave.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>gw5s19tvopaob4n23eqbv68vxvz95pr</sha1>
    </revision>
    <revision>
      <id>216018</id>
      <parentid>215777</parentid>
      <timestamp>2025-11-03T13:07:00Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"</comment>
      <origin>216018</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3442" sha1="e9j6k1nv2azv8faswnsuz7y4lav0nw1" xml:space="preserve">{{Infobox Interval
| Name = large undecimal neutral second, undecimal submajor second
| Color name = 1og2, logu 2nd
| Sound = jid_11_10_pluck_adu_dr220.mp3
}}
'''11/10''', the '''large undecimal neutral second''' or '''undecimal submajor second''', is the simplest submajor second. It is 15 cents sharp of [[12/11]] and 17 cents flat of [[10/9]]. When tuned [[just]] or near-just, it not only has the very exotic melodic role of being almost exactly a third of [[4/3]], leading to [[4000/3993]] being [[Fudging|fudged]], but is also very close in size to a stack consisting of an [[apotome]] and [[33/32]], leading to the [[schisma]] being fudged. Keeping 11/10 distinct from 12/11 ensures that 11/10 bridges [[quartertone]]-based chords with more typical [[5-limit]] and [[Pythagorean tuning|Pythagorean]] chords as a step between notes.  

11/10 is the [[octave-reduced]] form of [[11/5]], one of the three most [[concordant]] 11-limit intervals within the entire [[4/1|first two octaves]] along with [[11/4]] and [[11/3]].

== Approximation ==
11/10 is approximated extremely precisely by [[80edo]] and its multiples, with a chain of 80 11/10's failing to close at the octave by a mere third of a [[cent]], close enough that you could theoretically tune an instrument to 80edo by ear using it if you had the patience. 11/10 is also approximated within 2 cents by [[22edo]], and is 4c sharp of an octave-reduced stack of 9 generators in [[BPS]]. 
{{Interval edo approximation|11/10}}
== Temperaments ==
Using 11/10 as a generator tempering out 4000/3993 (as previously mentioned) leads to scales that look like [[porcupine]] but whose harmonies can more accurately be explained. A half-octave period is exceptionally natural when 11/10 is a generator, because by virtue of making the (extremely accurate) approximation of the half-octave by [[99/70]], [[9/7]] is found as the period-complement of the generator. Taking this approach, this gives us temperaments in the [[stearnsmic clan]] such as [[pogo]], [[supers]], or [[echidna]], all of which detemper [[100/99]] ~ [[121/120]] and accurately find [[11-limit]] and (no-13's) [[17-limit]] harmonies. Of these, echidna's mapping of the no-13's 17-limit is the simplest, though all three have the same mapping of the 2.3.7.11/10.17 subgroup so that they only differ on the mapping of 5 and 11. The complexity of 5 and 11 in pogo are used to increase accuracy, being a weak schismic extension. That leaves supers as the odd one out; if you are using an edo tuning for it, 58edo supports echidna while 94edo supports pogo, so it seems to exist as a portable alternate way of finding primes 5 and 11 across systems, unless you use the 152edo tuning, which requires using the second-best mapping of 13 (the 152f [[val]]).

Using sqrt(11/10) (22/21[[~]]21/20) as a generator leads to the low-complexity [[Nautilus]] with one period to the octave, and if you use two periods to the octave with this generator you get the high-accuracy temperament [[Harry]]; using cbrt(11/10) as a generator leads to [[Escapade]] with one period to the octave.

== See also ==
* [[20/11]] – its [[octave complement]]
* [[15/11]] – its [[fifth complement]]
* [[40/33]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Submajor second]]
[[Category:Over-5 intervals]]
[[Category:Equable heptatonic]]</text>
      <sha1>e9j6k1nv2azv8faswnsuz7y4lav0nw1</sha1>
    </revision>
  </page>
  <page>
    <title>11 10</title>
    <ns>0</ns>
    <id>69</id>
    <redirect title="11/10" />
    <revision>
      <id>69</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>69</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19" sha1="15w1doqhsx0ddmaca9iyqvehrc5i581" xml:space="preserve">#REDIRECT [[11/10]]</text>
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  </page>
  <page>
    <title>11/2</title>
    <ns>0</ns>
    <id>70</id>
    <revision>
      <id>70</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>70</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="214" sha1="t82wuy9ur61ebcp3l2yutx045fzk5tl" xml:space="preserve">'''11/2'''
|-1 0 0 0 1&amp;gt;

2951.3179 cents

[[File:jid_11_2_pluck_adu_dr110.mp3]] [[:File:jid_11_2_pluck_adu_dr110.mp3|sound sample]]

The ratio between the 2nd and 11th harmonics; two octaves above [[11/8|11/8]].</text>
      <sha1>t82wuy9ur61ebcp3l2yutx045fzk5tl</sha1>
    </revision>
    <revision>
      <id>4319</id>
      <timestamp>2014-06-08T14:00:48Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513249122 - Original comment: **</comment>
      <origin>4319</origin>
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      <text bytes="2980" sha1="dze1x1o51qmenxqbrk4gyw4akx37ftm" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-08 14:00:48 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513249122&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
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|-1 0 0 0 1&amp;gt;
2951.3179 cents
[[media type="file" key="jid_11_2_pluck_adu_dr110.mp3"]] [[file:xenharmonic/jid_11_2_pluck_adu_dr110.mp3|sound sample]]

The ratio between the 2nd and 11th harmonics; two octaves above [[11_8|11/8]].
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_2&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/2&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|-1 0 0 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
2951.3179 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_2_pluck_adu_dr110.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_2_pluck_adu_dr110.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_2_pluck_adu_dr110.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_2_pluck_adu_dr110.mp3/513246300/jid_11_2_pluck_adu_dr110.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_2_pluck_adu_dr110.mp3/513246300/jid_11_2_pluck_adu_dr110.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The ratio between the 2nd and 11th harmonics; two octaves above &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <text bytes="263" sha1="qwm1ncyhhrgmph9byqn616jhs7clnxk" xml:space="preserve">'''11/2'''
|-1 0 0 0 1&amp;gt;

2951.3179 cents

[[File:jid_11_2_pluck_adu_dr110.mp3]] [[:File:jid_11_2_pluck_adu_dr110.mp3|sound sample]]

The ratio between the 2nd and 11th harmonics; two octaves above [[11/8|11/8]].

[[Category:Pages with internal sound examples]]</text>
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      <timestamp>2022-02-22T23:46:39Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
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      <text bytes="214" sha1="t82wuy9ur61ebcp3l2yutx045fzk5tl" xml:space="preserve">'''11/2'''
|-1 0 0 0 1&amp;gt;

2951.3179 cents

[[File:jid_11_2_pluck_adu_dr110.mp3]] [[:File:jid_11_2_pluck_adu_dr110.mp3|sound sample]]

The ratio between the 2nd and 11th harmonics; two octaves above [[11/8|11/8]].</text>
      <sha1>t82wuy9ur61ebcp3l2yutx045fzk5tl</sha1>
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        <id>308</id>
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      <comment>+Infobox, name still missing</comment>
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| Ratio = 11/2
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| Name = 
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| FJS name = P18&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_2_pluck_adu_dr110.mp3
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'''11/2''' is the [[ratio]] between the 2nd and 11th [[harmonic]]s; two octaves above [[11/8]].

[[Category:11-limit]]
[[Category:Stub]]</text>
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      <contributor>
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        <id>365</id>
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      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
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'''11/2''' is the [[ratio]] between the 2nd and 11th [[harmonic]]s; two octaves above [[11/8]].

[[Category:Stub]]</text>
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      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Stub]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
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| Color name = cc1o4, cocolo 4th
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'''11/2''' is the [[ratio]] between the 2nd and 11th [[harmonic]]s; two octaves above [[11/8]].</text>
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      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
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      <minor/>
      <comment>Named</comment>
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| Color name = cc1o4, cocolo 4th
| Sound = jid_11_2_pluck_adu_dr110.mp3 1300 ninth
| Name = undecimal eighteenth
}}

'''11/2''' is the [[ratio]] between the 2nd and 11th [[harmonic]]s; two octaves above [[11/8]].</text>
      <sha1>mw1iin99xwshgl63aca9humebvgjs3l</sha1>
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      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
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      <minor/>
      <comment>[[Category:Taxicab-2 intervals]]</comment>
      <origin>168048</origin>
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| Color name = cc1o4, cocolo 4th
| Sound = jid_11_2_pluck_adu_dr110.mp3 1300 ninth
| Name = undecimal eighteenth
}}

'''11/2''' is the [[ratio]] between the 2nd and 11th [[harmonic]]s; two octaves above [[11/8]].

[[Category:Taxicab-2 intervals]]</text>
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  </page>
  <page>
    <title>11 2</title>
    <ns>0</ns>
    <id>71</id>
    <redirect title="11/2" />
    <revision>
      <id>71</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
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      <text bytes="18" sha1="1p7wo7v126w1vneg3o1xh4wj4abrsvp" xml:space="preserve">#REDIRECT [[11/2]]</text>
      <sha1>1p7wo7v126w1vneg3o1xh4wj4abrsvp</sha1>
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  </page>
  <page>
    <title>11/3</title>
    <ns>0</ns>
    <id>72</id>
    <revision>
      <id>72</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>72</origin>
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      <text bytes="213" sha1="nndpzicqh9ziljlwbumcwd53uulnmsg" xml:space="preserve">'''11/3'''
|0 -1 0 0 1&amp;gt;

2249.3629 cents

[[File:jid_11_3_pluck_adu_dr110.mp3]] [[:File:jid_11_3_pluck_adu_dr110.mp3|sound sample]]

The ratio between the 3rd and 11th harmonics; one octave above [[11/6|11/6]].</text>
      <sha1>nndpzicqh9ziljlwbumcwd53uulnmsg</sha1>
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    <revision>
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        <id>0</id>
      </contributor>
      <comment>**Imported revision 513248830 - Original comment: **</comment>
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      <text bytes="2978" sha1="7iar9lbh39628f7c8vjmkqiufe0dbw5" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-08 13:57:04 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513248830&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/3**
|0 -1 0 0 1&amp;gt;
2249.3629 cents
[[media type="file" key="jid_11_3_pluck_adu_dr110.mp3"]] [[file:xenharmonic/jid_11_3_pluck_adu_dr110.mp3|sound sample]]

The ratio between the 3rd and 11th harmonics; one octave above [[11_6|11/6]].
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_3&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/3&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|0 -1 0 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
2249.3629 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_3_pluck_adu_dr110.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_3_pluck_adu_dr110.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_3_pluck_adu_dr110.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_3_pluck_adu_dr110.mp3/513246460/jid_11_3_pluck_adu_dr110.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_3_pluck_adu_dr110.mp3/513246460/jid_11_3_pluck_adu_dr110.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The ratio between the 3rd and 11th harmonics; one octave above &amp;lt;a class="wiki_link" href="/11_6"&amp;gt;11/6&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>64442</id>
      <parentid>72</parentid>
      <timestamp>2021-03-27T04:02:50Z</timestamp>
      <contributor>
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      <text bytes="262" sha1="ebhe87g4oqhl7l5dn5aqeet2u7nmm2y" xml:space="preserve">'''11/3'''
|0 -1 0 0 1&amp;gt;

2249.3629 cents

[[File:jid_11_3_pluck_adu_dr110.mp3]] [[:File:jid_11_3_pluck_adu_dr110.mp3|sound sample]]

The ratio between the 3rd and 11th harmonics; one octave above [[11/6|11/6]].

[[Category:Pages with internal sound examples]]</text>
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    <revision>
      <id>86289</id>
      <parentid>64442</parentid>
      <timestamp>2022-02-22T23:46:39Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86289</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="213" sha1="nndpzicqh9ziljlwbumcwd53uulnmsg" xml:space="preserve">'''11/3'''
|0 -1 0 0 1&amp;gt;

2249.3629 cents

[[File:jid_11_3_pluck_adu_dr110.mp3]] [[:File:jid_11_3_pluck_adu_dr110.mp3|sound sample]]

The ratio between the 3rd and 11th harmonics; one octave above [[11/6|11/6]].</text>
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      <timestamp>2022-08-04T21:03:15Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
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      <comment>+Infobox, name still missing</comment>
      <origin>93925</origin>
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      <text bytes="322" sha1="3krf6pyjp63u9noiglwseb3ti9l1czi" xml:space="preserve">{{Infobox Interval
| Ratio = 11/3
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| Name = 
| Color name = c1o7, colo 7th
| FJS name = m14&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_3_pluck_adu_dr110.mp3
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'''11/3''' is the [[ratio]] between the 3rd and 11th [[harmonic]]s; one octave above [[11/6]].

[[Category:11-limit]]
[[Category:Stub]]</text>
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        <id>365</id>
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      <comment>Normalising usage of Infobox Interval</comment>
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'''11/3''' is the [[ratio]] between the 3rd and 11th [[harmonic]]s; one octave above [[11/6]].

[[Category:Stub]]</text>
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      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Stub]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>120003</origin>
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'''11/3''' is the [[ratio]] between the 3rd and 11th [[harmonic]]s; one octave above [[11/6]].</text>
      <sha1>nyny783b6uxezddwdttxsx6mrqi49dj</sha1>
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    <revision>
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      <contributor>
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        <id>359</id>
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      <minor/>
      <comment>Name</comment>
      <origin>130664</origin>
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| Color name = c1o7, colo 7th
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| Name = undecimal neutral fourteenth
}}

'''11/3''' is the [[ratio]] between the 3rd and 11th [[harmonic]]s; one octave above [[11/6]].</text>
      <sha1>1ro93j9slh41rtzyt6ri29rfe9ug6ja</sha1>
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    <revision>
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      <timestamp>2024-10-27T00:05:05Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>See also</comment>
      <origin>162429</origin>
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| Color name = c1o7, colo 7th
| Sound = jid_11_3_pluck_adu_dr110.mp3
| Name = undecimal neutral fourteenth
}}

'''11/3''' is the [[ratio]] between the 3rd and 11th [[harmonic]]s; one octave above [[11/6]].

== See also ==
* [[Ed11/3]]</text>
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    </revision>
    <revision>
      <id>168049</id>
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      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>[[Category:Taxicab-2 intervals]]</comment>
      <origin>168049</origin>
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'''11/3''' is the [[ratio]] between the 3rd and 11th [[harmonic]]s; one octave above [[11/6]].

== See also ==
* [[Ed11/3]]

[[Category:Taxicab-2 intervals]]</text>
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  <page>
    <title>11 3</title>
    <ns>0</ns>
    <id>73</id>
    <redirect title="11/3" />
    <revision>
      <id>73</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>73</origin>
      <model>wikitext</model>
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      <text bytes="18" sha1="bf5fcvg93ms18zclwkopd13a3t5bl5v" xml:space="preserve">#REDIRECT [[11/3]]</text>
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    </revision>
  </page>
  <page>
    <title>11/4</title>
    <ns>0</ns>
    <id>74</id>
    <revision>
      <id>74</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>74</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="213" sha1="7qrzo9rxqiz8jxad6w7ehi8zo3040mt" xml:space="preserve">'''11/4'''
|-2 0 0 0 1&amp;gt;

1751.3179 cents

[[File:jid_11_4_pluck_adu_dr220.mp3]] [[:File:jid_11_4_pluck_adu_dr220.mp3|sound sample]]

The ratio between the 4th and 11th harmonics; one octave above [[11/8|11/8]].</text>
      <sha1>7qrzo9rxqiz8jxad6w7ehi8zo3040mt</sha1>
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    <revision>
      <id>4321</id>
      <timestamp>2014-06-08T13:51:36Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513248400 - Original comment: **</comment>
      <origin>4321</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2977" sha1="g8w0q98om7rdry25lxb1u0aefgx3sun" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-08 13:51:36 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513248400&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/4**
|-2 0 0 0 1&amp;gt;
1751.3179 cents
[[media type="file" key="jid_11_4_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_11_4_pluck_adu_dr220.mp3|sound sample]]

The ratio between the 4th and 11th harmonics; one octave above [[11_8|11/8]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_4&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/4&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|-2 0 0 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
1751.3179 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_4_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_4_pluck_adu_dr220.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_4_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_4_pluck_adu_dr220.mp3/513246142/jid_11_4_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_4_pluck_adu_dr220.mp3/513246142/jid_11_4_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The ratio between the 4th and 11th harmonics; one octave above &amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>64443</id>
      <parentid>74</parentid>
      <timestamp>2021-03-27T04:03:00Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
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      <format>text/x-wiki</format>
      <text bytes="262" sha1="2ukehnfw83fkxd8ko1c5y56kgad00pr" xml:space="preserve">'''11/4'''
|-2 0 0 0 1&amp;gt;

1751.3179 cents

[[File:jid_11_4_pluck_adu_dr220.mp3]] [[:File:jid_11_4_pluck_adu_dr220.mp3|sound sample]]

The ratio between the 4th and 11th harmonics; one octave above [[11/8|11/8]].

[[Category:Pages with internal sound examples]]</text>
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    <revision>
      <id>86291</id>
      <parentid>64443</parentid>
      <timestamp>2022-02-22T23:46:39Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86291</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="213" sha1="7qrzo9rxqiz8jxad6w7ehi8zo3040mt" xml:space="preserve">'''11/4'''
|-2 0 0 0 1&amp;gt;

1751.3179 cents

[[File:jid_11_4_pluck_adu_dr220.mp3]] [[:File:jid_11_4_pluck_adu_dr220.mp3|sound sample]]

The ratio between the 4th and 11th harmonics; one octave above [[11/8|11/8]].</text>
      <sha1>7qrzo9rxqiz8jxad6w7ehi8zo3040mt</sha1>
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      <timestamp>2022-08-04T21:05:11Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox, name still missing</comment>
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| Ratio = 11/4
| Monzo = -2 0 0 0 1
| Cents = 1751.318
| Name = 
| Color name = 1o11, ilo 11th; c1o4, colo 4th
| FJS name = P11&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_4_pluck_adu_dr220.mp3
}}

'''11/4''' is the [[ratio]] between the 4th and 11th [[harmonic]]s; one octave above [[11/8]].

[[Category:11-limit]]
[[Category:Stub]]</text>
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      <parentid>93926</parentid>
      <timestamp>2022-10-25T12:04:37Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>98000</origin>
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      <format>text/x-wiki</format>
      <text bytes="221" sha1="ke79y5ou3y535f919k30u9p8cjfpc7h" xml:space="preserve">{{Infobox Interval
| Color name = 1o11, ilo 11th; c1o4, colo 4th
| Sound = jid_11_4_pluck_adu_dr220.mp3
}}

'''11/4''' is the [[ratio]] between the 4th and 11th [[harmonic]]s; one octave above [[11/8]].

[[Category:Stub]]</text>
      <sha1>ke79y5ou3y535f919k30u9p8cjfpc7h</sha1>
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      <parentid>98000</parentid>
      <timestamp>2023-01-11T20:17:01Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
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      <minor/>
      <origin>101258</origin>
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| Color name = 1o11, ilo 11th
| Sound = jid_11_4_pluck_adu_dr220.mp3
}}

'''11/4''' is the [[ratio]] between the 4th and 11th [[harmonic]]s; one octave above [[11/8]].

[[Category:Stub]]</text>
      <sha1>hlub1c257b4csfelceyfws3tsi4ha9i</sha1>
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    <revision>
      <id>120002</id>
      <parentid>101258</parentid>
      <timestamp>2023-07-11T21:49:17Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Stub]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>120002</origin>
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| Color name = 1o11, ilo 11th
| Sound = jid_11_4_pluck_adu_dr220.mp3
}}

'''11/4''' is the [[ratio]] between the 4th and 11th [[harmonic]]s; one octave above [[11/8]].</text>
      <sha1>sxtdc8je6ghxv5mtbs0q18xsblxqixp</sha1>
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    <revision>
      <id>130665</id>
      <parentid>120002</parentid>
      <timestamp>2024-01-04T03:13:57Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Name</comment>
      <origin>130665</origin>
      <model>wikitext</model>
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| Color name = 1o11, ilo 11th
| Sound = jid_11_4_pluck_adu_dr220.mp3
| Name = undecimal eleventh
}}

'''11/4''' is the [[ratio]] between the 4th and 11th [[harmonic]]s; one octave above [[11/8]].</text>
      <sha1>mbjxy91yxi7lie7vqfgsxs4onb2fmys</sha1>
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    <revision>
      <id>162427</id>
      <parentid>130665</parentid>
      <timestamp>2024-10-27T00:03:23Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Add see also</comment>
      <origin>162427</origin>
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      <text bytes="243" sha1="3q2mupit4b2vls9smznsruk1ji2vdws" xml:space="preserve">{{Infobox Interval
| Color name = 1o11, ilo 11th
| Sound = jid_11_4_pluck_adu_dr220.mp3
| Name = undecimal eleventh
}}

'''11/4''' is the [[ratio]] between the 4th and 11th [[harmonic]]s; one octave above [[11/8]].

== See also ==
* [[Ed11/4]]</text>
      <sha1>3q2mupit4b2vls9smznsruk1ji2vdws</sha1>
    </revision>
    <revision>
      <id>199647</id>
      <parentid>162427</parentid>
      <timestamp>2025-06-14T02:44:20Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <comment>probably the most notable association to this interval in particular</comment>
      <origin>199647</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="259" sha1="6xa0u1u1062yjrw6yrf9c7ai3kv7bqa" xml:space="preserve">{{Infobox Interval
| Color name = 1o11, ilo 11th
| Sound = jid_11_4_pluck_adu_dr220.mp3
| Name = undecimal eleventh
}}

'''11/4''' is the [[ratio]] between the 4th and 11th [[harmonic]]s; one octave above [[11/8]].

== See also ==
* [[Ed11/4]]
* [[L4MPLIGHT]]</text>
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  <page>
    <title>11 4</title>
    <ns>0</ns>
    <id>75</id>
    <redirect title="11/4" />
    <revision>
      <id>75</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>75</origin>
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      <format>text/x-wiki</format>
      <text bytes="18" sha1="4y18qckjufs3qlwlwnbbh1jjfqltw8r" xml:space="preserve">#REDIRECT [[11/4]]</text>
      <sha1>4y18qckjufs3qlwlwnbbh1jjfqltw8r</sha1>
    </revision>
  </page>
  <page>
    <title>11/5</title>
    <ns>0</ns>
    <id>76</id>
    <revision>
      <id>76</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>76</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="215" sha1="1si6we1e8iyw2zkfx318xw9d8t7cznz" xml:space="preserve">'''11/5'''
|0 0 -1 0 1&amp;gt;

1365.0042 cents

[[File:jid_11_5_pluck_adu_dr220.mp3]] [[:File:jid_11_5_pluck_adu_dr220.mp3|sound sample]]

The ratio between the 5th and 11th harmonics; one octave above [[11/10|11/10]].</text>
      <sha1>1si6we1e8iyw2zkfx318xw9d8t7cznz</sha1>
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    <revision>
      <id>4322</id>
      <timestamp>2014-06-08T13:47:59Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513248160 - Original comment: **</comment>
      <origin>4322</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2982" sha1="rhtvx3jru5mrrtqm9ctni0009qtieb6" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-08 13:47:59 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513248160&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/5**
|0 0 -1 0 1&amp;gt;
1365.0042 cents
[[media type="file" key="jid_11_5_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_11_5_pluck_adu_dr220.mp3|sound sample]]

The ratio between the 5th and 11th harmonics; one octave above [[11_10|11/10]].
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_5&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/5&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|0 0 -1 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
1365.0042 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_5_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_5_pluck_adu_dr220.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_5_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_5_pluck_adu_dr220.mp3/513246150/jid_11_5_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_5_pluck_adu_dr220.mp3/513246150/jid_11_5_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The ratio between the 5th and 11th harmonics; one octave above &amp;lt;a class="wiki_link" href="/11_10"&amp;gt;11/10&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>64445</id>
      <parentid>76</parentid>
      <timestamp>2021-03-27T04:03:10Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
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      <origin>64445</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="264" sha1="lpdldybje1ujmwewneu6ybt6o8xc99p" xml:space="preserve">'''11/5'''
|0 0 -1 0 1&amp;gt;

1365.0042 cents

[[File:jid_11_5_pluck_adu_dr220.mp3]] [[:File:jid_11_5_pluck_adu_dr220.mp3|sound sample]]

The ratio between the 5th and 11th harmonics; one octave above [[11/10|11/10]].

[[Category:Pages with internal sound examples]]</text>
      <sha1>lpdldybje1ujmwewneu6ybt6o8xc99p</sha1>
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    <revision>
      <id>86288</id>
      <parentid>64445</parentid>
      <timestamp>2022-02-22T23:46:39Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86288</origin>
      <model>wikitext</model>
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      <text bytes="215" sha1="1si6we1e8iyw2zkfx318xw9d8t7cznz" xml:space="preserve">'''11/5'''
|0 0 -1 0 1&amp;gt;

1365.0042 cents

[[File:jid_11_5_pluck_adu_dr220.mp3]] [[:File:jid_11_5_pluck_adu_dr220.mp3|sound sample]]

The ratio between the 5th and 11th harmonics; one octave above [[11/10|11/10]].</text>
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    <revision>
      <id>93927</id>
      <parentid>86288</parentid>
      <timestamp>2022-08-04T21:06:54Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
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      <comment>+Infobox</comment>
      <origin>93927</origin>
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      <text bytes="345" sha1="42ke8k1nzm7i4kgydsvjup28d6i7m7p" xml:space="preserve">{{Infobox Interval
| Ratio = 11/5
| Monzo = 0 0 -1 0 1
| Cents = 1365.004
| Name = neutral ninth
| Color name = 1o9, ilo 9th
| FJS name = m9&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;5&lt;/sub&gt;
| Sound = jid_11_5_pluck_adu_dr220.mp3
}}

'''11/5''' is the [[ratio]] between the 5th and 11th [[harmonic]]s; one octave above [[11/10]].

[[Category:11-limit]]
[[Category:Stub]]</text>
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    <revision>
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      <parentid>93927</parentid>
      <timestamp>2022-10-25T11:52:42Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
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      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97979</origin>
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      <format>text/x-wiki</format>
      <text bytes="227" sha1="r5wi0htm2l0iwnn19fz2coc3k7414ml" xml:space="preserve">{{Infobox Interval
| Name = neutral ninth
| Color name = 1o9, ilo 9th
| Sound = jid_11_5_pluck_adu_dr220.mp3
}}

'''11/5''' is the [[ratio]] between the 5th and 11th [[harmonic]]s; one octave above [[11/10]].

[[Category:Stub]]</text>
      <sha1>r5wi0htm2l0iwnn19fz2coc3k7414ml</sha1>
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    <revision>
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      <parentid>97979</parentid>
      <timestamp>2023-03-05T04:07:30Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
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      <origin>104505</origin>
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      <text bytes="430" sha1="1bypwbwc1ejfdfzl0gf410q7erebdx4" xml:space="preserve">{{Infobox Interval
| Name = neutral ninth
| Color name = 1o9, ilo 9th
| Sound = jid_11_5_pluck_adu_dr220.mp3
}}

'''11/5''' is the [[ratio]] between the 5th and 11th [[harmonic]]s; one octave above [[11/10]]. It is the all-odd voicing of [[11/10]] and thus the most consonant voicing. 11/5 is useful as an equivalence interval for "no-twos-or-three" music as [[5/1]] proves too wide.
== See also ==
* [[Ed11/5]]

[[Category:Stub]]</text>
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    <revision>
      <id>104506</id>
      <parentid>104505</parentid>
      <timestamp>2023-03-05T04:07:41Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
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      <origin>104506</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="431" sha1="cnfbidnnkg72mo6pncicsqzoaxrv4bv" xml:space="preserve">{{Infobox Interval
| Name = neutral ninth
| Color name = 1o9, ilo 9th
| Sound = jid_11_5_pluck_adu_dr220.mp3
}}

'''11/5''' is the [[ratio]] between the 5th and 11th [[harmonic]]s; one octave above [[11/10]]. It is the all-odd voicing of [[11/10]] and thus the most consonant voicing. 11/5 is useful as an equivalence interval for "no-twos-or-threes" music as [[5/1]] proves too wide.
== See also ==
* [[Ed11/5]]

[[Category:Stub]]</text>
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    <revision>
      <id>120005</id>
      <parentid>104506</parentid>
      <timestamp>2023-07-11T21:49:17Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Stub]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
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      <text bytes="412" sha1="2xb5k5c6q7jsw0s8npd1eul3vk0s7vv" xml:space="preserve">{{Infobox Interval
| Name = neutral ninth
| Color name = 1o9, ilo 9th
| Sound = jid_11_5_pluck_adu_dr220.mp3
}}

'''11/5''' is the [[ratio]] between the 5th and 11th [[harmonic]]s; one octave above [[11/10]]. It is the all-odd voicing of [[11/10]] and thus the most consonant voicing. 11/5 is useful as an equivalence interval for "no-twos-or-threes" music as [[5/1]] proves too wide.
== See also ==
* [[Ed11/5]]</text>
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    <revision>
      <id>158730</id>
      <parentid>120005</parentid>
      <timestamp>2024-10-06T00:37:16Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
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      <minor/>
      <comment>Line break</comment>
      <origin>158730</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="415" sha1="e0t4ybb1qy9l1yd8ztksamt0ynessvc" xml:space="preserve">{{Infobox Interval
| Name = neutral ninth
| Color name = 1o9, ilo 9th
| Sound = jid_11_5_pluck_adu_dr220.mp3
}}

'''11/5''' is the [[ratio]] between the 5th and 11th [[harmonic]]s; one octave above [[11/10]]. It is the all-odd voicing of [[11/10]] and thus the most consonant voicing. 

11/5 is useful as an equivalence interval for "no-twos-or-threes" music as [[5/1]] proves too wide.

== See also ==
* [[Ed11/5]]</text>
      <sha1>e0t4ybb1qy9l1yd8ztksamt0ynessvc</sha1>
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    <revision>
      <id>168050</id>
      <parentid>158730</parentid>
      <timestamp>2024-12-01T23:00:55Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
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      <minor/>
      <comment>[[Category:Taxicab-2 intervals]]</comment>
      <origin>168050</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="449" sha1="0nty5tkkfn8u3qezfsmhwsosdbk4111" xml:space="preserve">{{Infobox Interval
| Name = neutral ninth
| Color name = 1o9, ilo 9th
| Sound = jid_11_5_pluck_adu_dr220.mp3
}}

'''11/5''' is the [[ratio]] between the 5th and 11th [[harmonic]]s; one octave above [[11/10]]. It is the all-odd voicing of [[11/10]] and thus the most consonant voicing. 

11/5 is useful as an equivalence interval for "no-twos-or-threes" music as [[5/1]] proves too wide.

== See also ==
* [[Ed11/5]]

[[Category:Taxicab-2 intervals]]</text>
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    <revision>
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      <parentid>168050</parentid>
      <timestamp>2025-01-03T23:24:11Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Add links</comment>
      <origin>174467</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="457" sha1="bp20dva7p68rs7cdwyynpzptln122f4" xml:space="preserve">{{Infobox Interval
| Name = neutral ninth
| Color name = 1o9, ilo 9th
| Sound = jid_11_5_pluck_adu_dr220.mp3
}}

'''11/5''' is the [[ratio]] between the 5th and 11th [[harmonic]]s; one octave above [[11/10]]. It is the all-odd voicing of [[11/10]] and thus the most [[consonant]] voicing. 

11/5 is useful as an [[equivalence interval]] for "no-twos-or-threes" music if [[5/1]] proves too wide.

== See also ==
* [[Ed11/5]]

[[Category:Taxicab-2 intervals]]</text>
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  </page>
  <page>
    <title>11 5</title>
    <ns>0</ns>
    <id>77</id>
    <redirect title="11/5" />
    <revision>
      <id>77</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>77</origin>
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      <text bytes="18" sha1="t3dqtfy7jw1wwu6lco5oq9p6gwgmzlk" xml:space="preserve">#REDIRECT [[11/5]]</text>
      <sha1>t3dqtfy7jw1wwu6lco5oq9p6gwgmzlk</sha1>
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  </page>
  <page>
    <title>11/6</title>
    <ns>0</ns>
    <id>78</id>
    <revision>
      <id>78</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>78</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="774" sha1="bu7fjfcltgme5i8mj2jbd16py3d0764" xml:space="preserve">'''11/6'''
|-1 -1 0 0 1&amp;gt;

1049.3629 cents

[[File:jid_11_6_pluck_adu_dr220.mp3]] [[:File:jid_11_6_pluck_adu_dr220.mp3|sound sample]]

'''11/6''' is the undecimal neutral seventh of about 1049.4 [[cent|cent]]s. It can be treated as a consonance in [[11-limit|11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6|7/6]]-[[3/2|3/2]]-11/6) and the [[neutral_tetrad|neutral tetrad]] 18:22:27:33 (1-[[11/9|11/9]]-[[3/2|3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.
[[Category:11-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:neutral_seventh]]
[[Category:ratio]]
[[Category:seventh]]
[[Category:undecimal]]</text>
      <sha1>bu7fjfcltgme5i8mj2jbd16py3d0764</sha1>
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    <revision>
      <id>4323</id>
      <timestamp>2016-09-25T08:53:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Chartrekhan</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 593235136 - Original comment: **</comment>
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      <text bytes="4145" sha1="998qsitrni4thsejivkqz2na4s135pc" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Chartrekhan|Chartrekhan]] and made on &lt;tt&gt;2016-09-25 08:53:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;593235136&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/6**
|-1 -1 0 0 1&amp;gt;
1049.3629 cents
[[media type="file" key="jid_11_6_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_11_6_pluck_adu_dr220.mp3|sound sample]]

**11/6** is the undecimal neutral seventh of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7_6|7/6]]-[[3_2|3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11_9|11/9]]-[[3_2|3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_6&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/6&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|-1 -1 0 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
1049.3629 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_6_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_6_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_6_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_6_pluck_adu_dr220.mp3/513193564/jid_11_6_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_6_pluck_adu_dr220.mp3/513193564/jid_11_6_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;11/6&amp;lt;/strong&amp;gt; is the undecimal neutral seventh of about 1049.4 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be treated as a consonance in &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; harmony, forming a part of such chords as 6:7:9:11 (1-&amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt;-&amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;-11/6) and the &amp;lt;a class="wiki_link" href="/neutral%20tetrad"&amp;gt;neutral tetrad&amp;lt;/a&amp;gt; 18:22:27:33 (1-&amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;-&amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;-11/6).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>998qsitrni4thsejivkqz2na4s135pc</sha1>
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    <revision>
      <id>4324</id>
      <timestamp>2014-06-07T15:21:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513193698 - Original comment: **</comment>
      <origin>4324</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3681" sha1="76386e3z9v9drnadowh4qot9jeor2pl" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-07 15:21:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513193698&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/6**
|-1 -1 0 0 1&amp;gt;
1049.3629 cents
[[media type="file" key="jid_11_6_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_11_6_pluck_adu_dr220.mp3|sound sample]]

**11/6** is the undecimal neutral seventh of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7_6|7/6]]-[[3_2|3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11_9|11/9]]-[[3_2|3/2]]-11/6).&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_6&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/6&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|-1 -1 0 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
1049.3629 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_6_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_6_pluck_adu_dr220.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_6_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_6_pluck_adu_dr220.mp3/513193564/jid_11_6_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_6_pluck_adu_dr220.mp3/513193564/jid_11_6_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;11/6&amp;lt;/strong&amp;gt; is the undecimal neutral seventh of about 1049.4 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It can be treated as a consonance in &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; harmony, forming a part of such chords as 6:7:9:11 (1-&amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt;-&amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;-11/6) and the &amp;lt;a class="wiki_link" href="/neutral%20tetrad"&amp;gt;neutral tetrad&amp;lt;/a&amp;gt; 18:22:27:33 (1-&amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;-&amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;-11/6).&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>76386e3z9v9drnadowh4qot9jeor2pl</sha1>
    </revision>
    <revision>
      <id>4325</id>
      <timestamp>2012-02-24T15:07:31Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 304895624 - Original comment: cent size and links added**</comment>
      <origin>4325</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2167" sha1="48qbof7xtv64tjoax0w9v3kkexfv2cs" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2012-02-24 15:07:31 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;304895624&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;cent size and links added&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/6**, the undecimal neutral seventh o about 1049.36 [[cent]]. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 1-[[7_6|7/6]]-[[3_2|3/2]]-**11_6|11/6** and 1-[[11_9|11/9]]-[[3_2|3/2]]-**11/6** (the JI [[neutral tetrad]].)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_6&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/6&amp;lt;/strong&amp;gt;, the undecimal neutral seventh o about 1049.36 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;. It can be treated as a consonance in &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; harmony, forming a part of such chords as 1-&amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt;-&amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;-&amp;lt;strong&amp;gt;11_6|11/6&amp;lt;/strong&amp;gt; and 1-&amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;-&amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;-&amp;lt;strong&amp;gt;11/6&amp;lt;/strong&amp;gt; (the JI &amp;lt;a class="wiki_link" href="/neutral%20tetrad"&amp;gt;neutral tetrad&amp;lt;/a&amp;gt;.)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>48qbof7xtv64tjoax0w9v3kkexfv2cs</sha1>
    </revision>
    <revision>
      <id>4326</id>
      <timestamp>2012-02-23T23:41:34Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Cenobyte</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 304669514 - Original comment: **</comment>
      <origin>4326</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1645" sha1="4ttsaosiit1w8p9q38jh48aiop3i81s" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Cenobyte|Cenobyte]] and made on &lt;tt&gt;2012-02-23 23:41:34 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;304669514&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;11/6, the undecimal neutral seventh, can be treated as a consonance in 11-limit harmony, forming a part of such chords as 1-7/6-3/2-11/6 and 1-11/9-3/2-11/6 (the JI [[neutral tetrad]].)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_6&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;11/6, the undecimal neutral seventh, can be treated as a consonance in 11-limit harmony, forming a part of such chords as 1-7/6-3/2-11/6 and 1-11/9-3/2-11/6 (the JI &amp;lt;a class="wiki_link" href="/neutral%20tetrad"&amp;gt;neutral tetrad&amp;lt;/a&amp;gt;.)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>4ttsaosiit1w8p9q38jh48aiop3i81s</sha1>
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    <revision>
      <id>4327</id>
      <timestamp>2012-02-12T14:09:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 300960620 - Original comment: **</comment>
      <origin>4327</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1655" sha1="c86opiqdm18ff6ga43ywjb653h983on" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-02-12 14:09:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;300960620&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;11/9, the undecimal neutral seventh, can be treated as a consonance in 11-limit harmony, forming a part of such chords as 1-7/6-3/2-11/6 and 1-11/9-3/2-11/6 (the JI [[neutral tetrad]].)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_6&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;11/9, the undecimal neutral seventh, can be treated as a consonance in 11-limit harmony, forming a part of such chords as 1-7/6-3/2-11/6 and 1-11/9-3/2-11/6 (the JI &amp;lt;a class="wiki_link" href="/neutral%20tetrad"&amp;gt;neutral tetrad&amp;lt;/a&amp;gt;.)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>c86opiqdm18ff6ga43ywjb653h983on</sha1>
    </revision>
    <revision>
      <id>36586</id>
      <parentid>78</parentid>
      <timestamp>2018-10-11T22:27:59Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>switched to infobox interval; simplified links</comment>
      <origin>36586</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="760" sha1="ci25h40bwa3648h9wzjobevwviq53ii" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/6
| Monzo = -1 -1 0 0 1
| Cents = 1049.3629
| Name = undecimal neutral seventh
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the undecimal neutral seventh of about 1049.4 [[cent|cents]]. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.

[[Category:11-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:neutral_seventh]]
[[Category:ratio]]
[[Category:seventh]]
[[Category:undecimal]]</text>
      <sha1>ci25h40bwa3648h9wzjobevwviq53ii</sha1>
    </revision>
    <revision>
      <id>37583</id>
      <parentid>36586</parentid>
      <timestamp>2018-10-23T03:24:32Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <origin>37583</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="789" sha1="s1ii84p4lshvn24kyggmvx87d0tfn78" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/6
| Monzo = -1 -1 0 0 1
| Cents = 1049.3629
| Name = undecimal neutral seventh
| Color name = 1o7, lova 7th
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the undecimal neutral seventh of about 1049.4 [[cent|cents]]. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.

[[Category:11-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:neutral_seventh]]
[[Category:ratio]]
[[Category:seventh]]
[[Category:undecimal]]</text>
      <sha1>s1ii84p4lshvn24kyggmvx87d0tfn78</sha1>
    </revision>
    <revision>
      <id>37671</id>
      <parentid>37583</parentid>
      <timestamp>2018-10-23T19:04:22Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <origin>37671</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="788" sha1="0wfe4tsdzlglolzf9jdrtjm831qu6xk" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/6
| Monzo = -1 -1 0 0 1
| Cents = 1049.3629
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the undecimal neutral seventh of about 1049.4 [[cent|cents]]. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.

[[Category:11-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:neutral_seventh]]
[[Category:ratio]]
[[Category:seventh]]
[[Category:undecimal]]</text>
      <sha1>0wfe4tsdzlglolzf9jdrtjm831qu6xk</sha1>
    </revision>
    <revision>
      <id>38161</id>
      <parentid>37671</parentid>
      <timestamp>2018-10-29T15:25:18Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>replaced cent value via {{subst:#expr: 1200*ln({{subst:PAGENAME}})/ln(2) round 5}}</comment>
      <origin>38161</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="789" sha1="q13d41oih6kybp453zlboo7ylcsca0j" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/6
| Monzo = -1 -1 0 0 1
| Cents = 1049.36294
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the undecimal neutral seventh of about 1049.4 [[cent|cents]]. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.

[[Category:11-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:neutral_seventh]]
[[Category:ratio]]
[[Category:seventh]]
[[Category:undecimal]]</text>
      <sha1>q13d41oih6kybp453zlboo7ylcsca0j</sha1>
    </revision>
    <revision>
      <id>38224</id>
      <parentid>38161</parentid>
      <timestamp>2018-10-31T22:20:53Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>added typography (interval name bold), added see-also section</comment>
      <origin>38224</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="876" sha1="hfwgfm9me7xr9cba6dounuds8gqu0wo" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/6
| Monzo = -1 -1 0 0 1
| Cents = 1049.36294
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the '''undecimal neutral seventh''' of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.

== See also ==
* [[12/11]] its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval]]
[[Category:Ratio]]
[[Category:Just interval]]
[[Category:Neutral seventh]]
[[Category:Seventh]]
[[Category:Undecimal]]</text>
      <sha1>hfwgfm9me7xr9cba6dounuds8gqu0wo</sha1>
    </revision>
    <revision>
      <id>46751</id>
      <parentid>38224</parentid>
      <timestamp>2020-06-13T22:23:13Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cats</comment>
      <origin>46751</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="860" sha1="tnsrk6f726euw5hmk0lfid5llzjqa80" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/6
| Monzo = -1 -1 0 0 1
| Cents = 1049.36294
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the '''undecimal neutral seventh''' of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.

== See also ==
* [[12/11]] its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Neutral seventh]]
[[Category:Seventh]]
[[Category:Listen]]</text>
      <sha1>tnsrk6f726euw5hmk0lfid5llzjqa80</sha1>
    </revision>
    <revision>
      <id>47633</id>
      <parentid>46751</parentid>
      <timestamp>2020-07-12T18:55:22Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* See also */ +1</comment>
      <origin>47633</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="881" sha1="j06rc3rniuzgm9avtgm81gtynpz8dlf" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/6
| Monzo = -1 -1 0 0 1
| Cents = 1049.36294
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the '''undecimal neutral seventh''' of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.

== See also ==
* [[12/11]] its [[octave complement]]
* [[Iceface tuning]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Neutral seventh]]
[[Category:Seventh]]
[[Category:Listen]]</text>
      <sha1>j06rc3rniuzgm9avtgm81gtynpz8dlf</sha1>
    </revision>
    <revision>
      <id>49878</id>
      <parentid>47633</parentid>
      <timestamp>2020-09-20T06:39:37Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+FJS name; cleanup</comment>
      <origin>49878</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="914" sha1="ls38ky0t2afhcmcezpwg60kkwvdtyx0" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/6
| Monzo = -1 -1 0 0 1
| Cents = 1049.36294
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| FJS name = m7&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the '''undecimal neutral seventh''' of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.

== See also ==
* [[12/11]] – its [[octave complement]]
* [[Iceface tuning]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Neutral seventh]]
[[Category:Seventh]]
[[Category:Listen]]</text>
      <sha1>ls38ky0t2afhcmcezpwg60kkwvdtyx0</sha1>
    </revision>
    <revision>
      <id>64446</id>
      <parentid>49878</parentid>
      <timestamp>2021-03-27T04:03:28Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64446</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="962" sha1="axrip9bl0il3xw31bnu0k87jz30dcxn" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/6
| Monzo = -1 -1 0 0 1
| Cents = 1049.36294
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| FJS name = m7&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the '''undecimal neutral seventh''' of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.

== See also ==
* [[12/11]] – its [[octave complement]]
* [[Iceface tuning]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Neutral seventh]]
[[Category:Seventh]]
[[Category:Listen]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>axrip9bl0il3xw31bnu0k87jz30dcxn</sha1>
    </revision>
    <revision>
      <id>86243</id>
      <parentid>64446</parentid>
      <timestamp>2022-02-22T17:11:16Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Misc. edits, categories</comment>
      <origin>86243</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="830" sha1="b6nrvuo2pm24x0g6is4zhka9ovhyft3" xml:space="preserve">{{Infobox Interval
| Ratio = 11/6
| Monzo = -1 -1 0 0 1
| Cents = 1049.36294
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| FJS name = m7&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the '''undecimal neutral seventh''' of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.

== See also ==
* [[12/11]] – its [[octave complement]]
* [[Iceface tuning]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Seventh]]
[[Category:Neutral seventh]]</text>
      <sha1>b6nrvuo2pm24x0g6is4zhka9ovhyft3</sha1>
    </revision>
    <revision>
      <id>97998</id>
      <parentid>86243</parentid>
      <timestamp>2022-10-25T12:03:22Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97998</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="721" sha1="2y4qt2sgpewyer3ow2jcn7wtarb2cha" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the '''undecimal neutral seventh''' of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.

== See also ==
* [[12/11]] – its [[octave complement]]
* [[Iceface tuning]]
* [[Gallery of just intervals]]

[[Category:Seventh]]
[[Category:Neutral seventh]]</text>
      <sha1>2y4qt2sgpewyer3ow2jcn7wtarb2cha</sha1>
    </revision>
    <revision>
      <id>215907</id>
      <parentid>97998</parentid>
      <timestamp>2025-11-03T07:47:26Z</timestamp>
      <contributor>
        <username>Pailiaq</username>
        <id>507</id>
      </contributor>
      <origin>215907</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="780" sha1="otddwyzgff7ka8ns18ls3quhjwe7s7c" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the '''undecimal neutral seventh''' of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.
== Approximation ==
 {{Interval_Edo_Approximation | 11/6}}

== See also ==
* [[12/11]] – its [[octave complement]]
* [[Iceface tuning]]
* [[Gallery of just intervals]]

[[Category:Seventh]]
[[Category:Neutral seventh]]</text>
      <sha1>otddwyzgff7ka8ns18ls3quhjwe7s7c</sha1>
    </revision>
    <revision>
      <id>215993</id>
      <parentid>215907</parentid>
      <timestamp>2025-11-03T13:04:52Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"</comment>
      <origin>215993</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="777" sha1="b9le6i4phm9b4os289579kywh102ynu" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the '''undecimal neutral seventh''' of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency (60hz) is an 11/6, but two octaves up.
== Approximation ==
{{Interval edo approximation|11/6}}

== See also ==
* [[12/11]] – its [[octave complement]]
* [[Iceface tuning]]
* [[Gallery of just intervals]]

[[Category:Seventh]]
[[Category:Neutral seventh]]</text>
      <sha1>b9le6i4phm9b4os289579kywh102ynu</sha1>
    </revision>
    <revision>
      <id>226233</id>
      <parentid>215993</parentid>
      <timestamp>2026-03-17T10:49:14Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>60 Hz is not most common worldwide &amp;nbsp; 50 Hz dominates worldwide</comment>
      <origin>226233</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="794" sha1="jszhb1l79ri9pcu2gcriyodcttac1lt" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the '''undecimal neutral seventh''' of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency in the Americas (60hz) is an 11/6, but two octaves up.

== Approximation ==
{{Interval edo approximation|11/6}}

== See also ==
* [[12/11]] – its [[octave complement]]
* [[Iceface tuning]]
* [[Gallery of just intervals]]

[[Category:Seventh]]
[[Category:Neutral seventh]]</text>
      <sha1>jszhb1l79ri9pcu2gcriyodcttac1lt</sha1>
    </revision>
    <revision>
      <id>226234</id>
      <parentid>226233</parentid>
      <timestamp>2026-03-17T10:50:54Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>11/6 2 octaves up is 22/3</comment>
      <origin>226234</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="813" sha1="7sgspkn5peid2p3l7m4lp26dtgt364v" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral seventh
| Color name = 1o7, ilo 7th
| Sound = jid_11_6_pluck_adu_dr220.mp3
}}

'''11/6''' is the '''undecimal neutral seventh''' of about 1049.4 [[cent]]s. It can be treated as a consonance in [[11-limit]] harmony, forming a part of such chords as 6:7:9:11 (1-[[7/6]]-[[3/2]]-11/6) and the [[neutral tetrad]] 18:22:27:33 (1-[[11/9]]-[[3/2]]-11/6).

Coincidentally, the ratio between the most common musical tuning frequency (A440) and the most common electrical AC frequency in the Americas (60hz) is like 11/6, but two octaves up (thus, [[22/3]]).

== Approximation ==
{{Interval edo approximation|11/6}}

== See also ==
* [[12/11]] – its [[octave complement]]
* [[Iceface tuning]]
* [[Gallery of just intervals]]

[[Category:Seventh]]
[[Category:Neutral seventh]]</text>
      <sha1>7sgspkn5peid2p3l7m4lp26dtgt364v</sha1>
    </revision>
  </page>
  <page>
    <title>11 6</title>
    <ns>0</ns>
    <id>79</id>
    <redirect title="11/6" />
    <revision>
      <id>79</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>79</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18" sha1="8ctvtriaz7vqoj6xb7xr0nlr0h5oxah" xml:space="preserve">#REDIRECT [[11/6]]</text>
      <sha1>8ctvtriaz7vqoj6xb7xr0nlr0h5oxah</sha1>
    </revision>
  </page>
  <page>
    <title>11/7</title>
    <ns>0</ns>
    <id>80</id>
    <revision>
      <id>80</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>80</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="674" sha1="e1vhnsk1sp62meiju7l8mvx5tnt60a5" xml:space="preserve">'''11/7'''
|0 0 0 -1 1&amp;gt;

782.49204 cents

[[File:jid_11_7_pluck_adu_dr220.mp3]] [[:File:jid_11_7_pluck_adu_dr220.mp3|sound sample]]

In [[11-limit|11-limit]] [[Just_intonation|Just Intonation]], 11/7 is an undecimal (11-based) subminor sixth, measuring about 782.5¢ It is the inversion of [[14/11|14/11]], the undecimal supermajor third. Another subminor sixth is [[14/9|14/9]], at about 764.9¢; the interval between these is [[99/98|99/98]], about 17.6¢.

11/7 is [[22/21|22/21]] (about 80.5¢) above the [[3/2|3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]</text>
      <sha1>e1vhnsk1sp62meiju7l8mvx5tnt60a5</sha1>
    </revision>
    <revision>
      <id>4328</id>
      <timestamp>2014-06-07T15:25:07Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513193840 - Original comment: **</comment>
      <origin>4328</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4291" sha1="pvgygfm8n2vddib169rvfoyjo3pbeib" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-07 15:25:07 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513193840&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/7**
|0 0 0 -1 1&amp;gt;
782.49204 cents
[[media type="file" key="jid_11_7_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_11_7_pluck_adu_dr220.mp3|sound sample]]

In [[11-limit]] [[Just Intonation]], 11/7 is an undecimal (11-based) subminor sixth, measuring about 782.5¢ It is the inversion of [[14_11|14/11]], the undecimal supermajor third. Another subminor sixth is [[14_9|14/9]], at about 764.9¢; the interval between these is [[99_98|99/98]], about 17.6¢.

11/7 is [[22_21|22/21]] (about 80.5¢) above the [[3_2|3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_7&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|0 0 0 -1 1&amp;amp;gt;&amp;lt;br /&amp;gt;
782.49204 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_7_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_7_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_7_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_7_pluck_adu_dr220.mp3/513193782/jid_11_7_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_7_pluck_adu_dr220.mp3/513193782/jid_11_7_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 11/7 is an undecimal (11-based) subminor sixth, measuring about 782.5¢ It is the inversion of &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, the undecimal supermajor third. Another subminor sixth is &amp;lt;a class="wiki_link" href="/14_9"&amp;gt;14/9&amp;lt;/a&amp;gt;, at about 764.9¢; the interval between these is &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;, about 17.6¢.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11/7 is &amp;lt;a class="wiki_link" href="/22_21"&amp;gt;22/21&amp;lt;/a&amp;gt; (about 80.5¢) above the &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt; perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>pvgygfm8n2vddib169rvfoyjo3pbeib</sha1>
    </revision>
    <revision>
      <id>4329</id>
      <timestamp>2011-10-07T21:27:13Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 262753184 - Original comment: **</comment>
      <origin>4329</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2644" sha1="5ayz15ib1n6fzlou6gtn045d8wx5ik0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-10-07 21:27:13 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;262753184&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[11-limit]] [[Just Intonation]], 11/7 is an undecimal (11-based) subminor sixth, measuring about 782.5¢ It is the inversion of [[14_11|14/11]], the undecimal supermajor third. Another subminor sixth is [[14_9|14/9]], at about 764.9¢; the interval between these is [[99_98|99/98]], about 17.6¢.

11/7 is [[22_21|22/21]] (about 80.5¢) above the [[3_2|3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_7&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 11/7 is an undecimal (11-based) subminor sixth, measuring about 782.5¢ It is the inversion of &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, the undecimal supermajor third. Another subminor sixth is &amp;lt;a class="wiki_link" href="/14_9"&amp;gt;14/9&amp;lt;/a&amp;gt;, at about 764.9¢; the interval between these is &amp;lt;a class="wiki_link" href="/99_98"&amp;gt;99/98&amp;lt;/a&amp;gt;, about 17.6¢.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11/7 is &amp;lt;a class="wiki_link" href="/22_21"&amp;gt;22/21&amp;lt;/a&amp;gt; (about 80.5¢) above the &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt; perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>5ayz15ib1n6fzlou6gtn045d8wx5ik0</sha1>
    </revision>
    <revision>
      <id>48560</id>
      <parentid>80</parentid>
      <timestamp>2020-09-05T03:47:52Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>48560</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="760" sha1="1jet45onikplg15kwxxf64p0jz2rv3c" xml:space="preserve">'''11/7'''
|0 0 0 -1 1&amp;gt;

782.49204 cents

[[File:jid_11_7_pluck_adu_dr220.mp3]] [[:File:jid_11_7_pluck_adu_dr220.mp3|sound sample]]

In [[11-limit|11-limit]] [[Just_intonation|Just Intonation]], 11/7 is an undecimal (11-based) subminor sixth, measuring about 782.5¢ It is the inversion of [[14/11|14/11]], the undecimal supermajor third. Another subminor sixth is [[14/9|14/9]], at about 764.9¢; the interval between these is [[99/98|99/98]], about 17.6¢.

11/7 is [[22/21|22/21]] (about 80.5¢) above the [[3/2|3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]
[[Category:11-limit]]
[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]</text>
      <sha1>1jet45onikplg15kwxxf64p0jz2rv3c</sha1>
    </revision>
    <revision>
      <id>48601</id>
      <parentid>48560</parentid>
      <timestamp>2020-09-05T20:07:38Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>introduced infobox, simplified wiki markup</comment>
      <origin>48601</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="833" sha1="7yp4sgy0xwo5spo745fq18e52zwgwe5" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/7
| Monzo = 0 0 0 -1 1
| Cents = 782.49204
| Name = undecimal subminor sixth, &lt;br&gt; undecimal augmented fifth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal (11-based) subminor sixth''', measuring about 782.5¢ It is the inversion of [[14/11]], the undecimal supermajor third. Another subminor sixth is [[14/9]], at about 764.9¢; the interval between these is [[99/98]], about 17.6¢.

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== See also ==

* [[14/11]] -- its octave complement
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]</text>
      <sha1>7yp4sgy0xwo5spo745fq18e52zwgwe5</sha1>
    </revision>
    <revision>
      <id>48631</id>
      <parentid>48601</parentid>
      <timestamp>2020-09-06T12:16:33Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* See also */ +another sound example</comment>
      <origin>48631</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="925" sha1="ncjtwwsfsf66pnj8yh7qd3e3kzp1vzo" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/7
| Monzo = 0 0 0 -1 1
| Cents = 782.49204
| Name = undecimal subminor sixth, &lt;br&gt; undecimal augmented fifth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal (11-based) subminor sixth''', measuring about 782.5¢ It is the inversion of [[14/11]], the undecimal supermajor third. Another subminor sixth is [[14/9]], at about 764.9¢; the interval between these is [[99/98]], about 17.6¢.

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== See also ==

* [[14/11]] -- its octave complement
* [[Gallery of just intervals]]
* [[File:Ji-11-7-csound-foscil-220hz.mp3]] [[:File:Ji-11-7-csound-foscil-220hz.mp3|(info)]]

[[Category:11-limit]]
[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]</text>
      <sha1>ncjtwwsfsf66pnj8yh7qd3e3kzp1vzo</sha1>
    </revision>
    <revision>
      <id>49663</id>
      <parentid>48631</parentid>
      <timestamp>2020-09-16T20:44:24Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>linked file instead of embedding</comment>
      <origin>49663</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="901" sha1="sjqzig1qowurmmrno2vtq8rm8wmnjns" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/7
| Monzo = 0 0 0 -1 1
| Cents = 782.49204
| Name = undecimal subminor sixth, &lt;br&gt; undecimal augmented fifth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal (11-based) subminor sixth''', measuring about 782.5¢ It is the inversion of [[14/11]], the undecimal supermajor third. Another subminor sixth is [[14/9]], at about 764.9¢; the interval between these is [[99/98]], about 17.6¢.

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== See also ==

* [[14/11]] -- its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] - another sound example

[[Category:11-limit]]
[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]</text>
      <sha1>sjqzig1qowurmmrno2vtq8rm8wmnjns</sha1>
    </revision>
    <revision>
      <id>49892</id>
      <parentid>49663</parentid>
      <timestamp>2020-09-20T09:19:11Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+FJS name; cleanup; expansion</comment>
      <origin>49892</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1273" sha1="c01c5e9eeobobp60ioldqpakcdu4otm" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/7
| Monzo = 0 0 0 -1 1
| Cents = 782.49204
| Name = undecimal minor sixth, &lt;br&gt;undecimal augmented fifth
| Color name = 1or5, loru 5th
| FJS name = P5&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the undecimal major third. 

11/7 is flat of the Pythagorean minor sixth of [[128/81]] (about 792.2¢) by a pentacircle comma, [[896/891]]. It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== See also ==
* [[14/11]] – its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:11-limit]]
[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]</text>
      <sha1>c01c5e9eeobobp60ioldqpakcdu4otm</sha1>
    </revision>
    <revision>
      <id>51899</id>
      <parentid>49892</parentid>
      <timestamp>2020-10-25T12:20:52Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>added another table of EDO approximations, but is it really helpful?</comment>
      <origin>51899</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3167" sha1="mj1gztmd4gv5dsqvzmgll9qx3igg49u" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/7
| Monzo = 0 0 0 -1 1
| Cents = 782.49204
| Name = undecimal minor sixth, &lt;br&gt;undecimal augmented fifth
| Color name = 1or5, loru 5th
| FJS name = P5&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the undecimal major third. 

11/7 is flat of the Pythagorean minor sixth of [[128/81]] (about 792.2¢) by a pentacircle comma, [[896/891]]. It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== See also ==
* [[14/11]] – its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:11-limit]]
[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]</text>
      <sha1>mj1gztmd4gv5dsqvzmgll9qx3igg49u</sha1>
    </revision>
    <revision>
      <id>64447</id>
      <parentid>51899</parentid>
      <timestamp>2021-03-27T04:03:47Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64447</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3215" sha1="rlcnl7fq9caq6c7rw4vzfwkv6a0cije" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/7
| Monzo = 0 0 0 -1 1
| Cents = 782.49204
| Name = undecimal minor sixth, &lt;br&gt;undecimal augmented fifth
| Color name = 1or5, loru 5th
| FJS name = P5&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the undecimal major third. 

11/7 is flat of the Pythagorean minor sixth of [[128/81]] (about 792.2¢) by a pentacircle comma, [[896/891]]. It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== See also ==
* [[14/11]] – its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:11-limit]]
[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>rlcnl7fq9caq6c7rw4vzfwkv6a0cije</sha1>
    </revision>
    <revision>
      <id>86294</id>
      <parentid>64447</parentid>
      <timestamp>2022-02-22T23:46:39Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86294</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3167" sha1="mj1gztmd4gv5dsqvzmgll9qx3igg49u" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 11/7
| Monzo = 0 0 0 -1 1
| Cents = 782.49204
| Name = undecimal minor sixth, &lt;br&gt;undecimal augmented fifth
| Color name = 1or5, loru 5th
| FJS name = P5&lt;sup&gt;11&lt;/sup&gt;&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the undecimal major third. 

11/7 is flat of the Pythagorean minor sixth of [[128/81]] (about 792.2¢) by a pentacircle comma, [[896/891]]. It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== See also ==
* [[14/11]] – its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:11-limit]]
[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]</text>
      <sha1>mj1gztmd4gv5dsqvzmgll9qx3igg49u</sha1>
    </revision>
    <revision>
      <id>98013</id>
      <parentid>86294</parentid>
      <timestamp>2022-10-25T12:17:32Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>98013</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3034" sha1="kwlhzsuhnyxm8v4djhlqtueooweg9mn" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, undecimal augmented fifth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the undecimal major third. 

11/7 is flat of the Pythagorean minor sixth of [[128/81]] (about 792.2¢) by a pentacircle comma, [[896/891]]. It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== See also ==
* [[14/11]] – its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]</text>
      <sha1>kwlhzsuhnyxm8v4djhlqtueooweg9mn</sha1>
    </revision>
    <revision>
      <id>104697</id>
      <parentid>98013</parentid>
      <timestamp>2023-03-06T12:48:05Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <origin>104697</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3120" sha1="8vqpgbp9iqm7xpwg064bp44t7a0v5ix" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, undecimal augmented fifth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the undecimal major third. 

11/7 is flat of the Pythagorean minor sixth of [[128/81]] (about 792.2¢) by a pentacircle comma, [[896/891]]. It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== Proximity with Pi/2 ==
(11/7)/(Pi/2) is an unnoticeable comma of only +0.7 cents.

== See also ==
* [[14/11]] – its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]</text>
      <sha1>8vqpgbp9iqm7xpwg064bp44t7a0v5ix</sha1>
    </revision>
    <revision>
      <id>104698</id>
      <parentid>104697</parentid>
      <timestamp>2023-03-06T12:48:28Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <comment>/* Proximity with Pi/2 */</comment>
      <origin>104698</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3124" sha1="drtzowkiqam0gq9pn8pjx4cxb1etqmi" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, undecimal augmented fifth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the undecimal major third. 

11/7 is flat of the Pythagorean minor sixth of [[128/81]] (about 792.2¢) by a pentacircle comma, [[896/891]]. It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== Proximity with Pi/2 ==
(11/7)/(Pi/2) is an [[unnoticeable comma]] of only +0.7 cents.

== See also ==
* [[14/11]] – its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]</text>
      <sha1>drtzowkiqam0gq9pn8pjx4cxb1etqmi</sha1>
    </revision>
    <revision>
      <id>104709</id>
      <parentid>104698</parentid>
      <timestamp>2023-03-06T14:48:14Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <comment>/* Proximity with Pi/2 */</comment>
      <origin>104709</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3124" sha1="ajik5yq8151yaevpj6r4fgsux2qanu9" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, undecimal augmented fifth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the undecimal major third. 

11/7 is flat of the Pythagorean minor sixth of [[128/81]] (about 792.2¢) by a pentacircle comma, [[896/891]]. It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== Proximity with π/2 ==
(11/7)/(π/2) is an [[unnoticeable comma]] of only +0.7 cents.

== See also ==
* [[14/11]] – its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]</text>
      <sha1>ajik5yq8151yaevpj6r4fgsux2qanu9</sha1>
    </revision>
    <revision>
      <id>104905</id>
      <parentid>104709</parentid>
      <timestamp>2023-03-07T06:28:16Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Proximity with π/2 */</comment>
      <origin>104905</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3117" sha1="3tnilgnbuugrxyjwzywlej6snbtigu4" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, undecimal augmented fifth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the undecimal major third. 

11/7 is flat of the Pythagorean minor sixth of [[128/81]] (about 792.2¢) by a pentacircle comma, [[896/891]]. It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== Proximity with π/2 ==
22/7π is an [[unnoticeable comma]] of only +0.7 cents.

== See also ==
* [[14/11]] – its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]</text>
      <sha1>3tnilgnbuugrxyjwzywlej6snbtigu4</sha1>
    </revision>
    <revision>
      <id>104956</id>
      <parentid>104905</parentid>
      <timestamp>2023-03-07T17:17:08Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>/* Proximity with π/2 */ it's fine to include 22/7π now, but without also preserving the original formulation in terms of 11/7, it's not obvious why this fact is in this article</comment>
      <origin>104956</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3133" sha1="t9uxzgw0voq2sgobfjuev36h1v9zv2e" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, undecimal augmented fifth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the undecimal major third. 

11/7 is flat of the Pythagorean minor sixth of [[128/81]] (about 792.2¢) by a pentacircle comma, [[896/891]]. It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== Proximity with π/2 ==
(11/7)/(π/2) = 22/7π is an [[unnoticeable comma]] of only +0.7 cents.

== See also ==
* [[14/11]] – its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]</text>
      <sha1>t9uxzgw0voq2sgobfjuev36h1v9zv2e</sha1>
    </revision>
    <revision>
      <id>157034</id>
      <parentid>104956</parentid>
      <timestamp>2024-09-30T08:42:53Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Recategorize</comment>
      <origin>157034</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3143" sha1="oi6st6ii6z5ge7acyt7zen2dmc9pmyw" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, undecimal augmented fifth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the undecimal major third. 

11/7 is flat of the Pythagorean minor sixth of [[128/81]] (about 792.2¢) by a pentacircle comma, [[896/891]]. It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

11/7 is [[22/21]] (about 80.5¢) above the [[3/2]] perfect fifth, allowing the possibility of a resolution down by a step from 11/7 to 3/2.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== Proximity with π/2 ==
(11/7)/(π/2) = 22/7π is an [[unnoticeable comma]] of only +0.7 cents.

== See also ==
* [[14/11]] – its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7 intervals]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]</text>
      <sha1>oi6st6ii6z5ge7acyt7zen2dmc9pmyw</sha1>
    </revision>
    <revision>
      <id>162125</id>
      <parentid>157034</parentid>
      <timestamp>2024-10-25T09:08:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Rework; +"pentacircle minor sixth" and reasons; -"undecimal augmented fifth" (its octave complement isn't "undecimal diminished fourth" yet)</comment>
      <origin>162125</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3705" sha1="0wfu5czo05fbs02gl0r7p8awuwh7bp3" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, pentacircle minor sixth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the pentacircle major third, and represents the difference between the 7th and 11th harmonics of the [[harmonic series]]. 

In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fifth, as it is a [[3/2|perfect fifth (3/2)]] plus an instance of [[22/21]], which is a stack consisting of an [[33/32|undecimal quartertone (33/32)]] and a [[64/63|septimal comma (64/63)]], neither of which changes the [[scale|scale degree]] or [[interval quality|quality]]. However, it is only flat of the Pythagorean ([[3-limit]]) minor sixth of [[128/81]] (about 792.2¢) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a minor sixth, hence the names. 

It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

As 11/7 is 22/21 (about 80.5¢) above the perfect fifth, it can be resolved down by a step from 11/7 to 3/2.

== Approximations by edos ==

Following [[edo]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[Edo]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== Proximity with π/2 ==
(11/7)/(π/2) = 22/7π is an [[unnoticeable comma]] of only 0.7 cents.

== See also ==
* [[14/11]] – its octave complement
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7 intervals]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]</text>
      <sha1>0wfu5czo05fbs02gl0r7p8awuwh7bp3</sha1>
    </revision>
    <revision>
      <id>162420</id>
      <parentid>162125</parentid>
      <timestamp>2024-10-26T23:57:04Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* See also */ link</comment>
      <origin>162420</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3718" sha1="e6md4k4xsxg85nfozsazwytb37hevdr" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, pentacircle minor sixth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the pentacircle major third, and represents the difference between the 7th and 11th harmonics of the [[harmonic series]]. 

In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fifth, as it is a [[3/2|perfect fifth (3/2)]] plus an instance of [[22/21]], which is a stack consisting of an [[33/32|undecimal quartertone (33/32)]] and a [[64/63|septimal comma (64/63)]], neither of which changes the [[scale|scale degree]] or [[interval quality|quality]]. However, it is only flat of the Pythagorean ([[3-limit]]) minor sixth of [[128/81]] (about 792.2¢) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a minor sixth, hence the names. 

It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

As 11/7 is 22/21 (about 80.5¢) above the perfect fifth, it can be resolved down by a step from 11/7 to 3/2.

== Approximations by edos ==

Following [[edo]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[Edo]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== Proximity with π/2 ==
(11/7)/(π/2) = 22/7π is an [[unnoticeable comma]] of only 0.7 cents.

== See also ==
* [[14/11]] – its octave complement
* [[Ed11/7]]
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7 intervals]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]</text>
      <sha1>e6md4k4xsxg85nfozsazwytb37hevdr</sha1>
    </revision>
    <revision>
      <id>168051</id>
      <parentid>162420</parentid>
      <timestamp>2024-12-01T23:01:08Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>[[Category:Taxicab-2 intervals]]</comment>
      <origin>168051</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3751" sha1="0rgwvxo9l0tf4jaf8wtrev2dzxw2lnd" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, pentacircle minor sixth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the pentacircle major third, and represents the difference between the 7th and 11th harmonics of the [[harmonic series]]. 

In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fifth, as it is a [[3/2|perfect fifth (3/2)]] plus an instance of [[22/21]], which is a stack consisting of an [[33/32|undecimal quartertone (33/32)]] and a [[64/63|septimal comma (64/63)]], neither of which changes the [[scale|scale degree]] or [[interval quality|quality]]. However, it is only flat of the Pythagorean ([[3-limit]]) minor sixth of [[128/81]] (about 792.2¢) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a minor sixth, hence the names. 

It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

As 11/7 is 22/21 (about 80.5¢) above the perfect fifth, it can be resolved down by a step from 11/7 to 3/2.

== Approximations by edos ==

Following [[edo]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[Edo]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== Proximity with π/2 ==
(11/7)/(π/2) = 22/7π is an [[unnoticeable comma]] of only 0.7 cents.

== See also ==
* [[14/11]] – its octave complement
* [[Ed11/7]]
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7 intervals]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]
[[Category:Taxicab-2 intervals]]</text>
      <sha1>0rgwvxo9l0tf4jaf8wtrev2dzxw2lnd</sha1>
    </revision>
    <revision>
      <id>215651</id>
      <parentid>168051</parentid>
      <timestamp>2025-11-01T19:07:37Z</timestamp>
      <contributor>
        <username>Eufalesio</username>
        <id>581</id>
      </contributor>
      <comment>Made 22/7 to properly talk about pi approximants</comment>
      <origin>215651</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3848" sha1="aa2mddn52j1fzk8wo15tkj4flo0mwjd" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, pentacircle minor sixth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the pentacircle major third, and represents the difference between the 7th and 11th harmonics of the [[harmonic series]]. 

In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fifth, as it is a [[3/2|perfect fifth (3/2)]] plus an instance of [[22/21]], which is a stack consisting of an [[33/32|undecimal quartertone (33/32)]] and a [[64/63|septimal comma (64/63)]], neither of which changes the [[scale|scale degree]] or [[interval quality|quality]]. However, it is only flat of the Pythagorean ([[3-limit]]) minor sixth of [[128/81]] (about 792.2¢) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a minor sixth, hence the names. 

It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

As 11/7 is 22/21 (about 80.5¢) above the perfect fifth, it can be resolved down by a step from 11/7 to 3/2.

== Approximations by edos ==

Following [[edo]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/7. Errors are given by magnitude, the arrows in the table show if the edo representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[Edo]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[20edo|20]]  ||  13\20  || 2.4920 || 4.1534 || &amp;darr; ||
|-
|  [[23edo|23]]  ||  15\23  || 0.1167 || 0.2236 || &amp;uarr; ||  [[46edo|30\46]], [[69edo|45\69]], [[92edo|60\92]], [[115edo|75\115]], [[138edo|90\138]], [[161edo|105\161]], [[184edo|120\184]]
|-
|  [[26edo|26]]  ||  17\26  || 2.1233 || 4.6006 || &amp;uarr; ||
|-
|  [[43edo|43]]  ||  28\43  || 1.0967 || 3.9298 || &amp;darr; ||
|-
|  [[49edo|49]]  ||  32\49  || 1.1814 || 4.8242 || &amp;uarr; ||
|-
|  [[66edo|66]]  ||  43\66  || 0.6739 || 3.7062 || &amp;darr; ||
|-
|  [[72edo|72]]  ||  47\72  || 0.8413 || 5.0478 || &amp;uarr; ||
|-
|  [[89edo|89]]  ||  58\89  || 0.4696 || 3.4826 || &amp;darr; ||  [[178edo|116\178]]
|-
|  [[95edo|95]]  ||  62\95  || 0.6659 || 5.2714 || &amp;uarr; ||
|-
| [[112edo|112]] ||  73\112 || 0.3492 || 3.2590 || &amp;darr; ||
|-
| [[118edo|118]] ||  77\118 || 0.5588 || 5.4950 || &amp;uarr; ||
|-
| [[135edo|135]] ||  88\135 || 0.2698 || 3.0354 || &amp;darr; ||
|-
| [[141edo|141]] ||  92\141 || 0.4867 || 5.7186 || &amp;uarr; ||
|-
| [[158edo|158]] || 103\158 || 0.2136 || 2.8118 || &amp;darr; ||
|-
| [[164edo|164]] || 107\164 || 0.4348 || 5.9422 || &amp;uarr; ||
|-
| [[181edo|181]] || 118\181 || 0.1716 || 2.5882 || &amp;darr; ||
|-
| [[187edo|187]] || 122\187 || 0.3957 || 6.1658 || &amp;uarr; ||
|}

&lt;references/&gt;

== Proximity with π ==
[[22/7]], one octave higher, is a fraction convergent to the continued fraction of π. Such is the exactness, that 22/7π is an [[unnoticeable comma]] of only 0.7 cents.

== See also ==
* [[14/11]] – its octave complement
* [[Ed11/7]]
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7 intervals]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]
[[Category:Taxicab-2 intervals]]</text>
      <sha1>aa2mddn52j1fzk8wo15tkj4flo0mwjd</sha1>
    </revision>
    <revision>
      <id>215871</id>
      <parentid>215651</parentid>
      <timestamp>2025-11-03T07:36:09Z</timestamp>
      <contributor>
        <username>Pailiaq</username>
        <id>507</id>
      </contributor>
      <origin>215871</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2029" sha1="jse54yd5zje4m4398i0si3jktbii45g" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, pentacircle minor sixth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the pentacircle major third, and represents the difference between the 7th and 11th harmonics of the [[harmonic series]]. 

In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fifth, as it is a [[3/2|perfect fifth (3/2)]] plus an instance of [[22/21]], which is a stack consisting of an [[33/32|undecimal quartertone (33/32)]] and a [[64/63|septimal comma (64/63)]], neither of which changes the [[scale|scale degree]] or [[interval quality|quality]]. However, it is only flat of the Pythagorean ([[3-limit]]) minor sixth of [[128/81]] (about 792.2¢) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a minor sixth, hence the names. 

It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

As 11/7 is 22/21 (about 80.5¢) above the perfect fifth, it can be resolved down by a step from 11/7 to 3/2.

== Approximation ==
 {{Interval_Edo_Approximation | 11/7}}

&lt;references/&gt;

== Proximity with π ==
[[22/7]], one octave higher, is a fraction convergent to the continued fraction of π. Such is the exactness, that 22/7π is an [[unnoticeable comma]] of only 0.7 cents.

== See also ==
* [[14/11]] – its octave complement
* [[Ed11/7]]
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7 intervals]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]
[[Category:Taxicab-2 intervals]]</text>
      <sha1>jse54yd5zje4m4398i0si3jktbii45g</sha1>
    </revision>
    <revision>
      <id>216031</id>
      <parentid>215871</parentid>
      <timestamp>2025-11-03T13:07:47Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"</comment>
      <origin>216031</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2026" sha1="9zjq2902ubbtwhuitvsf21f2p0l70zx" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, pentacircle minor sixth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782.5¢. It is the inversion of [[14/11]], the pentacircle major third, and represents the difference between the 7th and 11th harmonics of the [[harmonic series]]. 

In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fifth, as it is a [[3/2|perfect fifth (3/2)]] plus an instance of [[22/21]], which is a stack consisting of an [[33/32|undecimal quartertone (33/32)]] and a [[64/63|septimal comma (64/63)]], neither of which changes the [[scale|scale degree]] or [[interval quality|quality]]. However, it is only flat of the Pythagorean ([[3-limit]]) minor sixth of [[128/81]] (about 792.2¢) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a minor sixth, hence the names. 

It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7¢) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9¢) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6¢) by a valinorsma, [[176/175]]. 

As 11/7 is 22/21 (about 80.5¢) above the perfect fifth, it can be resolved down by a step from 11/7 to 3/2.

== Approximation ==
{{Interval edo approximation|11/7}}

&lt;references/&gt;

== Proximity with π ==
[[22/7]], one octave higher, is a fraction convergent to the continued fraction of π. Such is the exactness, that 22/7π is an [[unnoticeable comma]] of only 0.7 cents.

== See also ==
* [[14/11]] – its octave complement
* [[Ed11/7]]
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7 intervals]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]
[[Category:Taxicab-2 intervals]]</text>
      <sha1>9zjq2902ubbtwhuitvsf21f2p0l70zx</sha1>
    </revision>
    <revision>
      <id>221696</id>
      <parentid>216031</parentid>
      <timestamp>2026-01-16T09:12:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Rework</comment>
      <origin>221696</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2401" sha1="7ac2eftli3boi83jz60lasulohx8abj" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, pentacircle minor sixth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782.5 [[cent]]s. It is the inversion of [[14/11]], the pentacircle major third, and represents the difference between the 7th and 11th harmonics of the [[harmonic series]]. 

In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fifth, as it is a [[3/2|perfect fifth (3/2)]] plus an instance of [[22/21]], which is a stack consisting of an [[33/32|undecimal quartertone (33/32)]] and a [[64/63|septimal comma (64/63)]], neither of which changes the [[scale|scale degree]] or [[interval quality|quality]]. However, it is only flat of the Pythagorean ([[3-limit]]) minor sixth of [[128/81]] (about 792.2{{c}}) by a [[896/891|pentacircle comma (896/891)]], which makes it function more often as a minor sixth, hence the names. It functions as such in voicings of the harmonic eleventh chord, [[4:5:6:7:9:11]]. 

However, it is only flat of the [[128/81|Pythagorean minor sixth]] (about 792.2{{c}}) by a [[896/891|pentacircle comma (896/891)]], which makes it function sometimes as a minor sixth, hence the names. For one thing, as it is 22/21 (about 80.5{{c}}) above the perfect fifth, it can be resolved down by a step to the perfect fifth. 

It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7{{c}}) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9{{c}}) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6{{c}}) by a valinorsma, [[176/175]]. 

== Approximation ==
{{Interval edo approximation|11/7}}

== Proximity with acoustic pi ==
[[22/7]], one octave higher, is a fraction convergent to the continued fraction of acoustic pi. Such is the exactness, that 22/7π is an [[unnoticeable comma]] of only 0.7 cents.

== See also ==
* [[14/11]] – its [[octave complement]]
* [[21/11]] – its [[twelfth complement]]
* [[Ed11/7]]
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7 intervals]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]
[[Category:Taxicab-2 intervals]]</text>
      <sha1>7ac2eftli3boi83jz60lasulohx8abj</sha1>
    </revision>
    <revision>
      <id>223080</id>
      <parentid>221696</parentid>
      <timestamp>2026-02-03T00:34:44Z</timestamp>
      <contributor>
        <username>Squib</username>
        <id>562</id>
      </contributor>
      <comment>removed duplicate text</comment>
      <origin>223080</origin>
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      <format>text/x-wiki</format>
      <text bytes="2180" sha1="rehsrp5cwj02xy24689sxspr7t51j8k" xml:space="preserve">{{Infobox Interval
| Name = undecimal minor sixth, pentacircle minor sixth
| Color name = 1or5, loru 5th
| Sound = jid_11_7_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/7''' is an '''undecimal minor sixth''', specifically the '''pentacircle minor sixth''', measuring about 782.5 [[cent]]s. It is the inversion of [[14/11]], the pentacircle major third, and represents the difference between the 7th and 11th harmonics of the [[harmonic series]]. 

In many notation systems (e.g. [[FJS]], [[HEJI]]), it is an imperfect fifth, as it is a [[3/2|perfect fifth (3/2)]] plus an instance of [[22/21]], which is a stack consisting of an [[33/32|undecimal quartertone (33/32)]] and a [[64/63|septimal comma (64/63)]], neither of which changes the [[scale|scale degree]] or [[interval quality|quality]]. It functions as such in voicings of the harmonic eleventh chord, [[4:5:6:7:9:11]].

However, it is only flat of the [[128/81|Pythagorean minor sixth]] (about 792.2{{c}}) by a [[896/891|pentacircle comma (896/891)]], which makes it function sometimes as a minor sixth, hence the names. For one thing, as it is 22/21 (about 80.5{{c}}) above the perfect fifth, it can be resolved down by a step to the perfect fifth. 

It is flat of the 5-limit minor sixth of [[8/5]] (about 813.7{{c}}) by [[56/55]]. It is sharp of the 7-limit subminor sixth of [[14/9]] (about 764.9{{c}}) by a mothwellsma, [[99/98]]. And finally, it is sharp of the classic augmented fifth of [[25/16]] (about 772.6{{c}}) by a valinorsma, [[176/175]].

== Approximation ==
{{Interval edo approximation|11/7}}

== Proximity with acoustic pi ==
[[22/7]], one octave higher, is a fraction convergent to the continued fraction of acoustic pi. Such is the exactness, that 22/7π is an [[unnoticeable comma]] of only 0.7 cents.

== See also ==
* [[14/11]] – its [[octave complement]]
* [[21/11]] – its [[twelfth complement]]
* [[Ed11/7]]
* [[Gallery of just intervals]]
* [[:File:Ji-11-7-csound-foscil-220hz.mp3]] – another sound example

[[Category:Over-7 intervals]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Subminor sixth]]
[[Category:Pentacircle]]
[[Category:Taxicab-2 intervals]]</text>
      <sha1>rehsrp5cwj02xy24689sxspr7t51j8k</sha1>
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  </page>
  <page>
    <title>11 7</title>
    <ns>0</ns>
    <id>81</id>
    <redirect title="11/7" />
    <revision>
      <id>81</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>81</origin>
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      <text bytes="18" sha1="fkudk1lkgygsdo9fgat97xtirept0qa" xml:space="preserve">#REDIRECT [[11/7]]</text>
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  </page>
  <page>
    <title>11/8</title>
    <ns>0</ns>
    <id>82</id>
    <revision>
      <id>82</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <text bytes="1114" sha1="2q0vnz799ds8tijltsuq5159lgy2fo6" xml:space="preserve">'''11/8'''
|-3 0 0 0 1&amp;gt;

551.31794 cents

[[File:jid_11_8_pluck_adu_dr220.mp3]] [[:File:jid_11_8_pluck_adu_dr220.mp3|sound sample]]

In [[11-limit|11-limit]] [[Just_intonation|Just Intonation]], 11/8 is an undecimal (11-based) [[Superfourth|superfourth]] of about 551.3¢. Falling about halfway between [[12edo|12edo]]'s [[Perfect_fourth|perfect fourth]] and [[tritone|tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo|24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]      [[Category:11-limit]]
[[Category:fourth]]
[[Category:interval]]
[[Category:ratio]]
[[Category:superfourth]]
[[Category:undecimal]]
[[Category:untwelve]]</text>
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    </revision>
    <revision>
      <id>4330</id>
      <timestamp>2014-06-07T15:33:47Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513194320 - Original comment: **</comment>
      <origin>4330</origin>
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      <text bytes="4686" sha1="7us7z41nvubaq0p2l9xrojp7gbxroft" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-07 15:33:47 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513194320&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/8**
|-3 0 0 0 1&amp;gt;
551.31794 cents
[[media type="file" key="jid_11_8_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_11_8_pluck_adu_dr220.mp3|sound sample]]

In [[11-limit]] [[Just Intonation]], 11/8 is an undecimal (11-based) [[superfourth]] of about 551.3¢. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_8&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/8&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|-3 0 0 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
551.31794 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_8_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_8_pluck_adu_dr220.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_8_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_8_pluck_adu_dr220.mp3/513194244/jid_11_8_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_8_pluck_adu_dr220.mp3/513194244/jid_11_8_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 11/8 is an undecimal (11-based) &amp;lt;a class="wiki_link" href="/superfourth"&amp;gt;superfourth&amp;lt;/a&amp;gt; of about 551.3¢. Falling about halfway between &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;'s &amp;lt;a class="wiki_link" href="/perfect%20fourth"&amp;gt;perfect fourth&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/tritone"&amp;gt;tritone&amp;lt;/a&amp;gt;, it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    </revision>
    <revision>
      <id>4331</id>
      <timestamp>2011-10-09T10:57:20Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 262969172 - Original comment: **</comment>
      <origin>4331</origin>
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      <text bytes="3139" sha1="bbnpnmkq0t0nedpoq5452ibzorcplm4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-10-09 10:57:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;262969172&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[11-limit]] [[Just Intonation]], 11/8 is an undecimal (11-based) [[superfourth]] of about 551.3¢. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_8&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 11/8 is an undecimal (11-based) &amp;lt;a class="wiki_link" href="/superfourth"&amp;gt;superfourth&amp;lt;/a&amp;gt; of about 551.3¢. Falling about halfway between &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;'s &amp;lt;a class="wiki_link" href="/perfect%20fourth"&amp;gt;perfect fourth&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/tritone"&amp;gt;tritone&amp;lt;/a&amp;gt;, it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4332</id>
      <timestamp>2011-09-14T20:31:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 254160566 - Original comment: **</comment>
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      <text bytes="2831" sha1="9hitoi2sw0wl8upnaqa5tn1fb5evgda" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-09-14 20:31:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;254160566&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[11-limit]] [[Just Intonation]], 11/8 is an interval of about 551.3¢. Falling about halfway between [[12edo]]'s perfect fourth and tritone, it is very xenharmonic. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_8&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 11/8 is an interval of about 551.3¢. Falling about halfway between &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;'s perfect fourth and tritone, it is very xenharmonic. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <parentid>82</parentid>
      <timestamp>2018-10-10T22:31:42Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
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      <comment>added interwiki:de; links simplified</comment>
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      <text bytes="1078" sha1="l626sx2bomkm854tuu81lqjlqp8c6ed" xml:space="preserve">'''11/8'''

{{Monzo| -3 0 0 0 1 }}

551.31794 cents

[[File:jid_11_8_pluck_adu_dr220.mp3]] [[:File:jid_11_8_pluck_adu_dr220.mp3|sound info]]

In [[11-limit]] [[Just Intonation]], 11/8 is an undecimal (11-based) [[superfourth]] of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

:''See also [[Gallery of Just Intervals]]''

[[Category:11-limit]]
[[Category:fourth]]
[[Category:interval]]
[[Category:ratio]]
[[Category:superfourth]]
[[Category:undecimal]]
[[Category:untwelve]]

&lt;!-- interwiki --&gt;
[[de:Alphorn-Fa]]</text>
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    <revision>
      <id>36587</id>
      <parentid>36531</parentid>
      <timestamp>2018-10-11T22:29:57Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>switched to infobox interval</comment>
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      <text bytes="1094" sha1="trlab6lhdiwuxwavjxnoe98ffoi6ntv" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], 11/8 is an undecimal (11-based) [[superfourth]] of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

:''See also [[Gallery of Just Intervals]]''

[[Category:11-limit]]
[[Category:fourth]]
[[Category:interval]]
[[Category:ratio]]
[[Category:superfourth]]
[[Category:undecimal]]
[[Category:untwelve]]

&lt;!-- interwiki --&gt;
[[de:Alphorn-Fa]]</text>
      <sha1>trlab6lhdiwuxwavjxnoe98ffoi6ntv</sha1>
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    <revision>
      <id>37584</id>
      <parentid>36587</parentid>
      <timestamp>2018-10-23T03:25:04Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
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      <text bytes="1123" sha1="nbdopgx2kn5wnjn1sfpibck2ougeaqj" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth
| Color name = 1o4, lova 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], 11/8 is an undecimal (11-based) [[superfourth]] of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

:''See also [[Gallery of Just Intervals]]''

[[Category:11-limit]]
[[Category:fourth]]
[[Category:interval]]
[[Category:ratio]]
[[Category:superfourth]]
[[Category:undecimal]]
[[Category:untwelve]]

&lt;!-- interwiki --&gt;
[[de:Alphorn-Fa]]</text>
      <sha1>nbdopgx2kn5wnjn1sfpibck2ougeaqj</sha1>
    </revision>
    <revision>
      <id>37672</id>
      <parentid>37584</parentid>
      <timestamp>2018-10-23T19:04:51Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
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      <minor/>
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      <text bytes="1122" sha1="gamqbuubkynt90zlzthq69cdmfucyv4" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], 11/8 is an undecimal (11-based) [[superfourth]] of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

:''See also [[Gallery of Just Intervals]]''

[[Category:11-limit]]
[[Category:fourth]]
[[Category:interval]]
[[Category:ratio]]
[[Category:superfourth]]
[[Category:undecimal]]
[[Category:untwelve]]

&lt;!-- interwiki --&gt;
[[de:Alphorn-Fa]]</text>
      <sha1>gamqbuubkynt90zlzthq69cdmfucyv4</sha1>
    </revision>
    <revision>
      <id>38180</id>
      <parentid>37672</parentid>
      <timestamp>2018-10-29T17:44:53Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>expanded see-also section, interwiki via template</comment>
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      <text bytes="1182" sha1="c7knv3tl7fn7cx0u2qe634zyloyj2n2" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an undecimal [[superfourth]] of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

== See also ==
* [[Gallery of just intervals]]
* [[16/11]] - its [[inverse interval]]

[[Category:11-limit]]
[[Category:Interval]]
[[Category:Ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Undecimal]]
[[Category:Untwelve]]</text>
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    </revision>
    <revision>
      <id>42973</id>
      <parentid>38180</parentid>
      <timestamp>2020-01-02T23:24:02Z</timestamp>
      <contributor>
        <username>Awelotta</username>
        <id>147</id>
      </contributor>
      <minor/>
      <comment>Major fourth as a possible name</comment>
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      <text bytes="1196" sha1="g0o0kr0ywwd8ss8d41j4xlrh0bbw5m2" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, major fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an undecimal [[superfourth]] of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

== See also ==
* [[Gallery of just intervals]]
* [[16/11]] - its [[inverse interval]]

[[Category:11-limit]]
[[Category:Interval]]
[[Category:Ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Undecimal]]
[[Category:Untwelve]]</text>
      <sha1>g0o0kr0ywwd8ss8d41j4xlrh0bbw5m2</sha1>
    </revision>
    <revision>
      <id>46687</id>
      <parentid>42973</parentid>
      <timestamp>2020-06-13T11:42:21Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>recat</comment>
      <origin>46687</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1183" sha1="4ots0ahuxd0r5j4zqablf561b63d8zk" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, major fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an undecimal [[superfourth]] of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

== See also ==
* [[Gallery of just intervals]]
* [[16/11]] - its [[inverse interval]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Undecimal]]
[[Category:Untwelve]]</text>
      <sha1>4ots0ahuxd0r5j4zqablf561b63d8zk</sha1>
    </revision>
    <revision>
      <id>46754</id>
      <parentid>46687</parentid>
      <timestamp>2020-06-13T22:26:08Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cats, lemma bold</comment>
      <origin>46754</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1186" sha1="6trnu2ug4bloxvsd35zuvr488f9mpd9" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, major fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

== See also ==
* [[Gallery of just intervals]]
* [[16/11]] - its [[inverse interval]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Listen]]
[[Category:Untwelve]]</text>
      <sha1>6trnu2ug4bloxvsd35zuvr488f9mpd9</sha1>
    </revision>
    <revision>
      <id>47619</id>
      <parentid>46754</parentid>
      <timestamp>2020-07-12T15:11:53Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>+cat</comment>
      <origin>47619</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1208" sha1="h1dnsczljw305jsb3ld6ua33zbvr1yg" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, major fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

== See also ==
* [[Gallery of just intervals]]
* [[16/11]] - its [[inverse interval]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]</text>
      <sha1>h1dnsczljw305jsb3ld6ua33zbvr1yg</sha1>
    </revision>
    <revision>
      <id>48545</id>
      <parentid>47619</parentid>
      <timestamp>2020-09-05T03:10:24Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* See also */</comment>
      <origin>48545</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1228" sha1="batokg6743fzzmks3pr5t8vzpe2h03j" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, major fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (5) and 12 (3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

== See also ==
* [[Gallery of just intervals]]
* [[16/11]] - its [[inverse interval]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]</text>
      <sha1>batokg6743fzzmks3pr5t8vzpe2h03j</sha1>
    </revision>
    <revision>
      <id>49875</id>
      <parentid>48545</parentid>
      <timestamp>2020-09-20T06:37:04Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+FJS name; cleanup</comment>
      <origin>49875</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1317" sha1="fapnem33q0hs73x6ut05eakph3ttofs" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

== See also ==
* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]</text>
      <sha1>fapnem33q0hs73x6ut05eakph3ttofs</sha1>
    </revision>
    <revision>
      <id>51912</id>
      <parentid>49875</parentid>
      <timestamp>2020-10-25T14:12:43Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>another table, I'm not sure if the decimal places should be reduced deltas can be very small sometimes</comment>
      <origin>51912</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3238" sha1="tt6ltpk2l15th33z4awrr6fc1pti4kw" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]</text>
      <sha1>tt6ltpk2l15th33z4awrr6fc1pti4kw</sha1>
    </revision>
    <revision>
      <id>51913</id>
      <parentid>51912</parentid>
      <timestamp>2020-10-25T14:13:58Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Approximations by EDOs */ footnotes added</comment>
      <origin>51913</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3253" sha1="t4qph1nr13u3grgzm7f7gs90z9jhr25" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]]. Falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic. It is the simplest superfourth in JI. As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references/&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]</text>
      <sha1>t4qph1nr13u3grgzm7f7gs90z9jhr25</sha1>
    </revision>
    <revision>
      <id>51988</id>
      <parentid>51913</parentid>
      <timestamp>2020-10-27T06:03:43Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>Added more information relating to my current studies</comment>
      <origin>51988</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3768" sha1="rh7a8ohh43c20ec0j6z26lo0iiujjoh" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Alpharabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval has [https://en.wikipedia.org/wiki/Major_fourth_and_minor_fifth also been referred to] as the '''major fourth'''.  Furthermore, as stacks of this interval form a core axis of Alpharabian tuning (see [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the "'''Alpharabian paramajor fourth'''" or even the "'''just paramajor fourth'''". 

This interval is the simplest superfourth in JI, and, falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references/&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]</text>
      <sha1>rh7a8ohh43c20ec0j6z26lo0iiujjoh</sha1>
    </revision>
    <revision>
      <id>51989</id>
      <parentid>51988</parentid>
      <timestamp>2020-10-27T06:04:34Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <minor/>
      <origin>51989</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3765" sha1="tfuit916xhp9tdn67tfcc4hns8d7eup" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Alpharabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval has [https://en.wikipedia.org/wiki/Major_fourth_and_minor_fifth also been referred to] as the '''major fourth'''.  Furthermore, as stacks of this interval form a core axis of Alpharabian tuning (see [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Alpharabian paramajor fourth''' or even the '''just paramajor fourth'''. 

This interval is the simplest superfourth in JI, and, falling about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3). It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]</text>
      <sha1>tfuit916xhp9tdn67tfcc4hns8d7eup</sha1>
    </revision>
    <revision>
      <id>51996</id>
      <parentid>51989</parentid>
      <timestamp>2020-10-27T15:24:52Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>Additional information added</comment>
      <origin>51996</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4190" sha1="o87li5wl16pbtrt15yuckawzshbhao0" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Alpharabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval has [https://en.wikipedia.org/wiki/Major_fourth_and_minor_fifth also been referred to] as the '''major fourth'''.  Furthermore, as stacks of this interval form a core axis of Alpharabian tuning (see [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Alpharabian paramajor fourth''' or even the '''just paramajor fourth'''.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).   It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]</text>
      <sha1>o87li5wl16pbtrt15yuckawzshbhao0</sha1>
    </revision>
    <revision>
      <id>51998</id>
      <parentid>51996</parentid>
      <timestamp>2020-10-27T18:01:58Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>51998</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4189" sha1="sks20svjuc8469soew0eyhr0n085gvb" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Alpharabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval has [https://en.wikipedia.org/wiki/Major_fourth_and_minor_fifth also been referred to] as the '''major fourth'''.  Furthermore, as stacks of this interval form a core axis of Alpharabian tuning (see [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Alpharabian paramajor fourth''' or even the '''just paramajor fourth'''.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]</text>
      <sha1>sks20svjuc8469soew0eyhr0n085gvb</sha1>
    </revision>
    <revision>
      <id>53153</id>
      <parentid>51998</parentid>
      <timestamp>2020-11-25T00:11:27Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>additional link</comment>
      <origin>53153</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4198" sha1="qjiblx7zinds55odafwhx1xh1dg2ghz" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Alpharabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval has [https://en.wikipedia.org/wiki/Major_fourth_and_minor_fifth also been referred to] as the '''major fourth'''.  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]] (see also [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Alpharabian paramajor fourth''' or even the '''just paramajor fourth'''.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]</text>
      <sha1>qjiblx7zinds55odafwhx1xh1dg2ghz</sha1>
    </revision>
    <revision>
      <id>64449</id>
      <parentid>53153</parentid>
      <timestamp>2021-03-27T04:04:02Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64449</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4246" sha1="rd03hy2bqolnkh5prg0d1tw0lscamtw" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Alpharabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval has [https://en.wikipedia.org/wiki/Major_fourth_and_minor_fifth also been referred to] as the '''major fourth'''.  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]] (see also [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Alpharabian paramajor fourth''' or even the '''just paramajor fourth'''.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>rd03hy2bqolnkh5prg0d1tw0lscamtw</sha1>
    </revision>
    <revision>
      <id>82621</id>
      <parentid>64449</parentid>
      <timestamp>2021-12-26T01:26:38Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>Adjusted the Alpharabian-tuning-based terminology</comment>
      <origin>82621</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4242" sha1="rwf3wbscuys5vb4qyyn9tlj55q0ozzy" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Axirabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval has [https://en.wikipedia.org/wiki/Major_fourth_and_minor_fifth also been referred to] as the '''major fourth'''.  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]] (see also [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Axirabian paramajor fourth''' or even the '''just paramajor fourth'''.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>rwf3wbscuys5vb4qyyn9tlj55q0ozzy</sha1>
    </revision>
    <revision>
      <id>83624</id>
      <parentid>82621</parentid>
      <timestamp>2022-01-10T07:11:32Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>83624</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4238" sha1="5fy0qsb9pt2msjq9679laaox6yfi6l5" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Axirabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval, or rather the tempered version found in [[24edo]]. was dubbed the '''major fourth''' by [[Ivan Wyschnegradsky]].  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]] (see also [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Axirabian paramajor fourth''' or even the '''just paramajor fourth'''.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>5fy0qsb9pt2msjq9679laaox6yfi6l5</sha1>
    </revision>
    <revision>
      <id>83625</id>
      <parentid>83624</parentid>
      <timestamp>2022-01-10T07:11:44Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>83625</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4238" sha1="i9ymepb4s4jm4cw3301xlc54kxbhjy7" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Axirabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval, or rather the tempered version found in [[24edo]], was dubbed the '''major fourth''' by [[Ivan Wyschnegradsky]].  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]] (see also [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Axirabian paramajor fourth''' or even the '''just paramajor fourth'''.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced overtone, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>i9ymepb4s4jm4cw3301xlc54kxbhjy7</sha1>
    </revision>
    <revision>
      <id>84518</id>
      <parentid>83625</parentid>
      <timestamp>2022-01-22T20:34:22Z</timestamp>
      <contributor>
        <username>Wolftune</username>
        <id>152</id>
      </contributor>
      <minor/>
      <comment>fixed "overtone series" vs "harmonic series"</comment>
      <origin>84518</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4238" sha1="8l7ugics9of1ppclxqcjhogwfge027s" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Icon =
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Axirabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval, or rather the tempered version found in [[24edo]], was dubbed the '''major fourth''' by [[Ivan Wyschnegradsky]].  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]] (see also [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Axirabian paramajor fourth''' or even the '''just paramajor fourth'''.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Superfourth]]
[[Category:Fourth]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Overtone]]
[[Category:Over-2]]
[[Category:Pages with internal sound examples]]</text>
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    </revision>
    <revision>
      <id>86246</id>
      <parentid>84518</parentid>
      <timestamp>2022-02-22T17:16:12Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Misc. edits, categories</comment>
      <origin>86246</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4155" sha1="3ol1ht89fxc9q4pokfa09h58htlmitf" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Axirabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval, or rather the tempered version found in [[24edo]], was dubbed the '''major fourth''' by [[Ivan Wyschnegradsky]].  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]] (see also [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Axirabian paramajor fourth''' or even the '''just paramajor fourth'''.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]
[[Category:Octave-reduced harmonics]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>3ol1ht89fxc9q4pokfa09h58htlmitf</sha1>
    </revision>
    <revision>
      <id>86296</id>
      <parentid>86246</parentid>
      <timestamp>2022-02-22T23:46:39Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86296</origin>
      <model>wikitext</model>
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| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Axirabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval, or rather the tempered version found in [[24edo]], was dubbed the '''major fourth''' by [[Ivan Wyschnegradsky]].  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]] (see also [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Axirabian paramajor fourth''' or even the '''just paramajor fourth'''.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]
[[Category:Octave-reduced harmonics]]</text>
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    </revision>
    <revision>
      <id>93304</id>
      <parentid>86296</parentid>
      <timestamp>2022-07-06T13:20:56Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>93304</origin>
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      <text bytes="4150" sha1="o39m15tmznilvf7mkr6xrxu1ubv2nia" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Axirabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval, or rather the tempered version found in [[24edo]], was dubbed the '''major fourth''' by [[Ivan Wyschnegradsky]].  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]] (see also [[User:Aura/Aura's Ideas on Tonality #11-limit Axis Functionality]]), it can also be somewhat similarly dubbed the '''Axirabian paramajor fourth''' or even the '''just paramajor fourth'''.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]
[[Category:Octave-reduced harmonics]]</text>
      <sha1>o39m15tmznilvf7mkr6xrxu1ubv2nia</sha1>
    </revision>
    <revision>
      <id>93306</id>
      <parentid>93304</parentid>
      <timestamp>2022-07-06T13:43:10Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
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| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;major fourth, &lt;br&gt;Axirabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval, or rather the tempered version found in [[24edo]], was dubbed the '''major fourth''' by [[Ivan Wyschnegradsky]].  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it can also be somewhat similarly dubbed the '''Axirabian paramajor fourth''' or even the '''just paramajor fourth'''- see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]
[[Category:Octave-reduced harmonics]]</text>
      <sha1>q0si8lwnvp2xmxco067yczv9t6iciw7</sha1>
    </revision>
    <revision>
      <id>93309</id>
      <parentid>93306</parentid>
      <timestamp>2022-07-06T14:20:31Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>93309</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4275" sha1="2e5c5ghfrvrf0wkeflrra552oih1gbg" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Ratio = 11/8
| Monzo = -3 0 0 0 1
| Cents = 551.31794
| Name = undecimal superfourth, &lt;br&gt;undecimal major fourth, &lt;br&gt;Axirabian paramajor fourth, &lt;br&gt;just paramajor fourth
| Color name = 1o4, ilo 4th
| FJS name = P4&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval can also be called the '''undecimal major fourth''' since the tempered version found in [[24edo]], was dubbed the "major fourth" by [[Ivan Wyschnegradsky]].  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it can also be somewhat similarly dubbed the '''Axirabian paramajor fourth''' or even the '''just paramajor fourth'''- see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:11-limit]]
[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]
[[Category:Octave-reduced harmonics]]</text>
      <sha1>2e5c5ghfrvrf0wkeflrra552oih1gbg</sha1>
    </revision>
    <revision>
      <id>98073</id>
      <parentid>93309</parentid>
      <timestamp>2022-10-25T12:59:05Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>98073</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4118" sha1="qfwz4q5sdrhspl1su3dmx4b2mtendb4" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, undecimal major fourth, Axirabian paramajor fourth, just paramajor fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]].  This interval can also be called the '''undecimal major fourth''' since the tempered version found in [[24edo]], was dubbed the "major fourth" by [[Ivan Wyschnegradsky]].  Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it can also be somewhat similarly dubbed the '''Axirabian paramajor fourth''' or even the '''just paramajor fourth'''- see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the much stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>qfwz4q5sdrhspl1su3dmx4b2mtendb4</sha1>
    </revision>
    <revision>
      <id>138841</id>
      <parentid>98073</parentid>
      <timestamp>2024-03-11T01:01:08Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <comment>add rigorous names based on generalised definition of "neutral" w.r.t all MOSSes applied to diatonic. calling this interval "major" is confusing, *especially* in light of the reasoning, so i put the name after "undecimal superfourth"</comment>
      <origin>138841</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5018" sha1="fje7dq0qjx1008m7bspl85qwnsut7sr" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic neutral fourth, harmonic semiaugmented fourth, harmonic semiperfect fourth, Axirabian paramajor fourth, just paramajor fourth, undecimal major fourth, harmonic fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]]. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' (a.k.a. ''imperfect'') version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic semiaugmented/semiperfect fourth''', or '''harmonic neutral fourth''' if you prefer to generalise the naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave-complements (which is also rigorous). Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or  more simply the '''just paramajor fourth'''- see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. More recently, [[Zhea Erose]] has suggested calling it something more simple: the ''harmonic fourth'' - under the idea that it is the simplest [[harmonic]] that is in the general range of "fourths" when octave-reduced.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>fje7dq0qjx1008m7bspl85qwnsut7sr</sha1>
    </revision>
    <revision>
      <id>138845</id>
      <parentid>138841</parentid>
      <timestamp>2024-03-11T01:27:02Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>was supposed to be bolded</comment>
      <origin>138845</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5020" sha1="pwuabz4cxaqtpk2aj4v188ualo5c5av" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic neutral fourth, harmonic semiaugmented fourth, harmonic semiperfect fourth, Axirabian paramajor fourth, just paramajor fourth, undecimal major fourth, harmonic fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]]. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' (a.k.a. ''imperfect'') version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic semiaugmented/semiperfect fourth''', or '''harmonic neutral fourth''' if you prefer to generalise the naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave-complements (which is also rigorous). Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or  more simply the '''just paramajor fourth'''- see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' - under the idea that it is the simplest [[harmonic]] that is in the general range of "fourths" when octave-reduced.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>pwuabz4cxaqtpk2aj4v188ualo5c5av</sha1>
    </revision>
    <revision>
      <id>138846</id>
      <parentid>138845</parentid>
      <timestamp>2024-03-11T01:27:28Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <origin>138846</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5033" sha1="je327v72mo32imsvo2ehx2hss5nmjij" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic neutral fourth, harmonic semiaugmented fourth, harmonic semiperfect fourth, Axirabian paramajor fourth, just paramajor fourth, undecimal major fourth, harmonic fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]]. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' (a.k.a. ''imperfect'') version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic semiaugmented/semiperfect fourth''', or '''harmonic neutral fourth''' if you prefer to generalise the naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave-complements (which is also rigorous). Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or  more simply the '''just paramajor fourth'''- see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' - under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>je327v72mo32imsvo2ehx2hss5nmjij</sha1>
    </revision>
    <revision>
      <id>138847</id>
      <parentid>138846</parentid>
      <timestamp>2024-03-11T01:37:40Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>consistency</comment>
      <origin>138847</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5069" sha1="1253weejrljvhuvsm0cmvjq5f86snjm" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, undecimal neutral fourth, harmonic neutral fourth, harmonic semiaugmented fourth, harmonic semiperfect fourth, Axirabian paramajor fourth, just paramajor fourth, undecimal major fourth, harmonic fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3[[cent|&amp;cent;]]. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' (a.k.a. ''imperfect'') version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic semiaugmented/semiperfect fourth''', or '''harmonic/undecimal neutral fourth''' if you prefer to generalise the naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave-complements (which is also rigorous). Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or  more simply the '''just paramajor fourth'''- see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' - under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].


== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>1253weejrljvhuvsm0cmvjq5f86snjm</sha1>
    </revision>
    <revision>
      <id>146450</id>
      <parentid>138847</parentid>
      <timestamp>2024-06-20T13:23:32Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup on this mess of names: "neutral" tried to convey the same thing as "semiaugmented" but less successfully. "Imperfect" generally means wolf rather than augmented, so "semiperfect" has no basis.</comment>
      <origin>146450</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5027" sha1="0qhk8x93xg42m4cvtdz1ujt6oq0zovi" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth, Axirabian paramajor fourth, just paramajor fourth, undecimal major fourth, harmonic fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic/undecimal semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional ''harmonic/undecimal neutral fourth''. Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth''' – see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>0qhk8x93xg42m4cvtdz1ujt6oq0zovi</sha1>
    </revision>
    <revision>
      <id>194851</id>
      <parentid>146450</parentid>
      <timestamp>2025-05-01T10:15:08Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>clean up names in infobox</comment>
      <origin>194851</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4995" sha1="igskn3rgwunxifd7rku9od3gfroa3w1" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic/undecimal semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional ''harmonic/undecimal neutral fourth''. Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth''' – see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref&gt;Super EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}

&lt;references /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>igskn3rgwunxifd7rku9od3gfroa3w1</sha1>
    </revision>
    <revision>
      <id>194852</id>
      <parentid>194851</parentid>
      <timestamp>2025-05-01T10:36:17Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>add undecimal tritone</comment>
      <origin>194852</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5275" sha1="skah3d0avitj0sn2rd537a1b4xd69qu" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic/undecimal semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. 
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.&lt;ref&gt;Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']&lt;/ref&gt;
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth''' – see [[User:Aura/Aura's Ideas on Functional Harmony #History|the history of Aura's Ideas on Functional Harmony]] for explanation of the modified names.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref group="note"&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref group="note"&gt;EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}
&lt;references group="note" /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

== References ==
&lt;references /&gt;

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>skah3d0avitj0sn2rd537a1b4xd69qu</sha1>
    </revision>
    <revision>
      <id>194890</id>
      <parentid>194852</parentid>
      <timestamp>2025-05-01T21:09:18Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>/* Terminology */ Don't promote your essays on random interval pages please.</comment>
      <origin>194890</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5122" sha1="7fv0sdyrafrs7bky2rlfbet8oflaxti" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic/undecimal semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. 
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.&lt;ref&gt;Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']&lt;/ref&gt;
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth'''..

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref group="note"&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref group="note"&gt;EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}
&lt;references group="note" /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

== References ==
&lt;references /&gt;

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>7fv0sdyrafrs7bky2rlfbet8oflaxti</sha1>
    </revision>
    <revision>
      <id>194891</id>
      <parentid>194890</parentid>
      <timestamp>2025-05-01T21:09:30Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <minor/>
      <comment>/* Terminology */ .</comment>
      <origin>194891</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5121" sha1="are09zgam1mtxwfscs8vijr48jhh44g" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic/undecimal semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting root motion both for chord progressions within a key and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. 
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.&lt;ref&gt;Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']&lt;/ref&gt;
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth'''.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref group="note"&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref group="note"&gt;EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}
&lt;references group="note" /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

== References ==
&lt;references /&gt;

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>are09zgam1mtxwfscs8vijr48jhh44g</sha1>
    </revision>
    <revision>
      <id>195802</id>
      <parentid>194891</parentid>
      <timestamp>2025-05-08T17:54:25Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>Added an example of how 11/8 might be used as a chord root in more tonal compositions</comment>
      <origin>195802</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5786" sha1="roviw25c4m4cx4chcosur9wihfauc0o" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic/undecimal semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting chord root both for chord progressions within a key, and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].  For example, in more tonal music, 11/8 relative to the tonic ends up being used as the chord root for what amounts to a voicing variation of a 1/1-9/8-225/176-3/2 chord, which, is preceded by a variation on a 1/1-5/4-3/2-225/128 chord built on [[16/15]] relative to the tonic (a variation on the [[Wikipedia: Neapolitan chord|Neapolitan chord]]) and, followed up by a variation on the 1/1-5/4-3/2-16/9 dominant seventh chord (or potentially even a 1/1-5/4-3/2-16/9-16/15 dominant ninth chord) built on [[3/2]] relative to the tonic for a special type of half cadence.  This is a dramatic musical gesture that [[User:Aura|Aura]] has named the "simul half cadence".

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in 24edo was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. 
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.&lt;ref&gt;Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']&lt;/ref&gt;
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth'''.

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref group="note"&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref group="note"&gt;EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}
&lt;references group="note" /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

== References ==
&lt;references /&gt;

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>roviw25c4m4cx4chcosur9wihfauc0o</sha1>
    </revision>
    <revision>
      <id>195803</id>
      <parentid>195802</parentid>
      <timestamp>2025-05-08T17:56:39Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>Decided to move the usage example to its own subsection, other examples of its usage by other composers are welcome in this section</comment>
      <origin>195803</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5796" sha1="dfghqnev0ape4gmk0ooacqcacrh8rel" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic/undecimal semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

It is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting chord root both for chord progressions within a key, and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in 24edo was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. 
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.&lt;ref&gt;Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']&lt;/ref&gt;
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth'''.

== Potential usage ==

In more tonal music, 11/8 relative to the tonic ends up being used as the chord root for what amounts to a voicing variation of a 1/1-9/8-225/176-3/2 chord, which, is preceded by a variation on a 1/1-5/4-3/2-225/128 chord built on [[16/15]] relative to the tonic (a variation on the [[Wikipedia: Neapolitan chord|Neapolitan chord]]) and, followed up by a variation on the 1/1-5/4-3/2-16/9 dominant seventh chord (or potentially even a 1/1-5/4-3/2-16/9-16/15 dominant ninth chord) built on [[3/2]] relative to the tonic for a special type of half cadence.  This is a dramatic musical gesture that [[User:Aura|Aura]] has named the "simul half cadence".

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref group="note"&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref group="note"&gt;EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}
&lt;references group="note" /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

== References ==
&lt;references /&gt;

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>dfghqnev0ape4gmk0ooacqcacrh8rel</sha1>
    </revision>
    <revision>
      <id>195804</id>
      <parentid>195803</parentid>
      <timestamp>2025-05-08T17:58:28Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>195804</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5807" sha1="cv9mqj12ffcmbdrbo0rl225rtyqcc81" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic/undecimal semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in 24edo was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. 
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.&lt;ref&gt;Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']&lt;/ref&gt;
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth'''.

== Potential usage ==

This interval is very well-represented in [[24edo]], making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting chord root both for chord progressions within a key, and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

In more tonal music, 11/8 relative to the tonic ends up being used as the chord root for what amounts to a voicing variation of a 1/1-9/8-225/176-3/2 chord, which, is preceded by a variation on a 1/1-5/4-3/2-225/128 chord built on [[16/15]] relative to the tonic (a variation on the [[Wikipedia: Neapolitan chord|Neapolitan chord]]) and, followed up by a variation on the 1/1-5/4-3/2-16/9 dominant seventh chord (or potentially even a 1/1-5/4-3/2-16/9-16/15 dominant ninth chord) built on [[3/2]] relative to the tonic for a special type of half cadence.  This is a dramatic musical gesture that [[User:Aura|Aura]] has named the "simul half cadence".

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref group="note"&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref group="note"&gt;EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}
&lt;references group="note" /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

== References ==
&lt;references /&gt;

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>cv9mqj12ffcmbdrbo0rl225rtyqcc81</sha1>
    </revision>
    <revision>
      <id>195805</id>
      <parentid>195804</parentid>
      <timestamp>2025-05-08T18:03:25Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>195805</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5807" sha1="lt2ffebdrjs1n49ow8uz1lfowqv2w1v" xml:space="preserve">{{interwiki
| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic/undecimal semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. 
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.&lt;ref&gt;Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']&lt;/ref&gt;
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth'''.

== Potential usage ==

This interval is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting chord root both for chord progressions within a key, and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

In more tonal music, 11/8 relative to the tonic ends up being used as the chord root for what amounts to a voicing variation of a 1/1-9/8-225/176-3/2 chord, which, is preceded by a variation on a 1/1-5/4-3/2-225/128 chord built on [[16/15]] relative to the tonic (a variation on the [[Wikipedia: Neapolitan chord|Neapolitan chord]]) and, followed up by a variation on the 1/1-5/4-3/2-16/9 dominant seventh chord (or potentially even a 1/1-5/4-3/2-16/9-16/15 dominant ninth chord) built on [[3/2]] relative to the tonic for a special type of half cadence.  This is a dramatic musical gesture that [[User:Aura|Aura]] has named the "simul half cadence".

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref group="note"&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref group="note"&gt;EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}
&lt;references group="note" /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

== References ==
&lt;references /&gt;

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>lt2ffebdrjs1n49ow8uz1lfowqv2w1v</sha1>
    </revision>
    <revision>
      <id>195806</id>
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| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic/undecimal semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. 
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.&lt;ref&gt;Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']&lt;/ref&gt;
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth'''.

== Potential usage ==

This interval is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting chord root both for chord progressions within a key, and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

In more tonal music, 11/8 relative to the tonic ends up being used as the chord root for what amounts to a voicing variation of a 1/1-9/8-225/176-3/2 chord, which, is preceded by a variation on a 1/1-5/4-3/2-225/128 chord built on [[16/15]] relative to the tonic (basically, a type of [[Wikipedia: Neapolitan chord|Neapolitan chord]]) and, followed up by a variation on the 1/1-5/4-3/2-16/9 dominant seventh chord (or potentially even a 1/1-5/4-3/2-16/9-16/15 dominant ninth chord) built on [[3/2]] relative to the tonic for a special type of half cadence.  This is a dramatic musical gesture that [[User:Aura|Aura]] has named the "simul half cadence".

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref group="note"&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref group="note"&gt;EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}
&lt;references group="note" /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

== References ==
&lt;references /&gt;

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>hwcox0nmnpm85xe7kh6eia79gy1r61f</sha1>
    </revision>
    <revision>
      <id>195840</id>
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      <timestamp>2025-05-09T01:09:02Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
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| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal [[superfourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic/undecimal semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. 
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.&lt;ref&gt;Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']&lt;/ref&gt;
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth'''.

== Potential usage ==

This interval is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting chord root both for chord progressions within a key, and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

In more tonal music, 11/8 relative to the tonic ends up being used as the chord root for what amounts to a voicing variation of a 1/1-9/8-225/176-3/2 chord, which, is preceded by a variation on a 1/1-5/4-3/2-225/128 chord built on [[16/15]] relative to the tonic (basically, a type of [[Wikipedia: Neapolitan chord|Neapolitan chord]]), and, followed up by a variation on the 1/1-5/4-3/2-16/9 dominant seventh chord (or potentially even a 1/1-5/4-3/2-16/9-16/15 dominant ninth chord) built on [[3/2]] relative to the tonic for a special type of half cadence.  This is a dramatic musical gesture that [[User:Aura|Aura]] has named the "simul half cadence".

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref group="note"&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref group="note"&gt;EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}
&lt;references group="note" /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

== References ==
&lt;references /&gt;

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>ihy4v7fhu5mbg8pgplfziqwv8pv8ire</sha1>
    </revision>
    <revision>
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| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal  [[superfourth|semiaugmented fourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. 
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.&lt;ref&gt;Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']&lt;/ref&gt;
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth'''.

== Potential usage ==

This interval is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting chord root both for chord progressions within a key, and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

In more tonal music, 11/8 relative to the tonic ends up being used as the chord root for what amounts to a voicing variation of a 1/1-9/8-225/176-3/2 chord, which, is preceded by a variation on a 1/1-5/4-3/2-225/128 chord built on [[16/15]] relative to the tonic (basically, a type of [[Wikipedia: Neapolitan chord|Neapolitan chord]]), and, followed up by a variation on the 1/1-5/4-3/2-16/9 dominant seventh chord (or potentially even a 1/1-5/4-3/2-16/9-16/15 dominant ninth chord) built on [[3/2]] relative to the tonic for a special type of half cadence.  This is a dramatic musical gesture that [[User:Aura|Aura]] has named the "simul half cadence".

== Approximations by EDOs ==

Following [[EDO]]s (up to 200) contain good approximations&lt;ref group="note"&gt;error magnitude below 7, both, absolute (in ¢) and relative (in r¢)&lt;/ref&gt; of the interval 11/8. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&amp;uarr;) or flat (&amp;darr;).

{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
|-
! [[EDO]]
! class="unsortable" | deg\edo
! Absolute &lt;br&gt; error ([[Cent|¢]])
! Relative &lt;br&gt; error ([[Relative cent|r¢]])
! &amp;#8597;
! class="unsortable" | Equally acceptable multiples &lt;ref group="note"&gt;EDOs up to 200 within the same error tolerance&lt;/ref&gt;
|-
|  [[11edo|11]]  ||   5\11  || 5.8634 || 5.3748 || &amp;darr; || 
|-
|  [[13edo|13]]  ||   6\13  || 2.5282 || 2.7389 || &amp;uarr; || [[26edo|12\26]]
|-
|  [[24edo|24]]  ||  11\24  || 1.3179 || 2.6359 || &amp;darr; || [[48edo|22\48]]
|-
|  [[37edo|37]]  ||  17\37  || 0.0334 || 0.1030 || &amp;uarr; || [[74edo|34\74]], [[111edo|51\111]], [[148edo|68\148]], [[185edo|85\185]]
|-
|  [[50edo|50]]  ||  23\50  || 0.6821 || 2.8419 || &amp;uarr; || [[100edo|46\100]]
|-
|  [[61edo|61]]  ||  28\61  || 0.4983 || 2.5329 || &amp;darr; || [[122edo|56\122]]
|-
|  [[63edo|63]]  ||  29\63  || 1.0630 || 5.5808 || &amp;uarr; || 
|-
|  [[85edo|85]]  ||  39\85  || 0.7297 || 5.1688 || &amp;darr; || 
|-
|  [[87edo|87]]  ||  40\87  || 0.4062 || 2.9449 || &amp;uarr; || [[174edo|80\174]]
|-
|  [[98edo|98]]  ||  45\98  || 0.2975 || 2.4299 || &amp;darr; || [[196edo|90\196]]
|-
| [[124edo|124]] ||  57\124 || 0.2950 || 3.0479 || &amp;uarr; || 
|-
| [[135edo|135]] ||  62\135 || 0.2068 || 2.3269 || &amp;darr; || 
|-
| [[137edo|137]] ||  63\137 || 0.5069 || 5.7868 || &amp;uarr; || 
|-
| [[159edo|159]] ||  73\159 || 0.3745 || 4.9627 || &amp;darr; || 
|-
| [[161edo|161]] ||  74\161 || 0.2349 || 3.1509 || &amp;uarr; || 
|-
| [[172edo|172]] ||  79\172 || 0.1552 || 2.2238 || &amp;darr; || 
|-
| [[198edo|198]] ||  91\198 || 0.1972 || 3.2540 || &amp;uarr; || 
|-
|}
&lt;references group="note" /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

== References ==
&lt;references /&gt;

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
      <sha1>mhnsw3idndwavm0ul7h63z0rxkmk7ca</sha1>
    </revision>
    <revision>
      <id>215843</id>
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| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal  [[superfourth|semiaugmented fourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. 
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.&lt;ref&gt;Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']&lt;/ref&gt;
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth'''.

== Potential usage ==

This interval is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting chord root both for chord progressions within a key, and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

In more tonal music, 11/8 relative to the tonic ends up being used as the chord root for what amounts to a voicing variation of a 1/1-9/8-225/176-3/2 chord, which, is preceded by a variation on a 1/1-5/4-3/2-225/128 chord built on [[16/15]] relative to the tonic (basically, a type of [[Wikipedia: Neapolitan chord|Neapolitan chord]]), and, followed up by a variation on the 1/1-5/4-3/2-16/9 dominant seventh chord (or potentially even a 1/1-5/4-3/2-16/9-16/15 dominant ninth chord) built on [[3/2]] relative to the tonic for a special type of half cadence.  This is a dramatic musical gesture that [[User:Aura|Aura]] has named the "simul half cadence".

== Approximations by EDOs ==
 {{Interval_Edo_Approximation | 11/8}}
&lt;references group="note" /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

== References ==
&lt;references /&gt;

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
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    <revision>
      <id>215965</id>
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      <contributor>
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| de = Alphorn-Fa
| en = 11/8
| es = 
| ja = 
}}
{{Infobox Interval
| Name = undecimal superfourth, harmonic fourth, undecimal tritone, undecimal major fourth, undecimal semiaugmented fourth, harmonic semiaugmented fourth
| Color name = 1o4, ilo 4th
| Sound = jid_11_8_pluck_adu_dr220.mp3
}}
{{Wikipedia|Major fourth and minor fifth}}

In [[11-limit]] [[just intonation]], '''11/8''' is an '''undecimal  [[superfourth|semiaugmented fourth]]''' of about 551.3{{cent}}. This interval is close (~3{{cent}}) to exactly between a [[4/3|perfect fourth]] and [[729/512|augmented fourth]], the latter of which is the ''augmented'' version of the [[Pythagorean tuning|Pythagorean]] [[diatonic]] generator, therefore may be called the '''harmonic semiaugmented fourth'''. 

This interval is the simplest superfourth in JI, and as it falls about halfway between [[12edo]]'s [[perfect fourth]] and [[tritone]], it is very xenharmonic.  As an octave-reduced harmonic, it is a basis of consonance in 11-limit JI, alongside the lower odd numbers 9, 7, 5 and 3. It can be found in harmonic series chords such as 4:5:6:7:8:9:10:11:12, sitting somewhere between the stronger and more familiar consonances of 10 (prime 5) and 12 (prime 3).

== Terminology ==
The naming pattern from [[11/9|undecimal neutral third]] and [[12/11|undecimal neutral second]] and their octave complements can be rigorously generalized and results in the somewhat unconventional '''harmonic/undecimal neutral fourth'''. This interval has also been termed the '''undecimal major fourth''' since the tempered version found in [[24edo]] was dubbed the "major fourth" by [[Ivan Wyschnegradsky]], although this may be confusing in diatonic contexts. 
Because it is right between the diatonic fourth and tritone, it may also be called the '''(lesser) undecimal tritone'''.&lt;ref&gt;Kyle Gann (1998) [https://www.kylegann.com/Octave.html ''Anatomy of an Octave'']&lt;/ref&gt;
More recently, [[Zhea Erose]] has suggested calling it something more simple: the '''harmonic fourth''' – under the idea that it is the simplest [[harmonic]] that is in the general (very) rough range of "fourths" when octave-reduced.
Furthermore, as stacks of this interval form a core axis of [[Alpharabian tuning]], it has also been dubbed the '''Axirabian paramajor fourth''' or more simply the '''just paramajor fourth'''.

== Potential usage ==

This interval is very well-represented in 24edo, making that system especially good for approximations of JI chords involving primes 3 and 11 such as 8:9:11:12.  Not only that, but composers who have experience with 24edo may find it very useful not only as a fantastic addition to major chords, but also as an interesting chord root both for chord progressions within a key, and for modulations to key signatures that are not in the same chain of fifths.  Furthermore, these same useful functions can carry over to higher EDOs with good 11-limit representation such as [[159edo]].

In more tonal music, 11/8 relative to the tonic ends up being used as the chord root for what amounts to a voicing variation of a 1/1-9/8-225/176-3/2 chord, which, is preceded by a variation on a 1/1-5/4-3/2-225/128 chord built on [[16/15]] relative to the tonic (basically, a type of [[Wikipedia: Neapolitan chord|Neapolitan chord]]), and, followed up by a variation on the 1/1-5/4-3/2-16/9 dominant seventh chord (or potentially even a 1/1-5/4-3/2-16/9-16/15 dominant ninth chord) built on [[3/2]] relative to the tonic for a special type of half cadence.  This is a dramatic musical gesture that [[User:Aura|Aura]] has named the "simul half cadence".

== Approximations by EDOs ==
{{Interval edo approximation|11/8}}
&lt;references group="note" /&gt;

== See also ==

* [[16/11]] – its [[octave complement]]
* [[12/11]] – its [[fifth complement]]
* [[Gallery of just intervals]]

== References ==
&lt;references /&gt;

[[Category:Fourth]]
[[Category:Superfourth]]
[[Category:Alpharabian]]</text>
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    <title>11 8</title>
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      <timestamp>2018-07-17T00:00:00Z</timestamp>
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      <text bytes="18" sha1="90c30hpapdoykim5wzltgjgs12hcgcp" xml:space="preserve">#REDIRECT [[11/8]]</text>
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  <page>
    <title>11/9</title>
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      <text bytes="1801" sha1="bc0yiuqneny50p76zdxuo14iawxm1v0" xml:space="preserve">'''11/9'''
|0 -2 0 0 1&amp;gt;

347.40794 cents

[[File:jid_11_9_pluck_adu_dr220.mp3]] [[:File:jid_11_9_pluck_adu_dr220.mp3|sound sample]]

In [[11-limit|11-limit]] [[Just_intonation|Just Intonation]], '''11/9''' is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of persony (others include [[16/13|16/13]], [[27/22|27/22]], [[49/40|49/40]] and [[60/49|60/49]]). It is nearly halfway between two intervals of [[12edo|12edo]], implying that it is both very xenharmonic and well-represented in [[24edo|24edo]].

In the 11-limit theyxad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Persony temperaments, including [[17edo|17edo]], [[24edo|24edo]], [[31edo|31edo]], [[41edo|41edo]], [[58edo|58edo]], [[72edo|72edo]], [[130edo|130edo]], [[202edo|202edo]], [[Gamelismic_clan#Miracle|miracle]], [[Breedsmic_temperaments#Harry|harry]], and [[Schismatic_family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed_family#Jove, aka Wonder|jove]].

See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]      [[Category:11-limit]]
[[Category:interval]]
[[Category:neutral_3rd]]
[[Category:ratio]]
[[Category:third]]
[[Category:undecimal]]
[[Category:untwelve]]</text>
      <sha1>bc0yiuqneny50p76zdxuo14iawxm1v0</sha1>
    </revision>
    <revision>
      <id>4333</id>
      <timestamp>2016-09-25T08:51:19Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Chartrekhan</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 593235044 - Original comment: **</comment>
      <origin>4333</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6783" sha1="23kxquu85jdhjcpjf7r72d2iu1yybrj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Chartrekhan|Chartrekhan]] and made on &lt;tt&gt;2016-09-25 08:51:19 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;593235044&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/9**
|0 -2 0 0 1&amp;gt;
347.40794 cents
[[media type="file" key="jid_11_9_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_11_9_pluck_adu_dr220.mp3|sound sample]]

In [[11-limit]] [[Just Intonation]], **11/9** is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of persony (others include [[16_13|16/13]], [[27_22|27/22]], [[49_40|49/40]] and [[60_49|60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit theyxad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Persony temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove,%20aka%20Wonder|jove]].

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/9&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|0 -2 0 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
347.40794 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_9_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_9_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_9_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_9_pluck_adu_dr220.mp3/513194394/jid_11_9_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_9_pluck_adu_dr220.mp3/513194394/jid_11_9_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, &amp;lt;strong&amp;gt;11/9&amp;lt;/strong&amp;gt; is a neutral third of about 347.4¢, falling in between &amp;amp;quot;major third&amp;amp;quot; and &amp;amp;quot;minor third&amp;amp;quot; territory. It is the simplest neutral third in just intonation, but of course, only one of persony (others include &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/49_40"&amp;gt;49/40&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/60_49"&amp;gt;60/49&amp;lt;/a&amp;gt;). It is nearly halfway between two intervals of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, implying that it is both very xenharmonic and well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit theyxad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Persony temperaments, including &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Breedsmic%20temperaments#Harry"&amp;gt;harry&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/Schismatic%20family#Sesquiquartififths"&amp;gt;sesquart&amp;lt;/a&amp;gt;, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament &amp;lt;a class="wiki_link" href="/Breed%20family#Jove,%20aka%20Wonder"&amp;gt;jove&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4334</id>
      <timestamp>2016-09-25T08:49:13Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Chartrekhan</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 593234990 - Original comment: **</comment>
      <origin>4334</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7109" sha1="hlh0tel9n4bldvocwt5aijryfq23sgx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Chartrekhan|Chartrekhan]] and made on &lt;tt&gt;2016-09-25 08:49:13 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;593234990&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/9**
|0 -2 0 0 1&amp;gt;
347.40794 cents
[[media type="file" key="jid_11_9_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_11_9_pluck_adu_dr220.mp3|sound sample]]

In [[11-limit]] [[Just Intonation]], **11/9** is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of persony (others include [[16_13|16/13]], [[27_22|27/22]], [[49_40|49/40]] and [[60_49|60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit theyxad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Persony temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove,%20aka%20Wonder|jove]].

By coincidence, the ratio between the common tuning frequency 440hz and the most common AC power frequency of 60hz is exactly 11/9 (but 6 octaves up)

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/9&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|0 -2 0 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
347.40794 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_9_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_9_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_9_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_9_pluck_adu_dr220.mp3/513194394/jid_11_9_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_9_pluck_adu_dr220.mp3/513194394/jid_11_9_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, &amp;lt;strong&amp;gt;11/9&amp;lt;/strong&amp;gt; is a neutral third of about 347.4¢, falling in between &amp;amp;quot;major third&amp;amp;quot; and &amp;amp;quot;minor third&amp;amp;quot; territory. It is the simplest neutral third in just intonation, but of course, only one of persony (others include &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/49_40"&amp;gt;49/40&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/60_49"&amp;gt;60/49&amp;lt;/a&amp;gt;). It is nearly halfway between two intervals of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, implying that it is both very xenharmonic and well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit theyxad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Persony temperaments, including &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Breedsmic%20temperaments#Harry"&amp;gt;harry&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/Schismatic%20family#Sesquiquartififths"&amp;gt;sesquart&amp;lt;/a&amp;gt;, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament &amp;lt;a class="wiki_link" href="/Breed%20family#Jove,%20aka%20Wonder"&amp;gt;jove&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
By coincidence, the ratio between the common tuning frequency 440hz and the most common AC power frequency of 60hz is exactly 11/9 (but 6 octaves up)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>hlh0tel9n4bldvocwt5aijryfq23sgx</sha1>
    </revision>
    <revision>
      <id>4335</id>
      <timestamp>2016-09-25T08:46:29Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Chartrekhan</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 593234932 - Original comment: **</comment>
      <origin>4335</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7119" sha1="ftj2myhspf944cz9pa4x4dugw6o05mk" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Chartrekhan|Chartrekhan]] and made on &lt;tt&gt;2016-09-25 08:46:29 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;593234932&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/9**
|0 -2 0 0 1&amp;gt;
347.40794 cents
[[media type="file" key="jid_11_9_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_11_9_pluck_adu_dr220.mp3|sound sample]]

In [[11-limit]] [[Just Intonation]], **11/9** is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of persony (others include [[16_13|16/13]], [[27_22|27/22]], [[49_40|49/40]] and [[60_49|60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit theyxad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Persony temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove,%20aka%20Wonder|jove]].

By coincidence, the ratio between the common tuning frequency 440hz and the most common AC power frequency of 60hz is exactly an 11/9 when octave-reduced.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/9&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|0 -2 0 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
347.40794 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_9_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_9_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_9_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_9_pluck_adu_dr220.mp3/513194394/jid_11_9_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_9_pluck_adu_dr220.mp3/513194394/jid_11_9_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, &amp;lt;strong&amp;gt;11/9&amp;lt;/strong&amp;gt; is a neutral third of about 347.4¢, falling in between &amp;amp;quot;major third&amp;amp;quot; and &amp;amp;quot;minor third&amp;amp;quot; territory. It is the simplest neutral third in just intonation, but of course, only one of persony (others include &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/49_40"&amp;gt;49/40&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/60_49"&amp;gt;60/49&amp;lt;/a&amp;gt;). It is nearly halfway between two intervals of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, implying that it is both very xenharmonic and well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit theyxad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Persony temperaments, including &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Breedsmic%20temperaments#Harry"&amp;gt;harry&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/Schismatic%20family#Sesquiquartififths"&amp;gt;sesquart&amp;lt;/a&amp;gt;, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament &amp;lt;a class="wiki_link" href="/Breed%20family#Jove,%20aka%20Wonder"&amp;gt;jove&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
By coincidence, the ratio between the common tuning frequency 440hz and the most common AC power frequency of 60hz is exactly an 11/9 when octave-reduced.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ftj2myhspf944cz9pa4x4dugw6o05mk</sha1>
    </revision>
    <revision>
      <id>4336</id>
      <timestamp>2014-06-07T15:36:33Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513194466 - Original comment: **</comment>
      <origin>4336</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6659" sha1="crtv872z3ko37ofx2knjqnurjsg73fe" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-07 15:36:33 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513194466&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**11/9**
|0 -2 0 0 1&amp;gt;
347.40794 cents
[[media type="file" key="jid_11_9_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_11_9_pluck_adu_dr220.mp3|sound sample]]

In [[11-limit]] [[Just Intonation]], **11/9** is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16_13|16/13]], [[27_22|27/22]], [[49_40|49/40]] and [[60_49|60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove,%20aka%20Wonder|jove]].

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;11/9&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|0 -2 0 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
347.40794 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_11_9_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_11_9_pluck_adu_dr220.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_11_9_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_11_9_pluck_adu_dr220.mp3/513194394/jid_11_9_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_11_9_pluck_adu_dr220.mp3/513194394/jid_11_9_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, &amp;lt;strong&amp;gt;11/9&amp;lt;/strong&amp;gt; is a neutral third of about 347.4¢, falling in between &amp;amp;quot;major third&amp;amp;quot; and &amp;amp;quot;minor third&amp;amp;quot; territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/49_40"&amp;gt;49/40&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/60_49"&amp;gt;60/49&amp;lt;/a&amp;gt;). It is nearly halfway between two intervals of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, implying that it is both very xenharmonic and well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Breedsmic%20temperaments#Harry"&amp;gt;harry&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/Schismatic%20family#Sesquiquartififths"&amp;gt;sesquart&amp;lt;/a&amp;gt;, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament &amp;lt;a class="wiki_link" href="/Breed%20family#Jove,%20aka%20Wonder"&amp;gt;jove&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>crtv872z3ko37ofx2knjqnurjsg73fe</sha1>
    </revision>
    <revision>
      <id>4337</id>
      <timestamp>2012-02-24T15:08:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 304896006 - Original comment: 11/9 bold**</comment>
      <origin>4337</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5093" sha1="qqd125lx1yjcxbte7do417ihusnwfxx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2012-02-24 15:08:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;304896006&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;11/9 bold&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[11-limit]] [[Just Intonation]], **11/9** is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16_13|16/13]], [[27_22|27/22]], [[49_40|49/40]] and [[60_49|60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, &amp;lt;strong&amp;gt;11/9&amp;lt;/strong&amp;gt; is a neutral third of about 347.4¢, falling in between &amp;amp;quot;major third&amp;amp;quot; and &amp;amp;quot;minor third&amp;amp;quot; territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/49_40"&amp;gt;49/40&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/60_49"&amp;gt;60/49&amp;lt;/a&amp;gt;). It is nearly halfway between two intervals of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, implying that it is both very xenharmonic and well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Breedsmic%20temperaments#Harry"&amp;gt;harry&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/Schismatic%20family#Sesquiquartififths"&amp;gt;sesquart&amp;lt;/a&amp;gt;, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament &amp;lt;a class="wiki_link" href="/Breed%20family#Jove, aka Wonder"&amp;gt;jove&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>qqd125lx1yjcxbte7do417ihusnwfxx</sha1>
    </revision>
    <revision>
      <id>4338</id>
      <timestamp>2011-09-21T17:30:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 256754480 - Original comment: **</comment>
      <origin>4338</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5063" sha1="c6lnl1pvpaignl9rsrfwnorenugzfs9" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-09-21 17:30:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;256754480&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[11-limit]] [[Just Intonation]], 11/9 is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16_13|16/13]], [[27_22|27/22]], [[49_40|49/40]] and [[60_49|60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 11/9 is a neutral third of about 347.4¢, falling in between &amp;amp;quot;major third&amp;amp;quot; and &amp;amp;quot;minor third&amp;amp;quot; territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/49_40"&amp;gt;49/40&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/60_49"&amp;gt;60/49&amp;lt;/a&amp;gt;). It is nearly halfway between two intervals of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, implying that it is both very xenharmonic and well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Breedsmic%20temperaments#Harry"&amp;gt;harry&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/Schismatic%20family#Sesquiquartififths"&amp;gt;sesquart&amp;lt;/a&amp;gt;, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament &amp;lt;a class="wiki_link" href="/Breed%20family#Jove, aka Wonder"&amp;gt;jove&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>c6lnl1pvpaignl9rsrfwnorenugzfs9</sha1>
    </revision>
    <revision>
      <id>4339</id>
      <timestamp>2011-09-21T17:16:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 256749878 - Original comment: **</comment>
      <origin>4339</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4918" sha1="lwhg5rpvy8yu6w1ughtk3om7nm28wfj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-09-21 17:16:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;256749878&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[11-limit]] [[Just Intonation]], 11/9 is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16_13|16/13]], [[27_22|27/22]], [[49_40|49/40]] and [[60_49|60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], and [[Breedsmic temperaments#Harry|harry]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 11/9 is a neutral third of about 347.4¢, falling in between &amp;amp;quot;major third&amp;amp;quot; and &amp;amp;quot;minor third&amp;amp;quot; territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/49_40"&amp;gt;49/40&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/60_49"&amp;gt;60/49&amp;lt;/a&amp;gt;). It is nearly halfway between two intervals of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, implying that it is both very xenharmonic and well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/Breedsmic%20temperaments#Harry"&amp;gt;harry&amp;lt;/a&amp;gt;, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament &amp;lt;a class="wiki_link" href="/Breed%20family#Jove, aka Wonder"&amp;gt;jove&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>lwhg5rpvy8yu6w1ughtk3om7nm28wfj</sha1>
    </revision>
    <revision>
      <id>4340</id>
      <timestamp>2011-09-21T17:15:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 256749618 - Original comment: **</comment>
      <origin>4340</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4940" sha1="57lc0o1oi0ab812gdtj98qk1wykdnak" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-09-21 17:15:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;256749618&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[11-limit]] [[Just Intonation]], 11/9 is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16_13|16/13]], [[27_22|27/22]], [[49_40|49/40]] and [[60_49|60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], to name a few, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 11/9 is a neutral third of about 347.4¢, falling in between &amp;amp;quot;major third&amp;amp;quot; and &amp;amp;quot;minor third&amp;amp;quot; territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/49_40"&amp;gt;49/40&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/60_49"&amp;gt;60/49&amp;lt;/a&amp;gt;). It is nearly halfway between two intervals of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, implying that it is both very xenharmonic and well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Gamelismic%20clan#Miracle"&amp;gt;miracle&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/Breedsmic%20temperaments#Harry"&amp;gt;harry&amp;lt;/a&amp;gt;, to name a few, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament &amp;lt;a class="wiki_link" href="/Breed%20family#Jove, aka Wonder"&amp;gt;jove&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4341</id>
      <timestamp>2011-09-21T17:14:56Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 256749524 - Original comment: **</comment>
      <origin>4341</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4897" sha1="414fxhlhwci1ug00rljm7v50euvphjg" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-09-21 17:14:56 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;256749524&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[11-limit]] [[Just Intonation]], 11/9 is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16_13|16/13]], [[27_22|27/22]], [[49_40|49/40]] and [[60_49|60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic+clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], to name a few, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 11/9 is a neutral third of about 347.4¢, falling in between &amp;amp;quot;major third&amp;amp;quot; and &amp;amp;quot;minor third&amp;amp;quot; territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/49_40"&amp;gt;49/40&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/60_49"&amp;gt;60/49&amp;lt;/a&amp;gt;). It is nearly halfway between two intervals of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, implying that it is both very xenharmonic and well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/41edo"&amp;gt;41edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202edo&amp;lt;/a&amp;gt;, [[Gamelismic+clan#Miracle|miracle]], &amp;lt;a class="wiki_link" href="/Breedsmic%20temperaments#Harry"&amp;gt;harry&amp;lt;/a&amp;gt;, to name a few, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament &amp;lt;a class="wiki_link" href="/Breed%20family#Jove, aka Wonder"&amp;gt;jove&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    </revision>
    <revision>
      <id>4342</id>
      <timestamp>2011-09-21T17:09:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 256747890 - Original comment: **</comment>
      <origin>4342</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4355" sha1="l21zy3wl261yy6uk92s3mhrxpynl51z" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-09-21 17:09:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;256747890&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[11-limit]] [[Just Intonation]], 11/9 is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16_13|16/13]], [[27_22|27/22]], [[49_40|49/40]] and [[60_49|60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]] and [[31edo]], to name a few, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 11/9 is a neutral third of about 347.4¢, falling in between &amp;amp;quot;major third&amp;amp;quot; and &amp;amp;quot;minor third&amp;amp;quot; territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/49_40"&amp;gt;49/40&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/60_49"&amp;gt;60/49&amp;lt;/a&amp;gt;). It is nearly halfway between two intervals of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, implying that it is both very xenharmonic and well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, to name a few, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament &amp;lt;a class="wiki_link" href="/Breed%20family#Jove, aka Wonder"&amp;gt;jove&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    </revision>
    <revision>
      <id>4343</id>
      <timestamp>2011-09-21T16:51:31Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 256741114 - Original comment: **</comment>
      <origin>4343</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3776" sha1="thtmtm3amk44mfajzyk5efbuwdy4nv6" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-09-21 16:51:31 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;256741114&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[11-limit]] [[Just Intonation]], 11/9 is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16_13|16/13]], [[27_22|27/22]], [[49_40|49/40]] and [[60_49|60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]] and [[31edo]], to name a few, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 11/9 is a neutral third of about 347.4¢, falling in between &amp;amp;quot;major third&amp;amp;quot; and &amp;amp;quot;minor third&amp;amp;quot; territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/49_40"&amp;gt;49/40&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/60_49"&amp;gt;60/49&amp;lt;/a&amp;gt;). It is nearly halfway between two intervals of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, implying that it is both very xenharmonic and well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, to name a few, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>thtmtm3amk44mfajzyk5efbuwdy4nv6</sha1>
    </revision>
    <revision>
      <id>4344</id>
      <timestamp>2011-09-14T20:23:13Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 254157862 - Original comment: **</comment>
      <origin>4344</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3660" sha1="3216x5ra9s4klxuqstt82h2g76t8rmc" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-09-14 20:23:13 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;254157862&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[11-limit]] [[Just Intonation]], 11/9 is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16_13|16/13]] and [[27_22|27/22]], to name two). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]] and [[31edo]], to name a few, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth.

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11_9&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 11/9 is a neutral third of about 347.4¢, falling in between &amp;amp;quot;major third&amp;amp;quot; and &amp;amp;quot;minor third&amp;amp;quot; territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include &amp;lt;a class="wiki_link" href="/16_13"&amp;gt;16/13&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/27_22"&amp;gt;27/22&amp;lt;/a&amp;gt;, to name two). It is nearly halfway between two intervals of &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt;, implying that it is both very xenharmonic and well-represented in &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including &amp;lt;a class="wiki_link" href="/17edo"&amp;gt;17edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, to name a few, conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>3216x5ra9s4klxuqstt82h2g76t8rmc</sha1>
    </revision>
    <revision>
      <id>36588</id>
      <parentid>84</parentid>
      <timestamp>2018-10-11T22:40:04Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>switched to infobox interval; simplified links</comment>
      <origin>36588</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1693" sha1="elx43urt60hgmaqlum4zkbkx3p5txnz" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of persony (others include [[16/13]], [[27/22]], [[49/40]] and [[60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit theyxad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Persony temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

:''See also [[Gallery of Just Intervals]]''

[[Category:11-limit]]
[[Category:interval]]
[[Category:neutral_3rd]]
[[Category:ratio]]
[[Category:third]]
[[Category:undecimal]]
[[Category:untwelve]]</text>
      <sha1>elx43urt60hgmaqlum4zkbkx3p5txnz</sha1>
    </revision>
    <revision>
      <id>37585</id>
      <parentid>36588</parentid>
      <timestamp>2018-10-23T03:25:40Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <origin>37585</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1722" sha1="o61mp877tdu1wszk2geog7eqomunu65" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third
| Color name = 1o3, lova 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of persony (others include [[16/13]], [[27/22]], [[49/40]] and [[60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit theyxad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Persony temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

:''See also [[Gallery of Just Intervals]]''

[[Category:11-limit]]
[[Category:interval]]
[[Category:neutral_3rd]]
[[Category:ratio]]
[[Category:third]]
[[Category:undecimal]]
[[Category:untwelve]]</text>
      <sha1>o61mp877tdu1wszk2geog7eqomunu65</sha1>
    </revision>
    <revision>
      <id>37673</id>
      <parentid>37585</parentid>
      <timestamp>2018-10-23T19:05:08Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <origin>37673</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1721" sha1="7jzjxkdttqovk8ev7mcgnwmxzzg0j17" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of persony (others include [[16/13]], [[27/22]], [[49/40]] and [[60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit theyxad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Persony temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

:''See also [[Gallery of Just Intervals]]''

[[Category:11-limit]]
[[Category:interval]]
[[Category:neutral_3rd]]
[[Category:ratio]]
[[Category:third]]
[[Category:undecimal]]
[[Category:untwelve]]</text>
      <sha1>7jzjxkdttqovk8ev7mcgnwmxzzg0j17</sha1>
    </revision>
    <revision>
      <id>41117</id>
      <parentid>37673</parentid>
      <timestamp>2019-03-22T02:03:45Z</timestamp>
      <contributor>
        <username>Bootmii</username>
        <id>104</id>
      </contributor>
      <minor/>
      <comment>scun thorpe problem</comment>
      <origin>41117</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1713" sha1="0aapg2e4w3o9stuin5hlhku072lu89g" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[49/40]] and [[60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

:''See also [[Gallery of Just Intervals]]''  [[Category:11-limit]] [[Category:interval]] [[Category:neutral_3rd]] [[Category:ratio]] [[Category:third]] [[Category:undecimal]] [[Category:untwelve]]</text>
      <sha1>0aapg2e4w3o9stuin5hlhku072lu89g</sha1>
    </revision>
    <revision>
      <id>45969</id>
      <parentid>41117</parentid>
      <timestamp>2020-06-07T13:43:45Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>recat</comment>
      <origin>45969</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1723" sha1="92dllb0oyqlxfa3pmxn4sv6kyxqsmzb" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[49/40]] and [[60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

:''See also [[Gallery of Just Intervals]]''  

[[Category:11-limit]] 
[[Category:Interval]] 
[[Category:Neutral third]] 
[[Category:Ratio]] 
[[Category:Third]] 
[[Category:Undecimal]] 
[[Category:Untwelve]]</text>
      <sha1>92dllb0oyqlxfa3pmxn4sv6kyxqsmzb</sha1>
    </revision>
    <revision>
      <id>45993</id>
      <parentid>45969</parentid>
      <timestamp>2020-06-07T14:55:38Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>expanded see-also section, moved from cat interval to cat just interval</comment>
      <origin>45993</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1763" sha1="tunkxfuh5nf1n6rt6ll3019846n3cw3" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[49/40]] and [[60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

== See also == 
* [[24edo]]
* [[Iceface tuning]]
* [[Gallery of Just Intervals]]

[[Category:11-limit]] 
[[Category:Just interval]] 
[[Category:Neutral third]] 
[[Category:Ratio]] 
[[Category:Third]] 
[[Category:Undecimal]] 
[[Category:Untwelve]]</text>
      <sha1>tunkxfuh5nf1n6rt6ll3019846n3cw3</sha1>
    </revision>
    <revision>
      <id>46757</id>
      <parentid>45993</parentid>
      <timestamp>2020-06-13T22:31:51Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>cats</comment>
      <origin>46757</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1769" sha1="lxlmuz6vydz0j3yur3lm77virzjmxdj" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[49/40]] and [[60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

== See also == 
* [[24edo]]
* [[Iceface tuning]]
* [[Gallery of Just Intervals]]

[[Category:11-limit]] 
[[Category:Just interval]] 
[[Category:Neutral third]] 
[[Category:Interval ratio]] 
[[Category:Third]] 
[[Category:Listen]] 
[[Category:Untwelve]]</text>
      <sha1>lxlmuz6vydz0j3yur3lm77virzjmxdj</sha1>
    </revision>
    <revision>
      <id>48344</id>
      <parentid>46757</parentid>
      <timestamp>2020-08-29T22:14:37Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <minor/>
      <comment>Note on this interval and 38edo</comment>
      <origin>48344</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1981" sha1="l6xht7wnw1o9o2auzjtxbq8ozbjbm5y" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[49/40]] and [[60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]]. It is approximated even more closely in [[38edo]], where the slight flatness of the 5th creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in this tuning.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

== See also == 
* [[24edo]]
* [[Iceface tuning]]
* [[Gallery of Just Intervals]]

[[Category:11-limit]] 
[[Category:Just interval]] 
[[Category:Neutral third]] 
[[Category:Interval ratio]] 
[[Category:Third]] 
[[Category:Listen]] 
[[Category:Untwelve]]</text>
      <sha1>l6xht7wnw1o9o2auzjtxbq8ozbjbm5y</sha1>
    </revision>
    <revision>
      <id>48350</id>
      <parentid>48344</parentid>
      <timestamp>2020-08-30T07:33:13Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Also 31edo</comment>
      <origin>48350</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2037" sha1="nqptc64ilzls4va1w2w5ritzmqrt9tn" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third
| Color name = 1o3, ilo 3rd
| FJS name = m3&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a neutral third of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]]. It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the 5th creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments#Harry|harry]], and [[Schismatic family#Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by 243/242, but also from 49/40 by 441/440 and 60/49 by 540/539, with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family#Jove, aka Wonder|jove]].

== See also == 
* [[24edo]]
* [[Iceface tuning]]
* [[Gallery of Just Intervals]]

[[Category:11-limit]] 
[[Category:Just interval]] 
[[Category:Neutral third]] 
[[Category:Interval ratio]] 
[[Category:Third]] 
[[Category:Listen]] 
[[Category:Untwelve]]</text>
      <sha1>nqptc64ilzls4va1w2w5ritzmqrt9tn</sha1>
    </revision>
    <revision>
      <id>49879</id>
      <parentid>48350</parentid>
      <timestamp>2020-09-20T06:44:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Cleanup</comment>
      <origin>49879</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2134" sha1="k7jyjyhas8thpk4f8095175giq6znwd" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third
| Color name = 1o3, ilo 3rd
| FJS name = m3&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a '''neutral third''' of about 347.4¢, falling in between "major third" and "minor third" territory. It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]). It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]]. It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the 5th creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== See also ==
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Iceface tuning]]

[[Category:11-limit]]
[[Category:Just interval]]
[[Category:Interval ratio]]
[[Category:Neutral third]]
[[Category:Third]]
[[Category:Listen]]
[[Category:Untwelve]]</text>
      <sha1>k7jyjyhas8thpk4f8095175giq6znwd</sha1>
    </revision>
    <revision>
      <id>53144</id>
      <parentid>49879</parentid>
      <timestamp>2020-11-24T23:31:31Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>Added more information and an additional name</comment>
      <origin>53144</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2362" sha1="45tlv7341h8imjibb296y0x3pwvbpsk" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third
| Color name = 1o3, ilo 3rd
| FJS name = m3&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a '''neutral third''' of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]).  As this is the smaller of two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''lesser Alpharabian neutral third''' in [[Alpharabian tuning]].  It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the 5th creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== See also ==
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Iceface tuning]]

[[Category:11-limit]]
[[Category:Just interval]]
[[Category:Interval ratio]]
[[Category:Neutral third]]
[[Category:Third]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]</text>
      <sha1>45tlv7341h8imjibb296y0x3pwvbpsk</sha1>
    </revision>
    <revision>
      <id>53145</id>
      <parentid>53144</parentid>
      <timestamp>2020-11-24T23:32:02Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>53145</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2401" sha1="f5ufl1436gj3hqy8s21u5usuu3az7es" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third, &lt;br&gt; lesser Alpharabian neutral third
| Color name = 1o3, ilo 3rd
| FJS name = m3&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a '''neutral third''' of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]).  As this is the smaller of two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''lesser Alpharabian neutral third''' in [[Alpharabian tuning]].  It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the 5th creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== See also ==
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Iceface tuning]]

[[Category:11-limit]]
[[Category:Just interval]]
[[Category:Interval ratio]]
[[Category:Neutral third]]
[[Category:Third]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]</text>
      <sha1>f5ufl1436gj3hqy8s21u5usuu3az7es</sha1>
    </revision>
    <revision>
      <id>64451</id>
      <parentid>53145</parentid>
      <timestamp>2021-03-27T04:04:17Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64451</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2449" sha1="16cybl0mtyn6q6pvqdhmxcbu1qfpomd" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third, &lt;br&gt; lesser Alpharabian neutral third
| Color name = 1o3, ilo 3rd
| FJS name = m3&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a '''neutral third''' of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]).  As this is the smaller of two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''lesser Alpharabian neutral third''' in [[Alpharabian tuning]].  It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the 5th creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== See also ==
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[Gallery of Just Intervals]]
* [[Iceface tuning]]

[[Category:11-limit]]
[[Category:Just interval]]
[[Category:Interval ratio]]
[[Category:Neutral third]]
[[Category:Third]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>16cybl0mtyn6q6pvqdhmxcbu1qfpomd</sha1>
    </revision>
    <revision>
      <id>72997</id>
      <parentid>64451</parentid>
      <timestamp>2021-06-23T17:42:12Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>72997</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2491" sha1="b9m4v2qf6ex8qlxr2vb664ibh5ipc6m" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third, &lt;br&gt; lesser Alpharabian neutral third
| Color name = 1o3, ilo 3rd
| FJS name = m3&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a '''neutral third''' of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]).  As this is the smaller of two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''lesser Alpharabian neutral third''' in [[Alpharabian tuning]].  It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the 5th creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing one neutral third interval to be stacked to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== See also ==
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[12/11]] – its [[fourth complement]]
* [[Gallery of Just Intervals]]
* [[Iceface tuning]]

[[Category:11-limit]]
[[Category:Just interval]]
[[Category:Interval ratio]]
[[Category:Neutral third]]
[[Category:Third]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>b9m4v2qf6ex8qlxr2vb664ibh5ipc6m</sha1>
    </revision>
    <revision>
      <id>83030</id>
      <parentid>72997</parentid>
      <timestamp>2022-01-01T02:14:25Z</timestamp>
      <contributor>
        <username>Morion</username>
        <id>322</id>
      </contributor>
      <comment>Clarified information.</comment>
      <origin>83030</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2497" sha1="jgyiptvqppnzf8jg4fhu2vpz34sze8s" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third, &lt;br&gt; lesser Alpharabian neutral third
| Color name = 1o3, ilo 3rd
| FJS name = m3&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a '''neutral third''' of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]).  As this is the smaller of two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''lesser Alpharabian neutral third''' in [[Alpharabian tuning]].  It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the 5th creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing either interval to be stacked on top of itself to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== See also ==
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[12/11]] – its [[fourth complement]]
* [[Gallery of Just Intervals]]
* [[Iceface tuning]]

[[Category:11-limit]]
[[Category:Just interval]]
[[Category:Interval ratio]]
[[Category:Neutral third]]
[[Category:Third]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>jgyiptvqppnzf8jg4fhu2vpz34sze8s</sha1>
    </revision>
    <revision>
      <id>85032</id>
      <parentid>83030</parentid>
      <timestamp>2022-01-29T23:36:09Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>Changing some of the nomenclature concerning Alpharabian tuning</comment>
      <origin>85032</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2491" sha1="mepulsczlh2gugkien34im9g1j5frfp" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third, &lt;br&gt; Alpharabian artoneutral third
| Color name = 1o3, ilo 3rd
| FJS name = m3&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[Just Intonation]], '''11/9''' is a '''neutral third''' of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]).  As this is the smaller of two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]].  It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the 5th creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing either interval to be stacked on top of itself to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== See also ==
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[12/11]] – its [[fourth complement]]
* [[Gallery of Just Intervals]]
* [[Iceface tuning]]

[[Category:11-limit]]
[[Category:Just interval]]
[[Category:Interval ratio]]
[[Category:Neutral third]]
[[Category:Third]]
[[Category:Alpharabian]]
[[Category:Listen]]
[[Category:Untwelve]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>mepulsczlh2gugkien34im9g1j5frfp</sha1>
    </revision>
    <revision>
      <id>86251</id>
      <parentid>85032</parentid>
      <timestamp>2022-02-22T17:22:08Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Misc. edits, categories</comment>
      <origin>86251</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2337" sha1="569xmiyky71cycom6967mx2j2k4is8h" xml:space="preserve">{{Infobox Interval
| Ratio = 11/9
| Monzo = 0 -2 0 0 1
| Cents = 347.40794
| Name = undecimal neutral third, &lt;br&gt; Alpharabian artoneutral third
| Color name = 1o3, ilo 3rd
| FJS name = m3&lt;sup&gt;11&lt;/sup&gt;
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/9''' is a '''neutral third''' of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]).  As this is the smaller of two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]].  It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the 5th creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing either interval to be stacked on top of itself to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== See also ==
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[12/11]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[Iceface tuning]]

[[Category:11-limit]]
[[Category:Third]]
[[Category:Neutral third]]
[[Category:Alpharabian]]</text>
      <sha1>569xmiyky71cycom6967mx2j2k4is8h</sha1>
    </revision>
    <revision>
      <id>97942</id>
      <parentid>86251</parentid>
      <timestamp>2022-10-25T11:29:04Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97942</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2225" sha1="oc18epi2n2acy60rs8d77zq6jcyvats" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral third, Alpharabian artoneutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/9''' is a '''neutral third''' of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]).  As this is the smaller of two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]].  It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the 5th creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing either interval to be stacked on top of itself to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== See also ==
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[12/11]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[Iceface tuning]]

[[Category:Third]]
[[Category:Neutral third]]
[[Category:Alpharabian]]</text>
      <sha1>oc18epi2n2acy60rs8d77zq6jcyvats</sha1>
    </revision>
    <revision>
      <id>112385</id>
      <parentid>97942</parentid>
      <timestamp>2023-05-07T01:02:11Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>112385</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2264" sha1="2nlyazxm0bbo5mch2qbqv1yd7fnsrdb" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral third, Alpharabian artoneutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/9''' is a '''neutral third''' of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but of course, only one of many (others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]]).  As this is the smaller of two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]].  It is nearly halfway between two intervals of [[12edo]], implying that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the 5th creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the harmonic 11th and the harmonic 9th. A triad can also be built with a 3/2 fifth and 11/9 third: this would be 18:22:27. This introduces a second neutral third, 27/22, which together make a perfect fifth. Many temperaments, including [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[72edo]], [[130edo]], [[202edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds, allowing either interval to be stacked on top of itself to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of them leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== See also ==
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[12/11]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[Iceface tuning]]

[[Category:Third]]
[[Category:Neutral third]]
[[Category:Alpharabian]]
[[Category:Tritave-reduced harmonics]]</text>
      <sha1>2nlyazxm0bbo5mch2qbqv1yd7fnsrdb</sha1>
    </revision>
    <revision>
      <id>206341</id>
      <parentid>112385</parentid>
      <timestamp>2025-08-05T08:30:44Z</timestamp>
      <contributor>
        <username>Fitzgerald Lee</username>
        <id>503</id>
      </contributor>
      <comment>changed phrasing, replaced big edo examples with small edo exmaples</comment>
      <origin>206341</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2305" sha1="jxzyju9php4izvry36ojpxytc19ig47" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral third, Alpharabian artoneutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/9''' is a [[neutral third]] of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but it is only one of many, of which others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]].  As this is the smaller of the two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]]. As it is nearly halfway between two intervals of [[12edo]], it implies that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the fifth creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the 11th harmonic and the 9th harmonic. A triad can also be built with 3/2 and 11/9, and the chord formed is 18:22:27. This introduces a second neutral third, 27/22, which is the difference between 3/2 and 11/9. Many temperaments, including [[7edo]], [[10edo]], [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds. The interval which represents both neutral thirds can then be stacked twice to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of these commas leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== See also ==
* [[7edo]]
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[12/11]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[Iceface tuning]]

[[Category:Third]]
[[Category:Neutral third]]
[[Category:Alpharabian]]
[[Category:Tritave-reduced harmonics]]</text>
      <sha1>jxzyju9php4izvry36ojpxytc19ig47</sha1>
    </revision>
    <revision>
      <id>206342</id>
      <parentid>206341</parentid>
      <timestamp>2025-08-05T08:49:54Z</timestamp>
      <contributor>
        <username>Fitzgerald Lee</username>
        <id>503</id>
      </contributor>
      <minor/>
      <origin>206342</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2308" sha1="jbpjadir0u9zoajjk5arj42g91r1yly" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral third, Alpharabian artoneutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/9''' is a [[neutral third]] of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but it is only one of many, of which others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]].  As this is the smaller of the two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]]. Since it is nearly halfway between two intervals of [[12edo]], it implies that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the fifth creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the 11th harmonic and the 9th harmonic. A triad can also be built with 3/2 and 11/9, and the chord formed is 18:22:27. This introduces a second neutral third, 27/22, which is the difference between 3/2 and 11/9. Many temperaments, including [[7edo]], [[10edo]], [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds. The interval which represents both neutral thirds can then be stacked twice to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of these commas leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].

== See also ==
* [[7edo]]
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[12/11]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[Iceface tuning]]

[[Category:Third]]
[[Category:Neutral third]]
[[Category:Alpharabian]]
[[Category:Tritave-reduced harmonics]]</text>
      <sha1>jbpjadir0u9zoajjk5arj42g91r1yly</sha1>
    </revision>
    <revision>
      <id>215808</id>
      <parentid>206342</parentid>
      <timestamp>2025-11-03T07:06:41Z</timestamp>
      <contributor>
        <username>Pailiaq</username>
        <id>507</id>
      </contributor>
      <minor/>
      <origin>215808</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2365" sha1="tub4bpw18n98850i8xxtmrz9uj7iu6c" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral third, Alpharabian artoneutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/9''' is a [[neutral third]] of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but it is only one of many, of which others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]].  As this is the smaller of the two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]]. Since it is nearly halfway between two intervals of [[12edo]], it implies that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the fifth creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the 11th harmonic and the 9th harmonic. A triad can also be built with 3/2 and 11/9, and the chord formed is 18:22:27. This introduces a second neutral third, 27/22, which is the difference between 3/2 and 11/9. Many temperaments, including [[7edo]], [[10edo]], [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds. The interval which represents both neutral thirds can then be stacked twice to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of these commas leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].
== Approximation ==
 {{Interval_Edo_Approximation | 11/9}
== See also ==
* [[7edo]]
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[12/11]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[Iceface tuning]]

[[Category:Third]]
[[Category:Neutral third]]
[[Category:Alpharabian]]
[[Category:Tritave-reduced harmonics]]</text>
      <sha1>tub4bpw18n98850i8xxtmrz9uj7iu6c</sha1>
    </revision>
    <revision>
      <id>215809</id>
      <parentid>215808</parentid>
      <timestamp>2025-11-03T07:07:09Z</timestamp>
      <contributor>
        <username>Pailiaq</username>
        <id>507</id>
      </contributor>
      <minor/>
      <origin>215809</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2366" sha1="qu6130hp61dcs8pxvtewph6c2i8k99d" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral third, Alpharabian artoneutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/9''' is a [[neutral third]] of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but it is only one of many, of which others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]].  As this is the smaller of the two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]]. Since it is nearly halfway between two intervals of [[12edo]], it implies that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the fifth creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the 11th harmonic and the 9th harmonic. A triad can also be built with 3/2 and 11/9, and the chord formed is 18:22:27. This introduces a second neutral third, 27/22, which is the difference between 3/2 and 11/9. Many temperaments, including [[7edo]], [[10edo]], [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds. The interval which represents both neutral thirds can then be stacked twice to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of these commas leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].
== Approximation ==
 {{Interval_Edo_Approximation | 11/9}}
== See also ==
* [[7edo]]
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[12/11]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[Iceface tuning]]

[[Category:Third]]
[[Category:Neutral third]]
[[Category:Alpharabian]]
[[Category:Tritave-reduced harmonics]]</text>
      <sha1>qu6130hp61dcs8pxvtewph6c2i8k99d</sha1>
    </revision>
    <revision>
      <id>216002</id>
      <parentid>215809</parentid>
      <timestamp>2025-11-03T13:05:26Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"</comment>
      <origin>216002</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2363" sha1="p855tikmprz8eyy5fw2sjwh1wo4ua5u" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral third, Alpharabian artoneutral third
| Color name = 1o3, ilo 3rd
| Sound = jid_11_9_pluck_adu_dr220.mp3
}}

In [[11-limit]] [[just intonation]], '''11/9''' is a [[neutral third]] of about 347.4¢, falling in between "major third" and "minor third" territory.  It is the simplest neutral third in just intonation, but it is only one of many, of which others include [[16/13]], [[27/22]], [[39/32]], [[49/40]] and [[60/49]].  As this is the smaller of the two [[11-limit]] neutral thirds obtained by modifying Pythagorean intervals by [[33/32]], it is dubbed the '''Alpharabian artoneutral third''' in [[Alpharabian tuning]]. Since it is nearly halfway between two intervals of [[12edo]], it implies that it is both very xenharmonic and well-represented in [[24edo]].  It is approximated even more closely in [[31edo]] and [[38edo]], where the slight flatness of the fifth creates two near perfect 11/9's when divided in half, making the neutral triad particularly stable sounding in these tunings.

In the 11-limit hexad 4:5:6:7:9:11, 11/9 appears between the 11th harmonic and the 9th harmonic. A triad can also be built with 3/2 and 11/9, and the chord formed is 18:22:27. This introduces a second neutral third, 27/22, which is the difference between 3/2 and 11/9. Many temperaments, including [[7edo]], [[10edo]], [[17edo]], [[24edo]], [[31edo]], [[41edo]], [[58edo]], [[Gamelismic clan#Miracle|miracle]], [[Breedsmic temperaments #Harry|harry]], and [[Schismatic family #Sesquiquartififths|sesquart]], conflate these two neutral thirds. The interval which represents both neutral thirds can then be stacked twice to generate a perfect fifth. 11/9 differs from 27/22 by [[243/242]], but also from 49/40 by [[441/440]] and 60/49 by [[540/539]], with varied consequences when one or more of them are tempered out. Tempering out all of these commas leads to the 11-limit rank three temperament [[Breed family #Jove, aka Wonder|jove]].
== Approximation ==
{{Interval edo approximation|11/9}}
== See also ==
* [[7edo]]
* [[24edo]]
* [[18/11]] – its [[octave complement]]
* [[27/22]] – its [[fifth complement]]
* [[12/11]] – its [[fourth complement]]
* [[Gallery of just intervals]]
* [[Iceface tuning]]

[[Category:Third]]
[[Category:Neutral third]]
[[Category:Alpharabian]]
[[Category:Tritave-reduced harmonics]]</text>
      <sha1>p855tikmprz8eyy5fw2sjwh1wo4ua5u</sha1>
    </revision>
  </page>
  <page>
    <title>11 9</title>
    <ns>0</ns>
    <id>85</id>
    <redirect title="11/9" />
    <revision>
      <id>85</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>85</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18" sha1="lu2z9q7zagzqr1vu0vlmbtqoijpapec" xml:space="preserve">#REDIRECT [[11/9]]</text>
      <sha1>lu2z9q7zagzqr1vu0vlmbtqoijpapec</sha1>
    </revision>
  </page>
  <page>
    <title>11ed5</title>
    <ns>0</ns>
    <id>86</id>
    <revision>
      <id>86</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>86</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="268" sha1="9i7c7nklaqtvu0yfn7qs46xpzrausjq" xml:space="preserve">0: 1/1 0.000 unison, perfect prime

1: 253.301 cents 8/7, 7/6

2: 506.602 cents 4/3

3: 759.904 cents 14/9

4: 1013.205 cents 9/5

5: 1266.506 cents

6: 1519.807 cents 12/5

7: 1773.109 cents 14/5

8: 2026.410 cents 16/5

9: 2279.711 cents

10: 2533.012 cents

11: 5/1</text>
      <sha1>9i7c7nklaqtvu0yfn7qs46xpzrausjq</sha1>
    </revision>
    <revision>
      <id>4345</id>
      <timestamp>2011-11-21T14:56:23Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Kosmorsky</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 277820172 - Original comment: **</comment>
      <origin>4345</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1866" sha1="tt8oi7othqbugnztdql8dkq3f3qn09n" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on &lt;tt&gt;2011-11-21 14:56:23 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;277820172&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;0: 1/1 0.000 unison, perfect prime
1: 253.301 cents 8/7, 7/6
2: 506.602 cents 4/3
3: 759.904 cents 14/9
4: 1013.205 cents 9/5
5: 1266.506 cents
6: 1519.807 cents 12/5
7: 1773.109 cents 14/5
8: 2026.410 cents 16/5
9: 2279.711 cents
10: 2533.012 cents
11: 5/1&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11ed5&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;0: 1/1 0.000 unison, perfect prime&amp;lt;br /&amp;gt;
1: 253.301 cents 8/7, 7/6&amp;lt;br /&amp;gt;
2: 506.602 cents 4/3&amp;lt;br /&amp;gt;
3: 759.904 cents 14/9&amp;lt;br /&amp;gt;
4: 1013.205 cents 9/5&amp;lt;br /&amp;gt;
5: 1266.506 cents&amp;lt;br /&amp;gt;
6: 1519.807 cents 12/5&amp;lt;br /&amp;gt;
7: 1773.109 cents 14/5&amp;lt;br /&amp;gt;
8: 2026.410 cents 16/5&amp;lt;br /&amp;gt;
9: 2279.711 cents&amp;lt;br /&amp;gt;
10: 2533.012 cents&amp;lt;br /&amp;gt;
11: 5/1&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>tt8oi7othqbugnztdql8dkq3f3qn09n</sha1>
    </revision>
    <revision>
      <id>39599</id>
      <parentid>86</parentid>
      <timestamp>2019-01-17T12:40:35Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>39599</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="946" sha1="oy8gra62utjjz4pevmd7v88jpty9si2" xml:space="preserve">'''11ED5''' is the [[Ed5|equal division of the 5th harmonic]] into 11 parts of 252.3012 [[cent|cents]] each. It is compared to every fourth step of [[19edo]], but with the 5/1 rather than 2/1 being just.

{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
| | 0
| | 0.0000
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 253.3012
| | [[8/7]], [[7/6]]
| | 
|-
| | 2
| | 506.6025
| | [[4/3]]
| | 
|-
| | 3
| | 759.9037
| | [[14/9]]
| | 
|-
| | 4
| | 1013.2050
| | [[9/5]]
| | 
|-
| | 5
| | 1266.5062
| | [[25/24|25/12]], [[21/20|21/10]]
| | 
|-
| | 6
| | 1519.8075
| | [[12/5]]
| | 
|-
| | 7
| | 1773.1087
| | [[14/5]]
| | 
|-
| | 8
| | 2026.4100
| | [[16/5]], 45/14
| | 
|-
| | 9
| | 2279.7112
| | [[15/4]]
| | 
|-
| | 10
| | 2533.0125
| | [[15/14|30/7]], [[35/32|35/8]]
| | 
|-
| | 11
| | 2786.3137
| | '''exact [[5/1]]'''
| | just major third plus two octaves
|}

[[Category:Ed5]]
[[Category:Edonoi]]</text>
      <sha1>oy8gra62utjjz4pevmd7v88jpty9si2</sha1>
    </revision>
    <revision>
      <id>39600</id>
      <parentid>39599</parentid>
      <timestamp>2019-01-17T13:02:36Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>39600</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1025" sha1="m50i7yj0xu17cjqp0x5iapl5hooubas" xml:space="preserve">'''11ED5''' is the [[Ed5|equal division of the 5th harmonic]] into 11 parts of 252.3012 [[cent|cents]] each. It is related to the [[Meantone family|godzilla temperament]], which tempers out 49/48, 81/80, and 91/90 in the 2.3.5.7.13 subgroup.

{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
| | 0
| | 0.0000
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 253.3012
| | [[8/7]], [[15/13]], [[7/6]]
| | 
|-
| | 2
| | 506.6025
| | [[4/3]]
| | 
|-
| | 3
| | 759.9037
| | [[14/9]]
| | 
|-
| | 4
| | 1013.2050
| | [[9/5]]
| | 
|-
| | 5
| | 1266.5062
| | [[27/26|27/13]], [[25/24|25/12]], [[21/20|21/10]]
| | 
|-
| | 6
| | 1519.8075
| | [[12/5]]
| | 
|-
| | 7
| | 1773.1087
| | [[14/5]]
| | 
|-
| | 8
| | 2026.4100
| | [[16/5]], [[13/4]]
| | 
|-
| | 9
| | 2279.7112
| | [[15/4]]
| | 
|-
| | 10
| | 2533.0125
| | [[15/14|30/7]], [[13/3]], [[35/32|35/8]]
| | 
|-
| | 11
| | 2786.3137
| | '''exact [[5/1]]'''
| | just major third plus two octaves
|}

[[Category:Ed5]]
[[Category:Edonoi]]</text>
      <sha1>m50i7yj0xu17cjqp0x5iapl5hooubas</sha1>
    </revision>
    <revision>
      <id>96773</id>
      <parentid>39600</parentid>
      <timestamp>2022-10-05T20:37:39Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
      <origin>96773</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1040" sha1="fppp3vuf2b7h8ky1qws53qp00kon0by" xml:space="preserve">{{Infobox ET}}
'''11ED5''' is the [[Ed5|equal division of the 5th harmonic]] into 11 parts of 252.3012 [[cent|cents]] each. It is related to the [[Meantone family|godzilla temperament]], which tempers out 49/48, 81/80, and 91/90 in the 2.3.5.7.13 subgroup.

{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
| | 0
| | 0.0000
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 253.3012
| | [[8/7]], [[15/13]], [[7/6]]
| | 
|-
| | 2
| | 506.6025
| | [[4/3]]
| | 
|-
| | 3
| | 759.9037
| | [[14/9]]
| | 
|-
| | 4
| | 1013.2050
| | [[9/5]]
| | 
|-
| | 5
| | 1266.5062
| | [[27/26|27/13]], [[25/24|25/12]], [[21/20|21/10]]
| | 
|-
| | 6
| | 1519.8075
| | [[12/5]]
| | 
|-
| | 7
| | 1773.1087
| | [[14/5]]
| | 
|-
| | 8
| | 2026.4100
| | [[16/5]], [[13/4]]
| | 
|-
| | 9
| | 2279.7112
| | [[15/4]]
| | 
|-
| | 10
| | 2533.0125
| | [[15/14|30/7]], [[13/3]], [[35/32|35/8]]
| | 
|-
| | 11
| | 2786.3137
| | '''exact [[5/1]]'''
| | just major third plus two octaves
|}

[[Category:Ed5]]
[[Category:Edonoi]]</text>
      <sha1>fppp3vuf2b7h8ky1qws53qp00kon0by</sha1>
    </revision>
    <revision>
      <id>158738</id>
      <parentid>96773</parentid>
      <timestamp>2024-10-06T01:12:32Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Harmonics</comment>
      <origin>158738</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1220" sha1="9y08akjwsj66a8lkl7pi9orftuzten6" xml:space="preserve">{{Infobox ET}}
'''11ED5''' is the [[Ed5|equal division of the 5th harmonic]] into 11 parts of 252.3012 [[cent|cents]] each. It is related to the [[Meantone family|godzilla temperament]], which tempers out 49/48, 81/80, and 91/90 in the 2.3.5.7.13 subgroup.

== Intervals ==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
| | 0
| | 0.0000
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 253.3012
| | [[8/7]], [[15/13]], [[7/6]]
| | 
|-
| | 2
| | 506.6025
| | [[4/3]]
| | 
|-
| | 3
| | 759.9037
| | [[14/9]]
| | 
|-
| | 4
| | 1013.2050
| | [[9/5]]
| | 
|-
| | 5
| | 1266.5062
| | [[27/26|27/13]], [[25/24|25/12]], [[21/20|21/10]]
| | 
|-
| | 6
| | 1519.8075
| | [[12/5]]
| | 
|-
| | 7
| | 1773.1087
| | [[14/5]]
| | 
|-
| | 8
| | 2026.4100
| | [[16/5]], [[13/4]]
| | 
|-
| | 9
| | 2279.7112
| | [[15/4]]
| | 
|-
| | 10
| | 2533.0125
| | [[15/14|30/7]], [[13/3]], [[35/32|35/8]]
| | 
|-
| | 11
| | 2786.3137
| | '''exact [[5/1]]'''
| | just major third plus two octaves
|}

== Harmonics ==
{{Harmonics in equal
| steps = 11
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 11
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}

[[Category:Ed5]]
[[Category:Edonoi]]</text>
      <sha1>9y08akjwsj66a8lkl7pi9orftuzten6</sha1>
    </revision>
    <revision>
      <id>205820</id>
      <parentid>158738</parentid>
      <timestamp>2025-08-01T18:39:04Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Ed5]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>205820</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1203" sha1="aag67wt1op3h577v3lsbhyjq1fkwmzg" xml:space="preserve">{{Infobox ET}}
'''11ED5''' is the [[Ed5|equal division of the 5th harmonic]] into 11 parts of 252.3012 [[cent|cents]] each. It is related to the [[Meantone family|godzilla temperament]], which tempers out 49/48, 81/80, and 91/90 in the 2.3.5.7.13 subgroup.

== Intervals ==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
| | 0
| | 0.0000
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 253.3012
| | [[8/7]], [[15/13]], [[7/6]]
| | 
|-
| | 2
| | 506.6025
| | [[4/3]]
| | 
|-
| | 3
| | 759.9037
| | [[14/9]]
| | 
|-
| | 4
| | 1013.2050
| | [[9/5]]
| | 
|-
| | 5
| | 1266.5062
| | [[27/26|27/13]], [[25/24|25/12]], [[21/20|21/10]]
| | 
|-
| | 6
| | 1519.8075
| | [[12/5]]
| | 
|-
| | 7
| | 1773.1087
| | [[14/5]]
| | 
|-
| | 8
| | 2026.4100
| | [[16/5]], [[13/4]]
| | 
|-
| | 9
| | 2279.7112
| | [[15/4]]
| | 
|-
| | 10
| | 2533.0125
| | [[15/14|30/7]], [[13/3]], [[35/32|35/8]]
| | 
|-
| | 11
| | 2786.3137
| | '''exact [[5/1]]'''
| | just major third plus two octaves
|}

== Harmonics ==
{{Harmonics in equal
| steps = 11
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 11
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}

[[Category:Edonoi]]</text>
      <sha1>aag67wt1op3h577v3lsbhyjq1fkwmzg</sha1>
    </revision>
    <revision>
      <id>205937</id>
      <parentid>205820</parentid>
      <timestamp>2025-08-01T19:20:36Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Edonoi]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>205937</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1182" sha1="exa0dsan91jlh3uwfo0o9q6mbur9gdd" xml:space="preserve">{{Infobox ET}}
'''11ED5''' is the [[Ed5|equal division of the 5th harmonic]] into 11 parts of 252.3012 [[cent|cents]] each. It is related to the [[Meantone family|godzilla temperament]], which tempers out 49/48, 81/80, and 91/90 in the 2.3.5.7.13 subgroup.

== Intervals ==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
| | 0
| | 0.0000
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 253.3012
| | [[8/7]], [[15/13]], [[7/6]]
| | 
|-
| | 2
| | 506.6025
| | [[4/3]]
| | 
|-
| | 3
| | 759.9037
| | [[14/9]]
| | 
|-
| | 4
| | 1013.2050
| | [[9/5]]
| | 
|-
| | 5
| | 1266.5062
| | [[27/26|27/13]], [[25/24|25/12]], [[21/20|21/10]]
| | 
|-
| | 6
| | 1519.8075
| | [[12/5]]
| | 
|-
| | 7
| | 1773.1087
| | [[14/5]]
| | 
|-
| | 8
| | 2026.4100
| | [[16/5]], [[13/4]]
| | 
|-
| | 9
| | 2279.7112
| | [[15/4]]
| | 
|-
| | 10
| | 2533.0125
| | [[15/14|30/7]], [[13/3]], [[35/32|35/8]]
| | 
|-
| | 11
| | 2786.3137
| | '''exact [[5/1]]'''
| | just major third plus two octaves
|}

== Harmonics ==
{{Harmonics in equal
| steps = 11
| num = 5
| denom = 1
}}
{{Harmonics in equal
| steps = 11
| num = 5
| denom = 1
| start = 12
| collapsed = 1
}}</text>
      <sha1>exa0dsan91jlh3uwfo0o9q6mbur9gdd</sha1>
    </revision>
  </page>
  <page>
    <title>11edf</title>
    <ns>0</ns>
    <id>87</id>
    <revision>
      <id>87</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>87</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="57" sha1="dfzl4d160ktauci9w9eadxwgl8mju96" xml:space="preserve">#REDIRECT [[Carlos_Beta]]      
Redirects to Carlos Beta.</text>
      <sha1>dfzl4d160ktauci9w9eadxwgl8mju96</sha1>
    </revision>
    <revision>
      <id>4346</id>
      <timestamp>2011-12-27T01:13:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;keenanpepper</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 288541992 - Original comment: **</comment>
      <origin>4346</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1276" sha1="i4jw1hgv1hqjm4jcr8bif6uzb2llnc0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on &lt;tt&gt;2011-12-27 01:13:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288541992&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;Redirects to Carlos Beta.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edf&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;Redirects to Carlos Beta.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>i4jw1hgv1hqjm4jcr8bif6uzb2llnc0</sha1>
    </revision>
    <revision>
      <id>96651</id>
      <parentid>87</parentid>
      <timestamp>2022-10-05T19:34:29Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <comment>11edf and Carlos Beta are different ETs (11 != 10.996753); replacing the redirect with a stub</comment>
      <origin>96651</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="75" sha1="4vgub4902wl54ag22k6875ck0vwzv7v" xml:space="preserve">{{Infobox ET}}

[[Category:edf]]
[[Category:Todo:expand]]
[[Category:Stub]]</text>
      <sha1>4vgub4902wl54ag22k6875ck0vwzv7v</sha1>
    </revision>
    <revision>
      <id>105077</id>
      <parentid>96651</parentid>
      <timestamp>2023-03-08T21:10:45Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>105077</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1863" sha1="kkxnnqf7mx1b4ot84ih3ajti9z4idod" xml:space="preserve">{{Infobox ET}}

'''22EDF''' is the [[EDF|equal division of the just perfect fifth]] into 22 parts of 31.907 [[cent|cents]] each, corresponding to 37.6092 [[edo]] (similar to every fifth step of [[188edo]]).

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
|  colspan="2"| 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 31.907
| | [[55/54]]
| | 
|-
| | 2
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 3
| | 95.7211
| | 
| | 
|-
| | 4
| | 127.6282
| | [[14/13]]
| | 
|-
| | 5
| | 159.5352
| | 57/52
| | 
|-
| | 6
| | 191.4423
| | 
| | 
|-
| | 7
| | 223.3493
| |8/7
| | 
|-
| | 8
| | 255.2564
| | 
| | 
|-
| | 9
| | 287.1634
| |13/11
| | 
|-
| | 10
| | 319.0705
| |6/5
| | 
|-
| | 11
| | 350.9775
| | 60/49, 49/40
| | 
|-
| | 12
| | 382.8845
| |5/4
| | 
|-
| | 13
| | 414.7916
| |14/11
| | 
|-
| | 14
| | 446.6986
| | 
| | 
|-
| | 15
| | 478.6057
| | 
| | 
|-
| | 16
| | 510.5127
| | 
| | 
|-
| | 17
| | 542.4198
| | [[26/19]]
| | 
|-
| | 18
| | 574.3268
| | 39/28
| | 
|-
| | 19
| | 606.2339
| |64/45
| | 
|-
| | 20
| | 638.1409
| | ([[13/9]])
| | 
|-
| | 21
| | 670.048
| | 81/55
| | 
|-
| | 22
| | 701.955
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|23
|733.862
|55/36
|
|-
|24
|765.7691
|14/9, 81/52
|
|-
|25
|797.6761
|
|
|-
|26
|828.5732
|21/13
|
|-
|27
|861.4902
|171/104
|
|-
|28
|893.3973
|
|
|-
|29
|925.3043
|12/7
|
|-
|30
|956.2114
|
|
|-
|31
|988.1184
|39/22
|
|-
|32
|1020.0255
|9/5
|
|-
|33
|1052.9235
|90/49, 147/80
|
|-
|34
|1084.8395
|15/8
|
|-
|35
|1116.7466
|21/11
|
|-
|36
|1148.6536
|
|
|-
|37
|1180.5607
|
|
|-
|38
|1211.4677
|
|
|-
|39
|1244.3748
|39/19
|
|-
|40
|1276.2816
|117/56
|
|-
|41
|1308.1889
|32/15
|
|-
|42
|1340.0959
|13/6
|
|-
|43
|1372.003
|243/110
|
|-
|44
|1403.91
|'''exact''' 9/4
|
|}

[[Category:edf]]
[[Category:Todo:expand]]
[[Category:Stub]]</text>
      <sha1>kkxnnqf7mx1b4ot84ih3ajti9z4idod</sha1>
    </revision>
    <revision>
      <id>105078</id>
      <parentid>105077</parentid>
      <timestamp>2023-03-08T21:17:43Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>105078</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1284" sha1="k1wqntejms13v0xlg9igt591inrfe23" xml:space="preserve">{{Infobox ET}}

'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]).

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
|  colspan="2"| 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| |[[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.0705
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| |39/28
| | 
|-
| | 10
| | 638.1409
| |([[13/9]])
| | 
|-
| | 11
| | 701.955
| |'''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| |14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| |21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| |'''exact''' 9/4
| | 
|}

[[Category:edf]]
[[Category:Todo:expand]]
[[Category:Stub]]</text>
      <sha1>k1wqntejms13v0xlg9igt591inrfe23</sha1>
    </revision>
    <revision>
      <id>105709</id>
      <parentid>105078</parentid>
      <timestamp>2023-03-16T23:39:12Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>105709</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5808" sha1="cjtn9vd52701d9zy390tdvixlg29qcv" xml:space="preserve">{{Infobox ET}}

'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]).

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
|  colspan="2"| 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| |[[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.0705
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| |39/28
| | 
|-
| | 10
| | 638.1409
| |([[13/9]])
| | 
|-
| | 11
| | 701.955
| |'''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| |14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| |21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| |'''exact''' 9/4
| | 
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 62.33766 cents (4\7/11 = 4\77) to 65.{{Overline|45}} cents (3\5/11 = 3\51)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || ||  || || ||62.3377||
|-
| ||  || || || || ||27\47||62.66925 ||
|-
|  || || ||  || ||23\40|| ||62.{{Overline|72}}||
|-
| || ||  || || || ||42\73||62.7646||
|-
|  || || || ||19\33|| || ||62.8099||
|-
| || || || || || || 53\92||62.84585 ||
|-
| || ||  || || || 34\59|| || 62.86595||
|-
| || || || ||  || || 49\85|| 62.8877||
|-
|  || ||  ||15\26|| ||  || ||62.9371||
|-
| || || || || ||  ||56\97 ||62.9803||The generator closest to a just [[28/27]] for EDOs less than 200
|-
| || ||  || ||  ||41\71 || ||62.9962 ||
|-
| || || || || || ||67\116||63.0094||
|-
| || || || ||26\45|| || ||63.{{Overline|03}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||63.0525||
|-
| || || || || ||37\64|| ||63.06{{Overline|81}}||
|-
| || || || || || ||48\83||63.0887||
|-
| || ||11\19|| || || || ||63.1579||
|-
| || || || || || ||51\88||63.2231||
|-
| || || || || ||40\69|| ||63.2411||
|-
| || || || || || ||69\119||63.2544||
|-
| || || || ||29\50|| || ||63.{{Overline|27}}||
|-
| || || || || || ||76\131||63.2893||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||63.2997||
|-
| || || || || || ||65\112||63.3117||
|-
| || || ||18\31|| || || ||63.3431||[[Meantone]] is in this region
|-
| || || || || || ||61\105||63.3766||
|-
| || || || || ||43\74|| ||63.3907||
|-
| || || || || || ||68\117||63.4033||
|-
| || || || ||25\43|| || ||63.42495||
|-
| || || || || || ||57\98||63.4508||
|-
| || || || || ||32\55|| ||63.{{Overline|45}}||
|-
| || || || || || ||39\67||63.5007||
|-
| ||7\12|| || || || || ||63.{{Overline|63}}||
|-
| || || || || || ||38\65||63.7762||
|-
| || || || || ||31\53|| ||63.8079||The generator closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||63.8298||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||63.8581||
|-
| || || || || || ||65\111||63.8821||
|-
| || || || || ||41\70|| ||63.8951||
|-
| || || || || || ||58\99||63.91185||
|-
| || || ||17\29|| || || ||63.9499||
|-
| || || || || || ||61\104||63.8960||
|-
| || || || || ||44\75|| ||64.000||
|-
| || || || || || ||71\121||64.0120||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||64.0316||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||64.0534||
|-
| || || || || ||37\63|| ||64.0693||
|-
| || || || || || ||47\80||64.{{Overline|09}}||
|-
| || ||10\17|| || || || ||64.1711||
|-
| || || || || || ||43\73||64.2590||
|-
| || || || || ||33\56|| ||64.2857||
|-
| || || || || || ||56\95||64.3062||
|-
| || || || ||23\39|| || ||64.3357||
|-
| || || || || || ||59\100||64.{{Overline|36}}||
|-
| || || || || ||36\61|| ||64.3815||
|-
| || || || || || ||49\83||64.4031||
|-
| || || ||13\22|| || || ||64.4628||[[Archy]] is in this region
|-
| || || || || || ||42\71||64.53265||
|-
| || || || || ||29\49|| ||64.5640||
|-
| || || || || || ||45\76||64.5933||
|-
| || || || ||16\27|| || ||64.{{Overline|64}}||
|-
| || || || || || ||35\59||64.71495||
|-
| || || || || ||19\32|| ||64.7{{Overline|72}}||
|-
| || || || || || ||22\37||64.{{Overline|864}}||
|-
|3\5|| || || || || || ||65.{{Overline|45}}||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:edf]]
[[Category:Todo:expand]]
[[Category:Stub]]</text>
      <sha1>cjtn9vd52701d9zy390tdvixlg29qcv</sha1>
    </revision>
    <revision>
      <id>105844</id>
      <parentid>105709</parentid>
      <timestamp>2023-03-20T00:50:53Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Scale tree */</comment>
      <origin>105844</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6029" sha1="cn5v7k3ais37gjalvdj8iy2idmhl2sz" xml:space="preserve">{{Infobox ET}}

'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]).

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
|  colspan="2"| 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | [[Tel:127.6282|127.6282]]
| |[[14/13]]
| | 
|-
| | 3
| | [[Tel:191.4423|191.4423]]
| | 
| | 
|-
| | 4
| | [[Tel:255.2564|255.2564]]
| | 
| | 
|-
| | 5
| | [[Tel:319.0705|319.0705]]
| | 6/5
| | 
|-
| | 6
| | [[Tel:382.8845|382.8845]]
| | 5/4
| | 
|-
| | 7
| | [[Tel:446.6986|446.6986]]
| |
| | 
|-
| | 8
| | [[Tel:510.5127|510.5127]]
| | 
| | 
|-
| | 9
| | [[Tel:574.3268|574.3268]]
| |39/28
| | 
|-
| | 10
| | [[Tel:638.1409|638.1409]]
| |([[13/9]])
| | 
|-
| | 11
| | 701.955
| |'''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | [[Tel:765.7691|765.7691]]
| |14/9, 81/52
| | 
|-
| | 13
| | [[Tel:828.5732|828.5732]]
| |21/13
| | 
|-
| | 14
| | [[Tel:893.3973|893.3973]]
| | 
| | 
|-
| | 15
| | [[Tel:956.2114|956.2114]]
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| |'''exact''' 9/4
| | 
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 62.33766 cents (4\7/11 = 4\77) to 65.{{Overline|45}} cents (3\5/11 = 3\55)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || ||  || || ||62.3377||
|-
| ||  || || || || ||27\47||62.66925 ||
|-
|  || || ||  || ||23\40|| ||62.{{Overline|72}}||
|-
| || ||  || || || ||42\73||62.7646||
|-
|  || || || ||19\33|| || ||62.8099||
|-
| || || || || || || 53\92||62.84585 ||
|-
| || ||  || || || 34\59|| || 62.86595||
|-
| || || || ||  || || 49\85|| 62.8877||
|-
|  || ||  ||15\26|| ||  || ||62.9371||
|-
| || || || || ||  ||56\97 ||62.9803||The generator closest to a just [[28/27]] for EDOs less than 200
|-
| || ||  || ||  ||41\71 || ||62.9962 ||
|-
| || || || || || ||67\116||63.0094||
|-
| || || || ||26\45|| || ||63.{{Overline|03}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||63.0525||
|-
| || || || || ||37\64|| ||63.06{{Overline|81}}||
|-
| || || || || || ||48\83||63.0887||
|-
| || ||11\19|| || || || ||63.1579||
|-
| || || || || || ||51\88||63.2231||
|-
| || || || || ||40\69|| ||63.2411||
|-
| || || || || || ||69\119||63.2544||
|-
| || || || ||29\50|| || ||63.{{Overline|27}}||
|-
| || || || || || ||76\131||63.2893||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||63.2997||
|-
| || || || || || ||65\112||63.3117||
|-
| || || ||18\31|| || || ||63.3431||[[Meantone]] is in this region
|-
| || || || || || ||61\105||63.3766||
|-
| || || || || ||43\74|| ||63.3907||
|-
| || || || || || ||68\117||63.4033||
|-
| || || || ||25\43|| || ||63.42495||
|-
| || || || || || ||57\98||63.4508||
|-
| || || || || ||32\55|| ||63.{{Overline|45}}||
|-
| || || || || || ||39\67||63.5007||
|-
| ||7\12|| || || || || ||63.{{Overline|63}}||
|-
| || || || || || ||38\65||63.7762||
|-
| || || || || ||31\53|| ||63.8079||The generator closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||63.8298||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||63.8581||
|-
| || || || || || ||65\111||63.8821||
|-
| || || || || ||41\70|| ||63.8951||
|-
| || || || || || ||58\99||63.91185||
|-
| || || ||17\29|| || || ||63.9499||
|-
| || || || || || ||61\104||63.8960||
|-
| || || || || ||44\75|| ||64.000||
|-
| || || || || || ||71\121||64.0120||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||64.0316||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||64.0534||
|-
| || || || || ||37\63|| ||64.0693||
|-
| || || || || || ||47\80||64.{{Overline|09}}||
|-
| || ||10\17|| || || || ||64.1711||
|-
| || || || || || ||43\73||64.2590||
|-
| || || || || ||33\56|| ||64.2857||
|-
| || || || || || ||56\95||64.3062||
|-
| || || || ||23\39|| || ||64.3357||
|-
| || || || || || ||59\100||64.{{Overline|36}}||
|-
| || || || || ||36\61|| ||64.3815||
|-
| || || || || || ||49\83||64.4031||
|-
| || || ||13\22|| || || ||64.4628||[[Archy]] is in this region
|-
| || || || || || ||42\71||64.53265||
|-
| || || || || ||29\49|| ||64.5640||
|-
| || || || || || ||45\76||64.5933||
|-
| || || || ||16\27|| || ||64.{{Overline|64}}||
|-
| || || || || || ||35\59||64.71495||
|-
| || || || || ||19\32|| ||64.7{{Overline|72}}||
|-
| || || || || || ||22\37||64.{{Overline|864}}||
|-
|3\5|| || || || || || ||65.{{Overline|45}}||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:edf]]
[[Category:Todo:expand]]
[[Category:Stub]]</text>
      <sha1>cn5v7k3ais37gjalvdj8iy2idmhl2sz</sha1>
    </revision>
    <revision>
      <id>105849</id>
      <parentid>105844</parentid>
      <timestamp>2023-03-20T01:13:04Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>105849</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5809" sha1="gw1ad09j84zqorgegjysxyhl8ibol9i" xml:space="preserve">{{Infobox ET}}

'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]).

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
|  colspan="2"| 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| |[[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.07045
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| |39/28
| | 
|-
| | 10
| | 638.1409
| |([[13/9]])
| | 
|-
| | 11
| | 701.955
| |'''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| |14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| |21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| |'''exact''' 9/4
| | 
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 62.33766 cents (4\7/11 = 4\77) to 65.{{Overline|45}} cents (3\5/11 = 3\55)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || ||  || || ||62.3377||
|-
| ||  || || || || ||27\47||62.66925 ||
|-
|  || || ||  || ||23\40|| ||62.{{Overline|72}}||
|-
| || ||  || || || ||42\73||62.7646||
|-
|  || || || ||19\33|| || ||62.8099||
|-
| || || || || || || 53\92||62.84585 ||
|-
| || ||  || || || 34\59|| || 62.86595||
|-
| || || || ||  || || 49\85|| 62.8877||
|-
|  || ||  ||15\26|| ||  || ||62.9371||
|-
| || || || || ||  ||56\97 ||62.9803||The generator closest to a just [[28/27]] for EDOs less than 200
|-
| || ||  || ||  ||41\71 || ||62.9962 ||
|-
| || || || || || ||67\116||63.0094||
|-
| || || || ||26\45|| || ||63.{{Overline|03}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||63.0525||
|-
| || || || || ||37\64|| ||63.06{{Overline|81}}||
|-
| || || || || || ||48\83||63.0887||
|-
| || ||11\19|| || || || ||63.1579||
|-
| || || || || || ||51\88||63.2231||
|-
| || || || || ||40\69|| ||63.2411||
|-
| || || || || || ||69\119||63.2544||
|-
| || || || ||29\50|| || ||63.{{Overline|27}}||
|-
| || || || || || ||76\131||63.2893||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||63.2997||
|-
| || || || || || ||65\112||63.3117||
|-
| || || ||18\31|| || || ||63.3431||[[Meantone]] is in this region
|-
| || || || || || ||61\105||63.3766||
|-
| || || || || ||43\74|| ||63.3907||
|-
| || || || || || ||68\117||63.4033||
|-
| || || || ||25\43|| || ||63.42495||
|-
| || || || || || ||57\98||63.4508||
|-
| || || || || ||32\55|| ||63.{{Overline|45}}||
|-
| || || || || || ||39\67||63.5007||
|-
| ||7\12|| || || || || ||63.{{Overline|63}}||
|-
| || || || || || ||38\65||63.7762||
|-
| || || || || ||31\53|| ||63.8079||The generator closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||63.8298||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||63.8581||
|-
| || || || || || ||65\111||63.8821||
|-
| || || || || ||41\70|| ||63.8951||
|-
| || || || || || ||58\99||63.91185||
|-
| || || ||17\29|| || || ||63.9499||
|-
| || || || || || ||61\104||63.8960||
|-
| || || || || ||44\75|| ||64.000||
|-
| || || || || || ||71\121||64.0120||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||64.0316||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||64.0534||
|-
| || || || || ||37\63|| ||64.0693||
|-
| || || || || || ||47\80||64.{{Overline|09}}||
|-
| || ||10\17|| || || || ||64.1711||
|-
| || || || || || ||43\73||64.2590||
|-
| || || || || ||33\56|| ||64.2857||
|-
| || || || || || ||56\95||64.3062||
|-
| || || || ||23\39|| || ||64.3357||
|-
| || || || || || ||59\100||64.{{Overline|36}}||
|-
| || || || || ||36\61|| ||64.3815||
|-
| || || || || || ||49\83||64.4031||
|-
| || || ||13\22|| || || ||64.4628||[[Archy]] is in this region
|-
| || || || || || ||42\71||64.53265||
|-
| || || || || ||29\49|| ||64.5640||
|-
| || || || || || ||45\76||64.5933||
|-
| || || || ||16\27|| || ||64.{{Overline|64}}||
|-
| || || || || || ||35\59||64.71495||
|-
| || || || || ||19\32|| ||64.7{{Overline|72}}||
|-
| || || || || || ||22\37||64.{{Overline|864}}||
|-
|3\5|| || || || || || ||65.{{Overline|45}}||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:edf]]
[[Category:Todo:expand]]
[[Category:Stub]]</text>
      <sha1>gw1ad09j84zqorgegjysxyhl8ibol9i</sha1>
    </revision>
    <revision>
      <id>106035</id>
      <parentid>105849</parentid>
      <timestamp>2023-03-22T18:02:50Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Scale tree */</comment>
      <origin>106035</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5805" sha1="drmv469onixj00ejcpx2tndygt9zigt" xml:space="preserve">{{Infobox ET}}

'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]).

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
|  colspan="2"| 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| |[[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.07045
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| |39/28
| | 
|-
| | 10
| | 638.1409
| |([[13/9]])
| | 
|-
| | 11
| | 701.955
| |'''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| |14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| |21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| |'''exact''' 9/4
| | 
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 62.33766 cents (4\7/11 = 4\77) to 65.{{Overline|45}} cents (3\5/11 = 3\55)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || ||  || || ||62.3377||
|-
| ||  || || || || ||27\47||62.66925 ||
|-
|  || || ||  || ||23\40|| ||62.{{Overline|72}}||
|-
| || ||  || || || ||42\73||62.7646||
|-
|  || || || ||19\33|| || ||62.8099||
|-
| || || || || || || 53\92||62.84585 ||
|-
| || ||  || || || 34\59|| || 62.86595||
|-
| || || || ||  || || 49\85|| 62.8877||
|-
|  || ||  ||15\26|| ||  || ||62.9371||
|-
| || || || || ||  ||56\97 ||62.9803||The generator closest to a just [[28/27]] for EDOs less than 200
|-
| || ||  || ||  ||41\71 || ||62.9962 ||
|-
| || || || || || ||67\116||63.0094||
|-
| || || || ||26\45|| || ||63.{{Overline|03}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||63.0525||
|-
| || || || || ||37\64|| ||63.06{{Overline|81}}||
|-
| || || || || || ||48\83||63.0887||
|-
| || ||11\19|| || || || ||63.1579||
|-
| || || || || || ||51\88||63.2231||
|-
| || || || || ||40\69|| ||63.2411||
|-
| || || || || || ||69\119||63.2544||
|-
| || || || ||29\50|| || ||63.{{Overline|27}}||
|-
| || || || || || ||76\131||63.2893||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||63.2997||
|-
| || || || || || ||65\112||63.3117||
|-
| || || ||18\31|| || || ||63.3431||[[Meantone]] is in this region
|-
| || || || || || ||61\105||63.3766||
|-
| || || || || ||43\74|| ||63.3907||
|-
| || || || || || ||68\117||63.4033||
|-
| || || || ||25\43|| || ||63.42495||
|-
| || || || || || ||57\98||63.4508||
|-
| || || || || ||32\55|| ||63.{{Overline|45}}||
|-
| || || || || || ||39\67||63.5007||
|-
| ||7\12|| || || || || ||63.{{Overline|63}}||
|-
| || || || || || ||38\65||63.7762||
|-
| || || || || ||31\53|| ||63.8079||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||63.8298||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||63.8581||
|-
| || || || || || ||65\111||63.8821||
|-
| || || || || ||41\70|| ||63.8951||
|-
| || || || || || ||58\99||63.91185||
|-
| || || ||17\29|| || || ||63.9499||
|-
| || || || || || ||61\104||63.8960||
|-
| || || || || ||44\75|| ||64.000||
|-
| || || || || || ||71\121||64.0120||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||64.0316||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||64.0534||
|-
| || || || || ||37\63|| ||64.0693||
|-
| || || || || || ||47\80||64.{{Overline|09}}||
|-
| || ||10\17|| || || || ||64.1711||
|-
| || || || || || ||43\73||64.2590||
|-
| || || || || ||33\56|| ||64.2857||
|-
| || || || || || ||56\95||64.3062||
|-
| || || || ||23\39|| || ||64.3357||
|-
| || || || || || ||59\100||64.{{Overline|36}}||
|-
| || || || || ||36\61|| ||64.3815||
|-
| || || || || || ||49\83||64.4031||
|-
| || || ||13\22|| || || ||64.4628||[[Archy]] is in this region
|-
| || || || || || ||42\71||64.53265||
|-
| || || || || ||29\49|| ||64.5640||
|-
| || || || || || ||45\76||64.5933||
|-
| || || || ||16\27|| || ||64.{{Overline|64}}||
|-
| || || || || || ||35\59||64.71495||
|-
| || || || || ||19\32|| ||64.7{{Overline|72}}||
|-
| || || || || || ||22\37||64.{{Overline|864}}||
|-
|3\5|| || || || || || ||65.{{Overline|45}}||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:edf]]
[[Category:Todo:expand]]
[[Category:Stub]]</text>
      <sha1>drmv469onixj00ejcpx2tndygt9zigt</sha1>
    </revision>
    <revision>
      <id>107010</id>
      <parentid>106035</parentid>
      <timestamp>2023-04-05T03:42:39Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>107010</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5882" sha1="h80ixyrtqwtvj91h5tdsejkn9swziry" xml:space="preserve">{{Infobox ET}}

'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]). It is almost identical to [[19edo]] and nearly identical to [[Carlos Beta]].

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
|  colspan="2"| 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| |[[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.07045
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| |39/28
| | 
|-
| | 10
| | 638.1409
| |([[13/9]])
| | 
|-
| | 11
| | 701.955
| |'''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| |14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| |21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| |'''exact''' 9/4
| | 
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 62.33766 cents (4\7/11 = 4\77) to 65.{{Overline|45}} cents (3\5/11 = 3\55)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || ||  || || ||62.3377||
|-
| ||  || || || || ||27\47||62.66925 ||
|-
|  || || ||  || ||23\40|| ||62.{{Overline|72}}||
|-
| || ||  || || || ||42\73||62.7646||
|-
|  || || || ||19\33|| || ||62.8099||
|-
| || || || || || || 53\92||62.84585 ||
|-
| || ||  || || || 34\59|| || 62.86595||
|-
| || || || ||  || || 49\85|| 62.8877||
|-
|  || ||  ||15\26|| ||  || ||62.9371||
|-
| || || || || ||  ||56\97 ||62.9803||The generator closest to a just [[28/27]] for EDOs less than 200
|-
| || ||  || ||  ||41\71 || ||62.9962 ||
|-
| || || || || || ||67\116||63.0094||
|-
| || || || ||26\45|| || ||63.{{Overline|03}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||63.0525||
|-
| || || || || ||37\64|| ||63.06{{Overline|81}}||
|-
| || || || || || ||48\83||63.0887||
|-
| || ||11\19|| || || || ||63.1579||
|-
| || || || || || ||51\88||63.2231||
|-
| || || || || ||40\69|| ||63.2411||
|-
| || || || || || ||69\119||63.2544||
|-
| || || || ||29\50|| || ||63.{{Overline|27}}||
|-
| || || || || || ||76\131||63.2893||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||63.2997||
|-
| || || || || || ||65\112||63.3117||
|-
| || || ||18\31|| || || ||63.3431||[[Meantone]] is in this region
|-
| || || || || || ||61\105||63.3766||
|-
| || || || || ||43\74|| ||63.3907||
|-
| || || || || || ||68\117||63.4033||
|-
| || || || ||25\43|| || ||63.42495||
|-
| || || || || || ||57\98||63.4508||
|-
| || || || || ||32\55|| ||63.{{Overline|45}}||
|-
| || || || || || ||39\67||63.5007||
|-
| ||7\12|| || || || || ||63.{{Overline|63}}||
|-
| || || || || || ||38\65||63.7762||
|-
| || || || || ||31\53|| ||63.8079||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||63.8298||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||63.8581||
|-
| || || || || || ||65\111||63.8821||
|-
| || || || || ||41\70|| ||63.8951||
|-
| || || || || || ||58\99||63.91185||
|-
| || || ||17\29|| || || ||63.9499||
|-
| || || || || || ||61\104||63.8960||
|-
| || || || || ||44\75|| ||64.000||
|-
| || || || || || ||71\121||64.0120||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||64.0316||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||64.0534||
|-
| || || || || ||37\63|| ||64.0693||
|-
| || || || || || ||47\80||64.{{Overline|09}}||
|-
| || ||10\17|| || || || ||64.1711||
|-
| || || || || || ||43\73||64.2590||
|-
| || || || || ||33\56|| ||64.2857||
|-
| || || || || || ||56\95||64.3062||
|-
| || || || ||23\39|| || ||64.3357||
|-
| || || || || || ||59\100||64.{{Overline|36}}||
|-
| || || || || ||36\61|| ||64.3815||
|-
| || || || || || ||49\83||64.4031||
|-
| || || ||13\22|| || || ||64.4628||[[Archy]] is in this region
|-
| || || || || || ||42\71||64.53265||
|-
| || || || || ||29\49|| ||64.5640||
|-
| || || || || || ||45\76||64.5933||
|-
| || || || ||16\27|| || ||64.{{Overline|64}}||
|-
| || || || || || ||35\59||64.71495||
|-
| || || || || ||19\32|| ||64.7{{Overline|72}}||
|-
| || || || || || ||22\37||64.{{Overline|864}}||
|-
|3\5|| || || || || || ||65.{{Overline|45}}||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:edf]]
[[Category:Todo:expand]]
[[Category:Stub]]</text>
      <sha1>h80ixyrtqwtvj91h5tdsejkn9swziry</sha1>
    </revision>
    <revision>
      <id>107011</id>
      <parentid>107010</parentid>
      <timestamp>2023-04-05T03:43:04Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>107011</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5873" sha1="i9k9epl511kx17uvi55lyz0sswondl5" xml:space="preserve">{{Infobox ET}}

'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]). It is similar to [[19edo]] and nearly identical to [[Carlos Beta]].

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
|  colspan="2"| 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| |[[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.07045
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| |39/28
| | 
|-
| | 10
| | 638.1409
| |([[13/9]])
| | 
|-
| | 11
| | 701.955
| |'''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| |14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| |21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| |'''exact''' 9/4
| | 
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 62.33766 cents (4\7/11 = 4\77) to 65.{{Overline|45}} cents (3\5/11 = 3\55)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || ||  || || ||62.3377||
|-
| ||  || || || || ||27\47||62.66925 ||
|-
|  || || ||  || ||23\40|| ||62.{{Overline|72}}||
|-
| || ||  || || || ||42\73||62.7646||
|-
|  || || || ||19\33|| || ||62.8099||
|-
| || || || || || || 53\92||62.84585 ||
|-
| || ||  || || || 34\59|| || 62.86595||
|-
| || || || ||  || || 49\85|| 62.8877||
|-
|  || ||  ||15\26|| ||  || ||62.9371||
|-
| || || || || ||  ||56\97 ||62.9803||The generator closest to a just [[28/27]] for EDOs less than 200
|-
| || ||  || ||  ||41\71 || ||62.9962 ||
|-
| || || || || || ||67\116||63.0094||
|-
| || || || ||26\45|| || ||63.{{Overline|03}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||63.0525||
|-
| || || || || ||37\64|| ||63.06{{Overline|81}}||
|-
| || || || || || ||48\83||63.0887||
|-
| || ||11\19|| || || || ||63.1579||
|-
| || || || || || ||51\88||63.2231||
|-
| || || || || ||40\69|| ||63.2411||
|-
| || || || || || ||69\119||63.2544||
|-
| || || || ||29\50|| || ||63.{{Overline|27}}||
|-
| || || || || || ||76\131||63.2893||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||63.2997||
|-
| || || || || || ||65\112||63.3117||
|-
| || || ||18\31|| || || ||63.3431||[[Meantone]] is in this region
|-
| || || || || || ||61\105||63.3766||
|-
| || || || || ||43\74|| ||63.3907||
|-
| || || || || || ||68\117||63.4033||
|-
| || || || ||25\43|| || ||63.42495||
|-
| || || || || || ||57\98||63.4508||
|-
| || || || || ||32\55|| ||63.{{Overline|45}}||
|-
| || || || || || ||39\67||63.5007||
|-
| ||7\12|| || || || || ||63.{{Overline|63}}||
|-
| || || || || || ||38\65||63.7762||
|-
| || || || || ||31\53|| ||63.8079||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||63.8298||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||63.8581||
|-
| || || || || || ||65\111||63.8821||
|-
| || || || || ||41\70|| ||63.8951||
|-
| || || || || || ||58\99||63.91185||
|-
| || || ||17\29|| || || ||63.9499||
|-
| || || || || || ||61\104||63.8960||
|-
| || || || || ||44\75|| ||64.000||
|-
| || || || || || ||71\121||64.0120||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||64.0316||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||64.0534||
|-
| || || || || ||37\63|| ||64.0693||
|-
| || || || || || ||47\80||64.{{Overline|09}}||
|-
| || ||10\17|| || || || ||64.1711||
|-
| || || || || || ||43\73||64.2590||
|-
| || || || || ||33\56|| ||64.2857||
|-
| || || || || || ||56\95||64.3062||
|-
| || || || ||23\39|| || ||64.3357||
|-
| || || || || || ||59\100||64.{{Overline|36}}||
|-
| || || || || ||36\61|| ||64.3815||
|-
| || || || || || ||49\83||64.4031||
|-
| || || ||13\22|| || || ||64.4628||[[Archy]] is in this region
|-
| || || || || || ||42\71||64.53265||
|-
| || || || || ||29\49|| ||64.5640||
|-
| || || || || || ||45\76||64.5933||
|-
| || || || ||16\27|| || ||64.{{Overline|64}}||
|-
| || || || || || ||35\59||64.71495||
|-
| || || || || ||19\32|| ||64.7{{Overline|72}}||
|-
| || || || || || ||22\37||64.{{Overline|864}}||
|-
|3\5|| || || || || || ||65.{{Overline|45}}||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:edf]]
[[Category:Todo:expand]]
[[Category:Stub]]</text>
      <sha1>i9k9epl511kx17uvi55lyz0sswondl5</sha1>
    </revision>
    <revision>
      <id>120010</id>
      <parentid>107011</parentid>
      <timestamp>2023-07-11T21:53:35Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Stub]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>120010</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5855" sha1="86sftuq1wxz24ok80p4p6r04v457bkx" xml:space="preserve">{{Infobox ET}}

'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]). It is similar to [[19edo]] and nearly identical to [[Carlos Beta]].

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
|  colspan="2"| 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| |[[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.07045
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| |39/28
| | 
|-
| | 10
| | 638.1409
| |([[13/9]])
| | 
|-
| | 11
| | 701.955
| |'''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| |14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| |21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| |'''exact''' 9/4
| | 
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 62.33766 cents (4\7/11 = 4\77) to 65.{{Overline|45}} cents (3\5/11 = 3\55)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7|| || || ||  || || ||62.3377||
|-
| ||  || || || || ||27\47||62.66925 ||
|-
|  || || ||  || ||23\40|| ||62.{{Overline|72}}||
|-
| || ||  || || || ||42\73||62.7646||
|-
|  || || || ||19\33|| || ||62.8099||
|-
| || || || || || || 53\92||62.84585 ||
|-
| || ||  || || || 34\59|| || 62.86595||
|-
| || || || ||  || || 49\85|| 62.8877||
|-
|  || ||  ||15\26|| ||  || ||62.9371||
|-
| || || || || ||  ||56\97 ||62.9803||The generator closest to a just [[28/27]] for EDOs less than 200
|-
| || ||  || ||  ||41\71 || ||62.9962 ||
|-
| || || || || || ||67\116||63.0094||
|-
| || || || ||26\45|| || ||63.{{Overline|03}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||63.0525||
|-
| || || || || ||37\64|| ||63.06{{Overline|81}}||
|-
| || || || || || ||48\83||63.0887||
|-
| || ||11\19|| || || || ||63.1579||
|-
| || || || || || ||51\88||63.2231||
|-
| || || || || ||40\69|| ||63.2411||
|-
| || || || || || ||69\119||63.2544||
|-
| || || || ||29\50|| || ||63.{{Overline|27}}||
|-
| || || || || || ||76\131||63.2893||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||63.2997||
|-
| || || || || || ||65\112||63.3117||
|-
| || || ||18\31|| || || ||63.3431||[[Meantone]] is in this region
|-
| || || || || || ||61\105||63.3766||
|-
| || || || || ||43\74|| ||63.3907||
|-
| || || || || || ||68\117||63.4033||
|-
| || || || ||25\43|| || ||63.42495||
|-
| || || || || || ||57\98||63.4508||
|-
| || || || || ||32\55|| ||63.{{Overline|45}}||
|-
| || || || || || ||39\67||63.5007||
|-
| ||7\12|| || || || || ||63.{{Overline|63}}||
|-
| || || || || || ||38\65||63.7762||
|-
| || || || || ||31\53|| ||63.8079||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||63.8298||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||63.8581||
|-
| || || || || || ||65\111||63.8821||
|-
| || || || || ||41\70|| ||63.8951||
|-
| || || || || || ||58\99||63.91185||
|-
| || || ||17\29|| || || ||63.9499||
|-
| || || || || || ||61\104||63.8960||
|-
| || || || || ||44\75|| ||64.000||
|-
| || || || || || ||71\121||64.0120||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||64.0316||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||64.0534||
|-
| || || || || ||37\63|| ||64.0693||
|-
| || || || || || ||47\80||64.{{Overline|09}}||
|-
| || ||10\17|| || || || ||64.1711||
|-
| || || || || || ||43\73||64.2590||
|-
| || || || || ||33\56|| ||64.2857||
|-
| || || || || || ||56\95||64.3062||
|-
| || || || ||23\39|| || ||64.3357||
|-
| || || || || || ||59\100||64.{{Overline|36}}||
|-
| || || || || ||36\61|| ||64.3815||
|-
| || || || || || ||49\83||64.4031||
|-
| || || ||13\22|| || || ||64.4628||[[Archy]] is in this region
|-
| || || || || || ||42\71||64.53265||
|-
| || || || || ||29\49|| ||64.5640||
|-
| || || || || || ||45\76||64.5933||
|-
| || || || ||16\27|| || ||64.{{Overline|64}}||
|-
| || || || || || ||35\59||64.71495||
|-
| || || || || ||19\32|| ||64.7{{Overline|72}}||
|-
| || || || || || ||22\37||64.{{Overline|864}}||
|-
|3\5|| || || || || || ||65.{{Overline|45}}||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:edf]]
[[Category:Todo:expand]]</text>
      <sha1>86sftuq1wxz24ok80p4p6r04v457bkx</sha1>
    </revision>
    <revision>
      <id>142513</id>
      <parentid>120010</parentid>
      <timestamp>2024-05-07T13:05:04Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>142513</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6224" sha1="gslyo5cmk26g3detbeiic7ndou1e4ke" xml:space="preserve">{{Infobox ET}}

'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]). It is similar to [[19edo]] and nearly identical to [[Carlos Beta]].

==Harmonics==
{{Harmonics in equal|11|3|2|prec=2}}

==Intervals==
{| class="wikitable"
|-
! | Degree
! | Cent value
! | Corresponding &lt;br&gt;JI intervals
! | Comments
|-
| colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| | [[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.07045
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| | 39/28
| | 
|-
| | 10
| | 638.1409
| | ([[13/9]])
| | 
|-
| | 11
| | 701.955
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| | 14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| | 21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| | '''exact''' 9/4
| | 
|}

==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 62.33766 cents (4\7/11 = 4\77) to 65.{{Overline|45}} cents (3\5/11 = 3\55)
{| class="wikitable center-all"
! colspan="7" | Fifth
! Cents
! Comments
|-
| 4\7 || || || ||  || || || 62.3377 || 
|-
| ||  || || || || || 27\47 || 62.66925 || 
|-
| || || ||  || || 23\4 0 || || 62.{{Overline|72}} || 
|-
| || ||  || || || || 42\73 || 62.7646 || 
|-
| || || || || 19\33 || || || 62.8099 || 
|-
| || || || || || || 53\92 || 62.84585 || 
|-
| || ||  || || || 34\59 || || 62.86595 || 
|-
| || || || ||  || || 49\85 || 62.8877 || 
|-
| || ||  || 15\26 || ||  || || 62.9371 || 
|-
| || || || || ||  || 56\97 || 62.9803 || The generator closest to a just [[28/27]] for EDOs less than 200
|-
| || ||  || ||  || 41\71 || || 62.9962 || 
|-
| || || || || || || 67\116 || 63.0094 || 
|-
| || || || || 26\45 || || || 63.{{Overline|03}} || [[Flattone]] is in this region
|-
| || || || || || || 63\109 || 63.0525 || 
|-
| || || || || || 37\64 || || 63.06{{Overline|81}} || 
|-
| || || || || || || 48\83 || 63.0887 || 
|-
| || || 11\19 || || || || || 63.1579 || 
|-
| || || || || || || 51\88 || 63.2231 || 
|-
| || || || || || 40\69 || || 63.2411 || 
|-
| || || || || || || 69\119 || 63.2544 || 
|-
| || || || || 29\50 || || || 63.{{Overline|27}} || 
|-
| || || || || || || 76\131 || 63.2893 || [[Golden meantone]] (696.2145¢)
|-
| || || || || || 47\81 || || 63.2997 || 
|-
| || || || || || || 65\112 || 63.3117 || 
|-
| || || || 18\31 || || || || 63.3431 || [[Meantone]] is in this region
|-
| || || || || || || 61\105 || 63.3766 || 
|-
| || || || || || 43\74 || || 63.3907 || 
|-
| || || || || || || 68\117 || 63.4033 || 
|-
| || || || || 25\43 || || || 63.42495 || 
|-
| || || || || || || 57\98 || 63.4508 || 
|-
| || || || || || 32\55 || || 63.{{Overline|45}} || 
|-
| || || || || || || 39\67 || 63.5007 || 
|-
| || 7\12 || || || || || || 63.{{Overline|63}} || 
|-
| || || || || || || 38\65 || 63.7762 || 
|-
| || || || || || 31\53 || || 63.8079 || The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || || 55\94 || 63.8298 || [[Garibaldi]] / [[Cassandra]]
|-
| || || || || 24\41 || || || 63.8581 || 
|-
| || || || || || || 65\111 || 63.8821 || 
|-
| || || || || || 41\70 || || 63.8951 || 
|-
| || || || || || || 58\99 || 63.91185 || 
|-
| || || || 17\29 || || || || 63.9499 || 
|-
| || || || || || || 61\104 || 63.8960 || 
|-
| || || || || || 44\75 || || 64.000 || 
|-
| || || || || || || 71\121 || 64.0120 || Golden neogothic (704.0956¢)
|-
| || || || || 27\46 || || || 64.0316 || [[Neogothic]] is in this region
|-
| || || || || || || 64\109 || 64.0534 || 
|-
| || || || || || 37\63 || || 64.0693 || 
|-
| || || || || || || 47\80 || 64.{{Overline|09}} || 
|-
| || || 10\17 || || || || || 64.1711 || 
|-
| || || || || || || 43\73 || 64.2590 || 
|-
| || || || || || 33\56 || || 64.2857 || 
|-
| || || || || || || 56\95 || 64.3062 || 
|-
| || || || || 23\39 || || || 64.3357 || 
|-
| || || || || || || 59\100 || 64.{{Overline|36}} || 
|-
| || || || || || 36\61 || || 64.3815 || 
|-
| || || || || || || 49\83 || 64.4031 || 
|-
| || || || 13\22 || || || || 64.4628 || [[Archy]] is in this region
|-
| || || || || || || 42\71 || 64.53265 || 
|-
| || || || || || 29\49 || || 64.5640 ||
|-
| || || || || || || 45\76 || 64.5933 ||
|-
| || || || || 16\27 || || || 64.{{Overline|64}} ||
|-
| || || || || || || 35\59 || 64.71495 ||
|-
| || || || || || 19\32 || || 64.7{{Overline|72}} ||
|-
| || || || || || || 22\37 || 64.{{Overline|864}} ||
|-
| 3\5 || || || || || || || 65.{{Overline|45}} ||
|}
Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:edf]]
[[Category:Todo:expand]]</text>
      <sha1>gslyo5cmk26g3detbeiic7ndou1e4ke</sha1>
    </revision>
    <revision>
      <id>142519</id>
      <parentid>142513</parentid>
      <timestamp>2024-05-07T13:10:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>-irrelevant shit</comment>
      <origin>142519</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1350" sha1="m4c0vlqaj19tm7qpmxapkua9zkyvqzr" xml:space="preserve">{{Infobox ET}}

'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]). It is similar to [[19edo]] and nearly identical to [[Carlos Beta]].

==Harmonics==
{{Harmonics in equal|11|3|2|prec=2}}

==Intervals==
{| class="wikitable"
|-
! | Degree
! | Cent value
! | Corresponding &lt;br&gt;JI intervals
! | Comments
|-
| colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| | [[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.07045
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| | 39/28
| | 
|-
| | 10
| | 638.1409
| | ([[13/9]])
| | 
|-
| | 11
| | 701.955
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| | 14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| | 21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| | '''exact''' 9/4
| | 
|}</text>
      <sha1>m4c0vlqaj19tm7qpmxapkua9zkyvqzr</sha1>
    </revision>
    <revision>
      <id>142972</id>
      <parentid>142519</parentid>
      <timestamp>2024-05-12T01:45:46Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>142972</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1495" sha1="k08czfn8oj660232un3pb0n11s93y26" xml:space="preserve">{{Infobox ET}}

'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]). It is similar to [[19edo]] and nearly identical to [[Carlos Beta]].

While the fifth is just, the fourth is much wider and less accurate than in 19edo, and is about four cents flat of that of [[7edo]].

==Harmonics==
{{Harmonics in equal|11|3|2|prec=2|columns=15}}

==Intervals==
{| class="wikitable"
|-
! | Degree
! | Cent value
! | Corresponding &lt;br&gt;JI intervals
! | Comments
|-
| colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| | [[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.07045
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| | 39/28
| | 
|-
| | 10
| | 638.1409
| | ([[13/9]])
| | 
|-
| | 11
| | 701.955
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| | 14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| | 21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| | '''exact''' 9/4
| | 
|}</text>
      <sha1>k08czfn8oj660232un3pb0n11s93y26</sha1>
    </revision>
    <revision>
      <id>143153</id>
      <parentid>142972</parentid>
      <timestamp>2024-05-14T19:13:57Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>143153</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1508" sha1="3vmxdmdowrhr0d96w4d1vhpuzpgoo7o" xml:space="preserve">{{Infobox ET}}

'''11EDF''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]). It is similar to [[19edo]] and nearly identical to [[Carlos Beta]].

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo, being about four cents flat of that of [[7edo]].

==Harmonics==
{{Harmonics in equal|11|3|2|prec=2|columns=15}}

==Intervals==
{| class="wikitable"
|-
! | Degree
! | Cent value
! | Corresponding &lt;br&gt;JI intervals
! | Comments
|-
| colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| | [[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.07045
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| | 39/28
| | 
|-
| | 10
| | 638.1409
| | ([[13/9]])
| | 
|-
| | 11
| | 701.955
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| | 14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| | 21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| | '''exact''' 9/4
| | 
|}</text>
      <sha1>3vmxdmdowrhr0d96w4d1vhpuzpgoo7o</sha1>
    </revision>
    <revision>
      <id>162443</id>
      <parentid>143153</parentid>
      <timestamp>2024-10-27T01:05:54Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Mention connection to phoenix</comment>
      <origin>162443</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1651" sha1="a50rqcnsn0pzcnkj76y0rmzxu858s0h" xml:space="preserve">{{Infobox ET}}

'''11edf''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]). It is similar to [[19edo]] and nearly identical to [[Carlos Beta]].

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo, being about four cents flat of that of [[7edo]].

11edf represents the upper bound of the [[phoenix]] tuning range. 11edf benefits from all the desirable properties of phoenix tuning systems.

==Harmonics==
{{Harmonics in equal|11|3|2|prec=2|columns=15}}

==Intervals==
{| class="wikitable"
|-
! | Degree
! | Cent value
! | Corresponding &lt;br&gt;JI intervals
! | Comments
|-
| colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| | [[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.07045
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| | 39/28
| | 
|-
| | 10
| | 638.1409
| | ([[13/9]])
| | 
|-
| | 11
| | 701.955
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| | 14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| | 21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| | '''exact''' 9/4
| | 
|}</text>
      <sha1>a50rqcnsn0pzcnkj76y0rmzxu858s0h</sha1>
    </revision>
    <revision>
      <id>171800</id>
      <parentid>162443</parentid>
      <timestamp>2024-12-19T08:56:36Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>{{todo|expand}}</comment>
      <origin>171800</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1668" sha1="bneoyc784vjblni5bvrwr01p43qdm2w" xml:space="preserve">{{Infobox ET}}

'''11edf''' is the [[EDF|equal division of the just perfect fifth]] into 11 parts of 63.8141 [[cent|cents]] each, corresponding to 18.8046 [[edo]] (similar to every fifth step of [[94edo]]). It is similar to [[19edo]] and nearly identical to [[Carlos Beta]].

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo, being about four cents flat of that of [[7edo]].

11edf represents the upper bound of the [[phoenix]] tuning range. 11edf benefits from all the desirable properties of phoenix tuning systems.

==Harmonics==
{{Harmonics in equal|11|3|2|prec=2|columns=15}}

==Intervals==
{| class="wikitable"
|-
! | Degree
! | Cent value
! | Corresponding &lt;br&gt;JI intervals
! | Comments
|-
| colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 63.8141
| | ([[28/27]]), ([[27/26]])
| | 
|-
| | 2
| | 127.6282
| | [[14/13]]
| | 
|-
| | 3
| | 191.4423
| | 
| | 
|-
| | 4
| | 255.2564
| | 
| | 
|-
| | 5
| | 319.07045
| | 6/5
| | 
|-
| | 6
| | 382.8845
| | 5/4
| | 
|-
| | 7
| | 446.6986
| |
| | 
|-
| | 8
| | 510.5127
| | 
| | 
|-
| | 9
| | 574.3268
| | 39/28
| | 
|-
| | 10
| | 638.1409
| | ([[13/9]])
| | 
|-
| | 11
| | 701.955
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
| | 12
| | 765.7691
| | 14/9, 81/52
| | 
|-
| | 13
| | 828.5732
| | 21/13
| | 
|-
| | 14
| | 893.3973
| | 
| | 
|-
| | 15
| | 956.2114
| | 
| | 
|-
| | 16
| | 1020.0255
| | 9/5
| | 
|-
| | 17
| | 1084.8395
| | 15/8
| | 
|-
| | 18
| | 1148.6536
| | 
| | 
|-
| | 19
| | 1211.4677
| |
| | 
|-
| | 20
| | 1276.2816
| | 117/56
| | 
|-
| | 21
| | 1340.0959
| | 13/6
| | 
|-
| | 22
| | 1403.91
| | '''exact''' 9/4
| | 
|}

{{todo|expand}}</text>
      <sha1>bneoyc784vjblni5bvrwr01p43qdm2w</sha1>
    </revision>
    <revision>
      <id>177521</id>
      <parentid>171800</parentid>
      <timestamp>2025-01-21T13:23:57Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>177521</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1343" sha1="0aamj2pz1t0wvb5eun0cmw0y3o256g4" xml:space="preserve">{{Infobox ET}}
{{ED intro}} It corresponds to 18.8046[[edo]], is is similar to [[19edo]], and nearly identical to [[Carlos Beta]].

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo, being about four cents flat of that of [[7edo]].

11edf represents the upper bound of the [[phoenix]] tuning range. 11edf benefits from all the desirable properties of phoenix tuning systems.

== Harmonics ==
{{Harmonics in equal|11|3|2|prec=2|columns=15}}

== Intervals ==
{| class="wikitable"
|-
! Degree
! Cent value
! Corresponding&lt;br /&gt;JI intervals
! Comments
|-
| colspan="2" | 0
| '''exact [[1/1]]'''
| 
|-
| 1
| 63.8141
| ([[28/27]]), ([[27/26]])
| 
|-
| 2
| 127.6282
| [[14/13]]
| 
|-
| 3
| 191.4423
| 
| 
|-
| 4
| 255.2564
| 
| 
|-
| 5
| 319.07045
| 6/5
| 
|-
| 6
| 382.8845
| 5/4
| 
|-
| 7
| 446.6986
| 
| 
|-
| 8
| 510.5127
| 
| 
|-
| 9
| 574.3268
| 39/28
| 
|-
| 10
| 638.1409
| ([[13/9]])
| 
|-
| 11
| 701.955
| '''exact [[3/2]]'''
| just perfect fifth
|-
| 12
| 765.7691
| 14/9, 81/52
| 
|-
| 13
| 828.5732
| 21/13
| 
|-
| 14
| 893.3973
| 
| 
|-
| 15
| 956.2114
| 
| 
|-
| 16
| 1020.0255
| 9/5
| 
|-
| 17
| 1084.8395
| 15/8
| 
|-
| 18
| 1148.6536
| 
| 
|-
| 19
| 1211.4677
| 
| 
|-
| 20
| 1276.2816
| 117/56
| 
|-
| 21
| 1340.0959
| 13/6
| 
|-
| 22
| 1403.91
| '''exact''' 9/4
| 
|}

{{todo|expand}}</text>
      <sha1>0aamj2pz1t0wvb5eun0cmw0y3o256g4</sha1>
    </revision>
    <revision>
      <id>184951</id>
      <parentid>177521</parentid>
      <timestamp>2025-03-06T20:02:37Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>184951</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1384" sha1="3i0i3ihmc0qmu42bikw60hjxlh2g2vq" xml:space="preserve">{{Infobox ET}}
{{ED intro}} It corresponds to 18.8046[[edo]], is is similar to [[19edo]], and nearly identical to [[Carlos Beta]].

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo. At 510.51{{c}}, it is 12.47{{c}} sharper than just and 3.7{{c}} flat of that of [[7edo]].

11edf represents the upper bound of the [[phoenix]] tuning range. 11edf benefits from all the desirable properties of phoenix tuning systems.

== Harmonics ==
{{Harmonics in equal|11|3|2|prec=2|columns=15}}

== Intervals ==
{| class="wikitable"
|-
! Degree
! Cent value
! Corresponding&lt;br /&gt;JI intervals
! Comments
|-
| colspan="2" | 0
| '''exact [[1/1]]'''
| 
|-
| 1
| 63.8141
| ([[28/27]]), ([[27/26]])
| 
|-
| 2
| 127.6282
| [[14/13]]
| 
|-
| 3
| 191.4423
| 
| 
|-
| 4
| 255.2564
| 
| 
|-
| 5
| 319.07045
| 6/5
| 
|-
| 6
| 382.8845
| 5/4
| 
|-
| 7
| 446.6986
| 
| 
|-
| 8
| 510.5127
| 
| 
|-
| 9
| 574.3268
| 39/28
| 
|-
| 10
| 638.1409
| ([[13/9]])
| 
|-
| 11
| 701.955
| '''exact [[3/2]]'''
| just perfect fifth
|-
| 12
| 765.7691
| 14/9, 81/52
| 
|-
| 13
| 828.5732
| 21/13
| 
|-
| 14
| 893.3973
| 
| 
|-
| 15
| 956.2114
| 
| 
|-
| 16
| 1020.0255
| 9/5
| 
|-
| 17
| 1084.8395
| 15/8
| 
|-
| 18
| 1148.6536
| 
| 
|-
| 19
| 1211.4677
| 
| 
|-
| 20
| 1276.2816
| 117/56
| 
|-
| 21
| 1340.0959
| 13/6
| 
|-
| 22
| 1403.91
| '''exact''' 9/4
| 
|}

{{todo|expand}}</text>
      <sha1>3i0i3ihmc0qmu42bikw60hjxlh2g2vq</sha1>
    </revision>
    <revision>
      <id>188914</id>
      <parentid>184951</parentid>
      <timestamp>2025-03-30T12:56:09Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>188914</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1534" sha1="em8tky5006r1rozzgh168j18z231nbk" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
11edf corresponds to 18.8046…[[edo]]. It is similar to [[19edo]], and nearly identical to [[Carlos Beta]].

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo. At 510.51{{c}}, it is 12.47{{c}} sharper than just and 3.7{{c}} flat of that of [[7edo]].

11edf represents the upper bound of the [[phoenix]] tuning range. It benefits from all the desirable properties of phoenix tuning systems.

=== Harmonics ===
{{Harmonics in equal|11|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edf (continued)}}

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! Approximate ratios
|-
| 0
| 0.0
| [[1/1]]
|-
| 1
| 63.8
| [[21/20]], [[25/24]], [[27/26]], [[28/27]]
|-
| 2
| 127.6
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
|-
| 3
| 191.4
| [[9/8]], [[10/9]]
|-
| 4
| 255.3
| [[7/6]], ''[[8/7]]''
|-
| 5
| 319.1
| [[6/5]]
|-
| 6
| 382.9
| [[5/4]]
|-
| 7
| 446.7
| [[9/7]]
|-
| 8
| 510.5
| [[4/3]]
|-
| 9
| 574.3
| [[7/5]]
|-
| 10
| 638.1
| [[13/9]]
|-
| 11
| 702.0
| [[3/2]]
|-
| 12
| 765.8
| [[14/9]]
|-
| 13
| 828.6
| [[8/5]], [[13/8]], [[21/13]]
|-
| 14
| 893.4
| [[5/3]]
|-
| 15
| 956.2
| [[7/4]]
|-
| 16
| 1020.0
| [[9/5]]
|-
| 17
| 1084.8
| [[15/8]]
|-
| 18
| 1148.7
| [[27/14]], [[35/18]]
|-
| 19
| 1211.5
| [[2/1]]
|-
| 20
| 1276.3
| [[21/10]], [[25/12]], [[27/13]]
|-
| 21
| 1340.1
| [[13/6]]
|-
| 22
| 1403.9
| [[9/4]]
|}</text>
      <sha1>em8tky5006r1rozzgh168j18z231nbk</sha1>
    </revision>
    <revision>
      <id>188915</id>
      <parentid>188914</parentid>
      <timestamp>2025-03-30T13:00:02Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ note consistency and +subsets and supersets</comment>
      <origin>188915</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1835" sha1="tlqixsjsoupxbfsne5d754hdedc8wmm" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
11edf corresponds to 18.8046…[[edo]]. It is similar to [[19edo]], and nearly identical to [[Carlos Beta]]. Unlike 19edo, which is [[consistent]] to the [[integer limit|10-integer-limit]], 11edf is only consistent to the 7-integer-limit. 

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo. At 510.51{{c}}, it is 12.47{{c}} sharper than just and 3.7{{c}} flat of that of [[7edo]].

11edf represents the upper bound of the [[phoenix]] tuning range. It benefits from all the desirable properties of phoenix tuning systems.

=== Harmonics ===
{{Harmonics in equal|11|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edf (continued)}}

=== Subsets and supersets ===
11edf is the fifth [[prime equal division|prime edf]], past [[7edf]] and before [[13edf]]. It does not contain any nontrivial subset edfs.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! Approximate ratios
|-
| 0
| 0.0
| [[1/1]]
|-
| 1
| 63.8
| [[21/20]], [[25/24]], [[27/26]], [[28/27]]
|-
| 2
| 127.6
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
|-
| 3
| 191.4
| [[9/8]], [[10/9]]
|-
| 4
| 255.3
| [[7/6]], ''[[8/7]]''
|-
| 5
| 319.1
| [[6/5]]
|-
| 6
| 382.9
| [[5/4]]
|-
| 7
| 446.7
| [[9/7]]
|-
| 8
| 510.5
| [[4/3]]
|-
| 9
| 574.3
| [[7/5]]
|-
| 10
| 638.1
| [[13/9]]
|-
| 11
| 702.0
| [[3/2]]
|-
| 12
| 765.8
| [[14/9]]
|-
| 13
| 828.6
| [[8/5]], [[13/8]], [[21/13]]
|-
| 14
| 893.4
| [[5/3]]
|-
| 15
| 956.2
| [[7/4]]
|-
| 16
| 1020.0
| [[9/5]]
|-
| 17
| 1084.8
| [[15/8]]
|-
| 18
| 1148.7
| [[27/14]], [[35/18]]
|-
| 19
| 1211.5
| [[2/1]]
|-
| 20
| 1276.3
| [[21/10]], [[25/12]], [[27/13]]
|-
| 21
| 1340.1
| [[13/6]]
|-
| 22
| 1403.9
| [[9/4]]
|}</text>
      <sha1>tlqixsjsoupxbfsne5d754hdedc8wmm</sha1>
    </revision>
    <revision>
      <id>188916</id>
      <parentid>188915</parentid>
      <timestamp>2025-03-30T13:00:49Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+see also</comment>
      <origin>188916</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2029" sha1="dqm2lsnh5gcwwosohn37uzp2uxmxyht" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
11edf corresponds to 18.8046…[[edo]]. It is similar to [[19edo]], and nearly identical to [[Carlos Beta]]. Unlike 19edo, which is [[consistent]] to the [[integer limit|10-integer-limit]], 11edf is only consistent to the 7-integer-limit. 

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo. At 510.51{{c}}, it is 12.47{{c}} sharper than just and 3.7{{c}} flat of that of [[7edo]].

11edf represents the upper bound of the [[phoenix]] tuning range. It benefits from all the desirable properties of phoenix tuning systems.

=== Harmonics ===
{{Harmonics in equal|11|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edf (continued)}}

=== Subsets and supersets ===
11edf is the fifth [[prime equal division|prime edf]], past [[7edf]] and before [[13edf]]. It does not contain any nontrivial subset edfs.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! Approximate ratios
|-
| 0
| 0.0
| [[1/1]]
|-
| 1
| 63.8
| [[21/20]], [[25/24]], [[27/26]], [[28/27]]
|-
| 2
| 127.6
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
|-
| 3
| 191.4
| [[9/8]], [[10/9]]
|-
| 4
| 255.3
| [[7/6]], ''[[8/7]]''
|-
| 5
| 319.1
| [[6/5]]
|-
| 6
| 382.9
| [[5/4]]
|-
| 7
| 446.7
| [[9/7]]
|-
| 8
| 510.5
| [[4/3]]
|-
| 9
| 574.3
| [[7/5]]
|-
| 10
| 638.1
| [[13/9]]
|-
| 11
| 702.0
| [[3/2]]
|-
| 12
| 765.8
| [[14/9]]
|-
| 13
| 828.6
| [[8/5]], [[13/8]], [[21/13]]
|-
| 14
| 893.4
| [[5/3]]
|-
| 15
| 956.2
| [[7/4]]
|-
| 16
| 1020.0
| [[9/5]]
|-
| 17
| 1084.8
| [[15/8]]
|-
| 18
| 1148.7
| [[27/14]], [[35/18]]
|-
| 19
| 1211.5
| [[2/1]]
|-
| 20
| 1276.3
| [[21/10]], [[25/12]], [[27/13]]
|-
| 21
| 1340.1
| [[13/6]]
|-
| 22
| 1403.9
| [[9/4]]
|}

== See also ==
* [[19edo]] – relative edo
* [[30edt]] – relative edt
* [[49ed6]] – relative ed6
* [[53ed7]] – relative ed7
* [[68ed12]] – relative ed12
* [[93ed30]] – relative ed30</text>
      <sha1>dqm2lsnh5gcwwosohn37uzp2uxmxyht</sha1>
    </revision>
    <revision>
      <id>198283</id>
      <parentid>188916</parentid>
      <timestamp>2025-05-31T19:34:49Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>+music</comment>
      <origin>198283</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2309" sha1="enohkaxxm3jw0n21bwlbfomhpgcudsx" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
11edf corresponds to 18.8046…[[edo]]. It is similar to [[19edo]], and nearly identical to [[Carlos Beta]]. Unlike 19edo, which is [[consistent]] to the [[integer limit|10-integer-limit]], 11edf is only consistent to the 7-integer-limit. 

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo. At 510.51{{c}}, it is 12.47{{c}} sharper than just and 3.7{{c}} flat of that of [[7edo]].

11edf represents the upper bound of the [[phoenix]] tuning range. It benefits from all the desirable properties of phoenix tuning systems.

=== Harmonics ===
{{Harmonics in equal|11|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edf (continued)}}

=== Subsets and supersets ===
11edf is the fifth [[prime equal division|prime edf]], past [[7edf]] and before [[13edf]]. It does not contain any nontrivial subset edfs.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! Approximate ratios
|-
| 0
| 0.0
| [[1/1]]
|-
| 1
| 63.8
| [[21/20]], [[25/24]], [[27/26]], [[28/27]]
|-
| 2
| 127.6
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
|-
| 3
| 191.4
| [[9/8]], [[10/9]]
|-
| 4
| 255.3
| [[7/6]], ''[[8/7]]''
|-
| 5
| 319.1
| [[6/5]]
|-
| 6
| 382.9
| [[5/4]]
|-
| 7
| 446.7
| [[9/7]]
|-
| 8
| 510.5
| [[4/3]]
|-
| 9
| 574.3
| [[7/5]]
|-
| 10
| 638.1
| [[13/9]]
|-
| 11
| 702.0
| [[3/2]]
|-
| 12
| 765.8
| [[14/9]]
|-
| 13
| 828.6
| [[8/5]], [[13/8]], [[21/13]]
|-
| 14
| 893.4
| [[5/3]]
|-
| 15
| 956.2
| [[7/4]]
|-
| 16
| 1020.0
| [[9/5]]
|-
| 17
| 1084.8
| [[15/8]]
|-
| 18
| 1148.7
| [[27/14]], [[35/18]]
|-
| 19
| 1211.5
| [[2/1]]
|-
| 20
| 1276.3
| [[21/10]], [[25/12]], [[27/13]]
|-
| 21
| 1340.1
| [[13/6]]
|-
| 22
| 1403.9
| [[9/4]]
|}

== Music ==
; [[Francium]]
* "McGarfyGarf" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/2iaicUkq6EcjcGM8RioFZj Spotify] | [https://francium223.bandcamp.com/track/mcgarfygarf Bandcamp] | [https://www.youtube.com/watch?v=sI8X6PNOiXE YouTube]

== See also ==
* [[19edo]] – relative edo
* [[30edt]] – relative edt
* [[49ed6]] – relative ed6
* [[53ed7]] – relative ed7
* [[68ed12]] – relative ed12
* [[93ed30]] – relative ed30</text>
      <sha1>enohkaxxm3jw0n21bwlbfomhpgcudsx</sha1>
    </revision>
    <revision>
      <id>214963</id>
      <parentid>198283</parentid>
      <timestamp>2025-10-28T01:47:25Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <origin>214963</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2440" sha1="sbxkq3iv7lwzm0sivf7txymb39rvv2l" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
11edf corresponds to 18.8046…[[edo]]. It is similar to [[19edo]], and nearly identical to [[Carlos Beta]]. Unlike 19edo, which is [[consistent]] to the [[integer limit|10-integer-limit]], 11edf is only consistent to the 7-integer-limit. 

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo. At 510.51{{c}}, it is 12.47{{c}} sharper than just and 3.7{{c}} flat of that of [[7edo]].

11edf represents the upper bound of the [[phoenix]] tuning range. It benefits from all the desirable properties of phoenix tuning systems.

=== Harmonics ===
{{Harmonics in equal|11|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edf (continued)}}

=== Subsets and supersets ===
11edf is the fifth [[prime equal division|prime edf]], past [[7edf]] and before [[13edf]]. It does not contain any nontrivial subset edfs.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! Approximate ratios
|-
| 0
| 0.0
| [[1/1]]
|-
| 1
| 63.8
| [[21/20]], [[25/24]], [[27/26]], [[28/27]]
|-
| 2
| 127.6
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
|-
| 3
| 191.4
| [[9/8]], [[10/9]]
|-
| 4
| 255.3
| [[7/6]], ''[[8/7]]''
|-
| 5
| 319.1
| [[6/5]]
|-
| 6
| 382.9
| [[5/4]]
|-
| 7
| 446.7
| [[9/7]]
|-
| 8
| 510.5
| [[4/3]]
|-
| 9
| 574.3
| [[7/5]]
|-
| 10
| 638.1
| [[13/9]]
|-
| 11
| 702.0
| [[3/2]]
|-
| 12
| 765.8
| [[14/9]]
|-
| 13
| 828.6
| [[8/5]], [[13/8]], [[21/13]]
|-
| 14
| 893.4
| [[5/3]]
|-
| 15
| 956.2
| [[7/4]]
|-
| 16
| 1020.0
| [[9/5]]
|-
| 17
| 1084.8
| [[15/8]]
|-
| 18
| 1148.7
| [[27/14]], [[35/18]]
|-
| 19
| 1211.5
| [[2/1]]
|-
| 20
| 1276.3
| [[21/10]], [[25/12]], [[27/13]]
|-
| 21
| 1340.1
| [[13/6]]
|-
| 22
| 1403.9
| [[9/4]]
|}

== Music ==
; [[Francium]]
* "McGarfyGarf" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/2iaicUkq6EcjcGM8RioFZj Spotify] | [https://francium223.bandcamp.com/track/mcgarfygarf Bandcamp] | [https://www.youtube.com/watch?v=sI8X6PNOiXE YouTube]

== See also ==
* [[19edo]] – relative edo
* [[30edt]] – relative edt
* [[49ed6]] – relative ed6
* [[53ed7]] – relative ed7
* [[68ed12]] – relative ed12
* [[93ed30]] – relative ed30

* [https://en.xen.wiki/w/Equal-step_tuning#Alpha-beta-gamma_family_of_equal_divisions Alpha-beta-gamma_family_of_equal_divisions]</text>
      <sha1>sbxkq3iv7lwzm0sivf7txymb39rvv2l</sha1>
    </revision>
    <revision>
      <id>224520</id>
      <parentid>214963</parentid>
      <timestamp>2026-02-22T07:10:08Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>224520</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2364" sha1="t9tet4c2p9kcljxj0cvtkm7eh5u8pp8" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
11edf corresponds to 18.8046…[[edo]]. It is similar to [[19edo]], and nearly identical to [[Carlos Beta]]. Unlike 19edo, which is [[consistent]] to the [[integer limit|10-integer-limit]], 11edf is only consistent to the 7-integer-limit. 

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo. At 510.51{{c}}, it is 12.47{{c}} sharper than just and 3.7{{c}} flat of that of [[7edo]].

11edf represents the upper bound of the [[phoenix]] tuning range. It benefits from all the desirable properties of phoenix tuning systems.

=== Harmonics ===
{{Harmonics in equal|11|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edf (continued)}}

=== Subsets and supersets ===
11edf is the fifth [[prime equal division|prime edf]], past [[7edf]] and before [[13edf]]. It does not contain any nontrivial subset edfs.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! Approximate ratios
|-
| 0
| 0.0
| [[1/1]]
|-
| 1
| 63.8
| [[21/20]], [[25/24]], [[27/26]], [[28/27]]
|-
| 2
| 127.6
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
|-
| 3
| 191.4
| [[9/8]], [[10/9]]
|-
| 4
| 255.3
| [[7/6]], ''[[8/7]]''
|-
| 5
| 319.1
| [[6/5]]
|-
| 6
| 382.9
| [[5/4]]
|-
| 7
| 446.7
| [[9/7]]
|-
| 8
| 510.5
| [[4/3]]
|-
| 9
| 574.3
| [[7/5]]
|-
| 10
| 638.1
| [[13/9]]
|-
| 11
| 702.0
| [[3/2]]
|-
| 12
| 765.8
| [[14/9]]
|-
| 13
| 828.6
| [[8/5]], [[13/8]], [[21/13]]
|-
| 14
| 893.4
| [[5/3]]
|-
| 15
| 956.2
| [[7/4]]
|-
| 16
| 1020.0
| [[9/5]]
|-
| 17
| 1084.8
| [[15/8]]
|-
| 18
| 1148.7
| [[27/14]], [[35/18]]
|-
| 19
| 1211.5
| [[2/1]]
|-
| 20
| 1276.3
| [[21/10]], [[25/12]], [[27/13]]
|-
| 21
| 1340.1
| [[13/6]]
|-
| 22
| 1403.9
| [[9/4]]
|}

== Music ==
; [[Francium]]
* "McGarfyGarf" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/2iaicUkq6EcjcGM8RioFZj Spotify] | [https://francium223.bandcamp.com/track/mcgarfygarf Bandcamp] | [https://www.youtube.com/watch?v=sI8X6PNOiXE YouTube]

== See also ==
* [[19edo]] – relative edo
* [[30edt]] – relative edt
* [[49ed6]] – relative ed6
* [[53ed7]] – relative ed7
* [[68ed12]] – relative ed12
* [[93ed30]] – relative ed30
* [[Alpha, beta, and gamma family of equal divisions]]</text>
      <sha1>t9tet4c2p9kcljxj0cvtkm7eh5u8pp8</sha1>
    </revision>
    <revision>
      <id>233386</id>
      <parentid>224520</parentid>
      <timestamp>2026-07-08T06:05:17Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Music */ Add Bryan Deister's ''9ed(3/2) + 10ed(3/2) + 11ed(3/2) locked at 6:8:11 improv'' (2026)</comment>
      <origin>233386</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2581" sha1="qxp16m6m49go6s9re74slmeapbcsah6" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
11edf corresponds to 18.8046…[[edo]]. It is similar to [[19edo]], and nearly identical to [[Carlos Beta]]. Unlike 19edo, which is [[consistent]] to the [[integer limit|10-integer-limit]], 11edf is only consistent to the 7-integer-limit. 

While the fifth is just, the fourth is very sharp and significantly less accurate than in 19edo. At 510.51{{c}}, it is 12.47{{c}} sharper than just and 3.7{{c}} flat of that of [[7edo]].

11edf represents the upper bound of the [[phoenix]] tuning range. It benefits from all the desirable properties of phoenix tuning systems.

=== Harmonics ===
{{Harmonics in equal|11|3|2|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edf (continued)}}

=== Subsets and supersets ===
11edf is the fifth [[prime equal division|prime edf]], past [[7edf]] and before [[13edf]]. It does not contain any nontrivial subset edfs.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents
! Approximate ratios
|-
| 0
| 0.0
| [[1/1]]
|-
| 1
| 63.8
| [[21/20]], [[25/24]], [[27/26]], [[28/27]]
|-
| 2
| 127.6
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
|-
| 3
| 191.4
| [[9/8]], [[10/9]]
|-
| 4
| 255.3
| [[7/6]], ''[[8/7]]''
|-
| 5
| 319.1
| [[6/5]]
|-
| 6
| 382.9
| [[5/4]]
|-
| 7
| 446.7
| [[9/7]]
|-
| 8
| 510.5
| [[4/3]]
|-
| 9
| 574.3
| [[7/5]]
|-
| 10
| 638.1
| [[13/9]]
|-
| 11
| 702.0
| [[3/2]]
|-
| 12
| 765.8
| [[14/9]]
|-
| 13
| 828.6
| [[8/5]], [[13/8]], [[21/13]]
|-
| 14
| 893.4
| [[5/3]]
|-
| 15
| 956.2
| [[7/4]]
|-
| 16
| 1020.0
| [[9/5]]
|-
| 17
| 1084.8
| [[15/8]]
|-
| 18
| 1148.7
| [[27/14]], [[35/18]]
|-
| 19
| 1211.5
| [[2/1]]
|-
| 20
| 1276.3
| [[21/10]], [[25/12]], [[27/13]]
|-
| 21
| 1340.1
| [[13/6]]
|-
| 22
| 1403.9
| [[9/4]]
|}

== Music ==
=== 21st century ===
; [[Francium]]
* "McGarfyGarf" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/2iaicUkq6EcjcGM8RioFZj Spotify] | [https://francium223.bandcamp.com/track/mcgarfygarf Bandcamp] | [https://www.youtube.com/watch?v=sI8X6PNOiXE YouTube]

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/U-XewMZiBC0 ''9ed(3/2) + 10ed(3/2) + 11ed(3/2) locked at 6:8:11 improv''] (2026, polymicrotonal, playing in 3 tuning systems simultaneously)

== See also ==
* [[19edo]] – relative edo
* [[30edt]] – relative edt
* [[49ed6]] – relative ed6
* [[53ed7]] – relative ed7
* [[68ed12]] – relative ed12
* [[93ed30]] – relative ed30
* [[Alpha, beta, and gamma family of equal divisions]]</text>
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  </page>
  <page>
    <title>11edo modes</title>
    <ns>0</ns>
    <id>88</id>
    <revision>
      <id>88</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
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        <id>0</id>
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      <text bytes="1042" sha1="3fg07xx9tivm9k8h1idtc8edi23e8rq" xml:space="preserve">Some modes of [[11edo|11edo]]. Available, of course, in [[22edo|22edo]], [[33edo|33edo]], [[44edo|44edo]], etc. Add yr very favorite ones here!

==11edo [[MOSNamingScheme|mish]] (MOS 4L+3s, generator: 3/11)==

{| class="wikitable"
|-
| | 2 1 2 1 2 1 2
|-
| | 1 2 1 2 1 2 2
|-
| | 2 1 2 1 2 2 1
|-
| | 1 2 1 2 2 1 2
|-
| | 2 1 2 2 1 2 1
|-
| | 1 2 2 1 2 1 2
|-
| | 2 2 1 2 1 2 1
|}

==[[MOSNamingScheme|father]] pentatonic (MOS 2L+3s, generator: 5/11)==

{| class="wikitable"
|-
| | 1 4 1 4 1
|-
| | 4 1 4 1 1
|-
| | 1 4 1 1 4
|-
| | 4 1 1 4 1
|-
| | 1 1 4 1 4
|}

==[[MOSNamingScheme|mavila]] heptatonic (MOS 2L+5s, generator: 5/11)==

{| class="wikitable"
|-
| | 1 1 1 3 1 1 3
|-
| | 1 1 3 1 1 3 1
|-
| | 1 3 1 1 3 1 1
|-
| | 3 1 1 3 1 1 1
|-
| | 1 1 3 1 1 1 3
|-
| | 1 3 1 1 1 3 1
|-
| | 3 1 1 1 3 1 1
|}

==Swooning Rushes ([[MOS_Cradle|MOS Cradle]])==

{| class="wikitable"
|-
| | 2 3 1 3 2
|-
| | 3 1 3 2 2
|-
| | 1 3 2 2 3
|-
| | 3 2 2 3 1
|-
| | 2 2 3 1 3
|}
[[Category:11edo]]
[[Category:catalog]]
[[Category:edo]]
[[Category:theory]]</text>
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    <revision>
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      <timestamp>2009-08-15T02:41:12Z</timestamp>
      <contributor>
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        <id>0</id>
      </contributor>
      <comment>**Imported revision 84345741 - Original comment: **</comment>
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      <text bytes="5952" sha1="qhh4u1yzmnh2anww582xi5vpu5f8rce" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2009-08-15 02:41:12 UTC&lt;/tt&gt;.&lt;br&gt;
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==11edo [[MOSNamingScheme|mish]] (MOS 4L+3s, generator: 3/11)== 

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|| 1 2 1 2 1 2 2 ||
|| 2 1 2 1 2 2 1 ||
|| 1 2 1 2 2 1 2 ||
|| 2 1 2 2 1 2 1 ||
|| 1 2 2 1 2 1 2 ||
|| 2 2 1 2 1 2 1 ||

==[[MOSNamingScheme|father]] pentatonic (MOS 2L+3s, generator: 5/11)== 

|| 1 4 1 4 1 ||
|| 4 1 4 1 1 ||
|| 1 4 1 1 4 ||
|| 4 1 1 4 1 ||
|| 1 1 4 1 4 ||

==[[MOSNamingScheme|mavila]] heptatonic (MOS 2L+5s, generator: 5/11)== 

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|| 1 3 1 1 3 1 1 ||
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==Swooning Rushes ([[MOS Cradle]])== 

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      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 84345719 - Original comment: **</comment>
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      <text bytes="5685" sha1="683irk1vihenac79ru9bmivkarr21t1" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2009-08-15 02:40:22 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;84345719&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
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==11edo [[MOSNamingScheme|mish]] (MOS 4L+3s, generator: 3/11)== 

|| 2 1 2 1 2 1 2 ||
|| 1 2 1 2 1 2 2 ||
|| 2 1 2 1 2 2 1 ||
|| 1 2 1 2 2 1 2 ||
|| 2 1 2 2 1 2 1 ||
|| 1 2 2 1 2 1 2 ||
|| 2 2 1 2 1 2 1 ||

==[[MOSNamingScheme|father]] pentatonic (MOS 2L+3s, generator: 5/11)== 

|| 1 4 1 4 1 ||
|| 4 1 4 1 1 ||
|| 1 4 1 1 4 ||
|| 4 1 1 4 1 ||
|| 1 1 4 1 4 ||

==[[MOSNamingScheme|mavila]] heptatonic (MOS 2L+5s, generator: 5/11)== 

|| 1 1 1 3 1 1 3 ||
|| 1 1 3 1 1 3 1 ||
|| 1 3 1 1 3 1 1 ||
|| 3 1 1 3 1 1 1 ||
|| 1 1 3 1 1 1 3 ||
|| 1 3 1 1 1 3 1 ||
|| 3 1 1 1 3 1 1 ||
==Swooning Rushes ([[MOS Cradle]])== 

|| 2 3 1 3 2 ||
|| 3 1 3 2 2 ||
|| 1 3 2 2 3 ||
|| 3 2 2 3 1 ||
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      <comment>**Imported revision 84165815 - Original comment: Link to MOSNamingScheme**</comment>
      <origin>4439</origin>
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      <text bytes="3461" sha1="4tnvno3ks293nnrfiysxplxupbdt0kd" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2009-08-13 10:16:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;84165815&lt;/tt&gt;.&lt;br&gt;
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&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;Some modes of [[11edo]]. Add yr very favorite ones here!

==11edo [[MOSNamingScheme|mish]] (MOS 4L+3s)== 

|| 2 1 2 1 2 1 2 ||
|| 1 2 1 2 1 2 2 ||
|| 2 1 2 1 2 2 1 ||
|| 1 2 1 2 2 1 2 ||
|| 2 1 2 2 1 2 1 ||
|| 1 2 2 1 2 1 2 ||
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==Swooning Rushes ([[MOS Cradle]])== 

|| 2 3 1 3 2 ||
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      <timestamp>2009-08-13T10:14:21Z</timestamp>
      <contributor>
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      <comment>**Imported revision 84165567 - Original comment: MOS type 4L+3s is mish**</comment>
      <origin>4440</origin>
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      <text bytes="3379" sha1="5ddak503bb59gui0krgqeoqn1kr7wvw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
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: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2009-08-13 10:14:21 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;84165567&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
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&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;Some modes of [[11edo]]. Add yr very favorite ones here!

==11edo mish (MOS 4L+3s)== 

|| 2 1 2 1 2 1 2 ||
|| 1 2 1 2 1 2 2 ||
|| 2 1 2 1 2 2 1 ||
|| 1 2 1 2 2 1 2 ||
|| 2 1 2 2 1 2 1 ||
|| 1 2 2 1 2 1 2 ||
|| 2 2 1 2 1 2 1 ||

==Swooning Rushes ([[MOS Cradle]])== 

|| 2 3 1 3 2 ||
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      <sha1>5ddak503bb59gui0krgqeoqn1kr7wvw</sha1>
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      <contributor>
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      <comment>**Imported revision 83648023 - Original comment: **</comment>
      <origin>4441</origin>
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      <text bytes="3371" sha1="kprrqcyzrv53oolm1yvwl1cz8kileh8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2009-08-08 15:49:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;83648023&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
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&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;Some modes of [[11edo]]. Add yr very favorite ones here!

==11edo mosh (MOS 4+3)== 

|| 2 1 2 1 2 1 2 ||
|| 1 2 1 2 1 2 2 ||
|| 2 1 2 1 2 2 1 ||
|| 1 2 1 2 2 1 2 ||
|| 2 1 2 2 1 2 1 ||
|| 1 2 2 1 2 1 2 ||
|| 2 2 1 2 1 2 1 ||

==Swooning Rushes ([[MOS Cradle]])== 

|| 2 3 1 3 2 ||
|| 3 1 3 2 2 ||
|| 1 3 2 2 3 ||
|| 3 2 2 3 1 ||
|| 2 2 3 1 3 ||&lt;/pre&gt;&lt;/div&gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Swooning Rushes (MOS Cradle)"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Swooning Rushes (&amp;lt;a class="wiki_link" href="/MOS%20Cradle"&amp;gt;MOS Cradle&amp;lt;/a&amp;gt;)&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2 3 1 3 2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3 1 3 2 2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1 3 2 2 3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3 2 2 3 1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2 2 3 1 3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>kprrqcyzrv53oolm1yvwl1cz8kileh8</sha1>
    </revision>
    <revision>
      <id>37762</id>
      <parentid>88</parentid>
      <timestamp>2018-10-24T21:16:51Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>simplified tables, corrected 2L+3s scales to classic pentatonic</comment>
      <origin>37762</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="978" sha1="jv9wfakb4q51kae93voxm63pdn7s2bv" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== 11edo [[MOSNamingScheme|mish]] (MOS 4L+3s, generator: 3/11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}

== [[MOSNamingScheme|classic]] pentatonic (MOS 2L+3s, generator: 5/11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[MOSNamingScheme|mavila]] heptatonic (MOS 2L+5s, generator: 5/11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>jv9wfakb4q51kae93voxm63pdn7s2bv</sha1>
    </revision>
    <revision>
      <id>37763</id>
      <parentid>37762</parentid>
      <timestamp>2018-10-24T21:17:13Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>Xenwolf moved page [[11edo Modes]] to [[11edo modes]]: changed into [[Sentence case]]</comment>
      <origin>37763</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="978" sha1="jv9wfakb4q51kae93voxm63pdn7s2bv" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== 11edo [[MOSNamingScheme|mish]] (MOS 4L+3s, generator: 3/11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}

== [[MOSNamingScheme|classic]] pentatonic (MOS 2L+3s, generator: 5/11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[MOSNamingScheme|mavila]] heptatonic (MOS 2L+5s, generator: 5/11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>jv9wfakb4q51kae93voxm63pdn7s2bv</sha1>
    </revision>
    <revision>
      <id>46976</id>
      <parentid>37763</parentid>
      <timestamp>2020-06-16T04:03:35Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>temperament names are more helpful (might add color names later)</comment>
      <origin>46976</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3810" sha1="8i46b20apasrivlt37wjdpyhrwtu9sv" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the ssLsssLss mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 109.09
| | C#v
| | 15/14, 16/15, '''17/16''', 18/17
| | -2
|-
| | 3
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -4
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -3
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}
== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>8i46b20apasrivlt37wjdpyhrwtu9sv</sha1>
    </revision>
    <revision>
      <id>46979</id>
      <parentid>46976</parentid>
      <timestamp>2020-06-16T04:29:39Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Joan nonatonic (MOS 2L+7s, generator: 5\11) */ Joan (11&amp;31) actually has a less complex mapping for 9</comment>
      <origin>46979</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3795" sha1="hoizpk6wpnrjazel0sn8fdrijlyxe1v" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the ssLsssLss mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 109.09
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>hoizpk6wpnrjazel0sn8fdrijlyxe1v</sha1>
    </revision>
    <revision>
      <id>46980</id>
      <parentid>46979</parentid>
      <timestamp>2020-06-16T04:30:30Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Joan nonatonic (MOS 2L+7s, generator: 5\11) */</comment>
      <origin>46980</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3795" sha1="1wfxzzpfs95egob9pxxsk08mqwo9a8l" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 109.09
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>1wfxzzpfs95egob9pxxsk08mqwo9a8l</sha1>
    </revision>
    <revision>
      <id>46981</id>
      <parentid>46980</parentid>
      <timestamp>2020-06-16T04:31:45Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Joan nonatonic (MOS 2L+7s, generator: 5\11) */</comment>
      <origin>46981</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3795" sha1="jmtdd4ofyvpqfv9m6x3934gc16or2m2" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode:
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>jmtdd4ofyvpqfv9m6x3934gc16or2m2</sha1>
    </revision>
    <revision>
      <id>47320</id>
      <parentid>46981</parentid>
      <timestamp>2020-06-22T05:44:56Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>47320</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3855" sha1="arpkk6l1rv68ir7yqtseqkro8m4u81e" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>arpkk6l1rv68ir7yqtseqkro8m4u81e</sha1>
    </revision>
    <revision>
      <id>48234</id>
      <parentid>47320</parentid>
      <timestamp>2020-08-19T04:54:05Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Orgone heptatonic (MOS 4L+3s, generator: 3\11) */</comment>
      <origin>48234</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4026" sha1="0kiq7spd2e85e5fdk7hlwts1xwgreq1" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>0kiq7spd2e85e5fdk7hlwts1xwgreq1</sha1>
    </revision>
    <revision>
      <id>50337</id>
      <parentid>48234</parentid>
      <timestamp>2020-09-27T02:25:06Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Orgone heptatonic (MOS 4L+3s, generator: 3\11) */</comment>
      <origin>50337</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4159" sha1="0ex2rgjgxlc6qdbi5zxi4jksx0kdq9i" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


===Modal harmony===
The seven modes are, from brightest to darkest:
*2212121
*2122121
*2121221
*2121212
*1221212
*1212212
*1212122

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>0ex2rgjgxlc6qdbi5zxi4jksx0kdq9i</sha1>
    </revision>
    <revision>
      <id>50338</id>
      <parentid>50337</parentid>
      <timestamp>2020-09-27T02:44:10Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>50338</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4377" sha1="j30fd3n8406fcyknzfcdbtj394h3a42" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


===Modal harmony===
The seven modes are, from brightest to darkest:
*2212121
*2122121
*2121221
*2121212
*1221212
*1212212
*1212122

Modally, the least to most "entropic" scale degrees (i.e. the modes that have this interval as major and minor respectively are a 50-50 split) are in order:
* 2 and 7 (equally)
* 4 and 5 (equally)
* 3 and 6 (equally)

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>j30fd3n8406fcyknzfcdbtj394h3a42</sha1>
    </revision>
    <revision>
      <id>50339</id>
      <parentid>50338</parentid>
      <timestamp>2020-09-27T03:16:12Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>50339</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4411" sha1="esi4fb6ux2q1z7dvh7xktv2jib905q6" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


===Modal harmony===
The seven modes are, from brightest to darkest:
*2212121
*2122121
*2121221
*2121212
*1221212
*1212212
*1212122

Modally, the least to most ''entropic'' scale degrees are (if a scale degree is highly entropic, the modes that have this interval as major and minor respectively are roughly a 50-50 split):
* 2 and 7 (equally)
* 4 and 5 (equally)
* 3 and 6 (equally)

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>esi4fb6ux2q1z7dvh7xktv2jib905q6</sha1>
    </revision>
    <revision>
      <id>50340</id>
      <parentid>50339</parentid>
      <timestamp>2020-09-27T03:47:29Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>50340</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4680" sha1="am4hc3r3ml1llx9dwrb8gelmyh0s8g8" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


===Modal harmony===
The seven modes are, from brightest to darkest:
*2212121
*2122121
*2121221
*2121212
*1221212
*1212212
*1212122

Modally, the least to most ''[https://en.wikipedia.org/wiki/Entropy_(information_theory) entropic]'' scale degrees are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is highly entropic it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split).

This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>am4hc3r3ml1llx9dwrb8gelmyh0s8g8</sha1>
    </revision>
    <revision>
      <id>50341</id>
      <parentid>50340</parentid>
      <timestamp>2020-09-27T03:49:48Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>50341</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4680" sha1="lm6jrepiko8mtprzim9fhx2m52g5bmp" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


===Modal harmony===
The seven modes are, from brightest to darkest:
*2212121
*2122121
*2121221
*2121212
*1221212
*1212212
*1212122

Modally, the most to least ''[https://en.wikipedia.org/wiki/Entropy_(information_theory) entropic]'' scale degrees are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is highly entropic it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split).

This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>lm6jrepiko8mtprzim9fhx2m52g5bmp</sha1>
    </revision>
    <revision>
      <id>50342</id>
      <parentid>50341</parentid>
      <timestamp>2020-09-27T04:04:15Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>50342</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4913" sha1="365e4pomhzc4oqdsxs1wyogponb6nrs" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


===Modal harmony===
The seven modes are, from brightest to darkest:
*2212121
*2122121
*2121221
*2121212
*1221212
*1212212
*1212122

Modally, the most to least ''[https://en.wikipedia.org/wiki/Entropy_(information_theory) entropic]'' scale degrees are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is highly entropic it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative. and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14, 16:17:22:28, and 32:34:41:56.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>365e4pomhzc4oqdsxs1wyogponb6nrs</sha1>
    </revision>
    <revision>
      <id>50343</id>
      <parentid>50342</parentid>
      <timestamp>2020-09-27T04:06:12Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>50343</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4951" sha1="bs5ba8d386bix9y9gf04dkmj1hgo063" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


===Modal harmony===
The seven modes are, from brightest to darkest:
*2212121
*2122121
*2121221
*2121212
*1221212
*1212212
*1212122

Modally, the most to least ''[https://en.wikipedia.org/wiki/Entropy_(information_theory) entropic]'' scale degrees are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is highly entropic it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative. and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14, 16:17:22:28, and (the most "minor" version of 1-2-4-7) 32:34:41:56.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>bs5ba8d386bix9y9gf04dkmj1hgo063</sha1>
    </revision>
    <revision>
      <id>50344</id>
      <parentid>50343</parentid>
      <timestamp>2020-09-27T04:06:41Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>50344</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4964" sha1="4141da3142fm1jgas4chnht0wzhy9k1" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


===Modal harmony===
The seven modes are, from brightest to darkest:
*2212121
*2122121
*2121221
*2121212
*1221212
*1212212
*1212122

Modally, the most to least ''[https://en.wikipedia.org/wiki/Entropy_(information_theory) entropic]'' scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is highly entropic it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative. and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14, 16:17:22:28, and (the most "minor" version of 1-2-4-7) 32:34:41:56.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>4141da3142fm1jgas4chnht0wzhy9k1</sha1>
    </revision>
    <revision>
      <id>50345</id>
      <parentid>50344</parentid>
      <timestamp>2020-09-27T04:11:07Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */ typo</comment>
      <origin>50345</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4964" sha1="dhcg5zmsnya101b8i88lnlrfkbwqx24" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


===Modal harmony===
The seven modes are, from brightest to darkest:
*2212121
*2122121
*2121221
*2121212
*1221212
*1212212
*1212122

Modally, the most to least ''[https://en.wikipedia.org/wiki/Entropy_(information_theory) entropic]'' scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is highly entropic it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14, 16:17:22:28, and (the most "minor" version of 1-2-4-7) 32:34:41:56.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>dhcg5zmsnya101b8i88lnlrfkbwqx24</sha1>
    </revision>
    <revision>
      <id>50346</id>
      <parentid>50345</parentid>
      <timestamp>2020-09-27T05:20:14Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>50346</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4967" sha1="howgh3wz5sayxb2i3huq06zllr8t5fh" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


===Modal harmony===
The seven modes are, from brightest to darkest:
*2212121
*2122121
*2121221
*2121212
*1221212
*1212212
*1212122

Modally, the highest to lowest ''[https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy]'' scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is highly entropic it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14, 16:17:22:28, and (the most "minor" version of 1-2-4-7) 32:34:41:56.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>howgh3wz5sayxb2i3huq06zllr8t5fh</sha1>
    </revision>
    <revision>
      <id>50347</id>
      <parentid>50346</parentid>
      <timestamp>2020-09-27T05:20:41Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>50347</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4963" sha1="qf72cbal7jl8rdlldx9nrmjh1woofro" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


===Modal harmony===
The seven modes are, from brightest to darkest:
*2212121
*2122121
*2121221
*2121212
*1221212
*1212212
*1212122

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is highly entropic it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14, 16:17:22:28, and (the most "minor" version of 1-2-4-7) 32:34:41:56.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>qf72cbal7jl8rdlldx9nrmjh1woofro</sha1>
    </revision>
    <revision>
      <id>50348</id>
      <parentid>50347</parentid>
      <timestamp>2020-09-27T05:40:47Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>50348</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4960" sha1="th8j6qpt7r8jwntdl9rfug0v6rzmzg6" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


===Modal harmony===
The seven modes are, from brightest to darkest:
*2212121
*2122121
*2121221
*2121212
*1221212
*1212212
*1212122

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14, 16:17:22:28, and (the most "minor" version of 1-2-4-7) 32:34:41:56.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>th8j6qpt7r8jwntdl9rfug0v6rzmzg6</sha1>
    </revision>
    <revision>
      <id>50772</id>
      <parentid>50348</parentid>
      <timestamp>2020-10-05T04:32:08Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>50772</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4027" sha1="2hiqu17vtxmao3nxgxeskk3pk0a7n39" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4


== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>2hiqu17vtxmao3nxgxeskk3pk0a7n39</sha1>
    </revision>
    <revision>
      <id>50773</id>
      <parentid>50772</parentid>
      <timestamp>2020-10-05T04:32:46Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Orgone heptatonic (MOS 4L+3s, generator: 3\11) */</comment>
      <origin>50773</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4042" sha1="ol0tyiwzpjd1ik0tk10gmeyd9gm7hci" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Orgone}}

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>ol0tyiwzpjd1ik0tk10gmeyd9gm7hci</sha1>
    </revision>
    <revision>
      <id>50774</id>
      <parentid>50773</parentid>
      <timestamp>2020-10-05T04:34:25Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Orgone heptatonic (MOS 4L+3s, generator: 3\11) */</comment>
      <origin>50774</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5031" sha1="2gjrwo6y6cv34tezfyl45wweho74omr" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Orgone}}

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4

===Modal harmony===
The seven modes are, from brightest to darkest:
*z mode: 2212121
*y mode: 2122121
*x mode: 2121221
*V mode: 2121212
*U mode: 1221212
*T mode: 1212212
*S mode: 1212122

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14, 16:17:22:28, and (the most "minor" version of 1-2-4-7) 32:34:41:56.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>2gjrwo6y6cv34tezfyl45wweho74omr</sha1>
    </revision>
    <revision>
      <id>51186</id>
      <parentid>50774</parentid>
      <timestamp>2020-10-10T23:54:57Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>51186</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5031" sha1="8pv5ca9i64umzulqs2tes267komunae" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Orgone}}

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4

===Modal harmony===
The seven modes are, from brightest to darkest:
*z mode: 2212121
*y mode: 2122121
*x mode: 2121221
*V mode: 2121212
*U mode: 1221212
*T mode: 1212212
*S mode: 1212122

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14, 16:17:22:28, and (the most "minor" version of 1-2-4-7) 14:15:18:32.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>8pv5ca9i64umzulqs2tes267komunae</sha1>
    </revision>
    <revision>
      <id>51187</id>
      <parentid>51186</parentid>
      <timestamp>2020-10-10T23:56:53Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>51187</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5031" sha1="ic8z8klgen124c8ihgw33v21o3a1beo" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Orgone}}

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*17:22:30 or 25:32:44 on degrees 2 and 4

===Modal harmony===
The seven modes are, from brightest to darkest:
*z mode: 2212121
*y mode: 2122121
*x mode: 2121221
*V mode: 2121212
*U mode: 1221212
*T mode: 1212212
*S mode: 1212122

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14, 16:17:22:28, and (the most "minor" version of 1-2-4-7) 28:30:32:49.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>ic8z8klgen124c8ihgw33v21o3a1beo</sha1>
    </revision>
    <revision>
      <id>53791</id>
      <parentid>51187</parentid>
      <timestamp>2020-11-29T18:46:25Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>53791</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5019" sha1="i8ybo3t2sy0xnkguvcsjdjy5667w9z2" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Orgone}}

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*25:32:44 on degrees 2 and 4

===Modal harmony===
The seven modes are, from brightest to darkest:
*z mode: 2212121
*y mode: 2122121
*x mode: 2121221
*V mode: 2121212
*U mode: 1221212
*T mode: 1212212
*S mode: 1212122

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14, 16:17:22:28, and (the most "minor" version of 1-2-4-7) 28:30:32:49.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>i8ybo3t2sy0xnkguvcsjdjy5667w9z2</sha1>
    </revision>
    <revision>
      <id>53792</id>
      <parentid>53791</parentid>
      <timestamp>2020-11-29T18:47:42Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>53792</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5083" sha1="lvbrn704v4aghchpyxk1c9hdx1rn3ts" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Orgone}}

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*25:32:44 on degrees 2 and 4

===Modal harmony===
The seven modes are, from brightest to darkest:
*z mode: 2212121
*y mode: 2122121
*x mode: 2121221
*V mode: 2121212
*U mode: 1221212
*T mode: 1212212
*S mode: 1212122

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14, 16:17:22:28, and (the most "minor" version of 1-2-4-7) 28:30:32:49. Those chords are therefore crucial for orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>lvbrn704v4aghchpyxk1c9hdx1rn3ts</sha1>
    </revision>
    <revision>
      <id>53793</id>
      <parentid>53792</parentid>
      <timestamp>2020-11-29T18:48:50Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>53793</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5031" sha1="5yycsy0in6hw5kxh9fhqtl2nmsncqbw" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Orgone}}

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*25:32:44 on degrees 2 and 4

===Modal harmony===
The seven modes are, from brightest to darkest:
*z mode: 2212121
*y mode: 2122121
*x mode: 2121221
*V mode: 2121212
*U mode: 1221212
*T mode: 1212212
*S mode: 1212122

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>5yycsy0in6hw5kxh9fhqtl2nmsncqbw</sha1>
    </revision>
    <revision>
      <id>54968</id>
      <parentid>53793</parentid>
      <timestamp>2020-12-04T23:13:06Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54968</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5057" sha1="5ux5sff2myhnpqyfu3as7n7zs1zr9wt" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Orgone}}

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*25:32:44 on degrees 2 and 4

===Modal harmony===
The seven modes are, from brightest to darkest:
*z mode: 2212121
*y mode: 2122121
*x mode: 2121221
*V mode: 2121212
*U mode: 1221212
*T mode: 1212212
*S mode: 1212122

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Equal divisions of the octave]]
[[Category:Theory]]</text>
      <sha1>5ux5sff2myhnpqyfu3as7n7zs1zr9wt</sha1>
    </revision>
    <revision>
      <id>55448</id>
      <parentid>54968</parentid>
      <timestamp>2020-12-05T10:48:56Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>+cat</comment>
      <origin>55448</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5077" sha1="q7mkv6et9zfskmteyd7umutu43ltksa" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Orgone}}

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*25:32:44 on degrees 2 and 4

===Modal harmony===
The seven modes are, from brightest to darkest:
*z mode: 2212121
*y mode: 2122121
*x mode: 2121221
*V mode: 2121212
*U mode: 1221212
*T mode: 1212212
*S mode: 1212122

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:11edo]]
[[Category:Catalog]]
[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Scales]]</text>
      <sha1>q7mkv6et9zfskmteyd7umutu43ltksa</sha1>
    </revision>
    <revision>
      <id>55449</id>
      <parentid>55448</parentid>
      <timestamp>2020-12-05T10:52:05Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cats</comment>
      <origin>55449</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5053" sha1="mgzjs2hram9rg7q4dnibl4j4d4hkgc7" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | '''11/8''', 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Orgone}}

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*25:32:44 on degrees 2 and 4

===Modal harmony===
The seven modes are, from brightest to darkest:
*z mode: 2212121
*y mode: 2122121
*x mode: 2121221
*V mode: 2121212
*U mode: 1221212
*T mode: 1212212
*S mode: 1212122

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | '''1/1'''
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, '''9/8''', 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | '''11/8''', 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | '''7/4''', 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | '''15/8''', 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:Theory]]
[[Category:Catalog]]
[[Category:11edo]]
[[Category:Scales]]
[[Category:Modes]]</text>
      <sha1>mgzjs2hram9rg7q4dnibl4j4d4hkgc7</sha1>
    </revision>
    <revision>
      <id>56046</id>
      <parentid>55449</parentid>
      <timestamp>2020-12-08T06:04:30Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>over-2 intervals shouldn't be given special treatment</comment>
      <origin>56046</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4975" sha1="bu3ky11kduva6dvqiu2io4c1pe8p4kt" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | 11/8, 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Orgone}}

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*25:32:44 on degrees 2 and 4

===Modal harmony===
The seven modes are, from brightest to darkest:
*z mode: 2212121
*y mode: 2122121
*x mode: 2121221
*V mode: 2121212
*U mode: 1221212
*T mode: 1212212
*S mode: 1212122

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | 15/8, 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:Theory]]
[[Category:Catalog]]
[[Category:11edo]]
[[Category:Scales]]
[[Category:Modes]]</text>
      <sha1>bu3ky11kduva6dvqiu2io4c1pe8p4kt</sha1>
    </revision>
    <revision>
      <id>65734</id>
      <parentid>56046</parentid>
      <timestamp>2021-04-05T20:52:08Z</timestamp>
      <contributor>
        <username>Ayceman</username>
        <id>284</id>
      </contributor>
      <comment>/* Orgone heptatonic (MOS 4L+3s, generator: 3\11) */</comment>
      <origin>65734</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5283" sha1="ptgve67zad4rs2j3dp8abuwgxx6c85b" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | 11/8, 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Orgone}}

[[File:11EDO_cheat_sheet.png|400px|thumb|right|11EDO Orgone smitonic cheat sheet using modified diatonic notation (G#=A, Ab=G), and smitonic interval names]]

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*44:56:77 (approx. 25:32:44) on degrees 2 and 4

===Modal harmony===
{{main|Smitonic}}
The seven modes are, from brightest to darkest:
{| class="wikitable"
|-
| Nerevarine mode
| 2212121
|-
| Vivecan mode
| 2122121
|-
| Lorkhanic mode
| 2121221
|-
| Sothic mode
| 2121212
|-
| Kagrenacan mode
| 1221212
|-
| Almalexian mode
| 1212212
|-
| Dagothic mode
| 1212122
|}

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | 15/8, 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:Theory]]
[[Category:Catalog]]
[[Category:11edo]]
[[Category:Scales]]
[[Category:Modes]]</text>
      <sha1>ptgve67zad4rs2j3dp8abuwgxx6c85b</sha1>
    </revision>
    <revision>
      <id>65735</id>
      <parentid>65734</parentid>
      <timestamp>2021-04-05T21:20:14Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Orgone heptatonic (MOS 4L+3s, generator: 3\11) */</comment>
      <origin>65735</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5285" sha1="4qumm7pekwdlwi16gov0g7bz3lv7crl" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | 11/8, 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Smitonic}}

[[File:11EDO_cheat_sheet.png|400px|thumb|right|11EDO Orgone smitonic cheat sheet using modified diatonic notation (G#=A, Ab=G), and smitonic interval names]]

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*44:56:77 (approx. 25:32:44) on degrees 2 and 4

===Modal harmony===
{{main|Smitonic}}
The seven modes are, from brightest to darkest:
{| class="wikitable"
|-
| Nerevarine mode
| 2212121
|-
| Vivecan mode
| 2122121
|-
| Lorkhanic mode
| 2121221
|-
| Sothic mode
| 2121212
|-
| Kagrenacan mode
| 1221212
|-
| Almalexian mode
| 1212212
|-
| Dagothic mode
| 1212122
|}

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
(if a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split). This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately  these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | 15/8, 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:Theory]]
[[Category:Catalog]]
[[Category:11edo]]
[[Category:Scales]]
[[Category:Modes]]</text>
      <sha1>4qumm7pekwdlwi16gov0g7bz3lv7crl</sha1>
    </revision>
    <revision>
      <id>65801</id>
      <parentid>65735</parentid>
      <timestamp>2021-04-07T01:49:39Z</timestamp>
      <contributor>
        <username>Ayceman</username>
        <id>284</id>
      </contributor>
      <minor/>
      <comment>/* Modal harmony */</comment>
      <origin>65801</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5283" sha1="77xn8g4u92gha2mn1rxbakcxxho7dug" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | 11/8, 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Smitonic}}

[[File:11EDO_cheat_sheet.png|400px|thumb|right|11EDO Orgone smitonic cheat sheet using modified diatonic notation (G#=A, Ab=G), and smitonic interval names]]

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*44:56:77 (approx. 25:32:44) on degrees 2 and 4

===Modal harmony===
{{main|Smitonic}}
The seven modes are, from brightest to darkest:
{| class="wikitable"
|-
| Nerevarine mode
| 2212121
|-
| Vivecan mode
| 2122121
|-
| Lorkhanic mode
| 2121221
|-
| Sothic mode
| 2121212
|-
| Kagrenacan mode
| 1221212
|-
| Almalexian mode
| 1212212
|-
| Dagothic mode
| 1212122
|}

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
If a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split. This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately, these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for Orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | 15/8, 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:Theory]]
[[Category:Catalog]]
[[Category:11edo]]
[[Category:Scales]]
[[Category:Modes]]</text>
      <sha1>77xn8g4u92gha2mn1rxbakcxxho7dug</sha1>
    </revision>
    <revision>
      <id>66955</id>
      <parentid>65801</parentid>
      <timestamp>2021-04-14T01:21:08Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>66955</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5297" sha1="pswhvx8o6dimdlq99nw2eqlc82zlv6v" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | 11/8, 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Smitonic}}

[[File:11EDO_cheat_sheet.png|400px|thumb|right|11EDO Orgone smitonic cheat sheet using modified diatonic notation (G#=A, Ab=G), and smitonic interval names]]

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*44:56:77 (approx. 25:32:44) on degrees 2 and 4

===Modal harmony===
{{main|Smitonic}}
The seven modes are, from brightest to darkest:
{| class="wikitable"
|-
| Nerevarine mode
| 2212121
|-
| Vivecan mode
| 2122121
|-
| Lorkhanic mode
| 2121221
|-
| Sothic mode
| 2121212
|-
| Kagrenacan mode
| 1221212
|-
| Almalexian mode
| 1212212
|-
| Dagothic mode
| 1212122
|}

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
If a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split. This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately, these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for Orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11, pathological) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | 15/8, 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:Theory]]
[[Category:Catalog]]
[[Category:11edo]]
[[Category:Scales]]
[[Category:Modes]]</text>
      <sha1>pswhvx8o6dimdlq99nw2eqlc82zlv6v</sha1>
    </revision>
    <revision>
      <id>87064</id>
      <parentid>66955</parentid>
      <timestamp>2022-02-25T14:54:41Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87064</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5277" sha1="0h4j8z4y4hc1nxp56tgpjicymzaw7kr" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | 11/8, 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Smitonic}}

[[File:11EDO_cheat_sheet.png|400px|thumb|right|11EDO Orgone smitonic cheat sheet using modified diatonic notation (G#=A, Ab=G), and smitonic interval names]]

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*44:56:77 (approx. 25:32:44) on degrees 2 and 4

===Modal harmony===
{{main|Smitonic}}
The seven modes are, from brightest to darkest:
{| class="wikitable"
|-
| Nerevarine mode
| 2212121
|-
| Vivecan mode
| 2122121
|-
| Lorkhanic mode
| 2121221
|-
| Sothic mode
| 2121212
|-
| Kagrenacan mode
| 1221212
|-
| Almalexian mode
| 1212212
|-
| Dagothic mode
| 1212122
|}

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
If a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split. This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately, these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for Orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11, pathological) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | 15/8, 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:Theory]]
[[Category:Catalog]]
[[Category:11edo]]
[[Category:Modes]]</text>
      <sha1>0h4j8z4y4hc1nxp56tgpjicymzaw7kr</sha1>
    </revision>
    <revision>
      <id>87438</id>
      <parentid>87064</parentid>
      <timestamp>2022-02-26T16:48:49Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87438</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5257" sha1="ma1wsu6ba91kbe31hgyfuka6l3mr4hj" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | 11/8, 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Smitonic}}

[[File:11EDO_cheat_sheet.png|400px|thumb|right|11EDO Orgone smitonic cheat sheet using modified diatonic notation (G#=A, Ab=G), and smitonic interval names]]

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*44:56:77 (approx. 25:32:44) on degrees 2 and 4

===Modal harmony===
{{main|Smitonic}}
The seven modes are, from brightest to darkest:
{| class="wikitable"
|-
| Nerevarine mode
| 2212121
|-
| Vivecan mode
| 2122121
|-
| Lorkhanic mode
| 2121221
|-
| Sothic mode
| 2121212
|-
| Kagrenacan mode
| 1221212
|-
| Almalexian mode
| 1212212
|-
| Dagothic mode
| 1212122
|}

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
If a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split. This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately, these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for Orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11, pathological) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | 15/8, 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:Catalog]]
[[Category:11edo]]
[[Category:Modes]]</text>
      <sha1>ma1wsu6ba91kbe31hgyfuka6l3mr4hj</sha1>
    </revision>
    <revision>
      <id>87710</id>
      <parentid>87438</parentid>
      <timestamp>2022-02-28T00:53:51Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Catalog]] to [[Category:Lists of scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87710</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5265" sha1="f1pnux1119e5mdmpd11m62riu9fy9to" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | 11/8, 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Smitonic}}

[[File:11EDO_cheat_sheet.png|400px|thumb|right|11EDO Orgone smitonic cheat sheet using modified diatonic notation (G#=A, Ab=G), and smitonic interval names]]

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*44:56:77 (approx. 25:32:44) on degrees 2 and 4

===Modal harmony===
{{main|Smitonic}}
The seven modes are, from brightest to darkest:
{| class="wikitable"
|-
| Nerevarine mode
| 2212121
|-
| Vivecan mode
| 2122121
|-
| Lorkhanic mode
| 2121221
|-
| Sothic mode
| 2121212
|-
| Kagrenacan mode
| 1221212
|-
| Almalexian mode
| 1212212
|-
| Dagothic mode
| 1212122
|}

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
If a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split. This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately, these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for Orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11, pathological) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | 15/8, 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

[[Category:Lists of scales]]
[[Category:11edo]]
[[Category:Modes]]</text>
      <sha1>f1pnux1119e5mdmpd11m62riu9fy9to</sha1>
    </revision>
    <revision>
      <id>168194</id>
      <parentid>87710</parentid>
      <timestamp>2024-12-02T04:49:54Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>{{Scale gallery}}</comment>
      <origin>168194</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5284" sha1="ivpk5zued8bir4uh1a6n18xzbaghi5l" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L+1s, generator: 2\11) ==

{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +1
|-
| | 3
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +2
|-
| | 4
| style="text-align:right;" | 545.45
| | Gb
| | 11/8, 15/11
| | -3
|-
| | 5
| style="text-align:right;" | 763.63
| | Ab
| | 11/7, 14/9, 17/11
| | -2
|-
| | 6
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Smitonic}}

[[File:11EDO_cheat_sheet.png|400px|thumb|right|11EDO Orgone smitonic cheat sheet using modified diatonic notation (G#=A, Ab=G), and smitonic interval names]]

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}


The intervals of the symmetric LsLsLsL mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree (1 = tonic)
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | -3
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +1
|-
| | 4
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | -2
|-
| | 5
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | +2
|-
| | 6
| style="text-align:right;" | 872.72
| | Av
| | 5/3, 18/11, 28/17
| | -1
|-
| | 7
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
*8:11:14 on degrees 1 and 6
*8:11:15 on degrees 3, 5, and 7
*44:56:77 (approx. 25:32:44) on degrees 2 and 4

===Modal harmony===
{{main|Smitonic}}
The seven modes are, from brightest to darkest:
{| class="wikitable"
|-
| Nerevarine mode
| 2212121
|-
| Vivecan mode
| 2122121
|-
| Lorkhanic mode
| 2121221
|-
| Sothic mode
| 2121212
|-
| Kagrenacan mode
| 1221212
|-
| Almalexian mode
| 1212212
|-
| Dagothic mode
| 1212122
|}

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
If a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split. This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately, these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for Orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11, pathological) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align:right" | Degree
! | Cents
! | Note name on C (22edo notation)
! | Approximate ratios
! | #Gens up
|-
| | 1
| style="text-align:right;" | 0
| | C
| | 1/1
| | 0
|-
| | 2
| style="text-align:right;" | 218.18
| | D
| | 8/7, 9/8, 17/15
| | +7
|-
| | 3
| style="text-align:right;" | 327.27
| | Eb^
| | 6/5, 11/9, 17/14
| | +5
|-
| | 4
| style="text-align:right;" | 436.36
| | E
| | 9/7, 14/11, 22/17
| | +3
|-
| | 5
| style="text-align:right;" | 545.45
| | F^
| | 11/8, 15/11
| | +1
|-
| | 6
| style="text-align:right;" | 654.54
| | Gv
| | 16/11, 22/15
| | -1
|-
| | 7
| style="text-align:right;" | 872.73
| | Av
| | 5/3, 18/11, 28/17
| | +6
|-
| | 8
| style="text-align:right;" | 981.81
| | Bb
| | 7/4, 16/9, 30/17
| | +4
|-
| | 9
| style="text-align:right;" | 1090.90
| | Bv
| | 15/8, 17/9, 28/15, 32/17 	
| | +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==

{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

{{Scale gallery}}

[[Category:Lists of scales]]
[[Category:11edo]]
[[Category:Modes]]</text>
      <sha1>ivpk5zued8bir4uh1a6n18xzbaghi5l</sha1>
    </revision>
    <revision>
      <id>180341</id>
      <parentid>168194</parentid>
      <timestamp>2025-02-11T19:43:53Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Formatting + mass-replace text</comment>
      <origin>180341</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5145" sha1="2v4k9cuhs138lfu089uusnjzhd9xu8x" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L&amp;nbsp;1s, generator: 2\11) ==
{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):

{| class="wikitable sortable sorted"
|-
! style="text-align: right;" | Degree
! Cents
! Note name on C (22edo notation)
! Approximate ratios
! #Gens up
|-
| 1
| style="text-align: right;" | 0
| C
| 1/1
| 0
|-
| 2
| style="text-align: right;" | 218.18
| D
| 8/7, 9/8, 17/15
|  +1
|-
| 3
| style="text-align: right;" | 436.36
| E
| 9/7, 14/11, 22/17
|  +2
|-
| 4
| style="text-align: right;" | 545.45
| Gb
| 11/8, 15/11
|  -3
|-
| 5
| style="text-align: right;" | 763.63
| Ab
| 11/7, 14/9, 17/11
|  -2
|-
| 6
| style="text-align: right;" | 981.81
| Bb
| 7/4, 16/9, 30/17
|  -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Smitonic}}

[[File:11EDO_cheat_sheet.png|400px|thumb|right|11EDO Orgone smitonic cheat sheet using modified diatonic notation (G#=A, Ab=G), and smitonic interval names]]

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}

The intervals of the symmetric LsLsLsL mode (harmonics in bold):

{| class="wikitable sortable sorted"
|-
! style="text-align: right;" | Degree (1 = tonic)
! Cents
! Note name on C (22edo notation)
! Approximate ratios
! #Gens up
|-
| 1
| style="text-align: right;" | 0
| C
| 1/1
| 0
|-
| 2
| style="text-align: right;" | 218.18
| D
| 8/7, 9/8, 17/15
|  -3
|-
| 3
| style="text-align: right;" | 327.27
| Eb^
| 6/5, 11/9, 17/14
|  +1
|-
| 4
| style="text-align: right;" | 545.45
| F^
| 11/8, 15/11
|  -2
|-
| 5
| style="text-align: right;" | 654.54
| Gv
| 16/11, 22/15
|  +2
|-
| 6
| style="text-align: right;" | 872.72
| Av
| 5/3, 18/11, 28/17
|  -1
|-
| 7
| style="text-align: right;" | 981.81
| Bb
| 7/4, 16/9, 30/17
|  +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
* 8:11:14 on degrees 1 and 6
* 8:11:15 on degrees 3, 5, and 7
* 44:56:77 (approx. 25:32:44) on degrees 2 and 4

=== Modal harmony ===
{{main|Smitonic}}
The seven modes are, from brightest to darkest:
{| class="wikitable"
|-
| Nerevarine mode
| 2212121
|-
| Vivecan mode
| 2122121
|-
| Lorkhanic mode
| 2121221
|-
| Sothic mode
| 2121212
|-
| Kagrenacan mode
| 1221212
|-
| Almalexian mode
| 1212212
|-
| Dagothic mode
| 1212122
|}

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
If a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split. This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately, these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for Orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==
{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==
{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11, pathological) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align: right;" | Degree
! Cents
! Note name on C (22edo notation)
! Approximate ratios
! #Gens up
|-
| 1
| style="text-align: right;" | 0
| C
| 1/1
| 0
|-
| 2
| style="text-align: right;" | 218.18
| D
| 8/7, 9/8, 17/15
|  +7
|-
| 3
| style="text-align: right;" | 327.27
| Eb^
| 6/5, 11/9, 17/14
|  +5
|-
| 4
| style="text-align: right;" | 436.36
| E
| 9/7, 14/11, 22/17
|  +3
|-
| 5
| style="text-align: right;" | 545.45
| F^
| 11/8, 15/11
|  +1
|-
| 6
| style="text-align: right;" | 654.54
| Gv
| 16/11, 22/15
|  -1
|-
| 7
| style="text-align: right;" | 872.73
| Av
| 5/3, 18/11, 28/17
|  +6
|-
| 8
| style="text-align: right;" | 981.81
| Bb
| 7/4, 16/9, 30/17
|  +4
|-
| 9
| style="text-align: right;" | 1090.90
| Bv
| 15/8, 17/9, 28/15, 32/17 	
|  +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==
{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

{{Navbox scale gallery}}

[[Category:Lists of scales]]
[[Category:11edo]]
[[Category:Modes]]</text>
      <sha1>2v4k9cuhs138lfu089uusnjzhd9xu8x</sha1>
    </revision>
    <revision>
      <id>192348</id>
      <parentid>180341</parentid>
      <timestamp>2025-04-16T09:08:23Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>Undo revision [[Special:Diff/66955|66955]] by [[Special:Contributions/Moremajorthanmajor|Moremajorthanmajor]] ([[User talk:Moremajorthanmajor|talk]])</comment>
      <origin>192348</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5131" sha1="087mybc1ki294sfs3i692pwdi3cjm46" xml:space="preserve">Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L&amp;nbsp;1s, generator: 2\11) ==
{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):

{| class="wikitable sortable sorted"
|-
! style="text-align: right;" | Degree
! Cents
! Note name on C (22edo notation)
! Approximate ratios
! #Gens up
|-
| 1
| style="text-align: right;" | 0
| C
| 1/1
| 0
|-
| 2
| style="text-align: right;" | 218.18
| D
| 8/7, 9/8, 17/15
|  +1
|-
| 3
| style="text-align: right;" | 436.36
| E
| 9/7, 14/11, 22/17
|  +2
|-
| 4
| style="text-align: right;" | 545.45
| Gb
| 11/8, 15/11
|  -3
|-
| 5
| style="text-align: right;" | 763.63
| Ab
| 11/7, 14/9, 17/11
|  -2
|-
| 6
| style="text-align: right;" | 981.81
| Bb
| 7/4, 16/9, 30/17
|  -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Smitonic}}

[[File:11EDO_cheat_sheet.png|400px|thumb|right|11EDO Orgone smitonic cheat sheet using modified diatonic notation (G#=A, Ab=G), and smitonic interval names]]

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}

The intervals of the symmetric LsLsLsL mode (harmonics in bold):

{| class="wikitable sortable sorted"
|-
! style="text-align: right;" | Degree (1 = tonic)
! Cents
! Note name on C (22edo notation)
! Approximate ratios
! #Gens up
|-
| 1
| style="text-align: right;" | 0
| C
| 1/1
| 0
|-
| 2
| style="text-align: right;" | 218.18
| D
| 8/7, 9/8, 17/15
|  -3
|-
| 3
| style="text-align: right;" | 327.27
| Eb^
| 6/5, 11/9, 17/14
|  +1
|-
| 4
| style="text-align: right;" | 545.45
| F^
| 11/8, 15/11
|  -2
|-
| 5
| style="text-align: right;" | 654.54
| Gv
| 16/11, 22/15
|  +2
|-
| 6
| style="text-align: right;" | 872.72
| Av
| 5/3, 18/11, 28/17
|  -1
|-
| 7
| style="text-align: right;" | 981.81
| Bb
| 7/4, 16/9, 30/17
|  +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
* 8:11:14 on degrees 1 and 6
* 8:11:15 on degrees 3, 5, and 7
* 44:56:77 (approx. 25:32:44) on degrees 2 and 4

=== Modal harmony ===
{{main|Smitonic}}
The seven modes are, from brightest to darkest:
{| class="wikitable"
|-
| Nerevarine mode
| 2212121
|-
| Vivecan mode
| 2122121
|-
| Lorkhanic mode
| 2121221
|-
| Sothic mode
| 2121212
|-
| Kagrenacan mode
| 1221212
|-
| Almalexian mode
| 1212212
|-
| Dagothic mode
| 1212122
|}

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
If a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split. This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately, these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for Orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==
{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==
{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align: right;" | Degree
! Cents
! Note name on C (22edo notation)
! Approximate ratios
! #Gens up
|-
| 1
| style="text-align: right;" | 0
| C
| 1/1
| 0
|-
| 2
| style="text-align: right;" | 218.18
| D
| 8/7, 9/8, 17/15
|  +7
|-
| 3
| style="text-align: right;" | 327.27
| Eb^
| 6/5, 11/9, 17/14
|  +5
|-
| 4
| style="text-align: right;" | 436.36
| E
| 9/7, 14/11, 22/17
|  +3
|-
| 5
| style="text-align: right;" | 545.45
| F^
| 11/8, 15/11
|  +1
|-
| 6
| style="text-align: right;" | 654.54
| Gv
| 16/11, 22/15
|  -1
|-
| 7
| style="text-align: right;" | 872.73
| Av
| 5/3, 18/11, 28/17
|  +6
|-
| 8
| style="text-align: right;" | 981.81
| Bb
| 7/4, 16/9, 30/17
|  +4
|-
| 9
| style="text-align: right;" | 1090.90
| Bv
| 15/8, 17/9, 28/15, 32/17 	
|  +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==
{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

{{Navbox scale gallery}}

[[Category:Lists of scales]]
[[Category:11edo]]
[[Category:Modes]]</text>
      <sha1>087mybc1ki294sfs3i692pwdi3cjm46</sha1>
    </revision>
    <revision>
      <id>234265</id>
      <parentid>192348</parentid>
      <timestamp>2026-07-17T03:27:21Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Interwiki</comment>
      <origin>234265</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5212" sha1="0yckdwxa7k0pbdzdi1yu3gootb7vi42" xml:space="preserve">{{Interwiki
| de = 
| en = 11edo modes
| es = 
| ja = 11平均律のモード
}}
Some '''modes of [[11edo]]'''. Available, of course, in [[22edo]], [[33edo]], [[44edo]], etc. Add your very favourite ones here!

== [[Machine]] hexatonic (MOS 5L&amp;nbsp;1s, generator: 2\11) ==
{| class="wikitable"
|-
| 2 2 2 2 2 1
|-
| 2 2 2 2 1 2
|-
| 2 2 2 1 2 2
|-
| 2 2 1 2 2 2
|-
| 2 1 2 2 2 2
|-
| 1 2 2 2 2 2
|}

The intervals of the LLsLLL mode (harmonics in bold):

{| class="wikitable sortable sorted"
|-
! style="text-align: right;" | Degree
! Cents
! Note name on C (22edo notation)
! Approximate ratios
! #Gens up
|-
| 1
| style="text-align: right;" | 0
| C
| 1/1
| 0
|-
| 2
| style="text-align: right;" | 218.18
| D
| 8/7, 9/8, 17/15
|  +1
|-
| 3
| style="text-align: right;" | 436.36
| E
| 9/7, 14/11, 22/17
|  +2
|-
| 4
| style="text-align: right;" | 545.45
| Gb
| 11/8, 15/11
|  -3
|-
| 5
| style="text-align: right;" | 763.63
| Ab
| 11/7, 14/9, 17/11
|  -2
|-
| 6
| style="text-align: right;" | 981.81
| Bb
| 7/4, 16/9, 30/17
|  -1
|}

== [[Orgone]] heptatonic (MOS 4L+3s, generator: 3\11) ==
{{main|Smitonic}}

[[File:11EDO_cheat_sheet.png|400px|thumb|right|11EDO Orgone smitonic cheat sheet using modified diatonic notation (G#=A, Ab=G), and smitonic interval names]]

{| class="wikitable"
|-
| 2 1 2 1 2 1 2
|-
| 1 2 1 2 1 2 2
|-
| 2 1 2 1 2 2 1
|-
| 1 2 1 2 2 1 2
|-
| 2 1 2 2 1 2 1
|-
| 1 2 2 1 2 1 2
|-
| 2 2 1 2 1 2 1
|}

The intervals of the symmetric LsLsLsL mode (harmonics in bold):

{| class="wikitable sortable sorted"
|-
! style="text-align: right;" | Degree (1 = tonic)
! Cents
! Note name on C (22edo notation)
! Approximate ratios
! #Gens up
|-
| 1
| style="text-align: right;" | 0
| C
| 1/1
| 0
|-
| 2
| style="text-align: right;" | 218.18
| D
| 8/7, 9/8, 17/15
|  -3
|-
| 3
| style="text-align: right;" | 327.27
| Eb^
| 6/5, 11/9, 17/14
|  +1
|-
| 4
| style="text-align: right;" | 545.45
| F^
| 11/8, 15/11
|  -2
|-
| 5
| style="text-align: right;" | 654.54
| Gv
| 16/11, 22/15
|  +2
|-
| 6
| style="text-align: right;" | 872.72
| Av
| 5/3, 18/11, 28/17
|  -1
|-
| 7
| style="text-align: right;" | 981.81
| Bb
| 7/4, 16/9, 30/17
|  +3
|}

Assuming the symmetric 2121212 mode, the 1-4-7 chord is:
* 8:11:14 on degrees 1 and 6
* 8:11:15 on degrees 3, 5, and 7
* 44:56:77 (approx. 25:32:44) on degrees 2 and 4

=== Modal harmony ===
{{main|Smitonic}}
The seven modes are, from brightest to darkest:
{| class="wikitable"
|-
| Nerevarine mode
| 2212121
|-
| Vivecan mode
| 2122121
|-
| Lorkhanic mode
| 2121221
|-
| Sothic mode
| 2121212
|-
| Kagrenacan mode
| 1221212
|-
| Almalexian mode
| 1212212
|-
| Dagothic mode
| 1212122
|}

Modally, the highest to lowest [https://en.wikipedia.org/wiki/Entropy_(information_theory) entropy] scale degrees (unison = 1) are:
* 2 and 7 (equally): .99 bits
* 4 and 5 (equally): .86 bits
* 3 and 6 (equally): .59 bits
If a scale degree is high entropy it tends to be the most informative on average; the modes that have this interval as major and minor respectively are roughly a 50-50 split. This tells us that unlike in the diatonic scale, thirds and sixths are the ''least'' informative in distinguishing orgone modes. On the other hand, seconds, fourths and sevenths are highly informative, and fortunately, these scale degrees appear in the most consonant triads and tetrads of 11edo orgone, namely 8:9:11:15, 8:9:11:14 and 16:17:22:28. Those chords are therefore crucial for Orgone[7] modal harmony.

== [[Joan]] pentatonic (MOS 2L+3s, generator: 5\11) ==
{| class="wikitable"
|-
| 1 4 1 4 1
|-
| 4 1 4 1 1
|-
| 1 4 1 1 4
|-
| 4 1 1 4 1
|-
| 1 1 4 1 4
|}

== [[Joan]] heptatonic (MOS 2L+5s, generator: 5\11) ==
{| class="wikitable"
|-
| 1 1 1 3 1 1 3
|-
| 1 1 3 1 1 3 1
|-
| 1 3 1 1 3 1 1
|-
| 3 1 1 3 1 1 1
|-
| 1 1 3 1 1 1 3
|-
| 1 3 1 1 1 3 1
|-
| 3 1 1 1 3 1 1
|}


== [[Joan]] nonatonic (MOS 2L+7s, generator: 5\11) ==

{| class="wikitable"
|-
| 1 1 1 2 1 1 1 2 1
|-
| 1 1 2 1 1 1 2 1 1
|-
| 1 2 1 1 1 2 1 1 1
|-
| 2 1 1 1 2 1 1 1 1
|-
| 1 1 1 2 1 1 1 1 2
|-
| 1 1 2 1 1 1 1 2 1
|-
| 1 2 1 1 1 1 2 1 1
|-
| 2 1 1 1 1 2 1 1 1
|-
| 1 1 1 1 2 1 1 1 2
|}

The intervals of the LssssLsss mode (harmonics in bold):
{| class="wikitable sortable sorted"
|-
! style="text-align: right;" | Degree
! Cents
! Note name on C (22edo notation)
! Approximate ratios
! #Gens up
|-
| 1
| style="text-align: right;" | 0
| C
| 1/1
| 0
|-
| 2
| style="text-align: right;" | 218.18
| D
| 8/7, 9/8, 17/15
|  +7
|-
| 3
| style="text-align: right;" | 327.27
| Eb^
| 6/5, 11/9, 17/14
|  +5
|-
| 4
| style="text-align: right;" | 436.36
| E
| 9/7, 14/11, 22/17
|  +3
|-
| 5
| style="text-align: right;" | 545.45
| F^
| 11/8, 15/11
|  +1
|-
| 6
| style="text-align: right;" | 654.54
| Gv
| 16/11, 22/15
|  -1
|-
| 7
| style="text-align: right;" | 872.73
| Av
| 5/3, 18/11, 28/17
|  +6
|-
| 8
| style="text-align: right;" | 981.81
| Bb
| 7/4, 16/9, 30/17
|  +4
|-
| 9
| style="text-align: right;" | 1090.90
| Bv
| 15/8, 17/9, 28/15, 32/17 	
|  +2
|}

== Swooning Rushes ([[MOS Cradle]]) ==
{| class="wikitable"
|-
| 2 3 1 3 2
|-
| 3 1 3 2 2
|-
| 1 3 2 2 3
|-
| 3 2 2 3 1
|-
| 2 2 3 1 3
|}

{{Navbox scale gallery}}

[[Category:Lists of scales]]
[[Category:11edo]]
[[Category:Modes]]</text>
      <sha1>0yckdwxa7k0pbdzdi1yu3gootb7vi42</sha1>
    </revision>
  </page>
  <page>
    <title>11edo Zine</title>
    <ns>0</ns>
    <id>89</id>
    <revision>
      <id>89</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
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      <format>text/x-wiki</format>
      <text bytes="902" sha1="gqkkfnndrsud5pt6x8lc4skkdqck0qb" xml:space="preserve">In the Spring of 2011, in the aftermath of the School for Designing a Society's first Microtonal Design Seminar, [[Andrew_Heathwaite|Andrew Heathwaite]] assembled a zine to teach and celebrate the virtues of 11edo. This particular page will be the site of some tuning files related to the zine.

If you have a copy of the zine and have come for this page looking for things to listen to, yay! There's nothing specific to the zine on here yet (I just finished slapping all the paper together, and I am tired, man!), but I recommend going over to [[11edo|11edo]] on this here wiki and taking a listen to the files there under "Compositions".

And if you're interested in some audio traces from the semester, see the blog post for the [http://oddmusicuc.wordpress.com/2011/02/17/11edo-band-on-ors/ Oddmusic Radio Shower on February 16, 2011].

More to come!

~Andrew
[[Category:11edo]]
[[Category:listen]]</text>
      <sha1>gqkkfnndrsud5pt6x8lc4skkdqck0qb</sha1>
    </revision>
    <revision>
      <id>4442</id>
      <timestamp>2011-05-03T15:12:22Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 225205772 - Original comment: **</comment>
      <origin>4442</origin>
      <model>wikitext</model>
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      <text bytes="3180" sha1="spf1y0ax3h9nby1dcgz9mut4551yx3d" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2011-05-03 15:12:22 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;225205772&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In the Spring of 2011, in the aftermath of the School for Designing a Society's first Microtonal Design Seminar, [[Andrew Heathwaite]] assembled a zine to teach and celebrate the virtues of 11edo. This particular page will be the site of some tuning files related to the zine.

If you have a copy of the zine and have come for this page looking for things to listen to, yay! There's nothing specific to the zine on here yet (I just finished slapping all the paper together, and I am tired, man!), but I recommend going over to [[11edo]] on this here wiki and taking a listen to the files there under "Compositions".

And if you're interested in some audio traces from the semester, see the blog post for the [[http://oddmusicuc.wordpress.com/2011/02/17/11edo-band-on-ors/|Oddmusic Radio Shower on February 16, 2011]].

More to come!

~Andrew&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo Zine&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In the Spring of 2011, in the aftermath of the School for Designing a Society's first Microtonal Design Seminar, &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; assembled a zine to teach and celebrate the virtues of 11edo. This particular page will be the site of some tuning files related to the zine.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
If you have a copy of the zine and have come for this page looking for things to listen to, yay! There's nothing specific to the zine on here yet (I just finished slapping all the paper together, and I am tired, man!), but I recommend going over to &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11edo&amp;lt;/a&amp;gt; on this here wiki and taking a listen to the files there under &amp;amp;quot;Compositions&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
And if you're interested in some audio traces from the semester, see the blog post for the &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/2011/02/17/11edo-band-on-ors/" rel="nofollow"&amp;gt;Oddmusic Radio Shower on February 16, 2011&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
More to come!&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
~Andrew&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2011-04-29T23:40:14Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 224245336 - Original comment: **</comment>
      <origin>4443</origin>
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      <text bytes="2717" sha1="pom2syvq0tlut1k8nv83cflxp7xrsdx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-04-29 23:40:14 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;224245336&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In the Spring of 2011, in the aftermath of the School for Designing a Society's first Microtonal Design Seminar, [[Andrew Heathwaite]] assembled a zine to teach and celebrate the virtues of 11edo. This particular page will be the site of some tuning files related to the zine.

If you have a copy of the zine and have come for this page looking for things to listen to, yay! There's nothing specific to the zine on here yet (I just finished slapping all the paper together, and I am tired, man!), but I recommend going over to [[11edo]] on this here wiki and taking a listen to the files there under "Compositions".

More to come!

~Andrew&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo Zine&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In the Spring of 2011, in the aftermath of the School for Designing a Society's first Microtonal Design Seminar, &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt; assembled a zine to teach and celebrate the virtues of 11edo. This particular page will be the site of some tuning files related to the zine.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
If you have a copy of the zine and have come for this page looking for things to listen to, yay! There's nothing specific to the zine on here yet (I just finished slapping all the paper together, and I am tired, man!), but I recommend going over to &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11edo&amp;lt;/a&amp;gt; on this here wiki and taking a listen to the files there under &amp;amp;quot;Compositions&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
More to come!&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
~Andrew&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>pom2syvq0tlut1k8nv83cflxp7xrsdx</sha1>
    </revision>
    <revision>
      <id>46086</id>
      <parentid>89</parentid>
      <timestamp>2020-06-08T19:28:22Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>simplified wiki markup</comment>
      <origin>46086</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="878" sha1="qkhf3ewvub7z14envds9584d3kqgln9" xml:space="preserve">In the Spring of 2011, in the aftermath of the School for Designing a Society's first Microtonal Design Seminar, [[Andrew Heathwaite]] assembled a zine to teach and celebrate the virtues of 11edo. This particular page will be the site of some tuning files related to the zine.

If you have a copy of the zine and have come for this page looking for things to listen to, yay! There's nothing specific to the zine on here yet (I just finished slapping all the paper together, and I am tired, man!), but I recommend going over to [[11edo]] on this here wiki and taking a listen to the files there under "Compositions".

And if you're interested in some audio traces from the semester, see the blog post for the [http://oddmusicuc.wordpress.com/2011/02/17/11edo-band-on-ors/ Oddmusic Radio Shower on February 16, 2011].

More to come!

~Andrew
[[Category:11edo]]
[[Category:Listen]]</text>
      <sha1>qkhf3ewvub7z14envds9584d3kqgln9</sha1>
    </revision>
    <revision>
      <id>46087</id>
      <parentid>46086</parentid>
      <timestamp>2020-06-08T19:29:58Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>link to section (title corrected)</comment>
      <origin>46087</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="888" sha1="4on3076fc75qcnz3a5pvzubcky7x3nb" xml:space="preserve">In the Spring of 2011, in the aftermath of the School for Designing a Society's first Microtonal Design Seminar, [[Andrew Heathwaite]] assembled a zine to teach and celebrate the virtues of 11edo. This particular page will be the site of some tuning files related to the zine.

If you have a copy of the zine and have come for this page looking for things to listen to, yay! There's nothing specific to the zine on here yet (I just finished slapping all the paper together, and I am tired, man!), but I recommend going over to [[11edo]] on this here wiki and taking a listen to the files there under [[11edo #Music|"Music"]].

And if you're interested in some audio traces from the semester, see the blog post for the [http://oddmusicuc.wordpress.com/2011/02/17/11edo-band-on-ors/ Oddmusic Radio Shower on February 16, 2011].

More to come!

~Andrew
[[Category:11edo]]
[[Category:Listen]]</text>
      <sha1>4on3076fc75qcnz3a5pvzubcky7x3nb</sha1>
    </revision>
  </page>
  <page>
    <title>11edo</title>
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      <id>90</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>90</origin>
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      <text bytes="19891" sha1="7722ic9u0rb36ojuarn8ct8ii74dn2w" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[11平均律|日本語]]&lt;/span&gt;

[[#toc0| ]] | 
__FORCETOC__
= =

-----

=11 tone equal temperament=

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

=Subgroup=
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=

{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

=Notation=

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, Cv
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; [http://tel.wikispaces.com/11_17_26_31_38_41 11 17 26 31 38 41] |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| | | -7 3 1 &amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| | | -68 18 17 &amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=11edo Instant Ensemble=
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

=Compositions=
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;''''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'' '') by [[Chris_Vaisvil|Chris Vaisvil]]''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;
==Instruments==

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]      [[Category:11-tone]]
[[Category:11edo]]
[[Category:edo]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:prime_edo]]
[[Category:scale]]
[[Category:subgroup]]
[[Category:todo:unify_precision]]</text>
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      <timestamp>2017-07-30T14:57:26Z</timestamp>
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      <comment>**Imported revision 616004535 - Original comment: added image (deorphaning)**</comment>
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      <text bytes="70456" sha1="dik0uphxi49kd51g5w82dhw9oo1x8zs" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2017-07-30 14:57:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;616004535&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;added image (deorphaning)&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
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[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=11 tone equal temperament= 

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
||~ Harmonic ||~ 8 ||~   ||~ 9 ||~   ||~ 11 ||~   ||~ 14 ||~   ||~ 16 ||
||~ JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
||~ nearest 11edo interval || 0\11edo = 0¢ ||   || 2\11 = 218¢ ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
||~ difference || 0 ||   || +14¢ ||   || -6¢ ||   || +13¢ ||   || 0¢ ||
||~ JI interval between ||   || 9:8 = 204¢ ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
||~ nearest 11edo interval ||   || 2\11 = 218¢ ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
||~ difference ||   || +14¢ ||   || -20¢ ||   || +18¢ ||   || -13¢ ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

=Notation= 

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the //melodic// meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the //harmonic// meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

||~ Degree ||~ Size in
[[cent|Cents]] ||~ Solfege ||~ Approximate Ratios* ||~ Sagittal
notation
(22edo subset) ||||~ [[Ups and Downs Notation|Up/down notation]]
with major wider
than minor ||||~ Up/down notation
with major narrower
than minor ||~ [[Tútim Dennsuul Wafiil|TDW]]
Machine
notation ||
||= 0 ||&amp;gt; 0.00 ||= **do** ||= 1/1 ||= A ||= P1 ||= A ||= P1 ||= A ||= Q\P# ||
||= 1 ||&amp;gt; 109.09 ||= **ra** ||= [[15_14|15/14]], [[16_15|16/15]], [[17_16|17/16]], [[18_17|18/17]] ||= AII\ or B!!/ ||= ^1, m2 ||= A^, B ||= ^1, M2 ||= A^, B ||= Q#\Rb ||
||= 2 ||&amp;gt; 218.18 ||= **re** ||= [[8_7|8/7]], [[9_8|9/8]], [[17_15|17/15]] ||= B ||= ~2, m3 ||= B^, Cb ||= ~2, M3 ||= B^, C# ||= R ||
||= 3 ||&amp;gt; 327.27 ||= **me** ||= [[6_5|6/5]], [[11_9|11/9]], [[17_14|17/14]] ||= C/I or BII\ or D\!!/ ||= M2, ~3 ||= B#, Cv ||= m2, ~3 ||= Bb, Cv ||= R#\Sb ||
||= 4 ||&amp;gt; 436.36 ||= **mo** ||= [[9_7|9/7]], [[14_11|14/11]], [[22_17|22/17]] ||= D\! or C/II\ ||= M3, v4 ||= C, Dv ||= m3, v4 ||= C, Dv ||= S ||
||= 5 ||&amp;gt; 545.45 ||= **fu** ||= [[11_8|11/8]], [[15_11|15/11]] ||= D/I or E\!!/ ||= P4, v5 ||= D, Ev ||= P4, v5 ||= D, Ev ||= S#\Tb ||
||= 6 ||&amp;gt; 654.55 ||= **su** ||= [[16_11|16/11]], [[22_15|22/15]] ||= E\! or D/II\ ||= ^4, P5 ||= D^, E ||= ^4, P5 ||= D^, E ||= T ||
||= 7 ||&amp;gt; 763.64 ||= **lo** ||= [[11_7|11/7]], [[14_9|14/9]], [[17_11|17/11]] ||= F ||= ^5, m6 ||= E^, Fb ||= ^5, M6 ||= E^, F# ||= T#\Ub ||
||= 8 ||&amp;gt; 872.73 ||= **la** ||= [[5_3|5/3]], [[18_11|18/11]], [[28_17|28/17]] ||= FII\ or G!!/ ||= ~6, m7 ||= Fv, Gb ||= ~6, M7 ||= Fv, G# ||= U ||
||= 9 ||&amp;gt; 981.82 ||= **ta** ||= [[7_4|7/4]], [[16_9|16/9]], [[30_17|30/17]] ||= G ||= M6, ~7 ||= F, Gv ||= m6, ~7 ||= F, Gv ||= U#\Pb ||
||= 10 ||&amp;gt; 1090.91 ||= **ti** ||= [[15_8|15/8]], [[17_9|17/9]], [[28_15|28/15]], [[32_17|32/17]] ||= GII\ or A!!/ ||= M7, v8 ||= G, Av ||= m7, v8 ||= G, Av ||= P\Qb ||
||= 11 ||&amp;gt; 1200.00 ||= **do** ||= 2/1 ||= A ||= P8 ||= A ||= P8 ||= A ||= Q\P# ||
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:
[[image:Sagittal11EDO.jpg]]

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; [[tel/11 17 26 31 38 41|11 17 26 31 38 41]] |.)

||~ Rational ||~ Monzo ||~ Size ([[cent|Cents]]) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= [[135_128|135/128]] || | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 || | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= [[225_224|225/224]] || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 || | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 || | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#Notation"&amp;gt;Notation&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:42: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:42 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:43: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:43 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:44: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:44 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11&amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cents&amp;lt;/a&amp;gt;. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, a tuning known as &amp;amp;quot;elevenplus&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:658:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:658 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Notation"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Notation&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the &amp;lt;em&amp;gt;melodic&amp;lt;/em&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The second approach preserves the &amp;lt;em&amp;gt;harmonic&amp;lt;/em&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo &amp;amp;quot;on the fly&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size in&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/cent"&amp;gt;Cents&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Solfege&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Approximate Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Sagittal&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
(22edo subset)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;Up/down notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
with major wider&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;Up/down notation&amp;lt;br /&amp;gt;
with major narrower&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
Machine&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/15_14"&amp;gt;15/14&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16_15"&amp;gt;16/15&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_16"&amp;gt;17/16&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/18_17"&amp;gt;18/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, m2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, M2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/8_7"&amp;gt;8/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_15"&amp;gt;17/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, Cb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, M3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, C#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_14"&amp;gt;17/14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B#, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Bb, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/9_7"&amp;gt;9/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22_17"&amp;gt;22/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/14_9"&amp;gt;14/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_11"&amp;gt;17/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, Fb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, M6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, F#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_17"&amp;gt;28/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, Gb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, M7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, G#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16_9"&amp;gt;16/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/30_17"&amp;gt;30/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/15_8"&amp;gt;15/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_9"&amp;gt;17/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_15"&amp;gt;28/15&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/32_17"&amp;gt;32/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1200.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11edo in &amp;lt;a class="wiki_link" href="/Sagittal%20notation"&amp;gt;Sagittal notation&amp;lt;/a&amp;gt;:&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:659:&amp;amp;lt;img src=&amp;amp;quot;/file/view/Sagittal11EDO.jpg/557089551/Sagittal11EDO.jpg&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/Sagittal11EDO.jpg/557089551/Sagittal11EDO.jpg" alt="Sagittal11EDO.jpg" title="Sagittal11EDO.jpg" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:659 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Supersharp&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic, octotonic and nonatonic fifth-generated) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic third-generated).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; &amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/11%2017%2026%2031%2038%2041"&amp;gt;11 17 26 31 38 41&amp;lt;/a&amp;gt; |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Rational&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size (&amp;lt;a class="wiki_link" href="/cent"&amp;gt;Cents&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/135_128"&amp;gt;135/128&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/225_224"&amp;gt;225/224&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc11"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:27:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc13"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:27 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:660:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:660 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>dik0uphxi49kd51g5w82dhw9oo1x8zs</sha1>
    </revision>
    <revision>
      <id>4348</id>
      <timestamp>2017-05-11T09:01:03Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jake.huryn</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 612649517 - Original comment: **</comment>
      <origin>4348</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="70086" sha1="tfvsgvn9h43huhemo6lkdl38juasykp" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jake.huryn|jake.huryn]] and made on &lt;tt&gt;2017-05-11 09:01:03 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;612649517&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
||~ Harmonic ||~ 8 ||~   ||~ 9 ||~   ||~ 11 ||~   ||~ 14 ||~   ||~ 16 ||
||~ JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
||~ nearest 11edo interval || 0\11edo = 0¢ ||   || 2\11 = 218¢ ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
||~ difference || 0 ||   || +14¢ ||   || -6¢ ||   || +13¢ ||   || 0¢ ||
||~ JI interval between ||   || 9:8 = 204¢ ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
||~ nearest 11edo interval ||   || 2\11 = 218¢ ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
||~ difference ||   || +14¢ ||   || -20¢ ||   || +18¢ ||   || -13¢ ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

=Notation= 

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the //melodic// meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the //harmonic// meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

||~ Degree ||~ Size in
[[cent|Cents]] ||~ Solfege ||~ Approximate Ratios* ||~ Sagittal
notation
(22edo subset) ||||~ [[Ups and Downs Notation|Up/down notation]]
with major wider
than minor ||||~ Up/down notation
with major narrower
than minor ||~ [[Tútim Dennsuul Wafiil|TDW]]
Machine
notation ||
||= 0 ||&amp;gt; 0.00 ||= **do** ||= 1/1 ||= A ||= P1 ||= A ||= P1 ||= A ||= Q\P# ||
||= 1 ||&amp;gt; 109.09 ||= **ra** ||= [[15_14|15/14]], [[16_15|16/15]], [[17_16|17/16]], [[18_17|18/17]] ||= AII\ or B!!/ ||= ^1, m2 ||= A^, B ||= ^1, M2 ||= A^, B ||= Q#\Rb ||
||= 2 ||&amp;gt; 218.18 ||= **re** ||= [[8_7|8/7]], [[9_8|9/8]], [[17_15|17/15]] ||= B ||= ~2, m3 ||= B^, Cb ||= ~2, M3 ||= B^, C# ||= R ||
||= 3 ||&amp;gt; 327.27 ||= **me** ||= [[6_5|6/5]], [[11_9|11/9]], [[17_14|17/14]] ||= C/I or BII\ or D\!!/ ||= M2, ~3 ||= B#, Cv ||= m2, ~3 ||= Bb, Cv ||= R#\Sb ||
||= 4 ||&amp;gt; 436.36 ||= **mo** ||= [[9_7|9/7]], [[14_11|14/11]], [[22_17|22/17]] ||= D\! or C/II\ ||= M3, v4 ||= C, Dv ||= m3, v4 ||= C, Dv ||= S ||
||= 5 ||&amp;gt; 545.45 ||= **fu** ||= [[11_8|11/8]], [[15_11|15/11]] ||= D/I or E\!!/ ||= P4, v5 ||= D, Ev ||= P4, v5 ||= D, Ev ||= S#\Tb ||
||= 6 ||&amp;gt; 654.55 ||= **su** ||= [[16_11|16/11]], [[22_15|22/15]] ||= E\! or D/II\ ||= ^4, P5 ||= D^, E ||= ^4, P5 ||= D^, E ||= T ||
||= 7 ||&amp;gt; 763.64 ||= **lo** ||= [[11_7|11/7]], [[14_9|14/9]], [[17_11|17/11]] ||= F ||= ^5, m6 ||= E^, Fb ||= ^5, M6 ||= E^, F# ||= T#\Ub ||
||= 8 ||&amp;gt; 872.73 ||= **la** ||= [[5_3|5/3]], [[18_11|18/11]], [[28_17|28/17]] ||= FII\ or G!!/ ||= ~6, m7 ||= Fv, Gb ||= ~6, M7 ||= Fv, G# ||= U ||
||= 9 ||&amp;gt; 981.82 ||= **ta** ||= [[7_4|7/4]], [[16_9|16/9]], [[30_17|30/17]] ||= G ||= M6, ~7 ||= F, Gv ||= m6, ~7 ||= F, Gv ||= U#\Pb ||
||= 10 ||&amp;gt; 1090.91 ||= **ti** ||= [[15_8|15/8]], [[17_9|17/9]], [[28_15|28/15]], [[32_17|32/17]] ||= GII\ or A!!/ ||= M7, v8 ||= G, Av ||= m7, v8 ||= G, Av ||= P\Qb ||
||= 11 ||&amp;gt; 1200.00 ||= **do** ||= 2/1 ||= A ||= P8 ||= A ||= P8 ||= A ||= Q\P# ||
*in 2.7.9.11.15.17 subgroup

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; [[tel/11 17 26 31 38 41|11 17 26 31 38 41]] |.)

||~ Rational ||~ Monzo ||~ Size ([[cent|Cents]]) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= [[135_128|135/128]] || | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 || | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= [[225_224|225/224]] || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 || | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 || | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#Notation"&amp;gt;Notation&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:42: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:42 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:43: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:43 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:44: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:44 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11&amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cents&amp;lt;/a&amp;gt;. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, a tuning known as &amp;amp;quot;elevenplus&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:658:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:658 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Notation"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Notation&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the &amp;lt;em&amp;gt;melodic&amp;lt;/em&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The second approach preserves the &amp;lt;em&amp;gt;harmonic&amp;lt;/em&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo &amp;amp;quot;on the fly&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size in&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/cent"&amp;gt;Cents&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Solfege&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Approximate Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Sagittal&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
(22edo subset)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;Up/down notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
with major wider&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;Up/down notation&amp;lt;br /&amp;gt;
with major narrower&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
Machine&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/15_14"&amp;gt;15/14&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16_15"&amp;gt;16/15&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_16"&amp;gt;17/16&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/18_17"&amp;gt;18/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, m2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, M2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/8_7"&amp;gt;8/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_15"&amp;gt;17/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, Cb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, M3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, C#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_14"&amp;gt;17/14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B#, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Bb, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/9_7"&amp;gt;9/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22_17"&amp;gt;22/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/14_9"&amp;gt;14/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_11"&amp;gt;17/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, Fb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, M6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, F#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_17"&amp;gt;28/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, Gb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, M7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, G#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16_9"&amp;gt;16/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/30_17"&amp;gt;30/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/15_8"&amp;gt;15/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_9"&amp;gt;17/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_15"&amp;gt;28/15&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/32_17"&amp;gt;32/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1200.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Supersharp&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic, octotonic and nonatonic fifth-generated) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic third-generated).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; &amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/11%2017%2026%2031%2038%2041"&amp;gt;11 17 26 31 38 41&amp;lt;/a&amp;gt; |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Rational&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size (&amp;lt;a class="wiki_link" href="/cent"&amp;gt;Cents&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/135_128"&amp;gt;135/128&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/225_224"&amp;gt;225/224&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc11"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:27:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc13"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:27 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:659:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:659 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>tfvsgvn9h43huhemo6lkdl38juasykp</sha1>
    </revision>
    <revision>
      <id>4349</id>
      <timestamp>2017-05-11T08:55:02Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jake.huryn</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 612649031 - Original comment: **</comment>
      <origin>4349</origin>
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      <format>text/x-wiki</format>
      <text bytes="70134" sha1="sq8oz8k9kxiozwciknbinr1x6bk2x18" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jake.huryn|jake.huryn]] and made on &lt;tt&gt;2017-05-11 08:55:02 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;612649031&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
||~ Harmonic ||~ 8 ||~   ||~ 9 ||~   ||~ 11 ||~   ||~ 14 ||~   ||~ 16 ||
||~ JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
||~ nearest 11edo interval || 0\11edo = 0¢ ||   || 2\11 = 218¢ ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
||~ difference || 0 ||   || +14¢ ||   || -6¢ ||   || +13¢ ||   || 0¢ ||
||~ JI interval between ||   || 9:8 = 204¢ ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
||~ nearest 11edo interval ||   || 2\11 = 218¢ ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
||~ difference ||   || +14¢ ||   || -20¢ ||   || +18¢ ||   || -13¢ ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

=Notation= 

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the //melodic// meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the //harmonic// meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

||~ Degree ||~ Size in
[[cent|Cents]] ||~ Solfege ||~ Approximate Ratios* ||~ Sagittal
notation
(22edo subset) ||||~ [[Ups and Downs Notation|Up/down notation]]
with major wider
than minor ||||~ Up/down notation
with major narrower
than minor ||~ [[Tútim Dennsuul Wafiil|TDW]]
Machine
notation ||
||= 0 ||&amp;gt; 0.00 ||= **do** ||= 1/1 ||= A ||= P1 ||= A ||= P1 ||= A ||= Q\P# ||
||= 1 ||&amp;gt; 109.09 ||= **ra** ||= [[15_14|15/14]], [[16_15|16/15]], [[17_16|17/16]], [[18_17|18/17]] ||= AII\ or B!!/ ||= ^1, m2 ||= A^, B ||= ^1, M2 ||= A^, B ||= Q#\Rb ||
||= 2 ||&amp;gt; 218.18 ||= **re** ||= [[8_7|8/7]], [[9_8|9/8]], [[17_15|17/15]] ||= B ||= ~2, m3 ||= B^, Cb ||= ~2, M3 ||= B^, C# ||= R ||
||= 3 ||&amp;gt; 327.27 ||= **me** ||= [[6_5|6/5]], [[11_9|11/9]], [[17_14|17/14]] ||= C/I or BII\ or D\!!/ ||= M2, ~3 ||= B#, Cv ||= m2, ~3 ||= Bb, Cv ||= R#\Sb ||
||= 4 ||&amp;gt; 436.36 ||= **mo** ||= [[9_7|9/7]], [[14_11|14/11]], [[22_17|22/17]] ||= D\! or C/II\ ||= M3, v4 ||= C, Dv ||= m3, v4 ||= C, Dv ||= S ||
||= 5 ||&amp;gt; 545.45 ||= **fu** ||= [[11_8|11/8]], [[15_11|15/11]] ||= D/I or E\!!/ ||= P4, v5 ||= D, Ev ||= P4, v5 ||= D, Ev ||= S#\Tb ||
||= 6 ||&amp;gt; 654.55 ||= **su** ||= [[16_11|16/11]], [[22_15|22/15]] ||= E\! or D/II\ ||= ^4, P5 ||= D^, E ||= ^4, P5 ||= D^, E ||= T ||
||= 7 ||&amp;gt; 763.64 ||= **lo** ||= [[11_7|11/7]], [[14_9|14/9]], [[17_11|17/11]] ||= F ||= ^5, m6 ||= E^, Fb ||= ^5, M6 ||= E^, F# ||= T#\Ub ||
||= 8 ||&amp;gt; 872.73 ||= **la** ||= [[5_3|5/3]], [[18_11|18/11]], [[28_17|28/17]] ||= FII\ or G!!/ ||= ~6, m7 ||= Fv, Gb ||= ~6, M7 ||= Fv, G# ||= U ||
||= 9 ||&amp;gt; 981.82 ||= **ta** ||= [[7_4|7/4]], [[16_9|16/9]], [[30_17|30/17]] ||= G ||= M6, ~7 ||= F, Gv ||= m6, ~7 ||= F, Gv ||= U#\Pb ||
||= 10 ||&amp;gt; 1090.91 ||= **ti** ||= [[15_8|15/8]], [[17_9|17/9]], [[28_15|28/15]], [[32_17|32/17]] ||= GII\ or A!!/ ||= M7, v8 ||= G, Av ||= m7, v8 ||= G, Av ||= P\Qb ||
||= 11 ||&amp;gt; 1200.00 ||= **do** ||= 2/1 ||= A ||= P8 ||= A ||= P8 ||= A ||= Q\P# ||
*in 2.7.9.11.15.17 subgroup

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; [[tel/11 17 26 31 38 41|11 17 26 31 38 41]] |.)

||~ Rational ||~ Monzo ||~ Size ([[cent|Cents]]) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= [[135_128|135/128]] || | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 || | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= [[tel:9859966/9733137|9859966/9733137]] || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= [[225_224|225/224]] || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 || | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 || | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#Notation"&amp;gt;Notation&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:42: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:42 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:43: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:43 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:44: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:44 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11&amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cents&amp;lt;/a&amp;gt;. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, a tuning known as &amp;amp;quot;elevenplus&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:658:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:658 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Notation"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Notation&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the &amp;lt;em&amp;gt;melodic&amp;lt;/em&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The second approach preserves the &amp;lt;em&amp;gt;harmonic&amp;lt;/em&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo &amp;amp;quot;on the fly&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size in&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/cent"&amp;gt;Cents&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Solfege&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Approximate Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Sagittal&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
(22edo subset)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;Up/down notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
with major wider&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;Up/down notation&amp;lt;br /&amp;gt;
with major narrower&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
Machine&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/15_14"&amp;gt;15/14&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16_15"&amp;gt;16/15&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_16"&amp;gt;17/16&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/18_17"&amp;gt;18/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, m2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, M2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/8_7"&amp;gt;8/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_15"&amp;gt;17/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, Cb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, M3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, C#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_14"&amp;gt;17/14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B#, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Bb, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/9_7"&amp;gt;9/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22_17"&amp;gt;22/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/14_9"&amp;gt;14/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_11"&amp;gt;17/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, Fb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, M6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, F#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_17"&amp;gt;28/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, Gb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, M7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, G#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16_9"&amp;gt;16/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/30_17"&amp;gt;30/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/15_8"&amp;gt;15/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_9"&amp;gt;17/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_15"&amp;gt;28/15&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/32_17"&amp;gt;32/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1200.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Supersharp&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic, octotonic and nonatonic fifth-generated) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic third-generated).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; &amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/11%2017%2026%2031%2038%2041"&amp;gt;11 17 26 31 38 41&amp;lt;/a&amp;gt; |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Rational&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size (&amp;lt;a class="wiki_link" href="/cent"&amp;gt;Cents&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/135_128"&amp;gt;135/128&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;[[tel:9859966/9733137|9859966/9733137]]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/225_224"&amp;gt;225/224&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc11"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:27:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc13"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:27 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:659:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:659 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>sq8oz8k9kxiozwciknbinr1x6bk2x18</sha1>
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    <revision>
      <id>4350</id>
      <timestamp>2017-05-11T08:53:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jake.huryn</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 612648925 - Original comment: **</comment>
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      <text bytes="70276" sha1="mb1coq9xp10pgx0v8wqvg2qbmj9beif" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jake.huryn|jake.huryn]] and made on &lt;tt&gt;2017-05-11 08:53:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;612648925&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
||~ Harmonic ||~ 8 ||~   ||~ 9 ||~   ||~ 11 ||~   ||~ 14 ||~   ||~ 16 ||
||~ JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
||~ nearest 11edo interval || 0\11edo = 0¢ ||   || 2\11 = 218¢ ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
||~ difference || 0 ||   || +14¢ ||   || -6¢ ||   || +13¢ ||   || 0¢ ||
||~ JI interval between ||   || 9:8 = 204¢ ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
||~ nearest 11edo interval ||   || 2\11 = 218¢ ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
||~ difference ||   || +14¢ ||   || -20¢ ||   || +18¢ ||   || -13¢ ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

=Notation= 

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the //melodic// meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the //harmonic// meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

||~ Degree ||~ Size in
[[cent|Cents]] ||~ Solfege ||~ Approximate Ratios* ||~ Sagittal
notation
(22edo subset) ||||~ [[Ups and Downs Notation|Up/down notation]]
with major wider
than minor ||||~ Up/down notation
with major narrower
than minor ||~ [[Tútim Dennsuul Wafiil|TDW]]
Machine
notation ||
||= 0 ||&amp;gt; 0.00 ||= **do** ||= 1/1 ||= A ||= P1 ||= A ||= P1 ||= A ||= Q\P# ||
||= 1 ||&amp;gt; 109.09 ||= **ra** ||= [[15_14|15/14]], [[16_15|16/15]], [[17_16|17/16]], [[18_17|18/17]] ||= AII\ or B!!/ ||= ^1, m2 ||= A^, B ||= ^1, M2 ||= A^, B ||= Q#\Rb ||
||= 2 ||&amp;gt; 218.18 ||= **re** ||= [[8_7|8/7]], [[9_8|9/8]], [[17_15|17/15]] ||= B ||= ~2, m3 ||= B^, Cb ||= ~2, M3 ||= B^, C# ||= R ||
||= 3 ||&amp;gt; 327.27 ||= **me** ||= [[6_5|6/5]], [[11_9|11/9]], [[17_14|17/14]] ||= C/I or BII\ or D\!!/ ||= M2, ~3 ||= B#, Cv ||= m2, ~3 ||= Bb, Cv ||= R#\Sb ||
||= 4 ||&amp;gt; 436.36 ||= **mo** ||= [[9_7|9/7]], [[14_11|14/11]], [[22_17|22/17]] ||= D\! or C/II\ ||= M3, v4 ||= C, Dv ||= m3, v4 ||= C, Dv ||= S ||
||= 5 ||&amp;gt; 545.45 ||= **fu** ||= [[11_8|11/8]], [[15_11|15/11]] ||= D/I or E\!!/ ||= P4, v5 ||= D, Ev ||= P4, v5 ||= D, Ev ||= S#\Tb ||
||= 6 ||&amp;gt; 654.55 ||= **su** ||= [[16_11|16/11]], [[22_15|22/15]] ||= E\! or D/II\ ||= ^4, P5 ||= D^, E ||= ^4, P5 ||= D^, E ||= T ||
||= 7 ||&amp;gt; 763.64 ||= **lo** ||= [[11_7|11/7]], [[14_9|14/9]], [[17_11|17/11]] ||= F ||= ^5, m6 ||= E^, Fb ||= ^5, M6 ||= E^, F# ||= T#\Ub ||
||= 8 ||&amp;gt; 872.73 ||= **la** ||= [[5_3|5/3]], [[18_11|18/11]], [[28_17|28/17]] ||= FII\ or G!!/ ||= ~6, m7 ||= Fv, Gb ||= ~6, M7 ||= Fv, G# ||= U ||
||= 9 ||&amp;gt; 981.82 ||= **ta** ||= [[7_4|7/4]], [[16_9|16/9]], [[30_17|30/17]] ||= G ||= M6, ~7 ||= F, Gv ||= m6, ~7 ||= F, Gv ||= U#\Pb ||
||= 10 ||&amp;gt; 1090.91 ||= **ti** ||= [[15_8|15/8]], [[17_9|17/9]], [[28_15|28/15]], [[32_17|32/17]] ||= GII\ or A!!/ ||= M7, v8 ||= G, Av ||= m7, v8 ||= G, Av ||= P\Qb ||
||= 11 ||&amp;gt; 1200.00 ||= **do** ||= 2/1 ||= A ||= P8 ||= A ||= P8 ||= A ||= Q\P# ||
*in 2.7.9.11.15.17 subgroup

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; [[tel/11 17 26 31 38 41|11 17 26 31 38 41]] |.)

||~ Rational ||~ Monzo ||~ Size ([[cent|Cents]]) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= [[135_128|135/128]] || | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= [[tel:9931568/9752117|9931568/9752117]] || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= [[tel:[[tel:1776337|1776337]]/1773750|1776337/1773750]] || | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= [[225_224|225/224]] || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 || | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 || | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#Notation"&amp;gt;Notation&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:42: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:42 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:43: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:43 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:44: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:44 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11&amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cents&amp;lt;/a&amp;gt;. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, a tuning known as &amp;amp;quot;elevenplus&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:658:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:658 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Notation"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Notation&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the &amp;lt;em&amp;gt;melodic&amp;lt;/em&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The second approach preserves the &amp;lt;em&amp;gt;harmonic&amp;lt;/em&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo &amp;amp;quot;on the fly&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size in&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/cent"&amp;gt;Cents&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Solfege&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Approximate Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Sagittal&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
(22edo subset)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;Up/down notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
with major wider&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;Up/down notation&amp;lt;br /&amp;gt;
with major narrower&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
Machine&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/15_14"&amp;gt;15/14&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16_15"&amp;gt;16/15&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_16"&amp;gt;17/16&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/18_17"&amp;gt;18/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, m2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, M2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/8_7"&amp;gt;8/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_15"&amp;gt;17/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, Cb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, M3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, C#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_14"&amp;gt;17/14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B#, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Bb, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/9_7"&amp;gt;9/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22_17"&amp;gt;22/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/14_9"&amp;gt;14/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_11"&amp;gt;17/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, Fb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, M6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, F#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_17"&amp;gt;28/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, Gb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, M7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, G#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16_9"&amp;gt;16/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/30_17"&amp;gt;30/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/15_8"&amp;gt;15/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_9"&amp;gt;17/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_15"&amp;gt;28/15&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/32_17"&amp;gt;32/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1200.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Supersharp&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic, octotonic and nonatonic fifth-generated) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic third-generated).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; &amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/11%2017%2026%2031%2038%2041"&amp;gt;11 17 26 31 38 41&amp;lt;/a&amp;gt; |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Rational&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size (&amp;lt;a class="wiki_link" href="/cent"&amp;gt;Cents&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/135_128"&amp;gt;135/128&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;[[tel:9931568/9752117|9931568/9752117]]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;[[tel:&amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/1776337"&amp;gt;1776337&amp;lt;/a&amp;gt;/1773750|1776337/1773750]]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/225_224"&amp;gt;225/224&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc11"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:27:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc13"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:27 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:659:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:659 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>mb1coq9xp10pgx0v8wqvg2qbmj9beif</sha1>
    </revision>
    <revision>
      <id>4351</id>
      <timestamp>2017-05-11T08:50:53Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jake.huryn</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 612648709 - Original comment: **</comment>
      <origin>4351</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="70230" sha1="4vylawj1ci017dfdl3vqo7qms9lnyyn" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jake.huryn|jake.huryn]] and made on &lt;tt&gt;2017-05-11 08:50:53 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;612648709&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
||~ Harmonic ||~ 8 ||~   ||~ 9 ||~   ||~ 11 ||~   ||~ 14 ||~   ||~ 16 ||
||~ JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
||~ nearest 11edo interval || 0\11edo = 0¢ ||   || 2\11 = 218¢ ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
||~ difference || 0 ||   || +14¢ ||   || -6¢ ||   || +13¢ ||   || 0¢ ||
||~ JI interval between ||   || 9:8 = 204¢ ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
||~ nearest 11edo interval ||   || 2\11 = 218¢ ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
||~ difference ||   || +14¢ ||   || -20¢ ||   || +18¢ ||   || -13¢ ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

=Notation= 

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the //melodic// meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the //harmonic// meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

||~ Degree ||~ Size in
[[cent|Cents]] ||~ Solfege ||~ Approximate Ratios* ||~ Sagittal
notation
(22edo subset) ||||~ [[Ups and Downs Notation|Up/down notation]]
with major wider
than minor ||||~ Up/down notation
with major narrower
than minor ||~ [[Tútim Dennsuul Wafiil|TDW]]
Machine
notation ||
||= 0 ||&amp;gt; 0.00 ||= **do** ||= 1/1 ||= A ||= P1 ||= A ||= P1 ||= A ||= Q\P# ||
||= 1 ||&amp;gt; 109.09 ||= **ra** ||= [[15_14|15/14]], [[16_15|16/15]], [[17_16|17/16]], [[18_17|18/17]] ||= AII\ or B!!/ ||= ^1, m2 ||= A^, B ||= ^1, M2 ||= A^, B ||= Q#\Rb ||
||= 2 ||&amp;gt; 218.18 ||= **re** ||= [[8_7|8/7]], [[9_8|9/8]], [[17_15|17/15]] ||= B ||= ~2, m3 ||= B^, Cb ||= ~2, M3 ||= B^, C# ||= R ||
||= 3 ||&amp;gt; 327.27 ||= **me** ||= [[6_5|6/5]], [[11_9|11/9]], [[17_14|17/14]] ||= C/I or BII\ or D\!!/ ||= M2, ~3 ||= B#, Cv ||= m2, ~3 ||= Bb, Cv ||= R#\Sb ||
||= 4 ||&amp;gt; 436.36 ||= **mo** ||= [[9_7|9/7]], [[14_11|14/11]], [[22_17|22/17]] ||= D\! or C/II\ ||= M3, v4 ||= C, Dv ||= m3, v4 ||= C, Dv ||= S ||
||= 5 ||&amp;gt; 545.45 ||= **fu** ||= [[11_8|11/8]], [[15_11|15/11]] ||= D/I or E\!!/ ||= P4, v5 ||= D, Ev ||= P4, v5 ||= D, Ev ||= S#\Tb ||
||= 6 ||&amp;gt; 654.55 ||= **su** ||= [[16_11|16/11]], [[22_15|22/15]] ||= E\! or D/II\ ||= ^4, P5 ||= D^, E ||= ^4, P5 ||= D^, E ||= T ||
||= 7 ||&amp;gt; 763.64 ||= **lo** ||= [[11_7|11/7]], [[14_9|14/9]], [[17_11|17/11]] ||= F ||= ^5, m6 ||= E^, Fb ||= ^5, M6 ||= E^, F# ||= T#\Ub ||
||= 8 ||&amp;gt; 872.73 ||= **la** ||= [[5_3|5/3]], [[18_11|18/11]], [[28_17|28/17]] ||= FII\ or G!!/ ||= ~6, m7 ||= Fv, Gb ||= ~6, M7 ||= Fv, G# ||= U ||
||= 9 ||&amp;gt; 981.82 ||= **ta** ||= [[7_4|7/4]], [[16_9|16/9]], [[30_17|30/17]] ||= G ||= M6, ~7 ||= F, Gv ||= m6, ~7 ||= F, Gv ||= U#\Pb ||
||= 10 ||&amp;gt; 1090.91 ||= **ti** ||= [[15_8|15/8]], [[17_9|17/9]], [[28_15|28/15]], [[32_17|32/17]] ||= GII\ or A!!/ ||= M7, v8 ||= G, Av ||= m7, v8 ||= G, Av ||= P\Qb ||
||= 11 ||&amp;gt; 1200.00 ||= **do** ||= 2/1 ||= A ||= P8 ||= A ||= P8 ||= A ||= Q\P# ||
*in 2.7.9.11.15.17 subgroup

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; [[tel:11 17 26 31 38 41|11 17 26 31 38 41]] |.)

||~ Rational ||~ Monzo ||~ Size ([[cent|Cents]]) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= [[135_128|135/128]] || | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= [[tel:9931568/9752117|9931568/9752117]] || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= [[tel:1776337/1773750|1776337/1773750]] || | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= [[tel:9859966/9733137|9859966/9733137]] || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= [[225_224|225/224]] || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 || | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 || | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#Notation"&amp;gt;Notation&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:42: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:42 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:43: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:43 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:44: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:44 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11&amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cents&amp;lt;/a&amp;gt;. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, a tuning known as &amp;amp;quot;elevenplus&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:658:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:658 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Notation"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Notation&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the &amp;lt;em&amp;gt;melodic&amp;lt;/em&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The second approach preserves the &amp;lt;em&amp;gt;harmonic&amp;lt;/em&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo &amp;amp;quot;on the fly&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size in&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/cent"&amp;gt;Cents&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Solfege&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Approximate Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Sagittal&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
(22edo subset)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;Up/down notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
with major wider&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;Up/down notation&amp;lt;br /&amp;gt;
with major narrower&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
Machine&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/15_14"&amp;gt;15/14&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16_15"&amp;gt;16/15&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_16"&amp;gt;17/16&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/18_17"&amp;gt;18/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, m2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, M2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/8_7"&amp;gt;8/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_15"&amp;gt;17/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, Cb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, M3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, C#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_14"&amp;gt;17/14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B#, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Bb, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/9_7"&amp;gt;9/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22_17"&amp;gt;22/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/14_9"&amp;gt;14/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_11"&amp;gt;17/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, Fb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, M6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, F#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_17"&amp;gt;28/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, Gb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, M7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, G#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16_9"&amp;gt;16/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/30_17"&amp;gt;30/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/15_8"&amp;gt;15/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_9"&amp;gt;17/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_15"&amp;gt;28/15&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/32_17"&amp;gt;32/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1200.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Supersharp&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic, octotonic and nonatonic fifth-generated) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic third-generated).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; &amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/11%2017%2026%2031%2038%2041"&amp;gt;11 17 26 31 38 41&amp;lt;/a&amp;gt; |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Rational&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size (&amp;lt;a class="wiki_link" href="/cent"&amp;gt;Cents&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/135_128"&amp;gt;135/128&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;[[tel:9931568/9752117|9931568/9752117]]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;[[tel:1776337/1773750|1776337/1773750]]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;[[tel:9859966/9733137|9859966/9733137]]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/225_224"&amp;gt;225/224&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc11"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:27:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc13"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:27 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:659:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:659 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>4vylawj1ci017dfdl3vqo7qms9lnyyn</sha1>
    </revision>
    <revision>
      <id>4352</id>
      <timestamp>2016-12-29T07:53:20Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602890256 - Original comment: **</comment>
      <origin>4352</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="69987" sha1="bfv8xesn5hexw1uxxwqub7c3oqxwycz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2016-12-29 07:53:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602890256&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
||~ Harmonic ||~ 8 ||~   ||~ 9 ||~   ||~ 11 ||~   ||~ 14 ||~   ||~ 16 ||
||~ JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
||~ nearest 11edo interval || 0\11edo = 0¢ ||   || 2\11 = 218¢ ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
||~ difference || 0 ||   || +14¢ ||   || -6¢ ||   || +13¢ ||   || 0¢ ||
||~ JI interval between ||   || 9:8 = 204¢ ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
||~ nearest 11edo interval ||   || 2\11 = 218¢ ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
||~ difference ||   || +14¢ ||   || -20¢ ||   || +18¢ ||   || -13¢ ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

=Notation= 

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the __melodic__ meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

||~ Degree ||~ Size in
[[cent|Cents]] ||~ Solfege ||~ Approximate Ratios* ||~ Sagittal
notation
(22edo subset) ||||~ [[Ups and Downs Notation|Up/down notation]]
with major wider
than minor ||||~ Up/down notation
with major narrower
than minor ||~ [[Tútim Dennsuul Wafiil|TDW]]
Machine
notation ||
||= 0 ||&amp;gt; 0.00 ||= **do** ||= 1/1 ||= A ||= P1 ||= A ||= P1 ||= A ||= Q\P# ||
||= 1 ||&amp;gt; 109.09 ||= **ra** ||= [[15_14|15/14]], [[16_15|16/15]], [[17_16|17/16]], [[18_17|18/17]] ||= AII\ or B!!/ ||= ^1, m2 ||= A^, B ||= ^1, M2 ||= A^, B ||= Q#\Rb ||
||= 2 ||&amp;gt; 218.18 ||= **re** ||= [[8_7|8/7]], [[9_8|9/8]], [[17_15|17/15]] ||= B ||= ~2, m3 ||= B^, Cb ||= ~2, M3 ||= B^, C# ||= R ||
||= 3 ||&amp;gt; 327.27 ||= **me** ||= [[6_5|6/5]], [[11_9|11/9]], [[17_14|17/14]] ||= C/I or BII\ or D\!!/ ||= M2, ~3 ||= B#, Cv ||= m2, ~3 ||= Bb, Cv ||= R#\Sb ||
||= 4 ||&amp;gt; 436.36 ||= **mo** ||= [[9_7|9/7]], [[14_11|14/11]], [[22_17|22/17]] ||= D\! or C/II\ ||= M3, v4 ||= C, Dv ||= m3, v4 ||= C, Dv ||= S ||
||= 5 ||&amp;gt; 545.45 ||= **fu** ||= [[11_8|11/8]], [[15_11|15/11]] ||= D/I or E\!!/ ||= P4, v5 ||= D, Ev ||= P4, v5 ||= D, Ev ||= S#\Tb ||
||= 6 ||&amp;gt; 654.55 ||= **su** ||= [[16_11|16/11]], [[22_15|22/15]] ||= E\! or D/II\ ||= ^4, P5 ||= D^, E ||= ^4, P5 ||= D^, E ||= T ||
||= 7 ||&amp;gt; 763.64 ||= **lo** ||= [[11_7|11/7]], [[14_9|14/9]], [[17_11|17/11]] ||= F ||= ^5, m6 ||= E^, Fb ||= ^5, M6 ||= E^, F# ||= T#\Ub ||
||= 8 ||&amp;gt; 872.73 ||= **la** ||= [[5_3|5/3]], [[18_11|18/11]], [[28_17|28/17]] ||= FII\ or G!!/ ||= ~6, m7 ||= Fv, Gb ||= ~6, M7 ||= Fv, G# ||= U ||
||= 9 ||&amp;gt; 981.82 ||= **ta** ||= [[7_4|7/4]], [[16_9|16/9]], [[30_17|30/17]] ||= G ||= M6, ~7 ||= F, Gv ||= m6, ~7 ||= F, Gv ||= U#\Pb ||
||= 10 ||&amp;gt; 1090.91 ||= **ti** ||= [[15_8|15/8]], [[17_9|17/9]], [[28_15|28/15]], [[32_17|32/17]] ||= GII\ or A!!/ ||= M7, v8 ||= G, Av ||= m7, v8 ||= G, Av ||= P\Qb ||
||= 11 ||&amp;gt; 1200.00 ||= **do** ||= 2/1 ||= A ||= P8 ||= A ||= P8 ||= A ||= Q\P# ||
*in 2.7.9.11.15.17 subgroup

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Rational ||~ Monzo ||~ Size ([[cent|Cents]]) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= [[135_128|135/128]] || | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 || | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 || | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137  || | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 || | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= [[225_224|225/224]] || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 || | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 || | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 || | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 || | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#Notation"&amp;gt;Notation&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:42: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:42 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:43: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:43 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:44: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:44 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:658:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:658 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Notation"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Notation&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the &amp;lt;u&amp;gt;melodic&amp;lt;/u&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The second approach preserves the &amp;lt;u&amp;gt;harmonic&amp;lt;/u&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo &amp;amp;quot;on the fly&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Degree&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size in&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/cent"&amp;gt;Cents&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Solfege&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Approximate Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Sagittal&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
(22edo subset)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;Up/down notation&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
with major wider&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;Up/down notation&amp;lt;br /&amp;gt;
with major narrower&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
Machine&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/15_14"&amp;gt;15/14&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16_15"&amp;gt;16/15&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_16"&amp;gt;17/16&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/18_17"&amp;gt;18/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, m2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, M2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/8_7"&amp;gt;8/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/9_8"&amp;gt;9/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_15"&amp;gt;17/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, Cb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, M3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, C#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/11_9"&amp;gt;11/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_14"&amp;gt;17/14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B#, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Bb, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/9_7"&amp;gt;9/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/14_11"&amp;gt;14/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22_17"&amp;gt;22/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/11_8"&amp;gt;11/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/15_11"&amp;gt;15/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/16_11"&amp;gt;16/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22_15"&amp;gt;22/15&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/11_7"&amp;gt;11/7&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/14_9"&amp;gt;14/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_11"&amp;gt;17/11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, Fb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, M6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, F#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/18_11"&amp;gt;18/11&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_17"&amp;gt;28/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, Gb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, M7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, G#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/16_9"&amp;gt;16/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/30_17"&amp;gt;30/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/15_8"&amp;gt;15/8&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/17_9"&amp;gt;17/9&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/28_15"&amp;gt;28/15&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/32_17"&amp;gt;32/17&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1200.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Supersharp&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic, octotonic and nonatonic fifth-generated) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic third-generated).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Rational&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size (&amp;lt;a class="wiki_link" href="/cent"&amp;gt;Cents&amp;lt;/a&amp;gt;)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/135_128"&amp;gt;135/128&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/225_224"&amp;gt;225/224&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc11"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:27:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc13"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:27 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:659:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:659 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>bfv8xesn5hexw1uxxwqub7c3oqxwycz</sha1>
    </revision>
    <revision>
      <id>4353</id>
      <timestamp>2016-12-25T22:18:01Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;TallKite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602809330 - Original comment: **</comment>
      <origin>4353</origin>
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      <text bytes="68345" sha1="92ctglx9esa17jwnhjpunxn0h6npj77" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:TallKite|TallKite]] and made on &lt;tt&gt;2016-12-25 22:18:01 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602809330&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0¢ ||   || 2\11 = 218¢ ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14¢ ||   || -6¢ ||   || +13¢ ||   || 0¢ ||
|| JI interval between ||   || 9:8 = 204¢ ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218¢ ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14¢ ||   || -20¢ ||   || +18¢ ||   || -13¢ ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

=Notation= 

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the __melodic__ meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

||= degree ||= cents ||= solfege ||= Approximate
Ratios* ||= Sagittal
notation
(22edo subset) ||||= [[Ups and Downs Notation|Up/down notation ]]
with major wider
than minor ||||= Up/down notation
with major narrower
than minor ||= [[Tútim Dennsuul Wafiil|TDW]]
Machine
notation ||
||= 0 ||= 0.00 ||= **do** ||= 1/1 ||= A ||= P1 ||= A ||= P1 ||= A ||= Q\P# ||
||= 1 ||= 109.09 ||= **ra** ||= 15/14, 16/15,
17/16, 18/17 ||= AII\ or B!!/ ||= ^1, m2 ||= A^, B ||= ^1, M2 ||= A^, B ||= Q#\Rb ||
||= 2 ||= 218.18 ||= **re** ||= 8/7, 9/8, 17/15 ||= B ||= ~2, m3 ||= B^, Cb ||= ~2, M3 ||= B^, C# ||= R ||
||= 3 ||= 327.27 ||= **me** ||= 6/5, 11/9, 17/14 ||= C/I or BII\ or D\!!/ ||= M2, ~3 ||= B#, Cv ||= m2, ~3 ||= Bb, Cv ||= R#\Sb ||
||= 4 ||= 436.36 ||= **mo** ||= 9/7, 14/11, 22/17 ||= D\! or C/II\ ||= M3, v4 ||= C, Dv ||= m3, v4 ||= C, Dv ||= S ||
||= 5 ||= 545.45 ||= **fu** ||= 11/8, 15/11 ||= D/I or E\!!/ ||= P4, v5 ||= D, Ev ||= P4, v5 ||= D, Ev ||= S#\Tb ||
||= 6 ||= 654.55 ||= **su** ||= 16/11, 22/15 ||= E\! or D/II\ ||= ^4, P5 ||= D^, E ||= ^4, P5 ||= D^, E ||= T ||
||= 7 ||= 763.64 ||= **lo** ||= 11/7, 14/9, 17/11 ||= F ||= ^5, m6 ||= E^, Fb ||= ^5, M6 ||= E^, F# ||= T#\Ub ||
||= 8 ||= 872.73 ||= **la** ||= 5/3, 18/11, 28/17 ||= FII\ or G!!/ ||= ~6, m7 ||= Fv, Gb ||= ~6, M7 ||= Fv, G# ||= U ||
||= 9 ||= 981.82 ||= **ta** ||= 7/4, 16/9, 30/17 ||= G ||= M6, ~7 ||= F, Gv ||= m6, ~7 ||= F, Gv ||= U#\Pb ||
||= 10 ||= 1090.91 ||= **ti** ||= 15/8, 17/9,
28/15, 32/17 ||= GII\ or A!!/ ||= M7, v8 ||= G, Av ||= m7, v8 ||= G, Av ||= P\Qb ||
||= 11 ||= 1200 ||= **do** ||= 2/1 ||= A ||= P8 ||= A ||= P8 ||= A ||= Q\P# ||
*in 2.7.9.11.15.17 subgroup

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||=   ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||=   ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||=   ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#Notation"&amp;gt;Notation&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:42: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:42 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:43: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:43 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:44: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:44 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:658:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:658 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Notation"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Notation&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the &amp;lt;u&amp;gt;melodic&amp;lt;/u&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The second approach preserves the &amp;lt;u&amp;gt;harmonic&amp;lt;/u&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo &amp;amp;quot;on the fly&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Sagittal&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
(22edo subset)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td colspan="2" style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;Up/down notation &amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
with major wider&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td colspan="2" style="text-align: center;"&amp;gt;Up/down notation&amp;lt;br /&amp;gt;
with major narrower&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
Machine&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;15/14, 16/15,&amp;lt;br /&amp;gt;
17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, m2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, M2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, Cb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, M3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, C#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B#, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Bb, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, Fb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, M6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, F#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, Gb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, M7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, G#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;15/8, 17/9,&amp;lt;br /&amp;gt;
28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Supersharp&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic, octotonic and nonatonic fifth-generated) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic third-generated).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc11"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:27:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc13"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:27 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:659:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:659 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>92ctglx9esa17jwnhjpunxn0h6npj77</sha1>
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    <revision>
      <id>4354</id>
      <timestamp>2016-12-25T21:43:34Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;TallKite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602809202 - Original comment: **</comment>
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      <text bytes="68333" sha1="lnmc68l9bbkc5y6k366udl76bp3sk7o" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:TallKite|TallKite]] and made on &lt;tt&gt;2016-12-25 21:43:34 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602809202&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0¢ ||   || 2\11 = 218¢ ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14¢ ||   || -6¢ ||   || +13¢ ||   || 0¢ ||
|| JI interval between ||   || 9:8 = 204¢ ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218¢ ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14¢ ||   || -20¢ ||   || +18¢ ||   || -13¢ ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

=Notation= 

11edo can be notated by adding ups and downs to conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the __melodic__ meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the __harmonic__ meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

||= degree ||= cents ||= solfege ||= Approximate
Ratios* ||= Sagittal
notation
(22edo subset) ||||= [[Ups and Downs Notation|Up/down notation ]]
with major wider
than minor ||||= Up/down notation
with major narrower
than minor ||= [[Tútim Dennsuul Wafiil|TDW]]
Machine
notation ||
||= 0 ||= 0.00 ||= **do** ||= 1/1 ||= A ||= P1 ||= A ||= P1 ||= A ||= Q\P# ||
||= 1 ||= 109.09 ||= **ra** ||= 15/14, 16/15,
17/16, 18/17 ||= AII\ or B!!/ ||= ^1, m2 ||= A^, B ||= ^1, M2 ||= A^, B ||= Q#\Rb ||
||= 2 ||= 218.18 ||= **re** ||= 8/7, 9/8, 17/15 ||= B ||= ~2, m3 ||= B^, Cb ||= ~2, M3 ||= B^, C# ||= R ||
||= 3 ||= 327.27 ||= **me** ||= 6/5, 11/9, 17/14 ||= C/I or BII\ or D\!!/ ||= M2, ~3 ||= B#, Cv ||= m2, ~3 ||= Bb, Cv ||= R#\Sb ||
||= 4 ||= 436.36 ||= **mo** ||= 9/7, 14/11, 22/17 ||= D\! or C/II\ ||= M3, v4 ||= C, Dv ||= m3, v4 ||= C, Dv ||= S ||
||= 5 ||= 545.45 ||= **fu** ||= 11/8, 15/11 ||= D/I or E\!!/ ||= P4, v5 ||= D, Ev ||= P4, v5 ||= D, Ev ||= S#\Tb ||
||= 6 ||= 654.55 ||= **su** ||= 16/11, 22/15 ||= E\! or D/II\ ||= ^4, P5 ||= D^, E ||= ^4, P5 ||= D^, E ||= T ||
||= 7 ||= 763.64 ||= **lo** ||= 11/7, 14/9, 17/11 ||= F ||= ^5, m6 ||= E^, Fb ||= ^5, M6 ||= E^, F# ||= T#\Ub ||
||= 8 ||= 872.73 ||= **la** ||= 5/3, 18/11, 28/17 ||= FII\ or G!!/ ||= ~6, m7 ||= Fv, Gb ||= ~6, M7 ||= Fv, G# ||= U ||
||= 9 ||= 981.82 ||= **ta** ||= 7/4, 16/9, 30/17 ||= G ||= M6, ~7 ||= F, Gv ||= m6, ~7 ||= F, Gv ||= U#\Pb ||
||= 10 ||= 1090.91 ||= **ti** ||= 15/8, 17/9,
28/15, 32/17 ||= GII\ or A!!/ ||= M7, v8 ||= G, Av ||= m7, v8 ||= G, Av ||= P\Qb ||
||= 11 ||= 1200 ||= **do** ||= 2/1 ||= A ||= P8 ||= A ||= P8 ||= A ||= Q\P# ||
*in 2.7.9.11.15.17 subgroup

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||=   ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||=   ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||=   ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#Notation"&amp;gt;Notation&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:42: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:42 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:43: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:43 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:44: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:44 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:658:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:658 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="Notation"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Notation&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11edo can be notated by adding ups and downs to conventional notation, including the staff, note names, relative notation, etc. in two ways. The first preserves the &amp;lt;u&amp;gt;melodic&amp;lt;/u&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The second approach preserves the &amp;lt;u&amp;gt;harmonic&amp;lt;/u&amp;gt; meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo &amp;amp;quot;on the fly&amp;amp;quot;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;degree&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Approximate&amp;lt;br /&amp;gt;
Ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Sagittal&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
(22edo subset)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td colspan="2" style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;Up/down notation &amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
with major wider&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td colspan="2" style="text-align: center;"&amp;gt;Up/down notation&amp;lt;br /&amp;gt;
with major narrower&amp;lt;br /&amp;gt;
than minor&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
Machine&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;15/14, 16/15,&amp;lt;br /&amp;gt;
17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, m2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^1, M2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A^, B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, m3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, Cb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~2, M3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B^, C#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;B#, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m2, ~3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Bb, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m3, v4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;C, Dv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P4, v5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D, Ev&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^4, P5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;D^, E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, m6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, Fb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;^5, M6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;E^, F#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, m7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, Gb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;~6, M7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Fv, G#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m6, ~7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;F, Gv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;15/8, 17/9,&amp;lt;br /&amp;gt;
28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;M7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;m7, v8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;G, Av&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;P8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Supersharp&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; (pentatonic, octotonic and nonatonic fifth-generated) and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt; (heptatonic third-generated).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc11"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:27:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc13"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:27 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:659:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:659 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>lnmc68l9bbkc5y6k366udl76bp3sk7o</sha1>
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    <revision>
      <id>4355</id>
      <timestamp>2016-12-22T03:57:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;TallKite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602738160 - Original comment: **</comment>
      <origin>4355</origin>
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      <text bytes="59670" sha1="o2wvdjkqfybtukatib9jew2qxfrmqqe" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:TallKite|TallKite]] and made on &lt;tt&gt;2016-12-22 03:57:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602738160&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0¢ ||   || 2\11 = 218¢ ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14¢ ||   || -6¢ ||   || +13¢ ||   || 0¢ ||
|| JI interval between ||   || 9:8 = 204¢ ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218¢ ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14¢ ||   || -20¢ ||   || +18¢ ||   || -13¢ ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents || solfege || ratios* || Sagittal
notation || [[Ups and Downs Notation|up/down notation ]]
with sharp
lower than flat || [[Tútim Dennsuul Wafiil|TDW]] Machine
notation ||
|| 0 || 0.00 || **do** || 1/1 || A || A || Q\P# ||
|| 1 || 109.09 || **ra** || 15/14, 16/15,
17/16, 18/17 || AII\ or B!!/ || B || Q#\Rb ||
|| 2 || 218.18 || **re** || 8/7, 9/8, 17/15 || B || B^, C# || R ||
|| 3 || 327.27 || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || Bb, Cv || R#\Sb ||
|| 4 || 436.36 || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || C || S ||
|| 5 || 545.45 || **fu** || 11/8, 15/11 || D/I or E\!!/ || D || S#\Tb ||
|| 6 || 654.55 || **su** || 16/11, 22/15 || E\! or D/II\ || E || T ||
|| 7 || 763.64 || **lo** || 11/7, 14/9, 17/11 || F || E^, F# || T#\Ub ||
|| 8 || 872.73 || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || Eb, Fv || U ||
|| 9 || 981.82 || **ta** || 7/4, 16/9, 30/17 || G || F || U#\Pb ||
|| 10 || 1090.91 || **ti** || 15/8, 17/9,
28/15, 32/17 || GII\ or A!!/ || G || P\Qb ||
*in 2.7.9.11.15.17 subgroup

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs|Ups and Downs Notation -"Superflat" EDOs]] and [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41/1 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||=   ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||=   ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||=   ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13¢&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:565:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:565 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;up/down notation &amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
with sharp&amp;lt;br /&amp;gt;
lower than flat&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15,&amp;lt;br /&amp;gt;
17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B^, C#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Bb, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E^, F#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Eb, Fv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9,&amp;lt;br /&amp;gt;
28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Superflat&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Supersharp&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41/1 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc10"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:566:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:566 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4356</id>
      <timestamp>2016-12-04T07:04:41Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 601321854 - Original comment: **</comment>
      <origin>4356</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="60336" sha1="czpgybhc6n02pe7b99tubw12i5ytto8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-12-04 07:04:41 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;601321854&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value (coarse/fine)
DMS value || solfege || ratios* || Sagittal
notation || [[Ups and Downs Notation|up/down notation ]]
with sharp
lower than flat || [[Tútim Dennsuul Wafiil|TDW]] Machine
notation ||
|| 0 || 0.00 || **do** || 1/1 || A || A || Q\P# ||
|| 1 || 109.09, 130.91
32°43'38" || **ra** || 15/14, 16/15,
17/16, 18/17 || AII\ or B!!/ || B || Q#\Rb ||
|| 2 || 218.18, 261.82
65°27'16" || **re** || 8/7, 9/8, 17/15 || B || B^, C# || R ||
|| 3 || 327.27, 392.73
98°10'55" || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || Bb, Cv || R#\Sb ||
|| 4 || 436.36, 523.64
120°54'33" || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || C || S ||
|| 5 || 545.455, 654.545
163°38'11" || **fu** || 11/8, 15/11 || D/I or E\!!/ || D || S#\Tb ||
|| 6 || 654.545, 785.455
196°21'49" || **su** || 16/11, 22/15 || E\! or D/II\ || E || T ||
|| 7 || 763.64, 916.36
229°5'27" || **lo** || 11/7, 14/9, 17/11 || F || E^, F# || T#\Ub ||
|| 8 || 872.73, 1047.27
261°49'5" || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || Eb, Fv || U ||
|| 9 || 981.82, 1178.18
294°31'44" || **ta** || 7/4, 16/9, 30/17 || G || F || U#\Pb ||
|| 10 || 1090.91, 1309.09
327°16'22" || **ti** || 15/8, 17/9,
28/15, 32/17 || GII\ or A!!/ || G || P\Qb ||
*in 2.7.9.11.15.17 subgroup

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs|Ups and Downs Notation -"Superflat" EDOs]] and [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41/1 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||=   ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||=   ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||=   ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt;
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&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:565:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:565 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value (coarse/fine)&amp;lt;br /&amp;gt;
DMS value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;up/down notation &amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
with sharp&amp;lt;br /&amp;gt;
lower than flat&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09, 130.91&amp;lt;br /&amp;gt;
32°43'38&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15,&amp;lt;br /&amp;gt;
17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18, 261.82&amp;lt;br /&amp;gt;
65°27'16&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B^, C#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27, 392.73&amp;lt;br /&amp;gt;
98°10'55&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Bb, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36, 523.64&amp;lt;br /&amp;gt;
120°54'33&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.455, 654.545&amp;lt;br /&amp;gt;
163°38'11&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.545, 785.455&amp;lt;br /&amp;gt;
196°21'49&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64, 916.36&amp;lt;br /&amp;gt;
229°5'27&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E^, F#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73, 1047.27&amp;lt;br /&amp;gt;
261°49'5&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Eb, Fv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82, 1178.18&amp;lt;br /&amp;gt;
294°31'44&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91, 1309.09&amp;lt;br /&amp;gt;
327°16'22&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9,&amp;lt;br /&amp;gt;
28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Superflat&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Supersharp&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41/1 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc10"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:566:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:566 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>czpgybhc6n02pe7b99tubw12i5ytto8</sha1>
    </revision>
    <revision>
      <id>4357</id>
      <timestamp>2016-11-21T07:17:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;TallKite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 599961072 - Original comment: **</comment>
      <origin>4357</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="60417" sha1="h9lisxkwiktw786t4tvzmx9t5sb6eqf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:TallKite|TallKite]] and made on &lt;tt&gt;2016-11-21 07:17:39 UTC&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value (coarse/fine)
DMS value || solfege || ratios* || Sagittal
notation || [[Ups and Downs Notation|up/down notation ]]
with sharp 
lower than flat || [[Tútim Dennsuul Wafiil|TDW]] Machine
notation ||
|| 0 || 0.00 || **do** || 1/1 || A || A || Q\P# ||
|| 1 || 109.09, 130.91
32°43'38" || **ra** || 15/14, 16/15,
17/16, 18/17 || AII\ or B!!/ || B || Q#\Rb ||
|| 2 || 218.18, 261.82
65°27'16" || **re** || 8/7, 9/8, 17/15 || B || B^, C# || R ||
|| 3 || 327.27, 392.73
98°10'55" || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || Bb, Cv || R#\Sb ||
|| 4 || 436.36, 523.64
120°54'33" || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || C || S ||
|| 5 || 545.455, 654.545
163°38'11" || **fu** || 11/8, 15/11 || D/I or E\!!/ || D || S#\Tb ||
|| 6 || 654.545, 785.455
196°21'49" || **su** || 16/11, 22/15 || E\! or D/II\ || E || T ||
|| 7 || 763.64, 916.36
229°5'27" || **lo** || 11/7, 14/9, 17/11 || F || E^, F# || T#\Ub ||
|| 8 || 872.73, 1047.27
261°49'5" || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || Eb, Fv || U ||
|| 9 || 981.82, 1178.18
294°31'44" || **ta** || 7/4, 16/9, 30/17 || G || F || U#\Pb ||
|| 10 || 1090.91, 1309.09
327°16'22" || **ti** || 15/8, 17/9,
28/15, 32/17 || GII\ or A!!/ || G || P\Qb ||
*in 2.7.9.11.15.17 subgroup

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs|Ups and Downs Notation -"Superflat" EDOs]] and [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:565:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:565 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value (coarse/fine)&amp;lt;br /&amp;gt;
DMS value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&amp;gt;up/down notation &amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
with sharp &amp;lt;br /&amp;gt;
lower than flat&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09, 130.91&amp;lt;br /&amp;gt;
32°43'38&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15,&amp;lt;br /&amp;gt;
17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18, 261.82&amp;lt;br /&amp;gt;
65°27'16&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B^, C#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27, 392.73&amp;lt;br /&amp;gt;
98°10'55&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Bb, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36, 523.64&amp;lt;br /&amp;gt;
120°54'33&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.455, 654.545&amp;lt;br /&amp;gt;
163°38'11&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.545, 785.455&amp;lt;br /&amp;gt;
196°21'49&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64, 916.36&amp;lt;br /&amp;gt;
229°5'27&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E^, F#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73, 1047.27&amp;lt;br /&amp;gt;
261°49'5&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Eb, Fv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82, 1178.18&amp;lt;br /&amp;gt;
294°31'44&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91, 1309.09&amp;lt;br /&amp;gt;
327°16'22&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9,&amp;lt;br /&amp;gt;
28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Superflat&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Supersharp&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc10"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:566:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:566 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>h9lisxkwiktw786t4tvzmx9t5sb6eqf</sha1>
    </revision>
    <revision>
      <id>4358</id>
      <timestamp>2016-11-21T05:08:02Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;TallKite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 599950052 - Original comment: **</comment>
      <origin>4358</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="60252" sha1="gclgfleif47sbbjic4ix566x52nls4o" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:TallKite|TallKite]] and made on &lt;tt&gt;2016-11-21 05:08:02 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;599950052&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value (coarse/fine)
DMS value || solfege || ratios* || Sagittal 
notation || up/down
notation || [[Tútim Dennsuul Wafiil|TDW]] Machine
notation ||
|| 0 || 0.00 || **do** || 1/1 || A || A || Q\P# ||
|| 1 || 109.09, 130.91
32°43'38" || **ra** || 15/14, 16/15, 
17/16, 18/17 || AII\ or B!!/ || B || Q#\Rb ||
|| 2 || 218.18, 261.82
65°27'16" || **re** || 8/7, 9/8, 17/15 || B || B^, C# || R ||
|| 3 || 327.27, 392.73
98°10'55" || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || Bb, Cv || R#\Sb ||
|| 4 || 436.36, 523.64
120°54'33" || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || C || S ||
|| 5 || 545.455, 654.545
163°38'11" || **fu** || 11/8, 15/11 || D/I or E\!!/ || D || S#\Tb ||
|| 6 || 654.545, 785.455
196°21'49" || **su** || 16/11, 22/15 || E\! or D/II\ || E || T ||
|| 7 || 763.64, 916.36
229°5'27" || **lo** || 11/7, 14/9, 17/11 || F || E^, F# || T#\Ub ||
|| 8 || 872.73, 1047.27
261°49'5" || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || Eb, Fv || U ||
|| 9 || 981.82, 1178.18
294°31'44" || **ta** || 7/4, 16/9, 30/17 || G || F || U#\Pb ||
|| 10 || 1090.91, 1309.09
327°16'22" || **ti** || 15/8, 17/9, 
28/15, 32/17 || GII\ or A!!/ || G || P\Qb ||
*in 2.7.9.11.15.17 subgroup

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs|Ups and Downs Notation -"Superflat" EDOs]] and [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:565:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:565 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value (coarse/fine)&amp;lt;br /&amp;gt;
DMS value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal &amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;up/down&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine&amp;lt;br /&amp;gt;
notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09, 130.91&amp;lt;br /&amp;gt;
32°43'38&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, &amp;lt;br /&amp;gt;
17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18, 261.82&amp;lt;br /&amp;gt;
65°27'16&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B^, C#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27, 392.73&amp;lt;br /&amp;gt;
98°10'55&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Bb, Cv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36, 523.64&amp;lt;br /&amp;gt;
120°54'33&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.455, 654.545&amp;lt;br /&amp;gt;
163°38'11&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.545, 785.455&amp;lt;br /&amp;gt;
196°21'49&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64, 916.36&amp;lt;br /&amp;gt;
229°5'27&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E^, F#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73, 1047.27&amp;lt;br /&amp;gt;
261°49'5&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Eb, Fv&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82, 1178.18&amp;lt;br /&amp;gt;
294°31'44&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91, 1309.09&amp;lt;br /&amp;gt;
327°16'22&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, &amp;lt;br /&amp;gt;
28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
For alternative notations, see &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Superflat&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&amp;gt;Ups and Downs Notation -&amp;amp;quot;Supersharp&amp;amp;quot; EDOs&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&amp;gt;Ups and Downs Notation - Natural Generators&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc10"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:566:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:566 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>gclgfleif47sbbjic4ix566x52nls4o</sha1>
    </revision>
    <revision>
      <id>4359</id>
      <timestamp>2016-11-01T18:22:32Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 597692160 - Original comment: removed tel links**</comment>
      <origin>4359</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="58439" sha1="a124qs0j358wi294zlcebqx1ywe5een" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2016-11-01 18:22:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;597692160&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;removed tel links&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value (coarse/fine)
DMS value || solfege || ratios* || Sagittal notation || [[Tútim Dennsuul Wafiil|TDW]] Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09, 130.91
32°43'38" || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18, 261.82
65°27'16" || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27, 392.73
98°10'55" || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36, 523.64
120°54'33" || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.455, 654.545
163°38'11" || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.545, 785.455
196°21'49" || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64, 916.36
229°5'27" || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73, 1047.27
261°49'5" || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82, 1178.18
294°31'44" || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91, 1309.09
327°16'22" || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

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&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:541:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:541 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value (coarse/fine)&amp;lt;br /&amp;gt;
DMS value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09, 130.91&amp;lt;br /&amp;gt;
32°43'38&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18, 261.82&amp;lt;br /&amp;gt;
65°27'16&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27, 392.73&amp;lt;br /&amp;gt;
98°10'55&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36, 523.64&amp;lt;br /&amp;gt;
120°54'33&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.455, 654.545&amp;lt;br /&amp;gt;
163°38'11&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.545, 785.455&amp;lt;br /&amp;gt;
196°21'49&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64, 916.36&amp;lt;br /&amp;gt;
229°5'27&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73, 1047.27&amp;lt;br /&amp;gt;
261°49'5&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82, 1178.18&amp;lt;br /&amp;gt;
294°31'44&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91, 1309.09&amp;lt;br /&amp;gt;
327°16'22&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc10"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:542:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:542 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>a124qs0j358wi294zlcebqx1ywe5een</sha1>
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    <revision>
      <id>4360</id>
      <timestamp>2016-10-30T21:09:01Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 597515996 - Original comment: **</comment>
      <origin>4360</origin>
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      <text bytes="58696" sha1="1ip5104724keqpdocrc0m2krmaurtwa" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-30 21:09:01 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;597515996&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value (coarse/fine)
DMS value || solfege || ratios* || Sagittal notation || [[Tútim Dennsuul Wafiil|TDW]] Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09, 130.91
32°43'38" || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18, 261.82
65°27'16" || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27, 392.73
98°10'55" || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36, 523.64
120°54'33" || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.455, 654.545
163°38'11" || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.545, 785.455
196°21'49" || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64, 916.36
229°5'27" || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73, 1047.27
261°49'5" || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82, 1178.18
294°31'44" || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91, 1309.09
327°16'22" || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; [[tel:11 17 26 31 38 41|11 17 26 31 38 41]] |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= [[tel:9931568/9752117|9931568/9752117]] ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= [[tel:1776337/1773750|1776337/1773750]] ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= [[tel:9859966/9733137|9859966/9733137]] ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:541:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:541 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value (coarse/fine)&amp;lt;br /&amp;gt;
DMS value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09, 130.91&amp;lt;br /&amp;gt;
32°43'38&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18, 261.82&amp;lt;br /&amp;gt;
65°27'16&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27, 392.73&amp;lt;br /&amp;gt;
98°10'55&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36, 523.64&amp;lt;br /&amp;gt;
120°54'33&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.455, 654.545&amp;lt;br /&amp;gt;
163°38'11&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.545, 785.455&amp;lt;br /&amp;gt;
196°21'49&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64, 916.36&amp;lt;br /&amp;gt;
229°5'27&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73, 1047.27&amp;lt;br /&amp;gt;
261°49'5&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82, 1178.18&amp;lt;br /&amp;gt;
294°31'44&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91, 1309.09&amp;lt;br /&amp;gt;
327°16'22&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; &amp;lt;a class="wiki_link" href="http://tel.wikispaces.com/11%2017%2026%2031%2038%2041"&amp;gt;11 17 26 31 38 41&amp;lt;/a&amp;gt; |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;[[tel:9931568/9752117|9931568/9752117]]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;[[tel:1776337/1773750|1776337/1773750]]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;[[tel:9859966/9733137|9859966/9733137]]&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc10"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:542:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:542 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>1ip5104724keqpdocrc0m2krmaurtwa</sha1>
    </revision>
    <revision>
      <id>4361</id>
      <timestamp>2016-10-30T12:07:06Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 597484860 - Original comment: **</comment>
      <origin>4361</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="58420" sha1="mzx4iaj1kpqobrza22g9xu8envo9he4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-30 12:07:06 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;597484860&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value (coarse/fine)
DMS value || solfege || ratios* || Sagittal notation || [[Tútim Dennsuul Wafiil|TDW]] Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09, 130.91
32°43'38" || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18, 261.82
65°27'16" || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27, 392.73
98°10'55" || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36, 523.64
120°54'33" || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45, 654.55
163°38'11" || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55, 785.45
196°21'49" || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64, 916.36
229°5'27" || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73, 1047.27
261°49'5" || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82, 1178.18
294°31'44" || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91, 1309.09
327°16'22" || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="/11%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:541:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:541 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value (coarse/fine)&amp;lt;br /&amp;gt;
DMS value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09, 130.91&amp;lt;br /&amp;gt;
32°43'38&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18, 261.82&amp;lt;br /&amp;gt;
65°27'16&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27, 392.73&amp;lt;br /&amp;gt;
98°10'55&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36, 523.64&amp;lt;br /&amp;gt;
120°54'33&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45, 654.55&amp;lt;br /&amp;gt;
163°38'11&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55, 785.45&amp;lt;br /&amp;gt;
196°21'49&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64, 916.36&amp;lt;br /&amp;gt;
229°5'27&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73, 1047.27&amp;lt;br /&amp;gt;
261°49'5&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82, 1178.18&amp;lt;br /&amp;gt;
294°31'44&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91, 1309.09&amp;lt;br /&amp;gt;
327°16'22&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc10"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:542:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:542 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4362</id>
      <timestamp>2015-08-04T07:37:47Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 556156107 - Original comment: **</comment>
      <origin>4362</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="58220" sha1="ojka0qc4i6ruh6sk9jg9jkhn46isx11" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2015-08-04 07:37:47 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;556156107&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[11平均律|日本語]]
&amp;lt;/span&amp;gt;
[[media type="custom" key="23473432"]][[toc|flat]]
= = 
----
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value
DMS value || solfege || ratios* || Sagittal notation || [[Tútim Dennsuul Wafiil|TDW]] Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09
32°43'38" || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18
65°27'16" || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27
98°10'55" || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36
120°54'33" || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45
163°38'11" || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55
196°21'49" || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64
229°5'27" || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73
261°49'5" || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82
294°31'44" || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91
327°16'22" || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;a href="#toc0"&amp;gt; &amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:39: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:39 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:40: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:40 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:41: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:41 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt; &amp;lt;/h1&amp;gt;
 &amp;lt;hr /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc4"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:541:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:541 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc5"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
DMS value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
32°43'38&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
65°27'16&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
98°10'55&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
120°54'33&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
163°38'11&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
196°21'49&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
229°5'27&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
261°49'5&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
294°31'44&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
327°16'22&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc10"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:25:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc12"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:25 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:542:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:542 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <id>4363</id>
      <timestamp>2015-03-24T15:36:32Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 545182904 - Original comment: **</comment>
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      <text bytes="57667" sha1="f99c5a32ezwb31m53zlln8y4sw7753v" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-03-24 15:36:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;545182904&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="23473432"]][[toc|flat]]
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value
DMS value || solfege || ratios* || Sagittal notation || [[Tútim Dennsuul Wafiil|TDW]] Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09
32°43'38" || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18
65°27'16" || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27
98°10'55" || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36
120°54'33" || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45
163°38'11" || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55
196°21'49" || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64
229°5'27" || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73
261°49'5" || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82
294°31'44" || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91
327°16'22" || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:537:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:537 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
DMS value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
32°43'38&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
65°27'16&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
98°10'55&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
120°54'33&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
163°38'11&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
196°21'49&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
229°5'27&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
261°49'5&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
294°31'44&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
327°16'22&amp;amp;quot;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:538:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:538 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>f99c5a32ezwb31m53zlln8y4sw7753v</sha1>
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    <revision>
      <id>4364</id>
      <timestamp>2013-09-21T14:01:46Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 452973368 - Original comment: **</comment>
      <origin>4364</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="57201" sha1="52y36rofmik8gz1qz5mlwqeq11c6uu9" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2013-09-21 14:01:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;452973368&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="23473432"]][[toc|flat]]
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value || solfege || ratios* || Sagittal notation || [[Tútim Dennsuul Wafiil|TDW]] Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09 || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27 || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73 || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82 || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//[[http://www.akjmusic.com/audio/black_ritual_dirge.mp3|Black Ritual Dirge]]// by [[Aaron Krister Johnson]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:537:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:537 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.akjmusic.com/audio/black_ritual_dirge.mp3" rel="nofollow"&amp;gt;Black Ritual Dirge&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Krister%20Johnson"&amp;gt;Aaron Krister Johnson&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:538:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:538 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>52y36rofmik8gz1qz5mlwqeq11c6uu9</sha1>
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    <revision>
      <id>4365</id>
      <timestamp>2013-07-13T23:16:22Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 441269318 - Original comment: **</comment>
      <origin>4365</origin>
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      <text bytes="56816" sha1="9a21lsti4w7n9hncqs1ldc8s1kz1gy1" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2013-07-13 23:16:22 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;441269318&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="23473432"]][[toc|flat]]
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value || solfege || ratios* || Sagittal notation || [[Tútim Dennsuul Wafiil|TDW]] Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09 || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27 || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73 || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82 || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/23473432?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;23473432&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player-beta.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:537:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:537 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:538:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:538 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9a21lsti4w7n9hncqs1ldc8s1kz1gy1</sha1>
    </revision>
    <revision>
      <id>4366</id>
      <timestamp>2013-05-06T23:37:05Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 429431406 - Original comment: **</comment>
      <origin>4366</origin>
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      <text bytes="56820" sha1="9i5y3yspmsq6jw2xxnybaxogqlbdnrw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2013-05-06 23:37:05 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;429431406&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value || solfege || ratios* || Sagittal notation || [[Tútim Dennsuul Wafiil|TDW]] Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09 || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27 || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73 || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82 || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:537:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:537 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia%20FAQ"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
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      <id>4367</id>
      <timestamp>2013-05-06T05:27:32Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
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      <comment>**Imported revision 429138506 - Original comment: **</comment>
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      <text bytes="56370" sha1="bj0mubs1lthunmko8yjrrpfoybnoshw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2013-05-06 05:27:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;429138506&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;= 

11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value || solfege || ratios* || Sagittal notation || [[Tútim Dennsuul Wafiil|TDW]] Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09 || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27 || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73 || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82 || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
////[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]//// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
////[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]//// by [[Jon Lyle Smith]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;//////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]//// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, [[Mike Battaglia]] - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;&amp;lt;span style="color: #000062; font-family: 'Times New Roman',Times,serif; font-size: 113%;"&amp;gt;11 tone equal temperament&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:537:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:537 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/T%C3%BAtim%20Dennsuul%20Wafiil"&amp;gt;TDW&amp;lt;/a&amp;gt; Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;a class="wiki_link" href="/Mike%20Battaglia"&amp;gt;Mike Battaglia&amp;lt;/a&amp;gt; - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:538:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:538 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>bj0mubs1lthunmko8yjrrpfoybnoshw</sha1>
    </revision>
    <revision>
      <id>4368</id>
      <timestamp>2013-04-01T01:33:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 419154506 - Original comment: **</comment>
      <origin>4368</origin>
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      <text bytes="56392" sha1="icisxdi7a92hnq6k1rbgks1bjr5c0rh" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2013-04-01 01:33:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;419154506&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value || solfege || ratios* || Sagittal notation || TDW Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09 || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27 || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73 || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82 || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (&amp;lt;/span&amp;gt;////[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// //) by [[Chris Vaisvil]]//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:537:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:537 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; &amp;lt;em&amp;gt;) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:538:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:538 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>icisxdi7a92hnq6k1rbgks1bjr5c0rh</sha1>
    </revision>
    <revision>
      <id>4369</id>
      <timestamp>2012-11-26T21:52:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 386346276 - Original comment: **</comment>
      <origin>4369</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="55949" sha1="p3rj60cydjyztmcj4fjari7vz5gqdv8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2012-11-26 21:52:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;386346276&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value || solfege || ratios* || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09 || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27 || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73 || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82 || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
////[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]//// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
////[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]//// by [[Jon Lyle Smith]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) (//[[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]]// ) by [[Chris Vaisvil]]&amp;lt;/span&amp;gt;//
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://soundcloud.com/vaisvil/the-execution-of-12-equal|The Execution of 12 Equal]] by [[@Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:537:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:537 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;/em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt;&amp;lt;em&amp;gt; ) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/vaisvil/the-execution-of-12-equal" rel="nofollow" target="_blank"&amp;gt;The Execution of 12 Equal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:538:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:538 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>p3rj60cydjyztmcj4fjari7vz5gqdv8</sha1>
    </revision>
    <revision>
      <id>4370</id>
      <timestamp>2012-09-29T09:12:24Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 368807408 - Original comment: **</comment>
      <origin>4370</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="55837" sha1="40qybh8ftx2q5cev8az1azjqukug0qq" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2012-09-29 09:12:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;368807408&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value || solfege || ratios* || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09 || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27 || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73 || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82 || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) ([[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3|audio only]] ) by [[Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:537:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:537 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) (&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3" rel="nofollow"&amp;gt;audio only&amp;lt;/a&amp;gt; ) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:538:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:538 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>40qybh8ftx2q5cev8az1azjqukug0qq</sha1>
    </revision>
    <revision>
      <id>4371</id>
      <timestamp>2012-09-29T09:05:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 368806938 - Original comment: **</comment>
      <origin>4371</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="55192" sha1="e9siuti8jd6vf0sgrd86y0002hy54zt" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2012-09-29 09:05:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;368806938&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value || solfege || ratios* || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09 || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27 || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73 || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82 || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
////[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]//// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
////[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]//// by [[Jon Lyle Smith]]
//&amp;lt;span class="wiki_link_ext"&amp;gt;[[@http://chrisvaisvil.com/?p=2701|Eleven Birds]] (video and music) by [[Chris Vaisvil]]&amp;lt;/span&amp;gt;//
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:537:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:537 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;span class="wiki_link_ext"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2701" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt; (video and music) by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:538:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:538 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>e9siuti8jd6vf0sgrd86y0002hy54zt</sha1>
    </revision>
    <revision>
      <id>4372</id>
      <timestamp>2012-09-29T09:02:48Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 368806730 - Original comment: **</comment>
      <origin>4372</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="55378" sha1="cf0vyq7kpf2houhype93nm4c8rvdynr" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2012-09-29 09:02:48 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;368806730&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value || solfege || ratios* || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09 || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27 || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73 || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82 || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
* [[@http://www.youtube.com/watch?v=mWSBoPK_m4s|Eleven Birds]]
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:539:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:539 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=mWSBoPK_m4s" rel="nofollow" target="_blank"&amp;gt;Eleven Birds&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:540:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:540 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>cf0vyq7kpf2houhype93nm4c8rvdynr</sha1>
    </revision>
    <revision>
      <id>4373</id>
      <timestamp>2012-07-25T22:05:37Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;phylingual</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 354841864 - Original comment: **</comment>
      <origin>4373</origin>
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      <format>text/x-wiki</format>
      <text bytes="54758" sha1="m5k0ekim1a2tm3miqzj4bki5yvi9m3y" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:phylingual|phylingual]] and made on &lt;tt&gt;2012-07-25 22:05:37 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;354841864&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value || solfege || ratios* || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09 || **ra** || 15/14, 16/15, 17/16, 18/17 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27 || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73 || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82 || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || 15/8, 17/9, 28/15, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
////[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]//// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
////[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]//// by [[Jon Lyle Smith]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;////[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]////&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:537:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:537 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16, 18/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 17/9, 28/15, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:538:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:538 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>m5k0ekim1a2tm3miqzj4bki5yvi9m3y</sha1>
    </revision>
    <revision>
      <id>4374</id>
      <timestamp>2012-07-25T21:58:21Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;phylingual</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 354840762 - Original comment: **</comment>
      <origin>4374</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="55126" sha1="92obh6sk2hcq18jzl5h9jgwb41yw6gv" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:phylingual|phylingual]] and made on &lt;tt&gt;2012-07-25 21:58:21 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;354840762&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la ta ti do**.

|| degrees of 11edo || cents value || solfege || ratios* || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || 1/1 || A || Q\P# ||
|| 1 || 109.09 || **ra** || 15/14, 16/15, 17/16 || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || 8/7, 9/8, 17/15 || B || R ||
|| 3 || 327.27 || **me** || 6/5, 11/9, 17/14 || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || 9/7, 14/11, 22/17 || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || 11/8, 15/11 || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || 16/11, 22/15 || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || 11/7, 14/9, 17/11 || F || T#\Ub ||
|| 8 || 872.73 || **la** || 5/3, 18/11, 28/17 || FII\ or G!!/ || U ||
|| 9 || 981.82 || **ta** || 7/4, 16/9, 30/17 || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || 15/8, 32/17 || GII\ or A!!/ || P\Qb ||
*in 2.7.9.11.15.17 subgroup
=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
//[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]]// by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
//[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]]// by [[Jon Lyle Smith]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;//[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]//&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites.&amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:537:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:537 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la ta ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;ratios*&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/14, 16/15, 17/16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/7, 9/8, 17/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/5, 11/9, 17/14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/7, 14/11, 22/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8, 15/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/11, 22/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/7, 14/9, 17/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/3, 18/11, 28/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ta&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4, 16/9, 30/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/8, 32/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

*in 2.7.9.11.15.17 subgroup&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:538:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:538 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>92obh6sk2hcq18jzl5h9jgwb41yw6gv</sha1>
    </revision>
    <revision>
      <id>4375</id>
      <timestamp>2012-05-29T16:39:07Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 340666722 - Original comment: **</comment>
      <origin>4375</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="53594" sha1="4zg8umtrul7mtnwjorveujqomkik0nt" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-05-29 16:39:07 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;340666722&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. 
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || A || Q\P# ||
|| 1 || 109.09 || **ra** || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || B || R ||
|| 3 || 327.27 || **me** || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || F || T#\Ub ||
|| 8 || 872.73 || **la** || FII\ or G!!/ || U ||
|| 9 || 981.82 || **te** || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || GII\ or A!!/ || P\Qb ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]] by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]
[[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3|Counterpoint in 11edo]] by [[Jon Lyle Smith]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. &amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:513:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:513 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3" rel="nofollow"&amp;gt;Counterpoint in 11edo&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Jon%20Lyle%20Smith"&amp;gt;Jon Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:514:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:514 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>4zg8umtrul7mtnwjorveujqomkik0nt</sha1>
    </revision>
    <revision>
      <id>4376</id>
      <timestamp>2012-04-12T00:40:34Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 319699376 - Original comment: **</comment>
      <origin>4376</origin>
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      <text bytes="53237" sha1="hps0m8z7tcna1tzkymz08aw5v72phvk" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-04-12 00:40:34 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;319699376&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. 
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || A || Q\P# ||
|| 1 || 109.09 || **ra** || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || B || R ||
|| 3 || 327.27 || **me** || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || F || T#\Ub ||
|| 8 || 872.73 || **la** || FII\ or G!!/ || U ||
|| 9 || 981.82 || **te** || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || GII\ or A!!/ || P\Qb ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3|The City Sleeps, A Madrigal]] by [[http://soundcloud.com/ninly/the-city-sleeps|Jason Conklin]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. &amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:513:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:513 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3" rel="nofollow"&amp;gt;The City Sleeps, A Madrigal&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/ninly/the-city-sleeps" rel="nofollow"&amp;gt;Jason Conklin&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:514:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:514 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>hps0m8z7tcna1tzkymz08aw5v72phvk</sha1>
    </revision>
    <revision>
      <id>4377</id>
      <timestamp>2012-03-15T21:30:54Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Kosmorsky</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 311582056 - Original comment: **</comment>
      <origin>4377</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="52697" sha1="6eycbtg6y8484aoraq0y8onzfel3rih" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on &lt;tt&gt;2012-03-15 21:30:54 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;311582056&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. 
(The superfluous Ab can be made a note of [[22edo]] and this tuning is known as "elevenplus".)

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || A || Q\P# ||
|| 1 || 109.09 || **ra** || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || B || R ||
|| 3 || 327.27 || **me** || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || F || T#\Ub ||
|| 8 || 872.73 || **la** || FII\ or G!!/ || U ||
|| 9 || 981.82 || **te** || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || GII\ or A!!/ || P\Qb ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Randy Winchester]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. &amp;lt;br /&amp;gt;
(The superfluous Ab can be made a note of &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt; and this tuning is known as &amp;amp;quot;elevenplus&amp;amp;quot;.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:513:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:513 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover ywp-page-img-link"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:514:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:514 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>6eycbtg6y8484aoraq0y8onzfel3rih</sha1>
    </revision>
    <revision>
      <id>4378</id>
      <timestamp>2011-12-31T01:43:56Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 288886441 - Original comment: **</comment>
      <origin>4378</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47676" sha1="3ibrbykc9jdbf7ji96sabywuv5lhug3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-12-31 01:43:56 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288886441&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s. It is the fifth [[prime numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || A || Q\P# ||
|| 1 || 109.09 || **ra** || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || B || R ||
|| 3 || 327.27 || **me** || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || F || T#\Ub ||
|| 8 || 872.73 || **la** || FII\ or G!!/ || U ||
|| 9 || 981.82 || **te** || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || GII\ or A!!/ || P\Qb ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s. It is the fifth &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo, after &amp;lt;a class="wiki_link" href="/2edo"&amp;gt;2edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/5edo"&amp;gt;5edo&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:513:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:513 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:514:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:514 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>3ibrbykc9jdbf7ji96sabywuv5lhug3</sha1>
    </revision>
    <revision>
      <id>4379</id>
      <timestamp>2011-12-30T13:24:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Ninly</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 288841017 - Original comment: **</comment>
      <origin>4379</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47235" sha1="9dka26oe6abw8m9kgf39joqz34f4msu" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Ninly|Ninly]] and made on &lt;tt&gt;2011-12-30 13:24:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288841017&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14:11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || A || Q\P# ||
|| 1 || 109.09 || **ra** || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || B || R ||
|| 3 || 327.27 || **me** || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || F || T#\Ub ||
|| 8 || 872.73 || **la** || FII\ or G!!/ || U ||
|| 9 || 981.82 || **te** || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || GII\ or A!!/ || P\Qb ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14:11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:513:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:513 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:514:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:514 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9dka26oe6abw8m9kgf39joqz34f4msu</sha1>
    </revision>
    <revision>
      <id>4380</id>
      <timestamp>2011-12-11T01:39:02Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 284599030 - Original comment: **</comment>
      <origin>4380</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47259" sha1="qf2yjcrqo31di02iv45vx05j8np63ec" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-12-11 01:39:02 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;284599030&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[image:11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || A || Q\P# ||
|| 1 || 109.09 || **ra** || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || B || R ||
|| 3 || 327.27 || **me** || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || F || T#\Ub ||
|| 8 || 872.73 || **la** || FII\ or G!!/ || U ||
|| 9 || 981.82 || **te** || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || GII\ or A!!/ || P\Qb ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:513:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17_2ndsave.png/284598982/11edo_approx_2-7-9-11-15-17_2ndsave.png" alt="11edo_approx_2-7-9-11-15-17_2ndsave.png" title="11edo_approx_2-7-9-11-15-17_2ndsave.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:513 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
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      <sha1>qf2yjcrqo31di02iv45vx05j8np63ec</sha1>
    </revision>
    <revision>
      <id>4381</id>
      <timestamp>2011-12-10T21:27:42Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 284566590 - Original comment: **</comment>
      <origin>4381</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47189" sha1="ogdcws6nk8zxy9z8x75w14qh8rvlhge" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-12-10 21:27:42 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;284566590&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:
[[image:11edo_approx_2-7-9-11-15-17.png]]

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || A || Q\P# ||
|| 1 || 109.09 || **ra** || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || B || R ||
|| 3 || 327.27 || **me** || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || F || T#\Ub ||
|| 8 || 872.73 || **la** || FII\ or G!!/ || U ||
|| 9 || 981.82 || **te** || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || GII\ or A!!/ || P\Qb ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:513:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11edo_approx_2-7-9-11-15-17.png/284566532/11edo_approx_2-7-9-11-15-17.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11edo_approx_2-7-9-11-15-17.png/284566532/11edo_approx_2-7-9-11-15-17.png" alt="11edo_approx_2-7-9-11-15-17.png" title="11edo_approx_2-7-9-11-15-17.png" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:513 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:514:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:514 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ogdcws6nk8zxy9z8x75w14qh8rvlhge</sha1>
    </revision>
    <revision>
      <id>4382</id>
      <timestamp>2011-12-08T02:13:31Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 283585702 - Original comment: **</comment>
      <origin>4382</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="46483" sha1="jcxuqoyu4avxskla7nwlburcx81cl8k" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-12-08 02:13:31 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;283585702&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || A || Q\P# ||
|| 1 || 109.09 || **ra** || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || B || R ||
|| 3 || 327.27 || **me** || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || F || T#\Ub ||
|| 8 || 872.73 || **la** || FII\ or G!!/ || U ||
|| 9 || 981.82 || **te** || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || GII\ or A!!/ || P\Qb ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/1L%204s"&amp;gt;1L 4s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:513:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:513 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>jcxuqoyu4avxskla7nwlburcx81cl8k</sha1>
    </revision>
    <revision>
      <id>4383</id>
      <timestamp>2011-12-08T02:08:16Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 283584958 - Original comment: **</comment>
      <origin>4383</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="46483" sha1="ibxiazwut7nck9kapay36m7gr7ipaql" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-12-08 02:08:16 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;283584958&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || A || Q\P# ||
|| 1 || 109.09 || **ra** || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || B || R ||
|| 3 || 327.27 || **me** || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || F || T#\Ub ||
|| 8 || 872.73 || **la** || FII\ or G!!/ || U ||
|| 9 || 981.82 || **te** || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || GII\ or A!!/ || P\Qb ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[4L 1s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[Machine]][6].
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone]][7].
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]]&amp;lt;/span&amp;gt; by [[Randy Winchester]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3, a &amp;lt;a class="wiki_link" href="/4L%201s"&amp;gt;4L 1s&amp;lt;/a&amp;gt; scale named Machine[5]; and 2 2 2 2 2 1, a &amp;lt;a class="wiki_link" href="/5L%201s"&amp;gt;5L 1s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Machine"&amp;gt;Machine&amp;lt;/a&amp;gt;[6].&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a &amp;lt;a class="wiki_link" href="/4L%203s"&amp;gt;4L 3s&amp;lt;/a&amp;gt; scale named &amp;lt;a class="wiki_link" href="/Orgone"&amp;gt;Orgone&amp;lt;/a&amp;gt;[7].&amp;lt;br /&amp;gt;
4\11 generates 4 4 3; 1 3 1 3 3, a &amp;lt;a class="wiki_link" href="/3L%202s"&amp;gt;3L 2s&amp;lt;/a&amp;gt; scale; and 1 1 2 1 1 2 1 2, a &amp;lt;a class="wiki_link" href="/3L%205s"&amp;gt;3L 5s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1; 1 4 1 4 1, a &amp;lt;a class="wiki_link" href="/2L%203s"&amp;gt;2L 3s&amp;lt;/a&amp;gt; scale; 1 1 3 1 1 3 1, a &amp;lt;a class="wiki_link" href="/2L%205s"&amp;gt;2L 5s&amp;lt;/a&amp;gt; scale; and 1 1 1 2 1 1 1 2 1, a &amp;lt;a class="wiki_link" href="/2L%207s"&amp;gt;2L 7s&amp;lt;/a&amp;gt; scale.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:513:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:513 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ibxiazwut7nck9kapay36m7gr7ipaql</sha1>
    </revision>
    <revision>
      <id>4384</id>
      <timestamp>2011-11-23T07:22:01Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 278435682 - Original comment: **</comment>
      <origin>4384</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45396" sha1="depnzi2hiu8lqop24cp6paclw53juvo" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-11-23 07:22:01 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;278435682&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || A || Q\P# ||
|| 1 || 109.09 || **ra** || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || B || R ||
|| 3 || 327.27 || **me** || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || F || T#\Ub ||
|| 8 || 872.73 || **la** || FII\ or G!!/ || U ||
|| 9 || 981.82 || **te** || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || GII\ or A!!/ || P\Qb ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3|Comets Over Flatland 10]] by [[Randy Winchester]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3" rel="nofollow"&amp;gt;Comets Over Flatland 10&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Randy%20Winchester"&amp;gt;Randy Winchester&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:513:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:513 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>depnzi2hiu8lqop24cp6paclw53juvo</sha1>
    </revision>
    <revision>
      <id>4385</id>
      <timestamp>2011-11-21T00:15:42Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 277560332 - Original comment: **</comment>
      <origin>4385</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="44977" sha1="a7v9377dz98vgjetbs0lpn1zs5ju17e" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2011-11-21 00:15:42 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;277560332&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege || Sagittal notation || TMDW Machine notation ||
|| 0 || 0.00 || **do** || A || Q\P# ||
|| 1 || 109.09 || **ra** || AII\ or B!!/ || Q#\Rb ||
|| 2 || 218.18 || **re** || B || R ||
|| 3 || 327.27 || **me** || C/I or BII\ or D\!!/ || R#\Sb ||
|| 4 || 436.36 || **mo** || D\! or C/II\ || S ||
|| 5 || 545.45 || **fu** || D/I or E\!!/ || S#\Tb ||
|| 6 || 654.55 || **su** || E\! or D/II\ || T ||
|| 7 || 763.64 || **lo** || F || T#\Ub ||
|| 8 || 872.73 || **la** || FII\ or G!!/ || U ||
|| 9 || 981.82 || **te** || G || U#\Pb ||
|| 10 || 1090.91 || **ti** || GII\ or A!!/ || P\Qb ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;TMDW Machine notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q\P#&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Q#\Rb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;R#\Sb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;S#\Tb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;T#\Ub&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;U#\Pb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;P\Qb&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-audio ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;span class="ywp-page-play-pause ywp-page-video ywp-link-hover"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:513:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:513 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>a7v9377dz98vgjetbs0lpn1zs5ju17e</sha1>
    </revision>
    <revision>
      <id>4386</id>
      <timestamp>2011-10-31T13:23:48Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenjacob</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 270330506 - Original comment: **</comment>
      <origin>4386</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="40801" sha1="28j5t0z063udr0w21h8hv2nyu4vv5ox" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenjacob|xenjacob]] and made on &lt;tt&gt;2011-10-31 13:23:48 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;270330506&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege || Sagittal notation ||
|| 0 || 0.00 || **do** || A ||
|| 1 || 109.09 || **ra** || AII\ or B!!/ ||
|| 2 || 218.18 || **re** || B ||
|| 3 || 327.27 || **me** || C/I or BII\ or D\!!/ ||
|| 4 || 436.36 || **mo** || D\! or C/II\ ||
|| 5 || 545.45 || **fu** || D/I or E\!!/ ||
|| 6 || 654.55 || **su** || E\! or D/II\ ||
|| 7 || 763.64 || **lo** || F ||
|| 8 || 872.73 || **la** || FII\ or G!!/ ||
|| 9 || 981.82 || **te** || G ||
|| 10 || 1090.91 || **ti** || GII\ or A!!/ ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Sagittal notation&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;A&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;AII\ or B!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;B&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;C/I or BII\ or D\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D\! or C/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;D/I or E\!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;E\! or D/II\&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;F&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;FII\ or G!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;G&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;GII\ or A!!/&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-dae72d78d6356b014681e07335ea2812"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:489:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:489 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>28j5t0z063udr0w21h8hv2nyu4vv5ox</sha1>
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    <revision>
      <id>4387</id>
      <timestamp>2011-09-09T16:57:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Cenobyte</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 252476470 - Original comment: **</comment>
      <origin>4387</origin>
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      <text bytes="39823" sha1="ghywjf8ag7k3qw8bgzv0ue5fux2kguq" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Cenobyte|Cenobyte]] and made on &lt;tt&gt;2011-09-09 16:57:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;252476470&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;, Mike Battaglia - youtube
==Instruments== 

[[image:11-edo-ukulele.JPG width="404" height="304"]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:36: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:36 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:37: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:37 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:38: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:38 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:23:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc11"&amp;gt;&amp;lt;a name="Compositions-Instruments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:23 --&amp;gt;Instruments&amp;lt;/h2&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextLocalImageRule:465:&amp;amp;lt;img src=&amp;amp;quot;/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 304px; width: 404px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/11-edo-ukulele.JPG/252473150/404x304/11-edo-ukulele.JPG" alt="11-edo-ukulele.JPG" title="11-edo-ukulele.JPG" style="height: 304px; width: 404px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:465 --&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ghywjf8ag7k3qw8bgzv0ue5fux2kguq</sha1>
    </revision>
    <revision>
      <id>4388</id>
      <timestamp>2011-09-05T01:02:22Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 250759594 - Original comment: **</comment>
      <origin>4388</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="38951" sha1="olaiw6v6o59944v67h4rw8mmlb5r7eg" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2011-09-05 01:02:22 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;250759594&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]]&amp;lt;/span&amp;gt; on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]]&amp;lt;/span&amp;gt; by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3|goopy]]&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3|ellie]]&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3|ziggy]]&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;[[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3|towelbear]]&amp;lt;/span&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]]&amp;lt;/span&amp;gt; by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]]&amp;lt;/span&amp;gt; by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]]&amp;lt;/span&amp;gt; by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
&amp;lt;span class="wpaudio-container"&amp;gt;[[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3|The Metamorphosis of Gregor]]&amp;lt;/span&amp;gt; by [[@xenharmonic/Chris Vaisvil|Chris Vaisvil]]
==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* &amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;[[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]]&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a35179d68c31b8294db3bcc94b8ba006"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-aedb7966e0a8a11202b283717c2d49e8"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;span class="ymp-btn-page-play ymp-media-b592d2a529add445b9863abc1db9a70b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-4cd9c2304367aa08aa2730dc83a625b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-948e3d93634c7086e5d35536a528c330"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-57c2e5d58d0554c5085a3e238dca0c7b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-d138ae1269aec7a6974e26d2ac0869c4"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-a8d1c66d2644eb4985929d27e345b034"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;span class="ymp-btn-page-play ymp-media-1eed7ec2aa83b410dc87688ca65cb250"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="wpaudio-container"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3" rel="nofollow"&amp;gt;The Metamorphosis of Gregor&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chris%20Vaisvil" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;span class="ymp-btn-page-play ymp-media-2b0ac7aee582cc2952d261655b7ab213"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>olaiw6v6o59944v67h4rw8mmlb5r7eg</sha1>
    </revision>
    <revision>
      <id>4389</id>
      <timestamp>2011-08-29T05:31:05Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 249055863 - Original comment: **</comment>
      <origin>4389</origin>
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      <format>text/x-wiki</format>
      <text bytes="36512" sha1="kd6yy3aamixxcwblwpz11m4jjat2rjy" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-08-29 05:31:05 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;249055863&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]] on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]] by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) [[http://music.columbia.edu/~chris/sounds/st.goopy.mp3|goopy]] [[http://music.columbia.edu/~chris/sounds/st.ellie.mp3|ellie]] [[http://music.columbia.edu/~chris/sounds/st.ziggy.mp3|ziggy]] [[http://music.columbia.edu/~chris/sounds/st.towelbear.mp3|towelbear]]
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]] by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]] by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; by [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

* [[http://www.youtube.com/watch?v=AhPjsCoMy-Q|11-equal Improvisation]], Mike Battaglia - youtube
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=AhPjsCoMy-Q" rel="nofollow"&amp;gt;11-equal Improvisation&amp;lt;/a&amp;gt;, Mike Battaglia - youtube&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>kd6yy3aamixxcwblwpz11m4jjat2rjy</sha1>
    </revision>
    <revision>
      <id>4390</id>
      <timestamp>2011-06-10T17:14:32Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 235849276 - Original comment: **</comment>
      <origin>4390</origin>
      <model>wikitext</model>
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      <text bytes="41612" sha1="sv58er5acibsqh325mn9at6ti1ra7zk" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-10 17:14:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;235849276&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
[[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3|She Is My Lilac-Hued Obsession]] on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]] by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) [[http://music.columbia.edu/~chris/sounds/st.goopy.mp3|goopy]] [[http://music.columbia.edu/~chris/sounds/st.ellie.mp3|ellie]] [[http://music.columbia.edu/~chris/sounds/st.ziggy.mp3|ziggy]] [[http://music.columbia.edu/~chris/sounds/st.towelbear.mp3|towelbear]]
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]] by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]] by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; by [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]


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&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3" rel="nofollow"&amp;gt;She Is My Lilac-Hued Obsession&amp;lt;/a&amp;gt; on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
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&amp;lt;/th&amp;gt;
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&amp;lt;/th&amp;gt;
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&amp;lt;/th&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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      <comment>**Imported revision 235056732 - Original comment: **</comment>
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      <text bytes="41261" sha1="7056ncufomesqt0py5ho70nep4dm7ur" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-08 00:10:35 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;235056732&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]] by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte) [[http://music.columbia.edu/~chris/sounds/st.goopy.mp3|goopy]] [[http://music.columbia.edu/~chris/sounds/st.ellie.mp3|ellie]] [[http://music.columbia.edu/~chris/sounds/st.ziggy.mp3|ziggy]] [[http://music.columbia.edu/~chris/sounds/st.towelbear.mp3|towelbear]]
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]] by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]] by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; by [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]


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&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte) &amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/sounds/st.goopy.mp3" rel="nofollow"&amp;gt;goopy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/sounds/st.ellie.mp3" rel="nofollow"&amp;gt;ellie&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/sounds/st.ziggy.mp3" rel="nofollow"&amp;gt;ziggy&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/sounds/st.towelbear.mp3" rel="nofollow"&amp;gt;towelbear&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/th&amp;gt;
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&amp;lt;/th&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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      <comment>**Imported revision 234976606 - Original comment: **</comment>
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      <text bytes="40493" sha1="ongci4uplgiynkxhrmspdbaa410y5lm" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-07 17:49:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234976606&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by [[David Hamill]], 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by [[Jacob Barton]], 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]] by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]] by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]] by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by [[Chris Vaisvil]]
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; by [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]


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&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
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&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

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        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/George%20Secor"&amp;gt;George Secor&amp;lt;/a&amp;gt;, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/David%20Hamill"&amp;gt;David Hamill&amp;lt;/a&amp;gt;, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by &amp;lt;a class="wiki_link" href="/Jacob%20Barton"&amp;gt;Jacob Barton&amp;lt;/a&amp;gt;, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Chris%20Vaisvil"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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      <id>4393</id>
      <timestamp>2011-06-07T02:20:30Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 234774118 - Original comment: **</comment>
      <origin>4393</origin>
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      <text bytes="40248" sha1="6dv0gadz7et9iqdjjhzpsvhp3cwyvtj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-07 02:20:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234774118&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]] by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]] by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]] by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by Chris Vaisvil
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; by [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]


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&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
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&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


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      <id>4394</id>
      <timestamp>2011-06-07T02:15:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 234773548 - Original comment: **</comment>
      <origin>4394</origin>
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      <text bytes="40266" sha1="1b2jax5g0obc9lwy1jvlaohueqo6dc9" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-07 02:15:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234773548&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
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=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]] by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt|Aaron Amdrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3|play]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=933772|Orange Clips on Sausages]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3|play]] by Andrew Heathwaite
[[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;songID=834492|Blue Gel]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3|play]] by Andrew Heathwaite
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by Chris Vaisvil
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; by [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]


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&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Amdrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=933772" rel="nofollow"&amp;gt;Orange Clips on Sausages&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;amp;amp;songID=834492" rel="nofollow"&amp;gt;Blue Gel&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by Andrew Heathwaite&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


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      <sha1>1b2jax5g0obc9lwy1jvlaohueqo6dc9</sha1>
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    <revision>
      <id>4395</id>
      <timestamp>2011-06-07T01:58:58Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 234771794 - Original comment: **</comment>
      <origin>4395</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="38683" sha1="2atyxf45ip64qqzkxvqtqa3o86z1gn6" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-07 01:58:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234771794&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]] by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt|Aaron Amdrew Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by Chris Vaisvil
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; by [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]


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&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Amdrew Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


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    <revision>
      <id>4396</id>
      <timestamp>2011-06-07T01:57:19Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 234771600 - Original comment: **</comment>
      <origin>4396</origin>
      <model>wikitext</model>
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      <text bytes="38973" sha1="9558n9ynv86s6za7040lc4zfivqbi8h" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-07 01:57:19 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;234771600&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]] by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by Chris Vaisvil
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; by [[John Lyle Smith]]
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude in 11et]] by [[Aaron Andrew Hunt]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]


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&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude in 11et&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Andrew Hunt&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


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      <id>4397</id>
      <timestamp>2011-06-06T21:12:32Z</timestamp>
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        <username>Wikispaces&gt;genewardsmith</username>
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      <comment>**Imported revision 234728726 - Original comment: **</comment>
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      <text bytes="38669" sha1="1dprrum2qpt5balkuuinhvqe8irdqln" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-06 21:12:32 UTC&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|The Turquoise Dabo Girl]] [[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3|play]] by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by Chris Vaisvil
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; ny [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]


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&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;The Turquoise Dabo Girl&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; ny &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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      <id>4398</id>
      <timestamp>2011-05-31T21:22:39Z</timestamp>
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      <comment>**Imported revision 233336272 - Original comment: **</comment>
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      <text bytes="38384" sha1="oixlbzjlyxrzwj1gv84haubv5vkqnnf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2011-05-31 21:22:39 UTC&lt;/tt&gt;.&lt;br&gt;
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&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 0 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 0 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 0 0 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||
=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by Chris Vaisvil
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; ny [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]


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&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc6"&amp;gt;&amp;lt;a name="MOS Scales-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 0 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 0 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 0 0 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc9"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; ny &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
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      <id>4399</id>
      <timestamp>2011-05-31T17:47:00Z</timestamp>
      <contributor>
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        <id>0</id>
      </contributor>
      <comment>**Imported revision 233290872 - Original comment: **</comment>
      <origin>4399</origin>
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      <text bytes="32987" sha1="qn7436h8jbjf6z6eefub03murgu6du2" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2011-05-31 17:47:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;233290872&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by Chris Vaisvil
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; ny [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 135/128 ||&amp;lt; | -7 3 1 &amp;gt; ||&amp;gt; 92.18 ||= Major Chroma ||= Major Limma ||= Pelogic Comma ||
||= 9931568/9752117 ||&amp;lt; | -25 7 6 &amp;gt; ||&amp;gt; 31.57 ||= Ampersand's Comma ||=   ||=   ||
||= 1776337/1773750 ||&amp;lt; | -68 18 17 &amp;gt; ||&amp;gt; 2.52 ||= Vavoom ||=   ||=   ||
||= 9859966/9733137 ||&amp;lt; | -10 7 8 -7 &amp;gt; ||&amp;gt; 22.41 ||= Blackjackisma ||=   ||=   ||
||= 1029/1024 ||&amp;lt; | -10 1 0 3 &amp;gt; ||&amp;gt; 8.43 ||= Gamelisma ||=   ||=   ||
||= 225/224 ||&amp;lt; | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 16875/16807 ||&amp;lt; | 3 4 -5 &amp;gt; ||&amp;gt; 6.99 ||= Mirkwai ||=   ||=   ||
||= 2401/2400 ||&amp;lt; | -5 -1 -2 4 &amp;gt; ||&amp;gt; 0.72 ||= Breedsma ||=   ||=   ||
||= 121/120 ||&amp;lt; | -3 -1 -1 2 &amp;gt; ||&amp;gt; 14.37 ||= Biyatisma ||=   ||=   ||
||= 65536/65219 ||&amp;lt; | 16 -2 -3 &amp;gt; ||&amp;gt; 8.39 ||= Orgonisma ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
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&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; ny &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| -3 -1 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;14.37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Biyatisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;65536/65219&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: left;"&amp;gt;| 16 -2 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Orgonisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>qn7436h8jbjf6z6eefub03murgu6du2</sha1>
    </revision>
    <revision>
      <id>4400</id>
      <timestamp>2011-05-31T17:45:56Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jdfreivald</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 233290636 - Original comment: Added comma table.**</comment>
      <origin>4400</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="31235" sha1="e32mtmne8tngqv8bmp2j38r8wwngmqw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2011-05-31 17:45:56 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;233290636&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;Added comma table.&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
&amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]]&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]]&amp;lt;/span&amp;gt; by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]]&amp;lt;/span&amp;gt; by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]]&amp;lt;/span&amp;gt; by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]]&amp;lt;/span&amp;gt; by Chris Vaisvil
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]]&amp;lt;/span&amp;gt; ny [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;[[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;[[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;[[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]]&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;[[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]]&amp;lt;/span&amp;gt; by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]

==Commas== 
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
|| 135/128 || | -7 3 1 &amp;gt; || 92.18 || Major Chroma || Major Limma || Pelogic Comma ||
|| 9931568/9752117 || | -25 7 6 &amp;gt; || 31.57 || Ampersand's Comma ||   ||   ||
|| 1776337/1773750 || | -68 18 17 &amp;gt; || 2.52 || Vavoom ||   ||   ||
|| 9859966/9733137 || | -10 7 8 -7 &amp;gt; || 22.41 || Blackjackisma ||   ||   ||
|| 1029/1024 || | -10 1 0 3 &amp;gt; || 8.43 || Gamelisma ||   ||   ||
|| 225/224 || | -5 2 2 -1 &amp;gt; || 7.71 || Septimal Kleisma || Marvel Comma ||   ||
|| 16875/16807 || | 3 4 -5 &amp;gt; || 6.99 || Mirkwai ||   ||   ||
|| 2401/2400 || | -5 -1 -2 4 &amp;gt; || 0.72 || Breedsma ||   ||   ||
|| 121/120 || | -3 -1 -1 2 &amp;gt; || 14.37 || Biyatisma ||   ||   ||
|| 65536/65219 || | 16 -2 -3 &amp;gt; || 8.39 || Orgonisma ||   ||   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:33: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:33 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:34: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:34 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:35: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:35 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;span class="ymp-btn-page-play ymp-media-c028d7e67c4c0d4e42d9a04ed9ed4a5e"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-426701bae21d6e77479d3ae7a1e7802b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a988c4ace8d436d7b5c28a911422c93b"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-4c992689bab8bac1e13e4b26da0151c7"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-a88698537238903579b56d7cd26d7111"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;span class="ymp-btn-page-play ymp-media-0ba2917b56073b373385642c54d40d75"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; ny &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;span class="ymp-btn-page-play ymp-media-93e47d957ab7ad517004a52411bd39b0"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1115622b253775296293be13693078a2"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-1a927c00e05a8682e1478c992ed79e42"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;, &amp;lt;span class="ymp-btn-page-play ymp-media-e625dd74ab1589511414e5c41097c0cd"&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:21:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc10"&amp;gt;&amp;lt;a name="Compositions-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:21 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 11 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 11 17 26 31 38 41 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;135/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -7 3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major Chroma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major Limma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Pelogic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9931568/9752117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -25 7 6 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;31.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Ampersand's Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1776337/1773750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -68 18 17 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2.52&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Vavoom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9859966/9733137&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 7 8 -7 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Blackjackisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1029/1024&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -10 1 0 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8.43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Gamelisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16875/16807&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -5 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Mirkwai&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2401/2400&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 -1 -2 4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.72&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Breedsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;121/120&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 -1 -1 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
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      <sha1>e32mtmne8tngqv8bmp2j38r8wwngmqw</sha1>
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    <revision>
      <id>4401</id>
      <timestamp>2011-05-28T22:21:53Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 232644840 - Original comment: **</comment>
      <origin>4401</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="24250" sha1="3jma9kug6qmko6o43p5hegelpr2brqq" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-05-28 22:21:53 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;232644840&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]] by George Secor, 1970. Apparently the first piece ever written for 11edo.
[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott 
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]] by Chris Vaisvil
[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]] ny [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the &amp;lt;a class="wiki_link" href="/k%2AN%20subgroups"&amp;gt;2*11 subgroup&amp;lt;/a&amp;gt; 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt; ny &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>3jma9kug6qmko6o43p5hegelpr2brqq</sha1>
    </revision>
    <revision>
      <id>4402</id>
      <timestamp>2011-05-17T20:19:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 229421938 - Original comment: **</comment>
      <origin>4402</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="24137" sha1="qnyr8u47pcmf5yexq74tr3vd1pnhqzj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2011-05-17 20:19:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;229421938&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="9458808"]][[toc|flat]]
=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]] by George Secor, 1970. Apparently the first piece ever written for 11edo.
[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott 
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]] by Chris Vaisvil
[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]] ny [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/9458808?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;9458808&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://webplayer.yahooapis.com/player.js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:32: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:32 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:9:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:9 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:11:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:11 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:13:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:13 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:15:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:15 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:17:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:17 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt; ny &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:19:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:19 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>qnyr8u47pcmf5yexq74tr3vd1pnhqzj</sha1>
    </revision>
    <revision>
      <id>4403</id>
      <timestamp>2011-04-29T23:35:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 224244990 - Original comment: **</comment>
      <origin>4403</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23580" sha1="i2aujsulewg8m2ktqvrxneijwf4ko3h" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2011-04-29 23:35:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;224244990&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[toc|flat]]

=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning= 
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup= 
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals= 
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege== 
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales= 
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine== 
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

=Compositions= 
[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]] by George Secor, 1970. Apparently the first piece ever written for 11edo.
[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott 
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]] by Chris Vaisvil
[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]] ny [[John Lyle Smith]]

==Videos== 
The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:29: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:29 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:30: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:30 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:31: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:31 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
 Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
 11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
 An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
 Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:14:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc7"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble-11edo Zine"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:14 --&amp;gt;11edo Zine&amp;lt;/h2&amp;gt;
 There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See &amp;lt;a class="wiki_link" href="/11edo%20Zine"&amp;gt;11edo Zine&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:16:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc8"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:16 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt; ny &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:18:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc9"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:18 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
 The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>i2aujsulewg8m2ktqvrxneijwf4ko3h</sha1>
    </revision>
    <revision>
      <id>4404</id>
      <timestamp>2011-03-31T03:59:56Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 215743152 - Original comment: **</comment>
      <origin>4404</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22932" sha1="an8lq5d7a1z675jqcurr435f3js83kx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-03-31 03:59:56 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;215743152&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[toc|flat]]


=11 tone equal temperament= 
11-tone equal temperament, or 11 [[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent]]s.

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup=
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege==
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

=Compositions=
[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]] by George Secor, 1970. Apparently the first piece ever written for 11edo.
[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott 
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]] by Chris Vaisvil
[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]] ny [[John Lyle Smith]]

==Videos==
The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11 &amp;lt;a class="wiki_link" href="/edo"&amp;gt;edo&amp;lt;/a&amp;gt;, divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into eleven equal steps of approximately 109.09 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:14:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:14 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt; ny &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:16:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:16 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>an8lq5d7a1z675jqcurr435f3js83kx</sha1>
    </revision>
    <revision>
      <id>4405</id>
      <timestamp>2011-03-30T19:39:43Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 215648442 - Original comment: **</comment>
      <origin>4405</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22779" sha1="hartr9fo4khpd6t66ccmmme154u7htr" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-30 19:39:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;215648442&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[toc|flat]]


=11 tone equal temperament= 
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup=
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege==
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

=Compositions=
[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]] by George Secor, 1970. Apparently the first piece ever written for 11edo.
[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott 
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]] by Chris Vaisvil
[[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3|Jaunt]] ny [[John Lyle Smith]]

==Videos==
The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:14:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:14 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3" rel="nofollow"&amp;gt;Jaunt&amp;lt;/a&amp;gt; ny &amp;lt;a class="wiki_link" href="/John%20Lyle%20Smith"&amp;gt;John Lyle Smith&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:16:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:16 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>hartr9fo4khpd6t66ccmmme154u7htr</sha1>
    </revision>
    <revision>
      <id>4406</id>
      <timestamp>2011-03-30T15:18:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 215566590 - Original comment: **</comment>
      <origin>4406</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22477" sha1="96eg8bzfjoqinxsh8fba36km8g2eano" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-30 15:18:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;215566590&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[toc|flat]]


=11 tone equal temperament= 
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup=
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege==
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

=Compositions=
[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]] by George Secor, 1970. Apparently the first piece ever written for 11edo.
[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott 
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
[[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3|Jeffrey Dahmer Cooks at 11edo]] by Chris Vaisvil

==Videos==
The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:14:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:14 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3" rel="nofollow"&amp;gt;Jeffrey Dahmer Cooks at 11edo&amp;lt;/a&amp;gt; by Chris Vaisvil&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:16:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:16 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>96eg8bzfjoqinxsh8fba36km8g2eano</sha1>
    </revision>
    <revision>
      <id>4407</id>
      <timestamp>2011-03-30T15:08:23Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 215562160 - Original comment: **</comment>
      <origin>4407</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22156" sha1="77lvdn1yefs4y90idg242eu2w78djwh" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-30 15:08:23 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;215562160&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[toc|flat]]


=11 tone equal temperament= 
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup=
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege==
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

=Compositions=
[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]] by George Secor, 1970. Apparently the first piece ever written for 11edo.
[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://soundclick.com/share.cfm?id=10267904|I]] [[http://soundclick.com/share.cfm?id=10267905|II]] [[http://soundclick.com/share.cfm?id=10267906|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott 
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

==Videos==
The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:14:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:14 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267904" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267905" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=10267906" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:16:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:16 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>77lvdn1yefs4y90idg242eu2w78djwh</sha1>
    </revision>
    <revision>
      <id>4408</id>
      <timestamp>2011-03-30T14:25:36Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 215544438 - Original comment: **</comment>
      <origin>4408</origin>
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      <text bytes="22534" sha1="elhsxq4lrt9zxr4s9w391b54aka649b" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-30 14:25:36 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;215544438&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[toc|flat]]


=11 tone equal temperament= 
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup=
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege==
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

=Compositions=
[[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3|First Piece Ever]] by George Secor, 1970. Apparently the first piece ever written for 11edo.
[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;q=hi&amp;amp;newref=1%27%29;|I]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;q=hi&amp;amp;newref=1%27%29;|II]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;q=hi&amp;amp;newref=1%27%29;|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott 
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

==Videos==
The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:14:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:14 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3" rel="nofollow"&amp;gt;First Piece Ever&amp;lt;/a&amp;gt; by George Secor, 1970. Apparently the first piece ever written for 11edo.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:16:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:16 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>elhsxq4lrt9zxr4s9w391b54aka649b</sha1>
    </revision>
    <revision>
      <id>4409</id>
      <timestamp>2011-03-29T01:45:04Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 214941036 - Original comment: **</comment>
      <origin>4409</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22127" sha1="en88jcmc71vhtgqvqd2w5nxj8mj33be" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-29 01:45:04 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214941036&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[toc|flat]]


=11 tone equal temperament= 
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup=
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege==
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

=Compositions=
[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;q=hi&amp;amp;newref=1%27%29;|I]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;q=hi&amp;amp;newref=1%27%29;|II]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;q=hi&amp;amp;newref=1%27%29;|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott [[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

==Videos==
The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Intervals-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:14:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:14 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:16:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:16 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>en88jcmc71vhtgqvqd2w5nxj8mj33be</sha1>
    </revision>
    <revision>
      <id>4410</id>
      <timestamp>2011-03-29T01:44:11Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 214940940 - Original comment: **</comment>
      <origin>4410</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22088" sha1="09at12xt0xt76sml5vwg4a0txkfggbx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-29 01:44:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214940940&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[toc|flat]]


=11 tone equal temperament= 
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup=
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Intervals==
|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==11 edo solfege==
An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

=11edo Instant Ensemble= 
In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

=Compositions=
[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;q=hi&amp;amp;newref=1%27%29;|I]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;q=hi&amp;amp;newref=1%27%29;|II]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;q=hi&amp;amp;newref=1%27%29;|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott [[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

==Videos==
The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Tuning"&amp;gt;Tuning&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt; | &amp;lt;a href="#MOS Scales"&amp;gt;MOS Scales&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt; | &amp;lt;a href="#x11edo Instant Ensemble"&amp;gt;11edo Instant Ensemble&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt; | &amp;lt;a href="#Compositions"&amp;gt;Compositions&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Tuning&amp;lt;/h1&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc3"&amp;gt;&amp;lt;a name="Subgroup-Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Intervals&amp;lt;/h2&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc4"&amp;gt;&amp;lt;a name="Subgroup-11 edo solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11 edo solfege&amp;lt;/h2&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc5"&amp;gt;&amp;lt;a name="MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;MOS Scales&amp;lt;/h1&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc6"&amp;gt;&amp;lt;a name="x11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;11edo Instant Ensemble&amp;lt;/h1&amp;gt;
 In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:14:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc7"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:14 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:16:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc8"&amp;gt;&amp;lt;a name="Compositions-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:16 --&amp;gt;Videos&amp;lt;/h2&amp;gt;
The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>09at12xt0xt76sml5vwg4a0txkfggbx</sha1>
    </revision>
    <revision>
      <id>4411</id>
      <timestamp>2011-03-29T01:38:18Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 214940224 - Original comment: **</comment>
      <origin>4411</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22136" sha1="d8aca1eg8nlvdo860dp6mpla802yi0o" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-29 01:38:18 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214940224&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[toc|flat]]


=11 tone equal temperament= 

11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

=Intervals=

Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

=Subgroup=
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==Intervals== 
|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||


===MOS Scales=== 

Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

===11edo Solfege=== 

An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

===11edo Instant Ensemble=== 

In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

===Compositions=== 

[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;q=hi&amp;amp;newref=1%27%29;|I]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;q=hi&amp;amp;newref=1%27%29;|II]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;q=hi&amp;amp;newref=1%27%29;|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott [[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.

===Videos=== 

The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:18:&amp;amp;lt;img id=&amp;amp;quot;wikitext@@toc@@flat&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaTocFlat&amp;amp;quot; title=&amp;amp;quot;Table of Contents&amp;amp;quot; src=&amp;amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;amp;h=16&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:18 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:19: --&amp;gt;&amp;lt;a href="#x11 tone equal temperament"&amp;gt;11 tone equal temperament&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:19 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:20: --&amp;gt; | &amp;lt;a href="#Intervals"&amp;gt;Intervals&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:20 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:21: --&amp;gt; | &amp;lt;a href="#Subgroup"&amp;gt;Subgroup&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:21 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:22: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:22 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:23: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:23 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:24: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:24 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:25: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:25 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:26: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:26 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:27: --&amp;gt;&amp;lt;!-- ws:end:WikiTextTocRule:27 --&amp;gt;&amp;lt;!-- ws:start:WikiTextTocRule:28: --&amp;gt;
&amp;lt;!-- ws:end:WikiTextTocRule:28 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
&amp;lt;br /&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Subgroup"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Subgroup&amp;lt;/h1&amp;gt;
11edo provides the same tuning on the 2.9.15.7.11 subgroup as 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc3"&amp;gt;&amp;lt;a name="Subgroup-Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Intervals&amp;lt;/h2&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="Subgroup-Intervals-MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;MOS Scales&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc5"&amp;gt;&amp;lt;a name="Subgroup-Intervals-11edo Solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;11edo Solfege&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc6"&amp;gt;&amp;lt;a name="Subgroup-Intervals-11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;11edo Instant Ensemble&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:14:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc7"&amp;gt;&amp;lt;a name="Subgroup-Intervals-Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:14 --&amp;gt;Compositions&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:16:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc8"&amp;gt;&amp;lt;a name="Subgroup-Intervals-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:16 --&amp;gt;Videos&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>d8aca1eg8nlvdo860dp6mpla802yi0o</sha1>
    </revision>
    <revision>
      <id>4412</id>
      <timestamp>2011-02-22T15:56:54Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 204106668 - Original comment: **</comment>
      <origin>4412</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20443" sha1="8w4gislkygz5cg2jy6l12rzq9k8zk6o" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-02-22 15:56:54 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;204106668&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

===Theory=== 

Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.

|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

==Intervals== 
|| degrees of 11edo || cents value || solfege ||
|| 0 || 0.00 || **do** ||
|| 1 || 109.09 || **ra** ||
|| 2 || 218.18 || **re** ||
|| 3 || 327.27 || **me** ||
|| 4 || 436.36 || **mo** ||
|| 5 || 545.45 || **fu** ||
|| 6 || 654.55 || **su** ||
|| 7 || 763.64 || **lo** ||
|| 8 || 872.73 || **la** ||
|| 9 || 981.82 || **te** ||
|| 10 || 1090.91 || **ti** ||


===MOS Scales=== 

Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

===11edo Solfege=== 

An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

===11edo Instant Ensemble=== 

In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

===Compositions=== 

[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;q=hi&amp;amp;newref=1%27%29;|I]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;q=hi&amp;amp;newref=1%27%29;|II]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;q=hi&amp;amp;newref=1%27%29;|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott [[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]].
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.

===Videos=== 

The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Theory"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Theory&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc2"&amp;gt;&amp;lt;a name="x11 tone equal temperament-Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Intervals&amp;lt;/h2&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;degrees of 11edo&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents value&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;solfege&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.00&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;do&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ra&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;re&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327.27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;me&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;mo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;fu&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;654.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;su&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;763.64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;lo&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;872.73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;la&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;981.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;te&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1090.91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;strong&amp;gt;ti&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x11 tone equal temperament-Intervals-MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;MOS Scales&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x11 tone equal temperament-Intervals-11edo Solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11edo Solfege&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc5"&amp;gt;&amp;lt;a name="x11 tone equal temperament-Intervals-11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;11edo Instant Ensemble&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc6"&amp;gt;&amp;lt;a name="x11 tone equal temperament-Intervals-Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;Compositions&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:14:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc7"&amp;gt;&amp;lt;a name="x11 tone equal temperament-Intervals-Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:14 --&amp;gt;Videos&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>8w4gislkygz5cg2jy6l12rzq9k8zk6o</sha1>
    </revision>
    <revision>
      <id>4413</id>
      <timestamp>2011-02-16T16:23:47Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 202503794 - Original comment: **</comment>
      <origin>4413</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17349" sha1="a1s8kvbf9nlh5kmhzmqh9f75ksjxwbf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-02-16 16:23:47 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;202503794&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

===Theory=== 

Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.

|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

===MOS Scales=== 

Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

===11edo Solfege=== 

An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

===11edo Instant Ensemble=== 

In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

===Compositions=== 

[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;q=hi&amp;amp;newref=1%27%29;|I]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;q=hi&amp;amp;newref=1%27%29;|II]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;q=hi&amp;amp;newref=1%27%29;|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share.cfm?id=955383|Angkor Wat, September 1066]] by X. J. Scott [[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.

===Videos=== 

The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Theory"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Theory&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x11 tone equal temperament--MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;MOS Scales&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x11 tone equal temperament--11edo Solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;11edo Solfege&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x11 tone equal temperament--11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11edo Instant Ensemble&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc5"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Compositions&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share.cfm?id=955383" rel="nofollow"&amp;gt;Angkor Wat, September 1066&amp;lt;/a&amp;gt; by X. J. Scott &amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc6"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;Videos&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>a1s8kvbf9nlh5kmhzmqh9f75ksjxwbf</sha1>
    </revision>
    <revision>
      <id>4414</id>
      <timestamp>2011-02-16T16:00:57Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 202496434 - Original comment: **</comment>
      <origin>4414</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17114" sha1="6klwqw0zokh8cx1gds2xyrfjhbgy7ot" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-02-16 16:00:57 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;202496434&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

===Theory=== 

Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.

|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

===MOS Scales=== 

Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

===11edo Solfege=== 

An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

===11edo Instant Ensemble=== 

In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

===Compositions=== 

[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;q=hi&amp;amp;newref=1%27%29;|I]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;q=hi&amp;amp;newref=1%27%29;|II]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;q=hi&amp;amp;newref=1%27%29;|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.

===Videos=== 

The Stuffed Ones: [[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;feature=related|Goopy]], [[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;feature=related|Ziggy]], [[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;feature=related|Ellie]], [[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;feature=related|Towelbear]] by [[http://www.youtube.com/user/zipzappoozoo|zipzappoozoo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Theory"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Theory&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x11 tone equal temperament--MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;MOS Scales&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x11 tone equal temperament--11edo Solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;11edo Solfege&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x11 tone equal temperament--11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11edo Instant Ensemble&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc5"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Compositions&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:12:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc6"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Videos"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:12 --&amp;gt;Videos&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
The Stuffed Ones: &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Goopy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ziggy&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=53IiHdXfJwI&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Ellie&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;amp;amp;feature=related" rel="nofollow"&amp;gt;Towelbear&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/user/zipzappoozoo" rel="nofollow"&amp;gt;zipzappoozoo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>6klwqw0zokh8cx1gds2xyrfjhbgy7ot</sha1>
    </revision>
    <revision>
      <id>4415</id>
      <timestamp>2011-02-16T15:41:32Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 202489228 - Original comment: **</comment>
      <origin>4415</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15784" sha1="phdndpavbylpz8n8kf7gi20v8pi8wt3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-02-16 15:41:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;202489228&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

===Theory=== 

Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.

|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

===MOS Scales=== 

Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

===11edo Solfege=== 

An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

===11edo Instant Ensemble=== 

In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

===Compositions=== 

[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;q=hi&amp;amp;newref=1%27%29;|I]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;q=hi&amp;amp;newref=1%27%29;|II]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;q=hi&amp;amp;newref=1%27%29;|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3|Icicle Caverns]] by Dr. Ozan Yarman
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Theory"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Theory&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x11 tone equal temperament--MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;MOS Scales&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x11 tone equal temperament--11edo Solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;11edo Solfege&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x11 tone equal temperament--11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11edo Instant Ensemble&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc5"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Compositions&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3" rel="nofollow"&amp;gt;Icicle Caverns&amp;lt;/a&amp;gt; by Dr. Ozan Yarman&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>phdndpavbylpz8n8kf7gi20v8pi8wt3</sha1>
    </revision>
    <revision>
      <id>4416</id>
      <timestamp>2011-02-13T17:06:14Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 201368368 - Original comment: **</comment>
      <origin>4416</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15523" sha1="8ovn6s6ry6hmcv0j5vbakhbqef4kvow" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-02-13 17:06:14 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;201368368&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

===Theory=== 

Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.

|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

===MOS Scales=== 

Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

===11edo Solfege=== 

An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

===11edo Instant Ensemble=== 

In February 2011, [[http://oddmusicuc.wordpress.com/|Oddmusic U-C]], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

===Compositions=== 

[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;q=hi&amp;amp;newref=1%27%29;|I]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;q=hi&amp;amp;newref=1%27%29;|II]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;q=hi&amp;amp;newref=1%27%29;|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Theory"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Theory&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x11 tone equal temperament--MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;MOS Scales&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x11 tone equal temperament--11edo Solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;11edo Solfege&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x11 tone equal temperament--11edo Instant Ensemble"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;11edo Instant Ensemble&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
In February 2011, &amp;lt;a class="wiki_link_ext" href="http://oddmusicuc.wordpress.com/" rel="nofollow"&amp;gt;Oddmusic U-C&amp;lt;/a&amp;gt;, as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:10:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc5"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:10 --&amp;gt;Compositions&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>8ovn6s6ry6hmcv0j5vbakhbqef4kvow</sha1>
    </revision>
    <revision>
      <id>4417</id>
      <timestamp>2011-02-11T17:36:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 201041940 - Original comment: **</comment>
      <origin>4417</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14515" sha1="pgp0bxdf44d0h1pow5lxlbvwcv064fv" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2011-02-11 17:36:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;201041940&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

===Theory=== 

Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.

|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

===MOS Scales=== 

Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

===11edo Solfege=== 

An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

===Compositions=== 

[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill, 2010
Hyperimprovisations Nuggetwarp ([[http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;q=hi&amp;amp;newref=1%27%29;|I]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;q=hi&amp;amp;newref=1%27%29;|II]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;q=hi&amp;amp;newref=1%27%29;|III]]) by Jacob Barton, 2009
She Is My Lilac-Hued Obsession on City of the Asleep, [[http://cityoftheasleep.com/music|Map of an Internal Landscape]] (2009)
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Theory"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Theory&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x11 tone equal temperament--MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;MOS Scales&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x11 tone equal temperament--11edo Solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;11edo Solfege&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Compositions&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill, 2010&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton, 2009&amp;lt;br /&amp;gt;
She Is My Lilac-Hued Obsession on City of the Asleep, &amp;lt;a class="wiki_link_ext" href="http://cityoftheasleep.com/music" rel="nofollow"&amp;gt;Map of an Internal Landscape&amp;lt;/a&amp;gt; (2009)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>pgp0bxdf44d0h1pow5lxlbvwcv064fv</sha1>
    </revision>
    <revision>
      <id>4418</id>
      <timestamp>2011-02-11T17:32:40Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 201041192 - Original comment: **</comment>
      <origin>4418</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14166" sha1="fentad16rk1frjf2x7tw3ctj204fw1d" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2011-02-11 17:32:40 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;201041192&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

===Theory=== 

Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.

|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

===MOS Scales=== 

Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

===11edo Solfege=== 

An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

===Compositions=== 

[[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3|Cool My Head]] by David Hamill
Hyperimprovisations Nuggetwarp ([[http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;q=hi&amp;amp;newref=1%27%29;|I]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;q=hi&amp;amp;newref=1%27%29;|II]] [[http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;q=hi&amp;amp;newref=1%27%29;|III]]) by Jacob Barton
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Theory"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Theory&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x11 tone equal temperament--MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;MOS Scales&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x11 tone equal temperament--11edo Solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;11edo Solfege&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Compositions&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3" rel="nofollow"&amp;gt;Cool My Head&amp;lt;/a&amp;gt; by David Hamill&amp;lt;br /&amp;gt;
Hyperimprovisations Nuggetwarp (&amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267904&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;I&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267905&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;II&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://javascript:Player%28%27../player/single_player.cfm?songid=10267906&amp;amp;amp;q=hi&amp;amp;amp;newref=1%27%29;" rel="nofollow"&amp;gt;III&amp;lt;/a&amp;gt;) by Jacob Barton&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>fentad16rk1frjf2x7tw3ctj204fw1d</sha1>
    </revision>
    <revision>
      <id>4419</id>
      <timestamp>2011-02-11T15:48:01Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 201014924 - Original comment: **</comment>
      <origin>4419</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12970" sha1="cfo1z56k33hs9qvpkjxjxosp0ws3wi5" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-02-11 15:48:01 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;201014924&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

===Theory=== 

Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.

|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

===MOS Scales=== 

Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

===11edo Solfege=== 

An 11edo solfege system can easily be applied from the [[22edo solfege]] system.
A chromatic scale would thus be sung: **do ra re me mo fu su lo la te ti do**.

===Compositions=== 

[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Theory"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Theory&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat with harmonic timbres.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x11 tone equal temperament--MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;MOS Scales&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of &amp;lt;a class="wiki_link" href="/MOSScales"&amp;gt;moment-of-symmetry scales&amp;lt;/a&amp;gt;, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x11 tone equal temperament--11edo Solfege"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;11edo Solfege&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
An 11edo solfege system can easily be applied from the &amp;lt;a class="wiki_link" href="/22edo%20solfege"&amp;gt;22edo solfege&amp;lt;/a&amp;gt; system.&amp;lt;br /&amp;gt;
A chromatic scale would thus be sung: &amp;lt;strong&amp;gt;do ra re me mo fu su lo la te ti do&amp;lt;/strong&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:8:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc4"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:8 --&amp;gt;Compositions&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>cfo1z56k33hs9qvpkjxjxosp0ws3wi5</sha1>
    </revision>
    <revision>
      <id>4420</id>
      <timestamp>2011-01-26T13:23:12Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 196242112 - Original comment: **</comment>
      <origin>4420</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12159" sha1="k9qooq058paaulbgfmtd807synl1hsl" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2011-01-26 13:23:12 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;196242112&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.

===Theory=== 

Compared to 12edo, the intervals of 11edo are stretched:
* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat quite a bit with harmonic timbres.

|| Harmonic || 8 ||   || 9 ||   || 11 ||   || 14 ||   || 16 ||
|| JI interval from 1/1 || 1/1 = 0 cents ||   || 9/8 = 204 ||   || 11/8 = 551 ||   || 7/4 = 969 ||   || 2/1 = 1200 ||
|| nearest 11edo interval || 0\11edo = 0 cents ||   || 2\11 = 218 ||   || 5\11 = 545 ||   || 9\11 = 982 ||   || 11\11 = 1200 ||
|| difference || 0 ||   || +14 ||   || -6 ||   || +13 ||   || 0 ||
|| JI interval between ||   || 9:8 = 204 cents ||   || 11:9 = 347 ||   || 14/11 = 418 ||   || 8:7 = 231 ||   ||
|| nearest 11edo interval ||   || 2\11 = 218 ||   || 3\11 = 327 ||   || 4\11 = 436 ||   || 2\11 = 218 ||   ||
|| difference ||   || +14 ||   || -20 ||   || +18 ||   || -13 ||   ||

===MOS Scales=== 

Although 11edo has one fewer interval in the octave than 12edo, in terms of moment-of-symmetry scales, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11, 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.

See [[11edo Modes]]

===Compositions=== 

[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
11-tone equal temperament, or 11edo, divides the octave into eleven equal steps of approximately 109.09 cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Theory"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Theory&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Compared to 12edo, the intervals of 11edo are stretched:&amp;lt;br /&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor second,&amp;amp;quot; at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major second,&amp;amp;quot; at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a &amp;amp;quot;bluesy&amp;amp;quot; seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;minor third,&amp;amp;quot; at 327.27 cents, is rather sharp and encroaching upon &amp;amp;quot;neutral third.&amp;amp;quot;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;major third,&amp;amp;quot; at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;The &amp;amp;quot;perfect fourth,&amp;amp;quot; at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
11edo can produce an approximation of an 11-limit JI chord, lacking a third and fifth harmonic: 8:9:11:14:16. The error is rather large, and this chord is sure to beat quite a bit with harmonic timbres.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;Harmonic&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval from 1/1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1/1 = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/8 = 204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/8 = 551&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/4 = 969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/1 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0\11edo = 0 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\11 = 545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\11 = 982&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11\11 = 1200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;JI interval between&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9:8 = 204 cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11:9 = 347&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/11 = 418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8:7 = 231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;nearest 11edo interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3\11 = 327&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4\11 = 436&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2\11 = 218&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;difference&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-20&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;+18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;-13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x11 tone equal temperament--MOS Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;MOS Scales&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
Although 11edo has one fewer interval in the octave than 12edo, in terms of moment-of-symmetry scales, it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11, 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
2\11 generates 2 2 2 2 3 and 2 2 2 2 2 1.&amp;lt;br /&amp;gt;
3\11 generates 3 3 3 2 and 1 2 1 2 1 2 2.&amp;lt;br /&amp;gt;
4\11 generates 4 4 3, 1 3 1 3 3, and 1 1 2 1 1 2 1 2.&amp;lt;br /&amp;gt;
5\11 generates 5 5 1, 1 4 1 4 1, 1 1 3 1 1 3 1, and 1 1 1 2 1 1 1 2 1.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc3"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Compositions&amp;lt;/h3&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>k9qooq058paaulbgfmtd807synl1hsl</sha1>
    </revision>
    <revision>
      <id>4421</id>
      <timestamp>2010-06-14T03:49:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 148685161 - Original comment: **</comment>
      <origin>4421</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3667" sha1="kcrlfjyht0qba0kk15mcmhhf1pfmiwi" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2010-06-14 03:49:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;148685161&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

===Theory=== 
See [[11edo Modes]]

===Compositions=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/mp3/Prelude11ET.mp3|Prelude11ET]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Theory"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Theory&amp;lt;/h3&amp;gt;
 See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Compositions&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Prelude11ET.mp3" rel="nofollow"&amp;gt;Prelude11ET&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>kcrlfjyht0qba0kk15mcmhhf1pfmiwi</sha1>
    </revision>
    <revision>
      <id>4422</id>
      <timestamp>2010-06-14T03:46:20Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 148684953 - Original comment: music link moved**</comment>
      <origin>4422</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3667" sha1="7ozek9hu1b2i3dy29tyl4wr0hrtsq5z" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2010-06-14 03:46:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;148684953&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;music link moved&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

===Theory=== 
See [[11edo Modes]]

===Compositions=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/musicFiles.html|Prelude]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc1"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Theory"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Theory&amp;lt;/h3&amp;gt;
 See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h3&amp;amp;gt; --&amp;gt;&amp;lt;h3 id="toc2"&amp;gt;&amp;lt;a name="x11 tone equal temperament--Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Compositions&amp;lt;/h3&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/musicFiles.html" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>7ozek9hu1b2i3dy29tyl4wr0hrtsq5z</sha1>
    </revision>
    <revision>
      <id>4423</id>
      <timestamp>2010-06-10T22:14:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 148292625 - Original comment: **</comment>
      <origin>4423</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3661" sha1="50uubcahf2heoqyh7mbf754e80ww9s6" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2010-06-10 22:14:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;148292625&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

===Theory=== 
See [[11edo Modes]]

===Compositions=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/downloads.html|Prelude]] by [[Aaron Andrew Hunt|Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 See &amp;lt;a class="wiki_link" href="/11edo%20Modes"&amp;gt;11edo Modes&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/downloads.html" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Andrew%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <contributor>
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      <comment>**Imported revision 148251087 - Original comment: composer's links**</comment>
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      <text bytes="3638" sha1="sqocyysejunpqx9yupuwtdt4l21ijab" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2010-06-10 17:57:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;148251087&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

===Theory=== 
See [[11edo Modes]]

===Compositions=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by [[Bill Sethares]] (spectrally bent synth ens.)
[[http://www.h-pi.com/downloads.html|Prelude]] by [[Aaron Hunt]] (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by [[Christopher Bailey]] (keyboards concréte)
[[http://soundclick.com/share?songid=8839070|conversation is]] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Bill%20Sethares"&amp;gt;Bill Sethares&amp;lt;/a&amp;gt; (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/downloads.html" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Aaron%20Hunt"&amp;gt;Aaron Hunt&amp;lt;/a&amp;gt; (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Christopher%20Bailey"&amp;gt;Christopher Bailey&amp;lt;/a&amp;gt; (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Andrew%20Heathwaite"&amp;gt;Andrew Heathwaite&amp;lt;/a&amp;gt;. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 139758519 - Original comment: **</comment>
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      <text bytes="3376" sha1="14b3duidb5cz1ama4yngv2p4flvoazw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2010-05-05 16:16:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;139758519&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

===Theory=== 
See [[11edo Modes]]

===Compositions=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by Bill Sethares (spectrally bent synth ens.)
[[http://www.h-pi.com/downloads.html|Prelude]] by Aaron Hunt (neo-Baroque)
[[http://music.columbia.edu/%7Echris/complist.html|The Stuffed Ones]] by Christopher Bailey (keyboards concréte)
[[http://soundclick.com/share?songid=8839070|conversation is]] by Andrew Heathwaite. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp; voice.&lt;/pre&gt;&lt;/div&gt;
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 &amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by Bill Sethares (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/downloads.html" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Aaron Hunt (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/%7Echris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by Christopher Bailey (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839070" rel="nofollow"&amp;gt;conversation is&amp;lt;/a&amp;gt; by Andrew Heathwaite. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar &amp;amp;amp; voice.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 83647779 - Original comment: **</comment>
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      <text bytes="2892" sha1="mob01pe8a0gp5xuwk3u53rss3yrh5rd" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
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===Theory=== 
See [[11edo Modes]]

===Compositions=== 
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by Bill Sethares (spectrally bent synth ens.)
[[http://www.h-pi.com/downloads.html|Prelude]] by Aaron Hunt (neo-Baroque)
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      <comment>**Imported revision 9403279 - Original comment: **</comment>
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      <text bytes="2374" sha1="pzrfbihe7obsk6a4a6my2g50uinchon" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
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**Compositions**
[[http://eceserv0.ece.wisc.edu/~sethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by Bill Sethares (spectrally bent synth ens.)
[[http://www.h-pi.com/downloads.html|Prelude]] by Aaron Hunt (neo-Baroque)
[[http://music.columbia.edu/~chris/complist.html|The Stuffed Ones]] by Christopher Bailey (keyboards concréte)
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 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;Compositions&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/~sethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by Bill Sethares (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/downloads.html" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Aaron Hunt (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by Christopher Bailey (keyboards concréte)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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        <id>0</id>
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      <comment>**Imported revision 9286387 - Original comment: **</comment>
      <origin>4428</origin>
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      <text bytes="2642" sha1="95fvtp70mwwiv5qclwk0prw5yenuqa3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2007-10-10 20:08:55 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;9286387&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

**Compositions**
[[http://eceserv0.ece.wisc.edu/~sethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by Bill Sethares (spectrally bent synth ens.)
[[http://www.h-pi.com/downloads.html|Prelude]] by Aaron Hunt (neo-Baroque)
[[http://music.columbia.edu/~chris/complist.html|The Stuffed Ones]] by Christopher Bailey (keyboards concréte)
[[http://zebox.com/cgi-bin/artists/dl.cgi?danstearns_2_-_11edo.m3u+qp+m4|over the rivers]] stearns&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;Compositions&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/~sethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by Bill Sethares (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/downloads.html" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Aaron Hunt (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by Christopher Bailey (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://zebox.com/cgi-bin/artists/dl.cgi?danstearns_2_-_11edo.m3u+qp+m4" rel="nofollow"&amp;gt;over the rivers&amp;lt;/a&amp;gt; stearns&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2007-10-10T20:06:45Z</timestamp>
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        <id>0</id>
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      <comment>**Imported revision 9286263 - Original comment: **</comment>
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      <text bytes="2648" sha1="p8699b73uh2jdakkvb9qya0wwb25vcx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2007-10-10 20:06:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;9286263&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

**Compositions**
[[http://eceserv0.ece.wisc.edu/~sethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by Bill Sethares (spectrally bent synth ens.)
[[http://www.h-pi.com/downloads.html|Prelude]] by Aaron Hunt (neo-Baroque)
[[http://music.columbia.edu/~chris/complist.html|The Stuffed Ones]] by Christopher Bailey (keyboards concréte)
[[http://zebox.com/cgi-bin/artists/dl.cgi?danstearns_2_-_11edo.m3u+qp+m4|over the rivers]] danstearns&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;Compositions&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/~sethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by Bill Sethares (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/downloads.html" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Aaron Hunt (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by Christopher Bailey (keyboards concréte)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://zebox.com/cgi-bin/artists/dl.cgi?danstearns_2_-_11edo.m3u+qp+m4" rel="nofollow"&amp;gt;over the rivers&amp;lt;/a&amp;gt; danstearns&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2007-10-08T20:15:50Z</timestamp>
      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 9159301 - Original comment: **</comment>
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:danstearns|danstearns]] and made on &lt;tt&gt;2007-10-08 20:15:50 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;9159301&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

**Compositions**
[[http://eceserv0.ece.wisc.edu/~sethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by Bill Sethares (spectrally bent synth ens.)
[[http://www.h-pi.com/downloads.html|Prelude]] by Aaron Hunt (neo-Baroque)
[[http://music.columbia.edu/~chris/complist.html|The Stuffed Ones]] by Christopher Bailey (keyboards concréte)
[[http://zebox.com/cgi-bin/artists/dl.cgi?danstearns_2_-_11edo.m3u+qp+m4|over the rivers]] [[http://www.myspace.com/danstearns|danstearns]]
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;Compositions&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/~sethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by Bill Sethares (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/downloads.html" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Aaron Hunt (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by Christopher Bailey (keyboards concréte)&amp;lt;br /&amp;gt;
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      <timestamp>2007-09-07T19:54:25Z</timestamp>
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        <id>0</id>
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      <comment>**Imported revision 7676063 - Original comment: **</comment>
      <origin>4431</origin>
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      <text bytes="2373" sha1="92si6m23w57p3k0n4np4lvxeh03p254" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2007-09-07 19:54:25 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;7676063&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

**Compositions**
[[http://eceserv0.ece.wisc.edu/~sethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by Bill Sethares (spectrally bent synth ens.)
[[http://www.h-pi.com/downloads.html|Prelude]] by Aaron Hunt (neo-Baroque)
[[http://music.columbia.edu/~chris/complist.html|The Stuffed Ones]] by Christopher Bailey (keyboards concréte)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x11 tone equal temperament"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;11 tone equal temperament&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;Compositions&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/~sethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by Bill Sethares (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/downloads.html" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Aaron Hunt (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by Christopher Bailey (keyboards concréte)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2007-06-05T20:26:40Z</timestamp>
      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 4799327 - Original comment: **</comment>
      <origin>4432</origin>
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      <text bytes="2821" sha1="4jayf1q2f7z1cv8tlq5zzza4aka8blo" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2007-06-05 20:26:40 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;4799327&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

**Compositions**
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by Bill Sethares (spectrally bent synth ens.)
[[http://zebox.com/cgi-bin/artists/dl.cgi?danstearns_2_-_11edo.m3u+qp+m4|Over the Rivers]] by [[http://www.myspace.com/danstearns|Dan Stearns]] (synth ens.)
[[http://www.h-pi.com/downloads.html|Prelude]] by Aaron Hunt (neo-Baroque)
[[http://music.columbia.edu/~chris/complist.html|The Stuffed Ones]] by Christopher Bailey (keyboards concréte)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;Compositions&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by Bill Sethares (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://zebox.com/cgi-bin/artists/dl.cgi?danstearns_2_-_11edo.m3u+qp+m4" rel="nofollow"&amp;gt;Over the Rivers&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.myspace.com/danstearns" rel="nofollow"&amp;gt;Dan Stearns&amp;lt;/a&amp;gt; (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/downloads.html" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Aaron Hunt (neo-Baroque)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://music.columbia.edu/~chris/complist.html" rel="nofollow"&amp;gt;The Stuffed Ones&amp;lt;/a&amp;gt; by Christopher Bailey (keyboards concréte)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4433</id>
      <timestamp>2007-05-26T10:51:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenjacob</username>
        <id>0</id>
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      <comment>**Imported revision 4629711 - Original comment: **</comment>
      <origin>4433</origin>
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      <text bytes="2532" sha1="4hn3olzqnq8di89szuer78q77usdsnv" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenjacob|xenjacob]] and made on &lt;tt&gt;2007-05-26 10:51:17 UTC&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament= 

**Compositions**
[[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html|Turquoise Dabo Girl]] by Bill Sethares (spectrally bent synth ens.)
[[http://zebox.com/cgi-bin/artists/dl.cgi?danstearns_2_-_11edo.m3u+qp+m4|Over the Rivers]] by [[http://www.myspace.com/danstearns|Dan Stearns]] (synth ens.)
[[http://www.h-pi.com/downloads.html|Prelude]] by Aaron Hunt (neo-Baroque)&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;a class="wiki_link_ext" href="http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html" rel="nofollow"&amp;gt;Turquoise Dabo Girl&amp;lt;/a&amp;gt; by Bill Sethares (spectrally bent synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://zebox.com/cgi-bin/artists/dl.cgi?danstearns_2_-_11edo.m3u+qp+m4" rel="nofollow"&amp;gt;Over the Rivers&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.myspace.com/danstearns" rel="nofollow"&amp;gt;Dan Stearns&amp;lt;/a&amp;gt; (synth ens.)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://www.h-pi.com/downloads.html" rel="nofollow"&amp;gt;Prelude&amp;lt;/a&amp;gt; by Aaron Hunt (neo-Baroque)&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 4466579 - Original comment: **</comment>
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      <text bytes="1958" sha1="0e2d8jnu3187izp5l4k1xz5meqjhmfc" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2007-05-17 16:20:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;4466579&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=11 tone equal temperament=

**Compositions**

[[http://zebox.com/cgi-bin/artists/dl.cgi?danstearns_2_-_11edo.m3u+qp+m4|Over the Rivers]] by [[http://www.myspace.com/danstearns|Dan Stearns]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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      <comment>**Imported revision 4458680 - Original comment: **</comment>
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:microtonaldan|microtonaldan]] and made on &lt;tt&gt;2007-05-17 11:11:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;4458680&lt;/tt&gt;.&lt;br&gt;
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**Compositions**&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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      <comment>**Imported revision 4458043 - Original comment: **</comment>
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: The original revision id was &lt;tt&gt;4458043&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
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&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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      <text bytes="19865" sha1="m5wucd6caeatd52lp6ft3w5htajzv9d" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[11平均律|日本語]]&lt;/span&gt;

__FORCETOC__

=11 tone equal temperament=

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

=Subgroup=
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=

{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

=Notation=

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, Cv
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; [http://tel.wikispaces.com/11_17_26_31_38_41 11 17 26 31 38 41] |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| | | -7 3 1 &amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| | | -68 18 17 &amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=11edo Instant Ensemble=
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

=Compositions=
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;''''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'' '') by [[Chris_Vaisvil|Chris Vaisvil]]''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;
==Instruments==

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]      [[Category:11-tone]]
[[Category:11edo]]
[[Category:edo]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:prime_edo]]
[[Category:scale]]
[[Category:subgroup]]
[[Category:todo:unify_precision]]</text>
      <sha1>m5wucd6caeatd52lp6ft3w5htajzv9d</sha1>
    </revision>
    <revision>
      <id>36639</id>
      <parentid>33207</parentid>
      <timestamp>2018-10-12T12:00:10Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>introduced [[Template:Japanese version page]]</comment>
      <origin>36639</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19837" sha1="7ho6nynpwuslxmage5i8c9h5ce86q9c" xml:space="preserve">{{Japanese version page| [[11平均律|日本語]] }}

__FORCETOC__

=11 tone equal temperament=

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

=Subgroup=
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=

{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

=Notation=

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, Cv
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; [http://tel.wikispaces.com/11_17_26_31_38_41 11 17 26 31 38 41] |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| | | -7 3 1 &amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| | | -68 18 17 &amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=11edo Instant Ensemble=
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

=Compositions=
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;''''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'' '') by [[Chris_Vaisvil|Chris Vaisvil]]''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;
==Instruments==

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]      [[Category:11-tone]]
[[Category:11edo]]
[[Category:edo]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:prime_edo]]
[[Category:scale]]
[[Category:subgroup]]
[[Category:todo:unify_precision]]</text>
      <sha1>7ho6nynpwuslxmage5i8c9h5ce86q9c</sha1>
    </revision>
    <revision>
      <id>36734</id>
      <parentid>36639</parentid>
      <timestamp>2018-10-12T22:19:39Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>replaced interwiki solution</comment>
      <origin>36734</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19841" sha1="5l78cygqgfatisjpd0eohz6el3ohvf4" xml:space="preserve">{{interwiki
| de = 
| en = 
| es = 
| ja = 11平均律
}}

__FORCETOC__

=11 tone equal temperament=

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

=Subgroup=
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=

{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

=Notation=

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, Cv
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; [http://tel.wikispaces.com/11_17_26_31_38_41 11 17 26 31 38 41] |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| | | -7 3 1 &amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| | | -68 18 17 &amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=11edo Instant Ensemble=
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

=Compositions=
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;''''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'' '') by [[Chris_Vaisvil|Chris Vaisvil]]''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;
==Instruments==

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]      [[Category:11-tone]]
[[Category:11edo]]
[[Category:edo]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:prime_edo]]
[[Category:scale]]
[[Category:subgroup]]
[[Category:todo:unify_precision]]</text>
      <sha1>5l78cygqgfatisjpd0eohz6el3ohvf4</sha1>
    </revision>
    <revision>
      <id>36755</id>
      <parentid>36734</parentid>
      <timestamp>2018-10-12T22:44:11Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>unified interwiki section for all related pages</comment>
      <origin>36755</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19847" sha1="p2z2qcyzhv0uiynb5zw7f1aycct87rp" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}

__FORCETOC__

=11 tone equal temperament=

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

=Subgroup=
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=

{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

=Notation=

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, Cv
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; [http://tel.wikispaces.com/11_17_26_31_38_41 11 17 26 31 38 41] |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| | | -7 3 1 &amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| | | -68 18 17 &amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=11edo Instant Ensemble=
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

=Compositions=
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;''''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'' '') by [[Chris_Vaisvil|Chris Vaisvil]]''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;
==Instruments==

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]      [[Category:11-tone]]
[[Category:11edo]]
[[Category:edo]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:prime_edo]]
[[Category:scale]]
[[Category:subgroup]]
[[Category:todo:unify_precision]]</text>
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      <text bytes="19844" sha1="l9obtozskrf6r4z5fisa30ypvaejv0c" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
=11 tone equal temperament=

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

=Subgroup=
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=

{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

=Notation=

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, Cv
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; [http://tel.wikispaces.com/11_17_26_31_38_41 11 17 26 31 38 41] |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| | | -7 3 1 &amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| | | -68 18 17 &amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=11edo Instant Ensemble=
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

=Compositions=
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;''''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'' '') by [[Chris_Vaisvil|Chris Vaisvil]]''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;
==Instruments==

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]      [[Category:11-tone]]
[[Category:11edo]]
[[Category:edo]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:prime_edo]]
[[Category:scale]]
[[Category:subgroup]]
[[Category:todo:unify_precision]]</text>
      <sha1>l9obtozskrf6r4z5fisa30ypvaejv0c</sha1>
    </revision>
    <revision>
      <id>37793</id>
      <parentid>37026</parentid>
      <timestamp>2018-10-24T22:54:23Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>removed tel link - once again :-)</comment>
      <origin>37793</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19824" sha1="ckapg3ahbmdirx6lblomnyxgma59cbr" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
=11 tone equal temperament=

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

=Subgroup=
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=

{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

=Notation=

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, Cv
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| | | -7 3 1 &amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| | | -68 18 17 &amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=11edo Instant Ensemble=
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

=Compositions=
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;''''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'' '') by [[Chris_Vaisvil|Chris Vaisvil]]''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;
==Instruments==

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]      [[Category:11-tone]]
[[Category:11edo]]
[[Category:edo]]
[[Category:listen]]
[[Category:macrotonal]]
[[Category:prime_edo]]
[[Category:scale]]
[[Category:subgroup]]
[[Category:todo:unify_precision]]</text>
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| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
=11 tone equal temperament=

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

=Subgroup=
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=

{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

=Notation=

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, Cv
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| | | -7 3 1 &amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| | | -68 18 17 &amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=11edo Instant Ensemble=
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

=Compositions=
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;''''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'' '') by [[Chris_Vaisvil|Chris Vaisvil]]''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;
==Instruments==

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
      <sha1>jqnmdqe9x50c4eq4fhzobr83a1vgxkx</sha1>
    </revision>
    <revision>
      <id>41209</id>
      <parentid>38317</parentid>
      <timestamp>2019-03-26T22:20:22Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
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      <comment>/* Notation */</comment>
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      <text bytes="20566" sha1="9qhc26f4xr7alqiadhwndbfjt3vhqze" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
=11 tone equal temperament=

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

=Subgroup=
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=

{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

=Notation=

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
!  rowspan="2"| Degree
!  colspan="3"| Size in
!  rowspan="2"| Solfege
!  rowspan="2"| Approximate Ratios*
!  rowspan="2"| Sagittal

notation

(22edo subset)
! colspan="2" rowspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" rowspan="2" | Up/down notation

with major narrower

than minor
!  rowspan="2"| [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
![[cent|Cents]]
!pions
!7mus
|-
| style="text-align:center;" | 0
| colspan="3" style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
|115.64
|139.64 (8B.A2&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
|231.27
|279.27 (117.46&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C\\
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
|346.91
|418.91 (1A2.E9&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bp, Cv
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
|462.545
|558.545 (22E.8C&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.455
|578.18
|698.18 (2BA.2F&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.545
|693.82
|837.82 (345.D1&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
|809.455
|977.455 (3D1.74&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F\\
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
|925.09
|1117.09 (45D.17&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G\\
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
|1040.73
|1256.73 (4E8.BA&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
|1156.36
|1396.36 (574.5D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
|1272
|1536 (600&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=11edo Instant Ensemble=
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

=Compositions=
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;'&lt;nowiki/&gt;'''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'''''&lt;nowiki/&gt;''' '') by [[Chris_Vaisvil|Chris Vaisvil]]'''''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;
==Instruments==

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
      <sha1>9qhc26f4xr7alqiadhwndbfjt3vhqze</sha1>
    </revision>
    <revision>
      <id>41311</id>
      <parentid>41209</parentid>
      <timestamp>2019-04-04T01:53:02Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Subgroup */</comment>
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      <text bytes="20566" sha1="ib48ox3q4ig4fqalnawgi1vasctk03f" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
=11 tone equal temperament=

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

=Subgroup=
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=

{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

=Notation=

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! rowspan="2" | Degree
! colspan="3" | Size in
! rowspan="2" | Solfege
! rowspan="2" | Approximate Ratios*
! rowspan="2" | Sagittal

notation

(22edo subset)
! colspan="2" rowspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" rowspan="2" | Up/down notation

with major narrower

than minor
! rowspan="2" | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
![[cent|Cents]]
!pions
!7mus
|-
| style="text-align:center;" | 0
| colspan="3" style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
|115.64
|139.64 (8B.A3&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
|231.27
|279.27 (117.46&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C\\
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
|346.91
|418.91 (1A2.E9&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bp, Cv
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
|462.545
|558.545 (22E.8C&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.455
|578.18
|698.18 (2BA.2F&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.545
|693.82
|837.82 (345.D1&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
|809.455
|977.455 (3D1.74&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F\\
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
|925.09
|1117.09 (45D.17&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G\\
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
|1040.73
|1256.73 (4E8.BA&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
|1156.36
|1396.36 (574.5D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
|1272
|1536 (600&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=11edo Instant Ensemble=
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

=Compositions=
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;'&lt;nowiki/&gt;'''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'''''&lt;nowiki/&gt;''' '') by [[Chris_Vaisvil|Chris Vaisvil]]'''''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;
==Instruments==

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
      <sha1>ib48ox3q4ig4fqalnawgi1vasctk03f</sha1>
    </revision>
    <revision>
      <id>41736</id>
      <parentid>41311</parentid>
      <timestamp>2019-05-18T19:38:21Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Subgroup */</comment>
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      <format>text/x-wiki</format>
      <text bytes="20534" sha1="01m5nchz0b7isiq85g6z430fvpdm8my" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
=11 tone equal temperament=

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.455 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

=Subgroup=
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=

{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

=Notation=

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! rowspan="2" | Degree
! colspan="3" | Size in
! rowspan="2" | Solfege
! rowspan="2" | Approximate Ratios*
! rowspan="2" | Sagittal

notation

(22edo subset)
! colspan="2" rowspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" rowspan="2" | Up/down notation

with major narrower

than minor
! rowspan="2" | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
![[cent|Cents]]
!pions
!7mus
|-
| colspan="4" style="text-align:center;" | 0
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
|115.64
|139.64 (8B.A3&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
|231.27
|279.27 (117.46&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C\\
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
|346.91
|418.91 (1A2.E88&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bp, Cv
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
|462.545
|558.545 (22E.8C&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.455
|578.18
|698.18 (2BA.2F&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.545
|693.82
|837.82 (345.D1&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
|809.455
|977.455 (3D1.74&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F\\
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
|925.09
|1117.09 (45D.178&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G\\
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
|1040.73
|1256.73 (4E8.BA&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
|1156.36
|1396.36 (574.5D&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
|1272
|1536 (600&lt;sub&gt;16&lt;/sub&gt;)
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=11edo Instant Ensemble=
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

=Compositions=
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;'&lt;nowiki/&gt;'''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'''''&lt;nowiki/&gt;''' '') by [[Chris_Vaisvil|Chris Vaisvil]]'''''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;
==Instruments==

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
      <sha1>01m5nchz0b7isiq85g6z430fvpdm8my</sha1>
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    <revision>
      <id>42589</id>
      <parentid>41736</parentid>
      <timestamp>2019-12-12T10:34:11Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <comment>Reverted edits by [[Special:Contributions/Moremajorthanmajor|Moremajorthanmajor]] ([[User talk:Moremajorthanmajor|talk]]) to last revision by [[User:Xenwolf|Xenwolf]]</comment>
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      <text bytes="19821" sha1="jqnmdqe9x50c4eq4fhzobr83a1vgxkx" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
=11 tone equal temperament=

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

=Tuning=
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

=Subgroup=
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

=Intervals=

{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

=Notation=

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | A^, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | A^, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | B^, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | B^, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, Cv
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, Cv
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, Dv
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, Ev
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | D^, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | E^, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | E^, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | Fv, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | Fv, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, Gv
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, Av
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

=MOS Scales=
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| | | -7 3 1 &amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| | | -25 7 6 &amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| | | -68 18 17 &amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| | | -10 7 8 -7 &amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| | | -10 1 0 3 &amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| | | 0 3 4 -5 &amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| | | -5 -1 -2 4 &amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| | | -3 -1 -1 0 2 &amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| | | 16 0 0 -2 -3 &amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=11edo Instant Ensemble=
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

=Compositions=
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;''''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'' '') by [[Chris_Vaisvil|Chris Vaisvil]]''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;
==Instruments==

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
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    <revision>
      <id>42590</id>
      <parentid>42589</parentid>
      <timestamp>2019-12-12T10:41:50Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
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      <comment>cleanup</comment>
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      <text bytes="20035" sha1="ofczltsv6o8rhcmzr4cu10up3l88k2r" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
==11 tone equal temperament==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

==Tuning==
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

==Subgroup==
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | ^A, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | ^A, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | ^B, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | ^B, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, vC
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, vC
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, vD
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, vD
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, vE
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, vE
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^D, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^D, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | ^E, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | ^E, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | vF, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | vF, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, vG
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, vG
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, vAv
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, vAv
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

For alternative notations, see [[Ups_and_Downs_Notation#Summary of EDO notation-"Supersharp" EDOs|Ups and Downs Notation -"Supersharp" EDOs]] (pentatonic, octotonic and nonatonic fifth-generated) and [[Ups_and_Downs_Notation#Natural Generators|Ups and Downs Notation - Natural Generators]] (heptatonic third-generated).

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Instruments==
[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

==11edo Instant Ensemble==
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

==Compositions==
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;'&lt;nowiki/&gt;'''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'''''&lt;nowiki/&gt;''' '') by [[Chris_Vaisvil|Chris Vaisvil]]'''''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
      <sha1>ofczltsv6o8rhcmzr4cu10up3l88k2r</sha1>
    </revision>
    <revision>
      <id>42750</id>
      <parentid>42590</parentid>
      <timestamp>2019-12-13T01:11:13Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>Added a section about alternative notations</comment>
      <origin>42750</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21873" sha1="e6q9fhac6rdihtgudekv5hjha0osxnp" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
==11 tone equal temperament==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

==Tuning==
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

==Subgroup==
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | ^A, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | ^A, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | ^B, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | ^B, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, vC
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, vC
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, vD
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, vD
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, vE
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, vE
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^D, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^D, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | ^E, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | ^E, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | vF, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | vF, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, vG
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, vG
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, vAv
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, vAv
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size ([[cent|Cents]])
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Instruments==
[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

==11edo Instant Ensemble==
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

==Compositions==
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;'&lt;nowiki/&gt;'''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'''''&lt;nowiki/&gt;''' '') by [[Chris_Vaisvil|Chris Vaisvil]]'''''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
      <sha1>e6q9fhac6rdihtgudekv5hjha0osxnp</sha1>
    </revision>
    <revision>
      <id>42759</id>
      <parentid>42750</parentid>
      <timestamp>2019-12-13T06:05:28Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>/* Commas */ color names</comment>
      <origin>42759</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22032" sha1="0xsdnbhfbw627fq3xugpal3yjszoh3q" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
==11 tone equal temperament==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

==Tuning==
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

==Subgroup==
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | ^A, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | ^A, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | ^B, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | ^B, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, vC
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, vC
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, vD
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, vD
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, vE
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, vE
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^D, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^D, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | ^E, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | ^E, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | vF, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | vF, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, vG
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, vG
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, vAv
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, vAv
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
|Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
|Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
|Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
|Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
|Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
|Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
|Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
|Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
|Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
|Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Instruments==
[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

==11edo Instant Ensemble==
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

==Compositions==
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;'&lt;nowiki/&gt;'''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'''''&lt;nowiki/&gt;''' '') by [[Chris_Vaisvil|Chris Vaisvil]]'''''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
      <sha1>0xsdnbhfbw627fq3xugpal3yjszoh3q</sha1>
    </revision>
    <revision>
      <id>42760</id>
      <parentid>42759</parentid>
      <timestamp>2019-12-13T06:08:22Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>/* Commas */  added color names</comment>
      <origin>42760</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22322" sha1="1ze7t2t0s73w7n2msnhwnlux33i5cda" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
==11 tone equal temperament==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

==Tuning==
Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;

==Subgroup==
11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | ^A, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | ^A, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | ^B, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | ^B, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, vC
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, vC
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, vD
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, vD
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, vE
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, vE
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^D, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^D, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | ^E, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | ^E, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | vF, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | vF, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, vG
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, vG
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, vAv
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, vAv
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==11edo solfege==
An 11edo solfege system can easily be applied from the [[22edo_Solfege|22edo solfege]] system.

A chromatic scale would thus be sung: '''do ra re me mo fu su lo la ta ti do'''.

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Instruments==
[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

==11edo Instant Ensemble==
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

==Compositions==
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;'&lt;nowiki/&gt;'''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'''''&lt;nowiki/&gt;''' '') by [[Chris_Vaisvil|Chris Vaisvil]]'''''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
      <sha1>1ze7t2t0s73w7n2msnhwnlux33i5cda</sha1>
    </revision>
    <revision>
      <id>42820</id>
      <parentid>42760</parentid>
      <timestamp>2019-12-15T07:24:55Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>more cleanup</comment>
      <origin>42820</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22152" sha1="fwz9ajvb14gxerq61d4tdh7ff94lrcw" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo_Solfege|22edo solfege]].

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | ^A, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | ^A, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | ^B, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | ^B, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, vC
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, vC
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, vD
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, vD
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, vE
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, vE
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^D, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^D, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | ^E, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | ^E, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | vF, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | vF, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, vG
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, vG
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, vAv
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, vAv
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].

3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].

4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.

5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.

See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==Music==
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;'&lt;nowiki/&gt;'''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'''''&lt;nowiki/&gt;''' '') by [[Chris_Vaisvil|Chris Vaisvil]]'''''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
      <sha1>fwz9ajvb14gxerq61d4tdh7ff94lrcw</sha1>
    </revision>
    <revision>
      <id>44000</id>
      <parentid>42820</parentid>
      <timestamp>2020-04-24T03:43:16Z</timestamp>
      <contributor>
        <username>Stephen Weigel</username>
        <id>121</id>
      </contributor>
      <minor/>
      <comment>Added horagrams to /* MOS Scales */</comment>
      <origin>44000</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22438" sha1="m75347vkjccqypvkfy3msbxx7vrxztd" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo_Solfege|22edo solfege]].

{| class="wikitable"
|-
! | Degree
! | Size in

[[cent|Cents]]
! | Solfege
! | Approximate Ratios*
! | Sagittal

notation

(22edo subset)
! colspan="2" | [[Ups_and_Downs_Notation|Up/down notation]]

with major wider

than minor
! colspan="2" | Up/down notation

with major narrower

than minor
! | [[Tútim_Dennsuul_Wafiil|TDW]]

Machine

notation
|-
| style="text-align:center;" | 0
| style="text-align:right;" | 0.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 1/1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | P1
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|-
| style="text-align:center;" | 1
| style="text-align:right;" | 109.09
| style="text-align:center;" | '''ra'''
| style="text-align:center;" | [[15/14|15/14]], [[16/15|16/15]], [[17/16|17/16]], [[18/17|18/17]]
| style="text-align:center;" | AII\ or B!!/
| style="text-align:center;" | ^1, m2
| style="text-align:center;" | ^A, B
| style="text-align:center;" | ^1, M2
| style="text-align:center;" | ^A, B
| style="text-align:center;" | Q#\Rb
|-
| style="text-align:center;" | 2
| style="text-align:right;" | 218.18
| style="text-align:center;" | '''re'''
| style="text-align:center;" | [[8/7|8/7]], [[9/8|9/8]], [[17/15|17/15]]
| style="text-align:center;" | B
| style="text-align:center;" | ~2, m3
| style="text-align:center;" | ^B, Cb
| style="text-align:center;" | ~2, M3
| style="text-align:center;" | ^B, C#
| style="text-align:center;" | R
|-
| style="text-align:center;" | 3
| style="text-align:right;" | 327.27
| style="text-align:center;" | '''me'''
| style="text-align:center;" | [[6/5|6/5]], [[11/9|11/9]], [[17/14|17/14]]
| style="text-align:center;" | C/I or BII\ or D\!!/
| style="text-align:center;" | M2, ~3
| style="text-align:center;" | B#, vC
| style="text-align:center;" | m2, ~3
| style="text-align:center;" | Bb, vC
| style="text-align:center;" | R#\Sb
|-
| style="text-align:center;" | 4
| style="text-align:right;" | 436.36
| style="text-align:center;" | '''mo'''
| style="text-align:center;" | [[9/7|9/7]], [[14/11|14/11]], [[22/17|22/17]]
| style="text-align:center;" | D\! or C/II\
| style="text-align:center;" | M3, v4
| style="text-align:center;" | C, vD
| style="text-align:center;" | m3, v4
| style="text-align:center;" | C, vD
| style="text-align:center;" | S
|-
| style="text-align:center;" | 5
| style="text-align:right;" | 545.45
| style="text-align:center;" | '''fu'''
| style="text-align:center;" | [[11/8|11/8]], [[15/11|15/11]]
| style="text-align:center;" | D/I or E\!!/
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, vE
| style="text-align:center;" | P4, v5
| style="text-align:center;" | D, vE
| style="text-align:center;" | S#\Tb
|-
| style="text-align:center;" | 6
| style="text-align:right;" | 654.55
| style="text-align:center;" | '''su'''
| style="text-align:center;" | [[16/11|16/11]], [[22/15|22/15]]
| style="text-align:center;" | E\! or D/II\
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^D, E
| style="text-align:center;" | ^4, P5
| style="text-align:center;" | ^D, E
| style="text-align:center;" | T
|-
| style="text-align:center;" | 7
| style="text-align:right;" | 763.64
| style="text-align:center;" | '''lo'''
| style="text-align:center;" | [[11/7|11/7]], [[14/9|14/9]], [[17/11|17/11]]
| style="text-align:center;" | F
| style="text-align:center;" | ^5, m6
| style="text-align:center;" | ^E, Fb
| style="text-align:center;" | ^5, M6
| style="text-align:center;" | ^E, F#
| style="text-align:center;" | T#\Ub
|-
| style="text-align:center;" | 8
| style="text-align:right;" | 872.73
| style="text-align:center;" | '''la'''
| style="text-align:center;" | [[5/3|5/3]], [[18/11|18/11]], [[28/17|28/17]]
| style="text-align:center;" | FII\ or G!!/
| style="text-align:center;" | ~6, m7
| style="text-align:center;" | vF, Gb
| style="text-align:center;" | ~6, M7
| style="text-align:center;" | vF, G#
| style="text-align:center;" | U
|-
| style="text-align:center;" | 9
| style="text-align:right;" | 981.82
| style="text-align:center;" | '''ta'''
| style="text-align:center;" | [[7/4|7/4]], [[16/9|16/9]], [[30/17|30/17]]
| style="text-align:center;" | G
| style="text-align:center;" | M6, ~7
| style="text-align:center;" | F, vG
| style="text-align:center;" | m6, ~7
| style="text-align:center;" | F, vG
| style="text-align:center;" | U#\Pb
|-
| style="text-align:center;" | 10
| style="text-align:right;" | 1090.91
| style="text-align:center;" | '''ti'''
| style="text-align:center;" | [[15/8|15/8]], [[17/9|17/9]], [[28/15|28/15]], [[32/17|32/17]]
| style="text-align:center;" | GII\ or A!!/
| style="text-align:center;" | M7, v8
| style="text-align:center;" | G, vAv
| style="text-align:center;" | m7, v8
| style="text-align:center;" | G, vAv
| style="text-align:center;" | P\Qb
|-
| style="text-align:center;" | 11
| style="text-align:right;" | 1200.00
| style="text-align:center;" | '''do'''
| style="text-align:center;" | 2/1
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | P8
| style="text-align:center;" | A
| style="text-align:center;" | Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal_notation|Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==Music==
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;'&lt;nowiki/&gt;'''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'''''&lt;nowiki/&gt;''' '') by [[Chris_Vaisvil|Chris Vaisvil]]'''''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
      <sha1>m75347vkjccqypvkfy3msbxx7vrxztd</sha1>
    </revision>
    <revision>
      <id>46081</id>
      <parentid>44000</parentid>
      <timestamp>2020-06-08T18:38:45Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Notation */ removed inline style, simplified wiki markup</comment>
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      <text bytes="18833" sha1="8zo5o9z086gicjx3am1mw2e33vxnj0x" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
|-
| 7
| 763.64
| '''lo'''
| [[11/7]], [[14/9]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

==Music==
[[File:11EDO-improv.mp3]]
&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[George_Secor|George Secor]], 1970. Apparently the first piece ever written for 11edo.

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[David_Hamill|David Hamill]], 2010

Hyperimprovisations Nuggetwarp ([http://soundclick.com/share.cfm?id=10267904 I] [http://soundclick.com/share.cfm?id=10267905 II] [http://soundclick.com/share.cfm?id=10267906 III]) by [[Jacob_Barton|Jacob Barton]], 2009

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; on City of the Asleep, [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)

[http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Bill_Sethares|Bill Sethares]] (spectrally bent synth ens.)

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Aaron_Andrew_Hunt|Aaron Andrew Hunt]] (neo-Baroque)

[http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher_Bailey|Christopher Bailey]] (keyboards concréte) &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Dr. Ozan Yarman

[http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by X. J. Scott

[http://soundclick.com/share?songid=8839070 conversation is] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Andrew_Heathwaite|Andrew Heathwaite]].

Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

[http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by Andrew Heathwaite

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[John_Lyle_Smith|John Lyle Smith]]

&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [[Chris_Vaisvil|Chris Vaisvil]]

&lt;span style=""&gt;&lt;span style=""&gt;''[http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]''&lt;/span&gt;&lt;/span&gt; by [[Randy_Winchester|Randy Winchester]]

''[http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]'' by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]

''[http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo]'' by [[Jon_Lyle_Smith|Jon Lyle Smith]]

''[http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]'' by [[Aaron_Krister_Johnson|Aaron Krister Johnson]]

''&lt;span style=""&gt;[http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) (&lt;/span&gt;'&lt;nowiki/&gt;'''[http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]'''''&lt;nowiki/&gt;''' '') by [[Chris_Vaisvil|Chris Vaisvil]]'''''

''&lt;span style=""&gt;[http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by [[Chris_Vaisvil|Chris Vaisvil]]&lt;/span&gt;''

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
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==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
|-
| 7
| 763.64
| '''lo'''
| [[11/7]], [[14/9]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever] by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque)
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
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==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[11/7]], [[14/9]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever] by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque)
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
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==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[11/7]], [[14/9]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque)
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
      <sha1>gtss9m8qtnc1g9eqev4so1v6k6epik5</sha1>
    </revision>
    <revision>
      <id>48805</id>
      <parentid>47423</parentid>
      <timestamp>2020-09-07T19:38:07Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Notation */ best approximation for 7\11 first</comment>
      <origin>48805</origin>
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      <format>text/x-wiki</format>
      <text bytes="17606" sha1="t58e91i0twt2vzb2xjizvq39ca5rza8" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque)
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]

[[Category:todo:unify_precision]]</text>
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==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque)
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
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      <timestamp>2020-10-21T20:53:47Z</timestamp>
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| de = 
| en = 11edo 
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==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>01omp5hriuyvin5uyt9r5tz9t1uwbc6</sha1>
    </revision>
    <revision>
      <id>54029</id>
      <parentid>51699</parentid>
      <timestamp>2020-12-03T01:38:35Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
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      <text bytes="18160" sha1="m51zu92u0s3k14suzqy2wqzbfpcwm49" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__

{{Infobox ET
| Prime factorization = 11
| Subgroup = 2.9.7.11.15.17
| Step size = 109.09¢
| Common uses = distorted 12edo, joan
| Important MOSes = orgone 4*2-3*1 (3\11, 1\1)&lt;br/&gt;machine 5*2-1*1 (2\11, 1\1)&lt;br/&gt;[[joan]] 2*3-5*1 (5\11, 1\1)&lt;br/&gt;[[joan]] 2*2-7*1 (5\11, 1\1)
}}
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>m51zu92u0s3k14suzqy2wqzbfpcwm49</sha1>
    </revision>
    <revision>
      <id>54030</id>
      <parentid>54029</parentid>
      <timestamp>2020-12-03T01:39:38Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>54030</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18168" sha1="829r4bcthakj45trxcjijohpehvgruk" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__

{{Infobox ET
| Prime factorization = 11
| Subgroup = 2.9.7.11.15.17
| Step size = 109.09¢
| Common uses = distorted 12edo, joan
| Important MOSes = [[orgone]] 4*2-3*1 (3\11, 1\1)&lt;br/&gt;[[machine]] 5*2-1*1 (2\11, 1\1)&lt;br/&gt;[[joan]] 2*3-5*1 (5\11, 1\1)&lt;br/&gt;[[joan]] 2*2-7*1 (5\11, 1\1)
}}
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>829r4bcthakj45trxcjijohpehvgruk</sha1>
    </revision>
    <revision>
      <id>54031</id>
      <parentid>54030</parentid>
      <timestamp>2020-12-03T01:40:10Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
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      <format>text/x-wiki</format>
      <text bytes="18162" sha1="sa7uy7ppd8fi15w4vrg4gjz3vmcf6x0" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__

{{Infobox ET
| Prime factorization = 11
| Subgroup = 2.9.7.11.15.17
| Step size = 109.09¢
| Common uses = distorted 12edo
| Important MOSes = [[orgone]] 4*2-3*1 (3\11, 1\1)&lt;br/&gt;[[machine]] 5*2-1*1 (2\11, 1\1)&lt;br/&gt;[[joan]] 2*3-5*1 (5\11, 1\1)&lt;br/&gt;[[joan]] 2*2-7*1 (5\11, 1\1)
}}
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>sa7uy7ppd8fi15w4vrg4gjz3vmcf6x0</sha1>
    </revision>
    <revision>
      <id>54103</id>
      <parentid>54031</parentid>
      <timestamp>2020-12-03T09:55:17Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>removed space on top of the article</comment>
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      <text bytes="18161" sha1="hll1wogx8wnuc3imrpsvb6mkxcwcgsa" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Subgroup = 2.9.7.11.15.17
| Step size = 109.09¢
| Common uses = distorted 12edo
| Important MOSes = [[orgone]] 4*2-3*1 (3\11, 1\1)&lt;br/&gt;[[machine]] 5*2-1*1 (2\11, 1\1)&lt;br/&gt;[[joan]] 2*3-5*1 (5\11, 1\1)&lt;br/&gt;[[joan]] 2*2-7*1 (5\11, 1\1)
}}
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>hll1wogx8wnuc3imrpsvb6mkxcwcgsa</sha1>
    </revision>
    <revision>
      <id>54186</id>
      <parentid>54103</parentid>
      <timestamp>2020-12-03T21:50:26Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>normalize ET template parameters</comment>
      <origin>54186</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18157" sha1="0qh9afqnk548mszzgju4pml4evxd00b" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Subgroup = 2.9.7.11.15.17
| Step size = 109.09
| Common uses = distorted 12edo
| Important MOS = [[orgone]] 4*2-3*1 (3\11, 1\1)&lt;br/&gt;[[machine]] 5*2-1*1 (2\11, 1\1)&lt;br/&gt;[[joan]] 2*3-5*1 (5\11, 1\1)&lt;br/&gt;[[joan]] 2*2-7*1 (5\11, 1\1)
}}
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>0qh9afqnk548mszzgju4pml4evxd00b</sha1>
    </revision>
    <revision>
      <id>54187</id>
      <parentid>54186</parentid>
      <timestamp>2020-12-03T21:52:02Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>Step size to 3 decimals</comment>
      <origin>54187</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18158" sha1="kc1gcuykd92n4iycubejxmln35b92fp" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Subgroup = 2.9.7.11.15.17
| Step size = 109.091
| Common uses = distorted 12edo
| Important MOS = [[orgone]] 4*2-3*1 (3\11, 1\1)&lt;br/&gt;[[machine]] 5*2-1*1 (2\11, 1\1)&lt;br/&gt;[[joan]] 2*3-5*1 (5\11, 1\1)&lt;br/&gt;[[joan]] 2*2-7*1 (5\11, 1\1)
}}
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>kc1gcuykd92n4iycubejxmln35b92fp</sha1>
    </revision>
    <revision>
      <id>54263</id>
      <parentid>54187</parentid>
      <timestamp>2020-12-04T17:45:57Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>standardize mos formatting</comment>
      <origin>54263</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18179" sha1="kbo3wx386wu8nm3epguw2npibct6qrg" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Subgroup = 2.9.7.11.15.17
| Step size = 109.091
| Common uses = distorted 12edo
| Important MOS = [[orgone]] 4L3s 2212121 (3\11, 1\1)&lt;br/&gt;[[machine]] 5L1s 222221 (2\11, 1\1)&lt;br/&gt;[[joan]] 2L5s 3113111 (5\11, 1\1)&lt;br/&gt;[[joan]] 2L7s 211121111 (5\11, 1\1)
}}
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Edo]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>kbo3wx386wu8nm3epguw2npibct6qrg</sha1>
    </revision>
    <revision>
      <id>54973</id>
      <parentid>54263</parentid>
      <timestamp>2020-12-04T23:13:07Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54973</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18205" sha1="5zo61lx491ez79mqx8lref56lggmm7o" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Subgroup = 2.9.7.11.15.17
| Step size = 109.091
| Common uses = distorted 12edo
| Important MOS = [[orgone]] 4L3s 2212121 (3\11, 1\1)&lt;br/&gt;[[machine]] 5L1s 222221 (2\11, 1\1)&lt;br/&gt;[[joan]] 2L5s 3113111 (5\11, 1\1)&lt;br/&gt;[[joan]] 2L7s 211121111 (5\11, 1\1)
}}
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo]]
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>5zo61lx491ez79mqx8lref56lggmm7o</sha1>
    </revision>
    <revision>
      <id>55791</id>
      <parentid>54973</parentid>
      <timestamp>2020-12-06T09:08:23Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cat sorting</comment>
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      <format>text/x-wiki</format>
      <text bytes="18229" sha1="0hb04y03dcp7oqxxxew1q6u63chxz7m" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Subgroup = 2.9.7.11.15.17
| Step size = 109.091
| Common uses = distorted 12edo
| Important MOS = [[orgone]] 4L3s 2212121 (3\11, 1\1)&lt;br/&gt;[[machine]] 5L1s 222221 (2\11, 1\1)&lt;br/&gt;[[joan]] 2L5s 3113111 (5\11, 1\1)&lt;br/&gt;[[joan]] 2L7s 211121111 (5\11, 1\1)
}}
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>0hb04y03dcp7oqxxxew1q6u63chxz7m</sha1>
    </revision>
    <revision>
      <id>56306</id>
      <parentid>55791</parentid>
      <timestamp>2020-12-09T05:56:19Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Music */</comment>
      <origin>56306</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18310" sha1="7d7pjmc9xxitahaqxz2swfm9p0fozct" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Subgroup = 2.9.7.11.15.17
| Step size = 109.091
| Common uses = distorted 12edo
| Important MOS = [[orgone]] 4L3s 2212121 (3\11, 1\1)&lt;br/&gt;[[machine]] 5L1s 222221 (2\11, 1\1)&lt;br/&gt;[[joan]] 2L5s 3113111 (5\11, 1\1)&lt;br/&gt;[[joan]] 2L7s 211121111 (5\11, 1\1)
}}
==Theory==

11-tone equal temperament, or 11[[EDO|edo]], divides the [[Octave|octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo|2edo]], [[3edo|3edo]], [[5edo|5edo]], and [[7edo|7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>7d7pjmc9xxitahaqxz2swfm9p0fozct</sha1>
    </revision>
    <revision>
      <id>56356</id>
      <parentid>56306</parentid>
      <timestamp>2020-12-09T09:37:23Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>added M2, m2 and A1 to the template, made the primes-error table</comment>
      <origin>56356</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18850" sha1="7ep3gc5lvx2pweonb4mw2s0x5xqs75q" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Subgroup = 2.9.7.11.15.17
| Step size = 109.091¢
| Fifth type = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
| Common uses = distorted 12edo
| Important MOS = [[orgone]] 4L3s 2212121 (3\11, 1\1)&lt;br/&gt;[[machine]] 5L1s 222221 (2\11, 1\1)&lt;br/&gt;[[joan]] 2L5s 3113111 (5\11, 1\1)&lt;br/&gt;[[joan]] 2L7s 211121111 (5\11, 1\1)
}}
==Theory==
{| class="wikitable"
! colspan="2" |
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
!prime 17
!prime 19
|-
! rowspan="2" |error
!absolute (¢)
|0
| -47.41
|50.0
|13.0
| -5.9
|32.2
|4.1
|29.8
|-
![[Relative error|relative]] (%)
|0
| -43
|46
|12
| -5
|30
|4
|27
|-
! colspan="2" |[[nearest edomapping]]
|11
|6
|4
|9
|5
|8
|1
|3
|-
! colspan="2" |[[fifthspan]]
|0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}
11-tone equal temperament, or 11[[EDO|edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>7ep3gc5lvx2pweonb4mw2s0x5xqs75q</sha1>
    </revision>
    <revision>
      <id>56401</id>
      <parentid>56356</parentid>
      <timestamp>2020-12-09T19:46:29Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>introduction missing</comment>
      <origin>56401</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18877" sha1="1fvykzb0qzah7rlypkugmhy7t0lwwxo" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Subgroup = 2.9.7.11.15.17
| Step size = 109.091¢
| Fifth type = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
| Common uses = distorted 12edo
| Important MOS = [[orgone]] 4L3s 2212121 (3\11, 1\1)&lt;br/&gt;[[machine]] 5L1s 222221 (2\11, 1\1)&lt;br/&gt;[[joan]] 2L5s 3113111 (5\11, 1\1)&lt;br/&gt;[[joan]] 2L7s 211121111 (5\11, 1\1)
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable"
! colspan="2" |
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
!prime 17
!prime 19
|-
! rowspan="2" |error
!absolute (¢)
|0
| -47.41
|50.0
|13.0
| -5.9
|32.2
|4.1
|29.8
|-
![[Relative error|relative]] (%)
|0
| -43
|46
|12
| -5
|30
|4
|27
|-
! colspan="2" |[[nearest edomapping]]
|11
|6
|4
|9
|5
|8
|1
|3
|-
! colspan="2" |[[fifthspan]]
|0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}
11-tone equal temperament, or 11[[EDO|edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>1fvykzb0qzah7rlypkugmhy7t0lwwxo</sha1>
    </revision>
    <revision>
      <id>56899</id>
      <parentid>56401</parentid>
      <timestamp>2020-12-18T05:57:11Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>template cleanup</comment>
      <origin>56899</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18661" sha1="kbgi0zjjkjwk2dbwoqy5ebg4z2z4oe1" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Step size = 109.091¢
| Fifth type = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable"
! colspan="2" |
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
!prime 17
!prime 19
|-
! rowspan="2" |error
!absolute (¢)
|0
| -47.41
|50.0
|13.0
| -5.9
|32.2
|4.1
|29.8
|-
![[Relative error|relative]] (%)
|0
| -43
|46
|12
| -5
|30
|4
|27
|-
! colspan="2" |[[nearest edomapping]]
|11
|6
|4
|9
|5
|8
|1
|3
|-
! colspan="2" |[[fifthspan]]
|0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}
11-tone equal temperament, or 11[[EDO|edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11-tone]]
[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>kbgi0zjjkjwk2dbwoqy5ebg4z2z4oe1</sha1>
    </revision>
    <revision>
      <id>56932</id>
      <parentid>56899</parentid>
      <timestamp>2020-12-18T08:44:32Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>remove cat 11-tone is better for scales</comment>
      <origin>56932</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18640" sha1="ore4ncar67brvl8zi87pl2c2zlj9ge2" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Step size = 109.091¢
| Fifth type = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable"
! colspan="2" |
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
!prime 17
!prime 19
|-
! rowspan="2" |error
!absolute (¢)
|0
| -47.41
|50.0
|13.0
| -5.9
|32.2
|4.1
|29.8
|-
![[Relative error|relative]] (%)
|0
| -43
|46
|12
| -5
|30
|4
|27
|-
! colspan="2" |[[nearest edomapping]]
|11
|6
|4
|9
|5
|8
|1
|3
|-
! colspan="2" |[[fifthspan]]
|0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}
11-tone equal temperament, or 11[[EDO|edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>ore4ncar67brvl8zi87pl2c2zlj9ge2</sha1>
    </revision>
    <revision>
      <id>56937</id>
      <parentid>56932</parentid>
      <timestamp>2020-12-18T09:38:38Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>ET parameter name, cleanup</comment>
      <origin>56937</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18682" sha1="bsp8uqchhgdvshlg3o1xtsyx4dg1hxk" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" |
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" |error
! absolute (¢)
| 0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" |[[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" |[[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}
11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>bsp8uqchhgdvshlg3o1xtsyx4dg1hxk</sha1>
    </revision>
    <revision>
      <id>56941</id>
      <parentid>56937</parentid>
      <timestamp>2020-12-18T09:45:49Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>consistent formatting, explicitly marked empty cell as such</comment>
      <origin>56941</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18704" sha1="g1iwkueloqc9kyo05lw3eqfzd8wywqj" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" |error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" |[[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" |[[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}
11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>g1iwkueloqc9kyo05lw3eqfzd8wywqj</sha1>
    </revision>
    <revision>
      <id>56948</id>
      <parentid>56941</parentid>
      <timestamp>2020-12-18T10:15:24Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>added some space</comment>
      <origin>56948</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18708" sha1="epyfik8pxi79om4cj12qoy4893hdlty" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>epyfik8pxi79om4cj12qoy4893hdlty</sha1>
    </revision>
    <revision>
      <id>56963</id>
      <parentid>56948</parentid>
      <timestamp>2020-12-18T11:28:37Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>State the primality in infobox</comment>
      <origin>56963</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18716" sha1="630hxipgq31qlh3j88w5ugeni4x9dit" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 11 17 26 31 38 41 |.)

{| class="wikitable"
|-
! | Ratio
! | Monzo
! | [[cent|Cents]]
![[Color notation/Temperament Names|Color Name]]
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | [[135/128|135/128]]
| |&lt;nowiki&gt; | -7 3 1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 92.18
| style="text-align:center;" |Layobi
| style="text-align:center;" | Major Chroma
| style="text-align:center;" | Major Limma
| style="text-align:center;" | Pelogic Comma
|-
| style="text-align:center;" | 9931568/9752117
| |&lt;nowiki&gt; | -25 7 6 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 31.57
| style="text-align:center;" |Lala-tribiyo
| style="text-align:center;" | Ampersand's Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1776337/1773750
| |&lt;nowiki&gt; | -68 18 17 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 2.52
| style="text-align:center;" |Quinla-seyo
| style="text-align:center;" | Vavoom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9859966/9733137
| |&lt;nowiki&gt; | -10 7 8 -7 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 22.41
| style="text-align:center;" |Lasepru-aquadbiyo
| style="text-align:center;" | Blackjackisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 1029/1024
| |&lt;nowiki&gt; | -10 1 0 3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.43
| style="text-align:center;" |Latrizo
| style="text-align:center;" | Gamelisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224|225/224]]
| |&lt;nowiki&gt; | -5 2 2 -1 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 16875/16807
| |&lt;nowiki&gt; | 0 3 4 -5 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 6.99
| style="text-align:center;" |Quinru-aquadyo
| style="text-align:center;" | Mirkwai
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2401/2400
| |&lt;nowiki&gt; | -5 -1 -2 4 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 0.72
| style="text-align:center;" |Bizozogu
| style="text-align:center;" | Breedsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 121/120
| |&lt;nowiki&gt; | -3 -1 -1 0 2 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 14.37
| style="text-align:center;" |Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 65536/65219
| |&lt;nowiki&gt; | 16 0 0 -2 -3 &lt;/nowiki&gt;&amp;gt;
| style="text-align:right;" | 8.39
| style="text-align:center;" |Satrilu-aruru
| style="text-align:center;" | Orgonisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>630hxipgq31qlh3j88w5ugeni4x9dit</sha1>
    </revision>
    <revision>
      <id>57145</id>
      <parentid>56963</parentid>
      <timestamp>2020-12-20T14:34:53Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Commas */ reworked table</comment>
      <origin>57145</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17102" sha1="lbmpsgkej9inilf1014zdffvty7n9mq" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>lbmpsgkej9inilf1014zdffvty7n9mq</sha1>
    </revision>
    <revision>
      <id>57147</id>
      <parentid>57145</parentid>
      <timestamp>2020-12-20T14:38:09Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Commas */ add class "commatable "</comment>
      <origin>57147</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17113" sha1="5zyfnyv3v6mq6r9ufr19n825dpmbz2k" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>5zyfnyv3v6mq6r9ufr19n825dpmbz2k</sha1>
    </revision>
    <revision>
      <id>64845</id>
      <parentid>57147</parentid>
      <timestamp>2021-03-28T04:03:52Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
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      <text bytes="17151" sha1="kiaku1w8dbg470z8u32ol86n0k9e0jr" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>kiaku1w8dbg470z8u32ol86n0k9e0jr</sha1>
    </revision>
    <revision>
      <id>65665</id>
      <parentid>64845</parentid>
      <timestamp>2021-04-05T01:12:42Z</timestamp>
      <contributor>
        <username>Ayceman</username>
        <id>284</id>
      </contributor>
      <comment>/* Music */</comment>
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      <text bytes="17337" sha1="626137opqzo8tc0fshyz8nhnyuujrip" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] (11edo) by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] (11edo) by [[User:Ayceman|Alexandru Ianu]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>626137opqzo8tc0fshyz8nhnyuujrip</sha1>
    </revision>
    <revision>
      <id>65668</id>
      <parentid>65665</parentid>
      <timestamp>2021-04-05T01:21:33Z</timestamp>
      <contributor>
        <username>Ayceman</username>
        <id>284</id>
      </contributor>
      <minor/>
      <comment>/* Music */</comment>
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      <format>text/x-wiki</format>
      <text bytes="17368" sha1="p5419dwkkrvllyr1hx73fx4a7zdo8rw" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>p5419dwkkrvllyr1hx73fx4a7zdo8rw</sha1>
    </revision>
    <revision>
      <id>66032</id>
      <parentid>65668</parentid>
      <timestamp>2021-04-09T09:08:50Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Theory */Concordant is less judgmental than harmonious</comment>
      <origin>66032</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17368" sha1="bgesssrd32p0jnb7gh5cyok1jqq69hy" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>bgesssrd32p0jnb7gh5cyok1jqq69hy</sha1>
    </revision>
    <revision>
      <id>66594</id>
      <parentid>66032</parentid>
      <timestamp>2021-04-13T06:54:39Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>66594</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17330" sha1="fk3u805wmhqf1mavuppa5hbqr2tnd5f" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>fk3u805wmhqf1mavuppa5hbqr2tnd5f</sha1>
    </revision>
    <revision>
      <id>66956</id>
      <parentid>66594</parentid>
      <timestamp>2021-04-14T01:23:42Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>66956</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17419" sha1="76npy9ktvwmglfjhe08sgxnwdblee5g" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 2 1 1 1 1 2L 7s MOS 

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>76npy9ktvwmglfjhe08sgxnwdblee5g</sha1>
    </revision>
    <revision>
      <id>66961</id>
      <parentid>66956</parentid>
      <timestamp>2021-04-14T01:34:26Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
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      <origin>66961</origin>
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      <text bytes="17457" sha1="5uomzvq8irdve89ml5c4fkzvdsmc0ti" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>5uomzvq8irdve89ml5c4fkzvdsmc0ti</sha1>
    </revision>
    <revision>
      <id>66974</id>
      <parentid>66961</parentid>
      <timestamp>2021-04-14T01:52:20Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Pathological Modes */</comment>
      <origin>66974</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17457" sha1="9cyeyq29liv3d5robxeiwkmpog9m542" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>9cyeyq29liv3d5robxeiwkmpog9m542</sha1>
    </revision>
    <revision>
      <id>67018</id>
      <parentid>66974</parentid>
      <timestamp>2021-04-14T05:51:50Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <comment>revert to before policy on march 27</comment>
      <origin>67018</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17113" sha1="5zyfnyv3v6mq6r9ufr19n825dpmbz2k" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less harmonious. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>5zyfnyv3v6mq6r9ufr19n825dpmbz2k</sha1>
    </revision>
    <revision>
      <id>67035</id>
      <parentid>67018</parentid>
      <timestamp>2021-04-14T06:04:11Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <comment>restore changes by another user</comment>
      <origin>67035</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17457" sha1="9cyeyq29liv3d5robxeiwkmpog9m542" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>9cyeyq29liv3d5robxeiwkmpog9m542</sha1>
    </revision>
    <revision>
      <id>71565</id>
      <parentid>67035</parentid>
      <timestamp>2021-06-02T03:53:22Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Notation */</comment>
      <origin>71565</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17564" sha1="l4xfrmgv1hznz7sr1d4pxkfsjjvyzlr" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>l4xfrmgv1hznz7sr1d4pxkfsjjvyzlr</sha1>
    </revision>
    <revision>
      <id>71676</id>
      <parentid>71565</parentid>
      <timestamp>2021-06-02T22:06:48Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Notation */</comment>
      <origin>71676</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17873" sha1="ncrk70loqfhqm21vw45m83q9iax450b" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
P0
| A
| P1
P0
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
^0, m1
| ^A, B
| ^1, M2
^0, M1
| ^A, B
| Q#\Rb
| Minor second
Minor prime
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
~1, m2
| ^B, Cb
| ~2, M3
~1, M2
| ^B, C#
| R
| Major second
Major prime
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
M1, ~2
| B#, vC
| m2, ~3
m1, ~2
| Bb, vC
| R#\Sb
| Minor third
Minor second
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
M2, v3
| C, vD
| m3, v4
m2, v3
| C, vD
| S
| Major third/Minor fourth
Major second/Minor third
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
P3, v4
| D, vE
| P4, v5
P3, v4
| D, vE
| S#\Tb
| Major fourth
Major third
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
^3, P4
| ^D, E
| ^4, P5
^3, P4
| ^D, E
| T
| Minor fifth
Minor fourth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
^4, m5
| ^E, Fb
| ^5, M6
^4, M5
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
Major fourth/Minor fifth 
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
~5, m6
| vF, Gb
| ~6, M7
~5, M6
| vF, G#
| U
| Major sixth
Major fifth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
M5, ~6
| F, vG
| m6, ~7
M5, ~6
| F, vG
| U#\Pb
| Minor seventh
Minor sixth
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
M6, v7
| G, vAv
| m7, v8
m6, v7
| G, vAv
| P\Qb
| Major seventh
Major sixth
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
P7
| A
| P8
P7
| A
| Q\P#
| Octave
Seventh
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
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      <id>71711</id>
      <parentid>71676</parentid>
      <timestamp>2021-06-03T04:05:48Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Undo revision 71676 by [[Special:Contributions/Moremajorthanmajor|Moremajorthanmajor]] ([[User talk:Moremajorthanmajor|talk]])</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17564" sha1="l4xfrmgv1hznz7sr1d4pxkfsjjvyzlr" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>l4xfrmgv1hznz7sr1d4pxkfsjjvyzlr</sha1>
    </revision>
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      <id>72836</id>
      <parentid>71711</parentid>
      <timestamp>2021-06-20T14:45:47Z</timestamp>
      <contributor>
        <username>Cerdded41</username>
        <id>300</id>
      </contributor>
      <comment>Added: "Make a Dream" --&gt; 11-EDO music by Sevish</comment>
      <origin>72836</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17659" sha1="6xdvcsv46lv7xnae173uxv1hb7podeg" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

11-tone equal temperament, or 11[[edo]], divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [https://youtu.be/kuwv7qH4s6U Make a Dream] by Sevish (from his 2015 album "Rhythm and Xen")

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>6xdvcsv46lv7xnae173uxv1hb7podeg</sha1>
    </revision>
    <revision>
      <id>73765</id>
      <parentid>72836</parentid>
      <timestamp>2021-07-06T13:44:13Z</timestamp>
      <contributor>
        <username>SupahstarSaga</username>
        <id>281</id>
      </contributor>
      <comment>Moved the intro line out of the Theory section and rewrote it</comment>
      <origin>73765</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17790" sha1="8wqw2kdxdpd3e6klceg1mo5tlrgfcz1" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Fifth = 6\11 = 654.545¢
| Major 2nd = 1\11 = 109¢
| Minor 2nd = 3\11 = 327¢
| Augmented 1sn = -2\11 = -218¢
}}
{{todo|add introduction}}
'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of approximately 109.09 [[cent|cents]]. It is the fifth [[prime_numbers|prime]] edo, after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [https://youtu.be/kuwv7qH4s6U Make a Dream] by Sevish (from his 2015 album "Rhythm and Xen")

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
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    <revision>
      <id>73768</id>
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      <timestamp>2021-07-06T14:29:52Z</timestamp>
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        <username>Xenwolf</username>
        <id>6</id>
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      <text bytes="17758" sha1="9dqyog14qtgs7aq3ru0nz7zabcxute2" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Minor 2nd = 3\11 (327¢)
| Augmented 1sn = -2\11 (-218¢)
}}
{{todo|add introduction}}
'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

==Theory==
{| class="wikitable center-all"
! colspan="2" | &lt;!-- empty cell --&gt;
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! rowspan="2" | error
! absolute (¢)
| 0.0
| -47.4
| +50.0
| +13.0
| -5.9
| +32.2
| +4.1
| +29.8
|-
! [[Relative error|relative]] (%)
| 0
| -43
| +46
| +12
| -5
| +30
| +4
| +27
|-
! colspan="2" | [[nearest edomapping]]
| 11
| 6
| 4
| 9
| 5
| 8
| 1
| 3
|-
! colspan="2" | [[fifthspan]]
| 0
| +1
| -3
| -4
| -1
| +5
| +2
| -5
|}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [https://youtu.be/kuwv7qH4s6U Make a Dream] by Sevish (from his 2015 album "Rhythm and Xen")

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>9dqyog14qtgs7aq3ru0nz7zabcxute2</sha1>
    </revision>
    <revision>
      <id>75185</id>
      <parentid>73768</parentid>
      <timestamp>2021-07-29T01:57:31Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17306" sha1="ij680s0vuln1ydrcwxajevgyil92hf0" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Minor 2nd = 3\11 (327¢)
| Augmented 1sn = -2\11 (-218¢)
}}
{{todo|add introduction}}
'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

==Theory==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [https://youtu.be/kuwv7qH4s6U Make a Dream] by Sevish (from his 2015 album "Rhythm and Xen")

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
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    <revision>
      <id>75455</id>
      <parentid>75185</parentid>
      <timestamp>2021-08-04T18:24:01Z</timestamp>
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      <text bytes="17681" sha1="o8d09445p8zuw6udz791kge7remqgox" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Minor 2nd = 3\11 (327¢)
| Augmented 1sn = -2\11 (-218¢)
}}
{{todo|add introduction}}
'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

==Theory==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

===Differences between distributionally-even scales and smaller edos===
{| class="wikitable"
|+
!N
!L-Nedo
! s-Nedo
|-
|2
|54.5455¢
| -54.5455¢
|-
|3
|36.364¢
| -72.727¢
|-
|4
|27.273¢
| -81.818¢
|-
|5
|87.273¢
|  -21.818¢
|-
|6
| 18.182¢
| -90.909¢
|-
|7
|46.753¢
| -63.339¢
|-
|8
|68.182¢
| -41.818¢
|-
|9
|84.8485¢
| -24.242¢
|-
|10
|98.182¢
|10.909¢
|}
==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [https://youtu.be/kuwv7qH4s6U Make a Dream] by Sevish (from his 2015 album "Rhythm and Xen")

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Scale]]
[[Category:Subgroup]]</text>
      <sha1>o8d09445p8zuw6udz791kge7remqgox</sha1>
    </revision>
    <revision>
      <id>75519</id>
      <parentid>75455</parentid>
      <timestamp>2021-08-05T04:21:09Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Recategorization</comment>
      <origin>75519</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17662" sha1="mcve4120574z0txdlu3echiiukhjhqb" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Minor 2nd = 3\11 (327¢)
| Augmented 1sn = -2\11 (-218¢)
}}
{{todo|add introduction}}
'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

==Theory==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

===Differences between distributionally-even scales and smaller edos===
{| class="wikitable"
|+
!N
!L-Nedo
! s-Nedo
|-
|2
|54.5455¢
| -54.5455¢
|-
|3
|36.364¢
| -72.727¢
|-
|4
|27.273¢
| -81.818¢
|-
|5
|87.273¢
|  -21.818¢
|-
|6
| 18.182¢
| -90.909¢
|-
|7
|46.753¢
| -63.339¢
|-
|8
|68.182¢
| -41.818¢
|-
|9
|84.8485¢
| -24.242¢
|-
|10
|98.182¢
|10.909¢
|}
==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; Machine &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| Q\P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| Q#\Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| R#\Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| S#\Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| T#\Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| U#\Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| P\Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| Q\P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [https://youtu.be/kuwv7qH4s6U Make a Dream] by Sevish (from his 2015 album "Rhythm and Xen")

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>mcve4120574z0txdlu3echiiukhjhqb</sha1>
    </revision>
    <revision>
      <id>75986</id>
      <parentid>75519</parentid>
      <timestamp>2021-08-10T15:00:42Z</timestamp>
      <contributor>
        <username>Ayceman</username>
        <id>284</id>
      </contributor>
      <comment>/* Notation */</comment>
      <origin>75986</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17773" sha1="rlet5wem17diuk99qq91pqutpq2psdf" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Minor 2nd = 3\11 (327¢)
| Augmented 1sn = -2\11 (-218¢)
}}
{{todo|add introduction}}
'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

==Theory==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

===Differences between distributionally-even scales and smaller edos===
{| class="wikitable"
|+
!N
!L-Nedo
! s-Nedo
|-
|2
|54.5455¢
| -54.5455¢
|-
|3
|36.364¢
| -72.727¢
|-
|4
|27.273¢
| -81.818¢
|-
|5
|87.273¢
|  -21.818¢
|-
|6
| 18.182¢
| -90.909¢
|-
|7
|46.753¢
| -63.339¢
|-
|8
|68.182¢
| -41.818¢
|-
|9
|84.8485¢
| -24.242¢
|-
|10
|98.182¢
|10.909¢
|}
==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
![[Smitonic]]
(3rd-gen)

notation
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
|A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
|A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
|B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
|C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
|C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
|D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
|D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
|E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
|F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
|F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
|G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
|A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

These are all heptatonic notations generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octotonic 5th-generated, nonatonic 5th-generated, and heptatonic 3rd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octotonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' (generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' (generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [https://youtu.be/kuwv7qH4s6U Make a Dream] by Sevish (from his 2015 album "Rhythm and Xen")

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>rlet5wem17diuk99qq91pqutpq2psdf</sha1>
    </revision>
    <revision>
      <id>75992</id>
      <parentid>75986</parentid>
      <timestamp>2021-08-10T15:15:12Z</timestamp>
      <contributor>
        <username>Ayceman</username>
        <id>284</id>
      </contributor>
      <comment>/* Notation */</comment>
      <origin>75992</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18296" sha1="b55u1axleixu3mytpban27n6lmwsiff" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Minor 2nd = 3\11 (327¢)
| Augmented 1sn = -2\11 (-218¢)
}}
{{todo|add introduction}}
'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

==Theory==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

===Differences between distributionally-even scales and smaller edos===
{| class="wikitable"
|+
!N
!L-Nedo
! s-Nedo
|-
|2
|54.5455¢
| -54.5455¢
|-
|3
|36.364¢
| -72.727¢
|-
|4
|27.273¢
| -81.818¢
|-
|5
|87.273¢
|  -21.818¢
|-
|6
| 18.182¢
| -90.909¢
|-
|7
|46.753¢
| -63.339¢
|-
|8
|68.182¢
| -41.818¢
|-
|9
|84.8485¢
| -24.242¢
|-
|10
|98.182¢
|10.909¢
|}
==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]] 
(3rd-gen) 

notation 
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
|A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
|A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
|B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
|C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
|C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
|D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
|D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
|E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
|F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
|F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
|G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
|A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (!3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A7/d8 - P8

genchain of seconds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of seconds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [https://youtu.be/kuwv7qH4s6U Make a Dream] by Sevish (from his 2015 album "Rhythm and Xen")

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>b55u1axleixu3mytpban27n6lmwsiff</sha1>
    </revision>
    <revision>
      <id>75993</id>
      <parentid>75992</parentid>
      <timestamp>2021-08-10T15:19:08Z</timestamp>
      <contributor>
        <username>Ayceman</username>
        <id>284</id>
      </contributor>
      <comment>/* Notation */  Genchain fixed</comment>
      <origin>75993</origin>
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      <format>text/x-wiki</format>
      <text bytes="18283" sha1="az9iq2tvf8eoe4wx5sa6amr1frsjz8i" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Minor 2nd = 3\11 (327¢)
| Augmented 1sn = -2\11 (-218¢)
}}
{{todo|add introduction}}
'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

==Theory==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

===Differences between distributionally-even scales and smaller edos===
{| class="wikitable"
|+
!N
!L-Nedo
! s-Nedo
|-
|2
|54.5455¢
| -54.5455¢
|-
|3
|36.364¢
| -72.727¢
|-
|4
|27.273¢
| -81.818¢
|-
|5
|87.273¢
|  -21.818¢
|-
|6
| 18.182¢
| -90.909¢
|-
|7
|46.753¢
| -63.339¢
|-
|8
|68.182¢
| -41.818¢
|-
|9
|84.8485¢
| -24.242¢
|-
|10
|98.182¢
|10.909¢
|}
==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]] 
(3rd-gen) 

notation 
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
|A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
|A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
|B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
|C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
|C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
|D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
|D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
|E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
|F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
|F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
|G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
|A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (!3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [https://youtu.be/kuwv7qH4s6U Make a Dream] by Sevish (from his 2015 album "Rhythm and Xen")

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>az9iq2tvf8eoe4wx5sa6amr1frsjz8i</sha1>
    </revision>
    <revision>
      <id>76024</id>
      <parentid>75993</parentid>
      <timestamp>2021-08-11T20:29:02Z</timestamp>
      <contributor>
        <username>Ayceman</username>
        <id>284</id>
      </contributor>
      <minor/>
      <comment>/* Notation */</comment>
      <origin>76024</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18283" sha1="77ji5vdvpthj82o73xfirduasjjgebr" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Minor 2nd = 3\11 (327¢)
| Augmented 1sn = -2\11 (-218¢)
}}
{{todo|add introduction}}
'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-tet''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

==Theory==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

===Differences between distributionally-even scales and smaller edos===
{| class="wikitable"
|+
!N
!L-Nedo
! s-Nedo
|-
|2
|54.5455¢
| -54.5455¢
|-
|3
|36.364¢
| -72.727¢
|-
|4
|27.273¢
| -81.818¢
|-
|5
|87.273¢
|  -21.818¢
|-
|6
| 18.182¢
| -90.909¢
|-
|7
|46.753¢
| -63.339¢
|-
|8
|68.182¢
| -41.818¢
|-
|9
|84.8485¢
| -24.242¢
|-
|10
|98.182¢
|10.909¢
|}
==Notation==

11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]] 
(3rd-gen) 

notation 
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
|A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
|A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
|B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
|C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
|C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
|D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
|D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
|E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
|F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
|F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
|G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
|A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

==Commas==
11 EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

==JI Intervals==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11edo = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

==MOS Scales==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L_4s|1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L_3s|4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L_2s|3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L_5s|3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L_3s|2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L_5s|2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L_7s|2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

==Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

==Instruments==
11-edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==

* [[File:11EDO-improv.mp3|link=Special:FilePath/11EDO-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11EDO-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11edo.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11edo] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11edo] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [https://youtu.be/kuwv7qH4s6U Make a Dream] by Sevish (from his 2015 album "Rhythm and Xen")

==Videos==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

==11edo Zine==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>77ji5vdvpthj82o73xfirduasjjgebr</sha1>
    </revision>
    <revision>
      <id>79498</id>
      <parentid>76024</parentid>
      <timestamp>2021-10-13T23:22:27Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
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      <text bytes="18240" sha1="4tvvqyfrcqvblicb0kt4sby10jiav63" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
}}

'''11 equal divisions of the octave''' ('''11EDO'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo|2EDO]], [[3edo|3EDO]], [[5edo|5EDO]], and [[7edo|7EDO]].

== Theory ==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11EDO maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22EDO]], a tuning known as "elevenplus".

Compared to 12EDO, the intervals of 11EDO are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12EDO.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12EDO, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12EDO's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11EDO provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22EDO, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11EDO is the largest EDO that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 2
| 54.5455¢
| -54.5455¢
|-
| 3
| 36.364¢
| -72.727¢
|-
| 4
| 27.273¢
| -81.818¢
|-
| 5
| 87.273¢
| -21.818¢
|-
| 6
| 18.182¢
| -90.909¢
|-
| 7
| 46.753¢
| -63.339¢
|-
| 8
| 68.182¢
| -41.818¢
|-
| 9
| 84.8485¢
| -24.242¢
|-
| 10
| 98.182¢
| 10.909¢
|}
== Notation ==
11EDO can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12EDO music can be directly translated to 11EDO "on the fly".

The 11EDO solfege in the table is derived from [[22edo Solfege|22EDO solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22EDO subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]] 
(3rd-gen) 

notation 
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11EDO in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Commas ==
11EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI Intervals ==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11EDO interval
| | 0\11 = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11EDO interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11EDO also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== MOS Scales ==
Although 11EDO has one fewer interval in the octave than 12EDO, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11EDO), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11EDO Modes]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11EDO ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11EDO. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
* [[File:11edo-improv.mp3|link=Special:FilePath/11edo-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [https://youtu.be/kuwv7qH4s6U Make a Dream] by Sevish (from his 2015 album "Rhythm and Xen")

== Videos ==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

== 11EDO Zine ==
There is an 11EDO Zine! As far as we know, 11EDO is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11EDO Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>4tvvqyfrcqvblicb0kt4sby10jiav63</sha1>
    </revision>
    <revision>
      <id>79546</id>
      <parentid>79498</parentid>
      <timestamp>2021-10-15T13:58:42Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
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      <text bytes="18258" sha1="9rxxplyl1uo1w4cg9xaevdsuiuz1wbk" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11EDO'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo|2EDO]], [[3edo|3EDO]], [[5edo|5EDO]], and [[7edo|7EDO]].

== Theory ==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11EDO maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22EDO]], a tuning known as "elevenplus".

Compared to 12EDO, the intervals of 11EDO are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12EDO.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12EDO, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12EDO's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11EDO provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22EDO, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11EDO is the largest EDO that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 2
| 54.5455¢
| -54.5455¢
|-
| 3
| 36.364¢
| -72.727¢
|-
| 4
| 27.273¢
| -81.818¢
|-
| 5
| 87.273¢
| -21.818¢
|-
| 6
| 18.182¢
| -90.909¢
|-
| 7
| 46.753¢
| -63.339¢
|-
| 8
| 68.182¢
| -41.818¢
|-
| 9
| 84.8485¢
| -24.242¢
|-
| 10
| 98.182¢
| 10.909¢
|}
== Notation ==
11EDO can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12EDO music can be directly translated to 11EDO "on the fly".

The 11EDO solfege in the table is derived from [[22edo Solfege|22EDO solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22EDO subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]] 
(3rd-gen) 

notation 
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11EDO in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Commas ==
11EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI Intervals ==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11EDO interval
| | 0\11 = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11EDO interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11EDO also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== MOS Scales ==
Although 11EDO has one fewer interval in the octave than 12EDO, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11EDO), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11EDO Modes]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11EDO ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11EDO. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
* [[File:11edo-improv.mp3|link=Special:FilePath/11edo-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=VjJgk9r_A_M Longwayaway People] by [[Sevish]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [https://youtu.be/kuwv7qH4s6U Make a Dream] by Sevish (from his 2015 album "Rhythm and Xen")

== Videos ==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

== 11EDO Zine ==
There is an 11EDO Zine! As far as we know, 11EDO is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11EDO Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>9rxxplyl1uo1w4cg9xaevdsuiuz1wbk</sha1>
    </revision>
    <revision>
      <id>80011</id>
      <parentid>79546</parentid>
      <timestamp>2021-10-27T16:04:25Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>/* Music */ 11edo tracks category</comment>
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      <text bytes="18118" sha1="98fi5kao3opqyrl1a8c9boqckjjsap4" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11EDO'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo|2EDO]], [[3edo|3EDO]], [[5edo|5EDO]], and [[7edo|7EDO]].

== Theory ==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11EDO maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22EDO]], a tuning known as "elevenplus".

Compared to 12EDO, the intervals of 11EDO are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12EDO.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12EDO, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12EDO's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11EDO provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22EDO, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11EDO is the largest EDO that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 2
| 54.5455¢
| -54.5455¢
|-
| 3
| 36.364¢
| -72.727¢
|-
| 4
| 27.273¢
| -81.818¢
|-
| 5
| 87.273¢
| -21.818¢
|-
| 6
| 18.182¢
| -90.909¢
|-
| 7
| 46.753¢
| -63.339¢
|-
| 8
| 68.182¢
| -41.818¢
|-
| 9
| 84.8485¢
| -24.242¢
|-
| 10
| 98.182¢
| 10.909¢
|}
== Notation ==
11EDO can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12EDO music can be directly translated to 11EDO "on the fly".

The 11EDO solfege in the table is derived from [[22edo Solfege|22EDO solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22EDO subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]] 
(3rd-gen) 

notation 
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11EDO in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Commas ==
11EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI Intervals ==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11EDO interval
| | 0\11 = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11EDO interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11EDO also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== MOS Scales ==
Although 11EDO has one fewer interval in the octave than 12EDO, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11EDO), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11EDO Modes]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11EDO ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11EDO. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [[File:11edo-improv.mp3|link=Special:FilePath/11edo-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] by [[User:Ayceman|Alexandru Ianu]] ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])

== Videos ==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

== 11EDO Zine ==
There is an 11EDO Zine! As far as we know, 11EDO is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11EDO Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>98fi5kao3opqyrl1a8c9boqckjjsap4</sha1>
    </revision>
    <revision>
      <id>80139</id>
      <parentid>80011</parentid>
      <timestamp>2021-11-02T17:53:21Z</timestamp>
      <contributor>
        <username>Ayceman</username>
        <id>284</id>
      </contributor>
      <comment>/* Music */</comment>
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      <text bytes="18495" sha1="8tttqcch46pivksnimy8jw8i2vi5ahh" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11EDO'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo|2EDO]], [[3edo|3EDO]], [[5edo|5EDO]], and [[7edo|7EDO]].

== Theory ==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11EDO maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22EDO]], a tuning known as "elevenplus".

Compared to 12EDO, the intervals of 11EDO are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12EDO.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12EDO, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12EDO's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11EDO provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22EDO, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11EDO is the largest EDO that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 2
| 54.5455¢
| -54.5455¢
|-
| 3
| 36.364¢
| -72.727¢
|-
| 4
| 27.273¢
| -81.818¢
|-
| 5
| 87.273¢
| -21.818¢
|-
| 6
| 18.182¢
| -90.909¢
|-
| 7
| 46.753¢
| -63.339¢
|-
| 8
| 68.182¢
| -41.818¢
|-
| 9
| 84.8485¢
| -24.242¢
|-
| 10
| 98.182¢
| 10.909¢
|}
== Notation ==
11EDO can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12EDO music can be directly translated to 11EDO "on the fly".

The 11EDO solfege in the table is derived from [[22edo Solfege|22EDO solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22EDO subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]] 
(3rd-gen) 

notation 
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11EDO in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Commas ==
11EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI Intervals ==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11EDO interval
| | 0\11 = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11EDO interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11EDO also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== MOS Scales ==
Although 11EDO has one fewer interval in the octave than 12EDO, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11EDO), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11EDO Modes]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11EDO ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11EDO. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [[File:11edo-improv.mp3|link=Special:FilePath/11edo-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
** [https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] ([[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]])

== Videos ==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

== 11EDO Zine ==
There is an 11EDO Zine! As far as we know, 11EDO is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11EDO Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>8tttqcch46pivksnimy8jw8i2vi5ahh</sha1>
    </revision>
    <revision>
      <id>80145</id>
      <parentid>80139</parentid>
      <timestamp>2021-11-02T19:33:26Z</timestamp>
      <contributor>
        <username>Ayceman</username>
        <id>284</id>
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      <text bytes="18628" sha1="kk7tzb1a0p6ylbk9f689v72rf4xf1fs" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11EDO'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo|2EDO]], [[3edo|3EDO]], [[5edo|5EDO]], and [[7edo|7EDO]].

== Theory ==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11EDO maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22EDO]], a tuning known as "elevenplus".

Compared to 12EDO, the intervals of 11EDO are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12EDO.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12EDO, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12EDO's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11EDO provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11 as does 22EDO, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11EDO is the largest EDO that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 2
| 54.5455¢
| -54.5455¢
|-
| 3
| 36.364¢
| -72.727¢
|-
| 4
| 27.273¢
| -81.818¢
|-
| 5
| 87.273¢
| -21.818¢
|-
| 6
| 18.182¢
| -90.909¢
|-
| 7
| 46.753¢
| -63.339¢
|-
| 8
| 68.182¢
| -41.818¢
|-
| 9
| 84.8485¢
| -24.242¢
|-
| 10
| 98.182¢
| 10.909¢
|}
== Notation ==
11EDO can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12EDO music can be directly translated to 11EDO "on the fly".

The 11EDO solfege in the table is derived from [[22edo Solfege|22EDO solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22EDO subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]] 
(3rd-gen) 

notation 
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11EDO in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Commas ==
11EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI Intervals ==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11EDO interval
| | 0\11 = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11EDO interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11EDO also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== MOS Scales ==
Although 11EDO has one fewer interval in the octave than 12EDO, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11EDO), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11EDO Modes]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11EDO ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11EDO. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [[File:11edo-improv.mp3|link=Special:FilePath/11edo-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

== 11EDO Zine ==
There is an 11EDO Zine! As far as we know, 11EDO is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11EDO Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>kk7tzb1a0p6ylbk9f689v72rf4xf1fs</sha1>
    </revision>
    <revision>
      <id>81434</id>
      <parentid>80145</parentid>
      <timestamp>2021-12-01T14:36:59Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ Add 17-limit to subgroup, as 11edo does that quite decently in the same way as 22edo.</comment>
      <origin>81434</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18635" sha1="cla543tbq92ss7ghtg5a5yry8du7p08" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11EDO'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo|2EDO]], [[3edo|3EDO]], [[5edo|5EDO]], and [[7edo|7EDO]].

== Theory ==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11EDO maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22EDO]], a tuning known as "elevenplus".

Compared to 12EDO, the intervals of 11EDO are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12EDO.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12EDO, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12EDO's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11EDO provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22EDO, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11EDO is the largest EDO that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
! N
! L-Nedo
! s-Nedo
|-
| 2
| 54.5455¢
| -54.5455¢
|-
| 3
| 36.364¢
| -72.727¢
|-
| 4
| 27.273¢
| -81.818¢
|-
| 5
| 87.273¢
| -21.818¢
|-
| 6
| 18.182¢
| -90.909¢
|-
| 7
| 46.753¢
| -63.339¢
|-
| 8
| 68.182¢
| -41.818¢
|-
| 9
| 84.8485¢
| -24.242¢
|-
| 10
| 98.182¢
| 10.909¢
|}

== Notation ==
11EDO can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12EDO music can be directly translated to 11EDO "on the fly".

The 11EDO solfege in the table is derived from [[22edo Solfege|22EDO solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22EDO subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]] 
(3rd-gen) 

notation 
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11EDO in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Commas ==
11EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI Intervals ==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11EDO interval
| | 0\11 = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11EDO interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11EDO also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== MOS Scales ==
Although 11EDO has one fewer interval in the octave than 12EDO, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11EDO), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11EDO Modes]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11EDO ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11EDO. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [[File:11edo-improv.mp3|link=Special:FilePath/11edo-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

== 11EDO Zine ==
There is an 11EDO Zine! As far as we know, 11EDO is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11EDO Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>cla543tbq92ss7ghtg5a5yry8du7p08</sha1>
    </revision>
    <revision>
      <id>83755</id>
      <parentid>81434</parentid>
      <timestamp>2022-01-13T00:03:10Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>remove table "Differences between distributionally-even scales and smaller edos"</comment>
      <origin>83755</origin>
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      <format>text/x-wiki</format>
      <text bytes="18238" sha1="fzjes4wrgtcvkerq5mnmhrix2c17iek" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}__FORCETOC__
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11EDO'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime EDO]], after [[2edo|2EDO]], [[3edo|3EDO]], [[5edo|5EDO]], and [[7edo|7EDO]].

== Theory ==
{{Odd harmonics in edo|edo=11}}

Being less than twelve, 11EDO maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo|22EDO]], a tuning known as "elevenplus".

Compared to 12EDO, the intervals of 11EDO are stretched:

&lt;ul&gt;&lt;li&gt;The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12EDO.&lt;/li&gt;&lt;li&gt;The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12EDO, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12EDO's 1000-cent interval, although it is still about 13 cents away from 7/4.&lt;/li&gt;&lt;li&gt;The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."&lt;/li&gt;&lt;li&gt;The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.&lt;/li&gt;&lt;li&gt;The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.&lt;/li&gt;&lt;/ul&gt;11EDO provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22EDO, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11EDO is the largest EDO that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Notation ==
11EDO can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12EDO music can be directly translated to 11EDO "on the fly".

The 11EDO solfege in the table is derived from [[22edo Solfege|22EDO solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22EDO subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]] 
(3rd-gen) 

notation 
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11EDO in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Commas ==
11EDO tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI Intervals ==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11EDO interval
| | 0\11 = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11EDO interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11EDO also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== MOS Scales ==
Although 11EDO has one fewer interval in the octave than 12EDO, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11EDO), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11EDO Modes]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11EDO ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11EDO. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [[File:11edo-improv.mp3|link=Special:FilePath/11edo-improv.mp3]] [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/City%20Of%20The%20Asleep%20-%20She%20Is%20My%20Lilac-Hued%20Obsession.mp3 She Is My Lilac-Hued Obsession] on [[City of the Asleep]], [http://cityoftheasleep.com/music Map of an Internal Landscape] (2009)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play] by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET] by [[Aaron Andrew Hunt]] (neo-Baroque) {{dead link}}
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt 
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones] by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy] 
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt] by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10] by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal] by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO] by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge] by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal] by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

== 11EDO Zine ==
There is an 11EDO Zine! As far as we know, 11EDO is the first xenharmonic tuning system to have its own zine. See [[11edo_Zine|11EDO Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>fzjes4wrgtcvkerq5mnmhrix2c17iek</sha1>
    </revision>
    <revision>
      <id>84027</id>
      <parentid>83755</parentid>
      <timestamp>2022-01-15T02:02:31Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Used new primes table, lowercase edo, fixed 1 dead link, marked 16 dead links, misc. edits</comment>
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| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{primes in equal|11}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]] 
(3rd-gen) 

notation 
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Commas ==
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI Intervals ==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11 = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== MOS Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

== Pathological Modes ==
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

== 11edo Zine ==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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| es = 
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{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Primes in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-1 right-2"
! Degree
! Size in &lt;br&gt; [[cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]] 
(3rd-gen) 

notation 
![[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Commas ==
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major Chroma, Major Limma, Pelogic Comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s Comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal Kleisma, Marvel Comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI Intervals ==
{| class="wikitable"
|-
! | Harmonic
! | 8
! | 
! | 9
! | 
! | 11
! | 
! | 14
! | 
! | 16
|-
! | JI interval from 1/1
| | 1/1 = 0 cents
| | 
| | 9/8 = 204
| | 
| | 11/8 = 551
| | 
| | 7/4 = 969
| | 
| | 2/1 = 1200
|-
! | nearest 11edo interval
| | 0\11 = 0¢
| | 
| | 2\11 = 218¢
| | 
| | 5\11 = 545
| | 
| | 9\11 = 982
| | 
| | 11\11 = 1200
|-
! | difference
| | 0
| | 
| | +14¢
| | 
| | -6¢
| | 
| | +13¢
| | 
| | 0¢
|-
! | JI interval between
| | 
| | 9:8 = 204¢
| | 
| | 11:9 = 347
| | 
| | 14:11 = 418
| | 
| | 8:7 = 231
| | 
|-
! | nearest 11edo interval
| | 
| | 2\11 = 218¢
| | 
| | 3\11 = 327
| | 
| | 4\11 = 436
| | 
| | 2\11 = 218
| | 
|-
! | difference
| | 
| | +14¢
| | 
| | -20¢
| | 
| | +18¢
| | 
| | -13¢
| | 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOSScales|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named Machine[5]; and 2 2 2 2 2 1, a [[5L_1s|5L 1s]] scale named [[Machine|Machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[Orgone|Orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo_Modes|11edo Modes]]

=== Pathological Modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
The Stuffed Ones: &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, &lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

&lt;ul&gt;&lt;li&gt;&lt;span style=""&gt;&lt;span style=""&gt;&lt;span style=""&gt;''[http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]''&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;, [[Mike_Battaglia_FAQ|Mike Battaglia]] - youtube&lt;/li&gt;&lt;/ul&gt;

== 11edo Zine ==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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| de = 
| en = 11edo 
| es = 
| ja = 11平均律
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{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Primes in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== 11edo Zine ==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [http://soundclick.com/share?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=933772 Orange Clips on Sausages]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] by Andrew Heathwaite
* [http://www.soundclick.com/bands/page_songInfo.cfm?bandID=122613&amp;songID=834492 Blue Gel]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== 11edo Zine ==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET
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'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== 11edo Zine ==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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| es = 
| ja = 11平均律
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{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
Relative Radian = 17.36236¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== 11edo Zine ==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET
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'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== 11edo Zine ==
There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine. See [[11edo Zine]].

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>bkn3p8m6x5rthiwxbtpwp8ssjdfvuuz</sha1>
    </revision>
    <revision>
      <id>87603</id>
      <parentid>84233</parentid>
      <timestamp>2022-02-26T18:38:51Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>see also section for lumatone mapping (integrated [[11edo Zine]] link here)</comment>
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      <text bytes="17655" sha1="hvo7c6syqen74vqj5lmhuzx0bmfgej5" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]]
== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  (generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' (generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
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'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
| Fifth = 6\11 (655¢)
| Major 2nd = 1\11 (109¢)
| Consistency = 3
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'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.09091¢
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| Major 2nd = 1\11 (109¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>t86zuxry2hygrzo757z21lnqr1itqz5</sha1>
    </revision>
    <revision>
      <id>94747</id>
      <parentid>92574</parentid>
      <timestamp>2022-08-19T18:52:30Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+dual fifths in infobox</comment>
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| de = 
| en = 11edo 
| es = 
| ja = 11平均律
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{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Sharp fifth = 7\11 (764¢)
| Flat fifth = 6\11 (655¢)
| Major 2nd = 2\11 (218¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{Dead link}} by [[George Secor]], 1970. Apparently the first piece ever written for 11EDO.
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 Cool My Head] by [[David Hamill]], 2010
* Hyperimprovisations Nuggetwarp by [[Jacob Barton]], 2009:
** [http://soundclick.com/share.cfm?id=10267904 Piece I]
** [http://soundclick.com/share.cfm?id=10267905 Piece II]
** [http://soundclick.com/share.cfm?id=10267906 Piece III]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession She is My Lilac-Hued Obsession] by [[City of the Asleep]] (2007)
* [http://eceserv0.ece.wisc.edu/%7Esethares/mp3s/dabo_girl.html The Turquoise Dabo Girl]{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Sethares/Turquoise_Dabo_Girl.mp3 play]{{dead link}} by [[Bill Sethares]] (spectrally bent synth ens.)
* [http://www.h-pi.com/mp3/Prelude11ET.mp3 Prelude11ET]{{dead link}} by [[Aaron Andrew Hunt]] (neo-Baroque)
* [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Adagio In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et Invention In 11ET &amp;#124; SoundCloud] by Aaron Andrew Hunt
* [http://music.columbia.edu/%7Echris/complist.html The Stuffed Ones]{{dead link}} by [[Christopher Bailey]] (keyboards concréte):
** [http://music.columbia.edu/%7Echris/sounds/st.goopy.mp3 goopy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ellie.mp3 ellie]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.ziggy.mp3 ziggy]{{dead link}}
** [http://music.columbia.edu/%7Echris/sounds/st.towelbear.mp3 towelbear]{{dead link}}
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 Icicle Caverns] by Dr. [[Ozan Yarman]]
* [http://soundclick.com/share.cfm?id=955383 Angkor Wat, September 1066] by [[X. J. Scott]]
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 conversation is] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play]{{dead link}} by [[Andrew Heathwaite]]. Text is a sentence borrowed from a paper by Larry Richards, set to an 11-tone row. For guitar and voice.
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 Orange Clips on Sausages] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 Blue Gel] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play]{{dead link}} by Andrew Heathwaite
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 Jeffrey Dahmer Cooks at 11EDO] by [[Chris Vaisvil]]
* [http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 Jaunt]{{dead link}} by [[Jon Lyle Smith]]
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 The Metamorphosis of Gregor] by Chris Vaisvil
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/10%20-%2010.%2011%20octave.mp3 Comets Over Flatland 10]{{dead link}} by [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 The City Sleeps, A Madrigal]{{dead link}} by [http://soundcloud.com/ninly/the-city-sleeps Jason Conklin]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 Counterpoint in 11EDO]{{dead link}} by [[Jon Lyle Smith]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 Black Ritual Dirge]{{dead link}} by [[Aaron Krister Johnson]]
* [http://chrisvaisvil.com/?p=2701 Eleven Birds] (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only]) by [[Chris Vaisvil]]
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal The Execution of 12 Equal]{{dead link}} by Chris Vaisvil
* [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] by [[User:Ks26|ks26]]
* 11edo pieces by  [[User:Ayceman|Alexandru Ianu]]:
**[https://www.youtube.com/watch?v=8x1f5WFkF4k Divertimento in 11 tone Orgone] ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
** [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
** [https://www.youtube.com/watch?v=CWU09fXXy1s Ocean of the Necrophages] (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>kiopuqe3tmso7q2s65ac3bye6vuzvhj</sha1>
    </revision>
    <revision>
      <id>95827</id>
      <parentid>94747</parentid>
      <timestamp>2022-09-30T14:18:53Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>/* Music */ Sorted by artist, a few new links, fixed some broken links</comment>
      <origin>95827</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19559" sha1="7fxydvitn3ke7c0xjwf3ekt6aj6pil9" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET
| Prime factorization = 11 (prime)
| Step size = 109.091¢
| Sharp fifth = 7\11 (764¢)
| Flat fifth = 6\11 (655¢)
| Major 2nd = 2\11 (218¢)
| Consistency = 3
}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}

'''[[Christopher Bailey]]'''
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

'''[[Jacob Barton]]'''
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

'''[[Andrew Heathwaite]]'''
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

'''[[User:Ayceman|Alexandru Ianu]]'''
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

'''[[Aaron Krister Johnson]]'''
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

'''[[User:ks26|ks26]]'''
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''[[X. J. Scott]]'''
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

'''[[George Secor]]'''
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

'''[[Jon Lyle Smith]]'''
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[Ozan Yarman]]'''
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
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      <text bytes="19397" sha1="2cl08og8zyg686hp8pg6hwcp18npisr" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
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| ja = 11平均律
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}

'''[[Christopher Bailey]]'''
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

'''[[Jacob Barton]]'''
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

'''[[Andrew Heathwaite]]'''
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

'''[[User:Ayceman|Alexandru Ianu]]'''
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

'''[[Aaron Krister Johnson]]'''
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

'''[[User:ks26|ks26]]'''
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''[[X. J. Scott]]'''
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

'''[[George Secor]]'''
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

'''[[Jon Lyle Smith]]'''
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[Ozan Yarman]]'''
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo does not have much to offer in terms of consonant chords, even if we look outside of "root-3rd-P5"-type chords. It does not even have a good approximation to 4:5:9 like [[13edo]] because of its lack of a 5/4. One possible chord that could be used in 11edo is the equal triad of 0-4-8 steps, approximated in 7-limit JI as 21:27:35.

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}

'''[[Christopher Bailey]]'''
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

'''[[Jacob Barton]]'''
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

'''[[Andrew Heathwaite]]'''
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

'''[[User:Ayceman|Alexandru Ianu]]'''
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

'''[[Aaron Krister Johnson]]'''
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

'''[[User:ks26|ks26]]'''
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''[[X. J. Scott]]'''
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

'''[[George Secor]]'''
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

'''[[Jon Lyle Smith]]'''
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[Ozan Yarman]]'''
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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      <text bytes="19781" sha1="mgbs8001y9pg7ftcwqu2jk5rqxz940o" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo does not have much to offer in terms of consonant chords, even if we look outside of "root-3rd-P5"-type chords. It does not even have a good approximation to 4:5:9 like [[13edo]] because of its lack of a 5/4. One possible chord that could be used in 11edo is the equal triad of 0-4-8 steps, approximated in 7-limit JI as 21:27:35 or less accurately in 11-limit JI as 11:14:18.

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}

'''[[Christopher Bailey]]'''
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

'''[[Jacob Barton]]'''
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

'''[[Andrew Heathwaite]]'''
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

'''[[User:Ayceman|Alexandru Ianu]]'''
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

'''[[Aaron Krister Johnson]]'''
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

'''[[User:ks26|ks26]]'''
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''[[X. J. Scott]]'''
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

'''[[George Secor]]'''
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

'''[[Jon Lyle Smith]]'''
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[Ozan Yarman]]'''
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo does not have much to offer in terms of consonant chords, even if we look outside of "root-3rd-P5"-type chords. It does not even have a good approximation to 4:5:9 like [[13edo]] because of its lack of a 5/4. One possible chord that could be used in 11edo is the equal triad of 0-4-8 steps, splitting the major sixth in half. It is approximated in 7-limit JI as 21:27:35 or less accurately in 11-limit JI as 11:14:18.

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}

'''[[Christopher Bailey]]'''
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

'''[[Jacob Barton]]'''
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

'''[[Andrew Heathwaite]]'''
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

'''[[User:Ayceman|Alexandru Ianu]]'''
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

'''[[Aaron Krister Johnson]]'''
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

'''[[User:ks26|ks26]]'''
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''[[X. J. Scott]]'''
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

'''[[George Secor]]'''
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

'''[[Jon Lyle Smith]]'''
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[Ozan Yarman]]'''
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>sjos6j8k04nr9upn0h44it7prs9qbxv</sha1>
    </revision>
    <revision>
      <id>106551</id>
      <parentid>106362</parentid>
      <timestamp>2023-03-31T19:34:53Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <comment>Undo revision 106362 by [[Special:Contributions/CompactStar|CompactStar]] ([[User talk:CompactStar|talk]]) New paragraph does not flow at all with previous ones, which just demonstrated how you can use 11edo in a somewhat harmonious way.</comment>
      <origin>106551</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19397" sha1="2cl08og8zyg686hp8pg6hwcp18npisr" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}

'''[[Christopher Bailey]]'''
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

'''[[Jacob Barton]]'''
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

'''[[Andrew Heathwaite]]'''
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

'''[[User:Ayceman|Alexandru Ianu]]'''
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

'''[[Aaron Krister Johnson]]'''
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

'''[[User:ks26|ks26]]'''
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''[[X. J. Scott]]'''
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

'''[[George Secor]]'''
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

'''[[Jon Lyle Smith]]'''
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[Ozan Yarman]]'''
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}

'''[[Christopher Bailey]]'''
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

'''[[Jacob Barton]]'''
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

'''[[Andrew Heathwaite]]'''
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

'''[[User:Ayceman|Alexandru Ianu]]'''
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

'''[[Aaron Krister Johnson]]'''
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

'''[[User:ks26|ks26]]'''
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]

'''[[X. J. Scott]]'''
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

'''[[George Secor]]'''
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

'''[[Jon Lyle Smith]]'''
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[Ozan Yarman]]'''
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}

'''[[Christopher Bailey]]'''
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

'''[[Jacob Barton]]'''
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

'''[[Andrew Heathwaite]]'''
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

'''[[User:Ayceman|Alexandru Ianu]]'''
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

'''[[Aaron Krister Johnson]]'''
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

'''[[User:ks26|ks26]]'''
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

'''[[X. J. Scott]]'''
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

'''[[George Secor]]'''
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

'''[[Jon Lyle Smith]]'''
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[Ozan Yarman]]'''
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}

'''5 hideya'''
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3'']

'''[[Christopher Bailey]]'''
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

'''[[Jacob Barton]]'''
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

'''[[Andrew Heathwaite]]'''
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

'''[[User:Ayceman|Alexandru Ianu]]'''
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

'''[[Aaron Krister Johnson]]'''
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

'''[[User:ks26|ks26]]'''
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

'''[[X. J. Scott]]'''
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

'''[[George Secor]]'''
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

'''[[Jon Lyle Smith]]'''
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[Ozan Yarman]]'''
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
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'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}

'''5 hideya'''
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)

'''[[Christopher Bailey]]'''
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

'''[[Jacob Barton]]'''
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

'''[[Andrew Heathwaite]]'''
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

'''[[User:Ayceman|Alexandru Ianu]]'''
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

'''[[Aaron Krister Johnson]]'''
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

'''[[User:ks26|ks26]]'''
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

'''[[X. J. Scott]]'''
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

'''[[George Secor]]'''
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

'''[[Jon Lyle Smith]]'''
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[Ozan Yarman]]'''
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{See also|:Category:11edo tracks}}

'''5 hideya'''
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

'''[[Christopher Bailey]]'''
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

'''[[Jacob Barton]]'''
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

'''[[City of the Asleep]]'''
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

'''[[Andrew Heathwaite]]'''
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

'''[[Aaron Andrew Hunt]]'''
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

'''[[User:Ayceman|Alexandru Ianu]]'''
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

'''[[Aaron Krister Johnson]]'''
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

'''[[User:ks26|ks26]]'''
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

'''[[X. J. Scott]]'''
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

'''[[George Secor]]'''
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

'''[[Bill Sethares]]'''
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

'''[[Jon Lyle Smith]]'''
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

'''[[Randy Winchester]]'''
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

'''[[Ozan Yarman]]'''
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo has acceptable approximations of harmonics 7 and 11, so the simplest accurate interpretation of it is as a 2.7.11 subgroup temperament. 11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
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[[Category:Macrotonal]]
[[Category:Prime EDO]]
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo does not have acceptable approximations of harmonics 3 and 5, so the simplest accurate interpretation of it is as a 2.7.11 subgroup temperament. On this subgroup it tempers out [[352/343]] allowing for 3 [[7/4]] to equal [[11/8]]. 11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo does not have acceptable approximations of harmonics 3 and 5, so the simplest accurate interpretation of it using only primes is as a 2.7.11 subgroup temperament. On this subgroup it tempers out [[352/343]] allowing for 3 [[7/4]] to equal [[11/8]], relating to the [[5L 1s]] scale of 11edo. 11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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      <text bytes="20090" sha1="hhxnji5x0iorerzbznf3mny3cmh9ec6" xml:space="preserve">{{interwiki
| de = 
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| es = 
| ja = 11平均律
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo does not have acceptable approximations of harmonics 3 and 5, so the simplest accurate interpretation of it using only primes is as a 2.7.11 subgroup temperament. On this subgroup it tempers out [[352/343]] allowing for 3 [[7/4]] (the generator of 11edo's [[5L 1s]] MOS) to equal [[11/8]]. 11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
      <sha1>hhxnji5x0iorerzbznf3mny3cmh9ec6</sha1>
    </revision>
    <revision>
      <id>118329</id>
      <parentid>118328</parentid>
      <timestamp>2023-07-04T02:23:45Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <comment>Removing the thing I added because its just a lower-prime limit version of Machine temperament.</comment>
      <origin>118329</origin>
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      <text bytes="19794" sha1="o61t6h9d4y4mjyyzdzqvm6k7n25zbf0" xml:space="preserve">{{interwiki
| de = 
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
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'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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      <id>119182</id>
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      <text bytes="20009" sha1="9wwpn6wpc2sekbvgacces98vbrkt47c" xml:space="preserve">{{interwiki
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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      <id>119188</id>
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; Yeah Gore
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
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[[Category:Macrotonal]]
[[Category:Prime EDO]]
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'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically and harmonically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [http://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [http://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([http://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [http://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [http://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; Yeah Gore
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[http://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[http://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[http://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[http://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [http://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [http://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([https://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; Yeah Gore
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([https://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; Yeah Gore
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([https://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; Yeah Gore
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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      <id>124038</id>
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      <timestamp>2023-09-13T02:48:45Z</timestamp>
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      <text bytes="20886" sha1="0rve4e9msnrckqsrno2s19oc41pafhv" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; 5 hideya
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

'''Jason Conklin'''
* [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 ''The City Sleeps, A Madrigal''] (2011) ([https://soundcloud.com/ninly/the-city-sleeps SoundCloud])

'''David Hamill'''
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

'''Mundoworld'''
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; Yeah Gore
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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    <revision>
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      <text bytes="21006" sha1="aglqillmzzmx6yf1vufj8dcxj1aio7q" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; [[Amano Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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      <text bytes="21114" sha1="cxkbsxrizt7l8ktmfslw0rt7ncp4vo9" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:11edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Prime EDO]]
[[Category:Subgroup]]</text>
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{{Infobox ET}}

'''11 equal divisions of the octave''' ('''11edo'''), or '''11-tone equal temperament''' ('''11-TET''', '''11ET''') when viewed from a [[regular temperament]] perspective, is the tuning that divides the [[octave]] into eleven equal steps of about 109.09 [[cent]]s. It is the fifth [[prime edo]], after [[2edo]], [[3edo]], [[5edo]], and [[7edo]].

== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]</text>
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    </revision>
    <revision>
      <id>129204</id>
      <parentid>128106</parentid>
      <timestamp>2023-12-17T00:06:43Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
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      <text bytes="20625" sha1="3xh4asgwfh1cj0u7corsobek96a9joy" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
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{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
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{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]</text>
      <sha1>to6n0u8iujyg63i1eeohcz3k14ho6a3</sha1>
    </revision>
    <revision>
      <id>131578</id>
      <parentid>129425</parentid>
      <timestamp>2024-01-10T12:06:06Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added todo: add rank 2 temperaments table</comment>
      <origin>131578</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20773" sha1="0xxsj1thyidgkevhrxden4s7pqorw63" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
11edo ukulele:

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>0xxsj1thyidgkevhrxden4s7pqorw63</sha1>
    </revision>
    <revision>
      <id>134660</id>
      <parentid>131578</parentid>
      <timestamp>2024-02-07T04:57:22Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>/* Instruments */ Added link to Lumatone mapping</comment>
      <origin>134660</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20852" sha1="581ma0aahjuk2gnmyyc1tnf0ine196e" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''
[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

'''Ensembles'''
In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

'''Lumatone'''
There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
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      <text bytes="20857" sha1="l36vdndvhh56b7ma0sqmw3oqqgq43cm" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|ks26]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
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| de = 11-EDO
| en = 11edo 
| es = 
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{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
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| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
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{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== JI approximation ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* [https://www.youtube.com/watch?v=28SOTJvT6sw ''Tostadosto''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
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{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* [https://www.youtube.com/watch?v=28SOTJvT6sw ''Tostadosto''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
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      <timestamp>2024-04-12T11:39:34Z</timestamp>
      <contributor>
        <username>Francium</username>
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      <text bytes="21121" sha1="hqofka0nz5iqosrpvcwx840ggbstp60" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
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{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>hqofka0nz5iqosrpvcwx840ggbstp60</sha1>
    </revision>
    <revision>
      <id>142423</id>
      <parentid>140980</parentid>
      <timestamp>2024-05-06T08:18:39Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Music */ Added a section for well-tempered derivatives of 11EDO, and added Bryan Deister's 11 Tone March (YouTube link) to start it off.</comment>
      <origin>142423</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21268" sha1="qpifazzn51i2m3iwib2hycmbssdvtoi" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
===Strict 11EDO===
;[[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

===Well-Tempered Derivatives of 11EDO===
;[[Bryan Deister]]
;* 11 [https://www.youtube.com/watch?v=z0lWcguNsNs Tone March] (2024)
== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>qpifazzn51i2m3iwib2hycmbssdvtoi</sha1>
    </revision>
    <revision>
      <id>142426</id>
      <parentid>142423</parentid>
      <timestamp>2024-05-06T08:36:20Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Well-Tempered Derivatives of 11EDO */ Changed to Unequal Derivatives of 11EDO -- deserves more testing to determine how truly circulating this temperament is.</comment>
      <origin>142426</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21262" sha1="smkm38qqqhjs77bueh6xzf9oreelqbp" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
===Strict 11EDO===
;[[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

===Unequal Derivatives of 11EDO===
;[[Bryan Deister]]
;* 11 [https://www.youtube.com/watch?v=z0lWcguNsNs Tone March] (2024)
== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>smkm38qqqhjs77bueh6xzf9oreelqbp</sha1>
    </revision>
    <revision>
      <id>147138</id>
      <parentid>142426</parentid>
      <timestamp>2024-07-07T09:53:29Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Strict 11EDO */ +music</comment>
      <origin>147138</origin>
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      <format>text/x-wiki</format>
      <text bytes="21357" sha1="5j3t8jhmfxtmhgxwvg5loxhwm9baze9" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11EDO ===
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

===Unequal Derivatives of 11EDO===
;[[Bryan Deister]]
;* 11 [https://www.youtube.com/watch?v=z0lWcguNsNs Tone March] (2024)
== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
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{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11EDO ===
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

===Unequal Derivatives of 11EDO===
;[[Bryan Deister]]
;* 11 [https://www.youtube.com/watch?v=z0lWcguNsNs Tone March] (2024)
== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>n2ytu6vc1g66m6wjria5cc4kvrs862s</sha1>
    </revision>
    <revision>
      <id>147181</id>
      <parentid>147140</parentid>
      <timestamp>2024-07-08T09:44:58Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Strict 11EDO */ +music</comment>
      <origin>147181</origin>
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      <format>text/x-wiki</format>
      <text bytes="21534" sha1="7f1bm72ztu0c38g9x7ak3dl7dd91ox6" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

2\11 generates 2 2 2 2 3, a [[1L 4s]] scale named machine[5]; and 2 2 2 2 2 1, a [[5L 1s]] scale named [[machine]][6].
[[File:Screen Shot 2020-04-23 at 11.32.40 PM.png|none|thumb|1003x1003px]]
3\11 generates 3 3 3 2; and 1 2 1 2 1 2 2, a [[4L 3s]] scale named [[orgone]][7].
[[File:Screen Shot 2020-04-23 at 11.33.13 PM.png|none|thumb|987x987px]]
4\11 generates 4 4 3; 1 3 1 3 3, a [[3L 2s]] scale; and 1 1 2 1 1 2 1 2, a [[3L 5s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.29 PM.png|none|thumb|970x970px]]
5\11 generates [[joan]] scales 5 5 1; 1 4 1 4 1, a [[2L 3s]] scale; 1 1 3 1 1 3 1, a [[2L 5s]] scale; and 1 1 1 2 1 1 1 2 1, a [[2L 7s]] scale.
[[File:Screen Shot 2020-04-23 at 11.33.44 PM.png|none|thumb|995x995px]]
See [[11edo Modes]]

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11EDO ===
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star'']

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

===Unequal Derivatives of 11EDO===
;[[Bryan Deister]]
;* 11 [https://www.youtube.com/watch?v=z0lWcguNsNs Tone March] (2024)
== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>7f1bm72ztu0c38g9x7ak3dl7dd91ox6</sha1>
    </revision>
    <revision>
      <id>148140</id>
      <parentid>147181</parentid>
      <timestamp>2024-07-18T18:33:26Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>/* Scales */ Move MOS scales to [[List of 11edo MOS scales]], reorganize section</comment>
      <origin>148140</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20861" sha1="qbw4wsdfv2kl2ij8wu5oa6dj9rmnbkc" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}
&lt;references/&gt;

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11EDO ===
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star'']

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

===Unequal Derivatives of 11EDO===
;[[Bryan Deister]]
;* 11 [https://www.youtube.com/watch?v=z0lWcguNsNs Tone March] (2024)
== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>qbw4wsdfv2kl2ij8wu5oa6dj9rmnbkc</sha1>
    </revision>
    <revision>
      <id>151186</id>
      <parentid>148140</parentid>
      <timestamp>2024-08-21T01:55:46Z</timestamp>
      <contributor>
        <username>Tristanbay</username>
        <id>418</id>
      </contributor>
      <comment>/* Commas */Changed ref tag to note group</comment>
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      <format>text/x-wiki</format>
      <text bytes="20858" sha1="m97he08bzk45slzswv9bfsqw0ybggn5" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11EDO ===
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star'']

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

===Unequal Derivatives of 11EDO===
;[[Bryan Deister]]
;* 11 [https://www.youtube.com/watch?v=z0lWcguNsNs Tone March] (2024)
== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>m97he08bzk45slzswv9bfsqw0ybggn5</sha1>
    </revision>
    <revision>
      <id>151187</id>
      <parentid>151186</parentid>
      <timestamp>2024-08-21T01:57:06Z</timestamp>
      <contributor>
        <username>Tristanbay</username>
        <id>418</id>
      </contributor>
      <comment>/* See also */Added notes section with note</comment>
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      <text bytes="20896" sha1="gt4zxsdyfevey06ob5gu2p4d122wfzr" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11EDO ===
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star'']

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

===Unequal Derivatives of 11EDO===
;[[Bryan Deister]]
;* 11 [https://www.youtube.com/watch?v=z0lWcguNsNs Tone March] (2024)
== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>gt4zxsdyfevey06ob5gu2p4d122wfzr</sha1>
    </revision>
    <revision>
      <id>153990</id>
      <parentid>151187</parentid>
      <timestamp>2024-09-04T12:16:40Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Strict 11EDO */ +music</comment>
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      <text bytes="21099" sha1="6bz1q7ijx76kyo9qxch0m7gg4e8ykh9" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11EDO ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star'']

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

===Unequal Derivatives of 11EDO===
;[[Bryan Deister]]
;* 11 [https://www.youtube.com/watch?v=z0lWcguNsNs Tone March] (2024)
== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>6bz1q7ijx76kyo9qxch0m7gg4e8ykh9</sha1>
    </revision>
    <revision>
      <id>156156</id>
      <parentid>153990</parentid>
      <timestamp>2024-09-22T21:21:42Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Unequal Derivatives of 11EDO */ Fix formatting error</comment>
      <origin>156156</origin>
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      <format>text/x-wiki</format>
      <text bytes="21100" sha1="pyj6uki4dfa27siod6yye7i5z0fu8cv" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11EDO ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star'']

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

===Unequal Derivatives of 11EDO===
;[[Bryan Deister]]
;* [https://www.youtube.com/watch?v=z0lWcguNsNs 11 Tone March] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>pyj6uki4dfa27siod6yye7i5z0fu8cv</sha1>
    </revision>
    <revision>
      <id>160296</id>
      <parentid>156156</parentid>
      <timestamp>2024-10-15T18:11:29Z</timestamp>
      <contributor>
        <username>Phanomium</username>
        <id>511</id>
      </contributor>
      <minor/>
      <comment>/* Strict 11EDO */ I added my song to the list of 11edo music.</comment>
      <origin>160296</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21193" sha1="98olzxngr4aj4d2za8ktq8ez014adka" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11EDO ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — Apparently the first piece ever written for 11EDO.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"], from [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) ([http://christopherbaileymusic.com/composition-list/ details])
* [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"], from ''The Stuffed Ones'' (2004)
* [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"], from ''The Stuffed Ones'' (2004)

; [[Jacob Barton]]
* [https://soundclick.com/share.cfm?id=10267904 "Piece I"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267905 "Piece II"], from ''Hyperimprovisations Nuggetwarp'' (2009)
* [https://soundclick.com/share.cfm?id=10267906 "Piece III"], from ''Hyperimprovisations Nuggetwarp'' (2009)

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=933772 ''Orange Clips on Sausages''] (2004) ([http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songID=834492 ''Blue Gel''] (2004) ([http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 archived])
* [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 ''conversation is''] (2010) ([http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 archived])

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
* [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et "Adagio in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])
* [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et "Invention in 11ET"], from ''The Equal-Tempered Keyboard'' (1999-2022) ([https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud])

; [[User:Ayceman|Alexandru Ianu]]
* [https://youtu.be/8x1f5WFkF4k ''Divertimento in 11 tone Orgone''] (2021) ([[:File:Divertimento in 11 tone Orgone.pdf|sheet music]])
* [https://youtu.be/81uZbsmbet8 ''Sylvian Moon Dance''] (2021) ([[:File:SylvianMoonDance.ogg|audio]], [[:File:Sylvian_Moon_Dance.pdf|sheet music]]) 
* [https://youtu.be/CWU09fXXy1s ''Ocean of the Necrophages''] (2021) (piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]], [[:File:Ocean of the Necrophages.pdf|sheet music]]; strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]], [[:File:Ocean of the Necrophages (strings).pdf|sheet music]])

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo'']
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello'']
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)'']

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg Fire Memes (with Anthony "Pomp" Pompliano)]
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 Theory of Creation]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star'']

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s ''Edolian - Overcoming''] (2020)

; [[User:Phanomium|Phanomium]]
;* [https://www.youtube.com/watch?v=y939ciE9MQY 33322] (2024)
; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) ([http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 archived]) ([https://www.youtube.com/watch?v=HKULte3WhuE YouTube])
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* [http://chrisvaisvil.com/?p=2701 ''Eleven Birds''] (2012) (video and music) ([http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 audio only])
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

===Unequal Derivatives of 11EDO===
;[[Bryan Deister]]
;* [https://www.youtube.com/watch?v=z0lWcguNsNs 11 Tone March] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>98olzxngr4aj4d2za8ktq8ez014adka</sha1>
    </revision>
    <revision>
      <id>160526</id>
      <parentid>160296</parentid>
      <timestamp>2024-10-16T12:06:16Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Music */ -bold, +italic; misc. cleanup</comment>
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      <text bytes="21399" sha1="3d62o1xpcsya8vf0do8o4livgi446vy" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>3d62o1xpcsya8vf0do8o4livgi446vy</sha1>
    </revision>
    <revision>
      <id>173585</id>
      <parentid>160526</parentid>
      <timestamp>2024-12-29T04:29:19Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>/* Intervals and notation */ add more Sagittal notation</comment>
      <origin>173585</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22240" sha1="fbh1vnypo2ah00zo34nrnj937c5m5gq" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and notation ==
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! Sagittal &lt;br&gt; notation &lt;br&gt;(22edo subset)
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| A
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| AII\ or B!!/
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| B
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| C/I or BII\ or D\!!/
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| D\! or C/II\
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| D/I or E\!!/
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| E\! or D/II\
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| F
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| FII\ or G!!/
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| G
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| GII\ or A!!/
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| A
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

11edo in [[Sagittal notation]]:

[[File:Sagittal11EDO.jpg|alt=Sagittal11EDO.jpg|Sagittal11EDO.jpg]]

Sagittal and up/down notations are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>fbh1vnypo2ah00zo34nrnj937c5m5gq</sha1>
    </revision>
    <revision>
      <id>173626</id>
      <parentid>173585</parentid>
      <timestamp>2024-12-29T07:29:46Z</timestamp>
      <contributor>
        <username>Dave Keenan</username>
        <id>218</id>
      </contributor>
      <comment>/* Intervals and notation */ Remove Sagittal duplication. Add Ups and downs heading.</comment>
      <origin>173626</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21982" sha1="9knredphaz0ktdi5v0wwhknw6npp9ec" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving Orgone[7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "elevenplus".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less concordant. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from 7/4.
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "neutral third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the supermajor third of frequency ratio 9/7 than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the 11/8 superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it tempers out the same commas as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 chord and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating JI chords.

11edo is the largest edo that patently alternates with an undivided 9/8 in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>9knredphaz0ktdi5v0wwhknw6npp9ec</sha1>
    </revision>
    <revision>
      <id>174983</id>
      <parentid>173626</parentid>
      <timestamp>2025-01-07T08:57:14Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ Add links</comment>
      <origin>174983</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22034" sha1="4rtvsixyq3tcnsezwht0qzlpjgza20v" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo is the largest edo that patently alternates with an undivided [[9/8]] in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11edo tempers out the following [[comma]]s. (Note: This assumes val {{val| 11 17 26 31 38 41 }}.)

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>4rtvsixyq3tcnsezwht0qzlpjgza20v</sha1>
    </revision>
    <revision>
      <id>176659</id>
      <parentid>174983</parentid>
      <timestamp>2025-01-17T12:14:24Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ cleanup</comment>
      <origin>176659</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22037" sha1="5ehytgk6xynt5pcchsbkdh9iges4vwx" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second," at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second," at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third," at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third."
* The "major third," at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth," at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N_subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo is the largest edo that patently alternates with an undivided [[9/8]] in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0-8-16-20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11et tempers out the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>5ehytgk6xynt5pcchsbkdh9iges4vwx</sha1>
    </revision>
    <revision>
      <id>176660</id>
      <parentid>176659</parentid>
      <timestamp>2025-01-17T12:18:02Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ cleanup</comment>
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      <text bytes="22052" sha1="pw4oa4zs5woh8blfh9mh9wwlncebhmp" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo is the largest edo that patently alternates with an undivided [[9/8]] in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11et tempers out the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand]]'s comma
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>pw4oa4zs5woh8blfh9mh9wwlncebhmp</sha1>
    </revision>
    <revision>
      <id>176890</id>
      <parentid>176660</parentid>
      <timestamp>2025-01-18T08:53:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ ampersand -&gt; ampersand comma to distinguish it from the temp</comment>
      <origin>176890</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22068" sha1="eiz8pys0th085eee6ttmrlnrc1mhkh0" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo is the largest edo that patently alternates with an undivided [[9/8]] in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>eiz8pys0th085eee6ttmrlnrc1mhkh0</sha1>
    </revision>
    <revision>
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      <parentid>176890</parentid>
      <timestamp>2025-02-02T13:08:05Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
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      <text bytes="22302" sha1="dvslno9gkl5mtbgeuttt6yp5a024fxu" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
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{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo is the largest edo that patently alternates with an undivided [[9/8]] in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* [https://www.youtube.com/watch?v=hM0BAC_YZnQ ''Sleep Slope''] (2024)

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>dvslno9gkl5mtbgeuttt6yp5a024fxu</sha1>
    </revision>
    <revision>
      <id>179263</id>
      <parentid>179262</parentid>
      <timestamp>2025-02-02T13:11:37Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* 21st century */ +links</comment>
      <origin>179263</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22462" sha1="3jb7zzomu63vyp72q3s3cx27rijq9ad" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo is the largest edo that patently alternates with an undivided [[9/8]] in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>3jb7zzomu63vyp72q3s3cx27rijq9ad</sha1>
    </revision>
    <revision>
      <id>180560</id>
      <parentid>179263</parentid>
      <timestamp>2025-02-13T09:28:14Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>todo:check maths</comment>
      <origin>180560</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22585" sha1="g95qyxk7180lb1uoqvzk99g9xxosmau" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo is the largest edo that patently alternates with an undivided [[9/8]] in a [[Well tempered nonet|wtn]].

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;


{{todo|inline=1|correct maths|comment=Please check mapping of 9-limit intervals: see talk page for exact instructions.}}

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>g95qyxk7180lb1uoqvzk99g9xxosmau</sha1>
    </revision>
    <revision>
      <id>180622</id>
      <parentid>180560</parentid>
      <timestamp>2025-02-13T23:02:37Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>/* Theory */ Idiosyncratic nonsense</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22474" sha1="28d36xbg8otnp72imyohyzzitlkbp4w" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
*in 2.7.9.11.15.17 subgroup

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;


{{todo|inline=1|correct maths|comment=Please check mapping of 9-limit intervals: see talk page for exact instructions.}}

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
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{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;


{{todo|inline=1|correct maths|comment=Please check mapping of 9-limit intervals: see talk page for exact instructions.}}

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>krzx2xghherqkguptrcl04dofg99jt3</sha1>
    </revision>
    <revision>
      <id>180626</id>
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      <timestamp>2025-02-14T02:07:48Z</timestamp>
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        <username>BudjarnLambeth</username>
        <id>359</id>
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      <text bytes="22369" sha1="pggjv085aud2tff5fbu1o481pp51daz" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.
* [[Lumatone mapping for 11edo]]

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>pggjv085aud2tff5fbu1o481pp51daz</sha1>
    </revision>
    <revision>
      <id>180627</id>
      <parentid>180626</parentid>
      <timestamp>2025-02-14T02:08:11Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* See also */ Lumatone mapping already linked earlier in instruments section</comment>
      <origin>180627</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22336" sha1="6sucjdrx481j0p77mwk567dhzjwriuy" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{EDO intro|11}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
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| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
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{{Infobox ET}}
{{ED intro}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
[[Category:Macrotonal]]
[[Category:Todo:add rank 2 temperaments table]]</text>
      <sha1>pht0hsps2stlwzsfztti7siuot4cfc5</sha1>
    </revision>
    <revision>
      <id>182803</id>
      <parentid>181325</parentid>
      <timestamp>2025-02-23T13:29:40Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
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      <text bytes="22299" sha1="h81xemur5aisfonu3cxp5vbuhpki337" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|13|10.5|11.5}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
      <sha1>h81xemur5aisfonu3cxp5vbuhpki337</sha1>
    </revision>
    <revision>
      <id>188767</id>
      <parentid>182803</parentid>
      <timestamp>2025-03-29T17:04:20Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>/* Uniform maps */ use new args</comment>
      <origin>188767</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22293" sha1="fbvoqsp7ctwacbxinw8znrt3d2gxk42" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

=== Pathological modes ===
2 1 1 1 2 1 1 1 1 [[2L 7s]] MOS 

3 1 1 1 1 1 1 1 1 [[1L 8s]] MOS 

2 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
      <sha1>fbvoqsp7ctwacbxinw8znrt3d2gxk42</sha1>
    </revision>
    <revision>
      <id>192302</id>
      <parentid>188767</parentid>
      <timestamp>2025-04-15T22:18:06Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>/* Scales */ -Pathological</comment>
      <origin>192302</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22163" sha1="apok4lbj16bsurwji9zx4rtwoubbdqd" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
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{{Infobox ET}}
{{ED intro}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and Downs Notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
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    <revision>
      <id>207618</id>
      <parentid>198791</parentid>
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      <contributor>
        <username>Fredg999</username>
        <id>308</id>
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      <text bytes="22402" sha1="30v87sodicxfx6nmjwztn4bm6kn0b1e" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only 0.3 cents. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
      <sha1>30v87sodicxfx6nmjwztn4bm6kn0b1e</sha1>
    </revision>
    <revision>
      <id>208116</id>
      <parentid>207618</parentid>
      <timestamp>2025-08-23T15:56:19Z</timestamp>
      <contributor>
        <username>Jerdle</username>
        <id>159</id>
      </contributor>
      <comment>Added link to the chlorisma.</comment>
      <origin>208116</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22416" sha1="pfr8fckxi65t0eg8tbgyvhejpec2j9y" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
      <sha1>pfr8fckxi65t0eg8tbgyvhejpec2j9y</sha1>
    </revision>
    <revision>
      <id>209985</id>
      <parentid>208116</parentid>
      <timestamp>2025-09-16T08:13:40Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Temporary until full version is done</comment>
      <origin>209985</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23009" sha1="nbnl68rd5tryel5syp4axjmul44gwlo" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[ed9|35ed9]] and [[ed7|31ed7]].

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[ed10|37ed10]] is a very compressed version of 11edo.

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
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      <text bytes="23018" sha1="qmb6djdz8zfopfmdpphgudygt4n0qg6" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[ed10|37ed10]] is a very compressed version of 11edo.

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
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    </revision>
    <revision>
      <id>210191</id>
      <parentid>210187</parentid>
      <timestamp>2025-09-19T08:46:01Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
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| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
{{Harmonics in equal|11|intervals=odd}}

Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".
[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[equal tuning|37ed10]] is a very compressed version of 11edo.

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
      <sha1>7d6xdapfgnxjcw58mebr5u9ln39r9e4</sha1>
    </revision>
    <revision>
      <id>218994</id>
      <parentid>210191</parentid>
      <timestamp>2025-12-14T04:46:49Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <minor/>
      <comment>move harmonics table later</comment>
      <origin>218994</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23013" sha1="fb7c2dwuli4uug8wofaea90exug784a" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
{{Harmonics in equal|11}}

11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[equal tuning|37ed10]] is a very compressed version of 11edo.

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== Strict 11edo ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
      <sha1>fb7c2dwuli4uug8wofaea90exug784a</sha1>
    </revision>
    <revision>
      <id>219127</id>
      <parentid>218994</parentid>
      <timestamp>2025-12-16T11:13:06Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Strict 11edo */ Make heading consistent with the corresponding section in 5edo/Music</comment>
      <origin>219127</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23048" sha1="4e9m66vd3i65wkfwl51sb48wtj24sae" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
{{Harmonics in equal|11}}

11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[equal tuning|37ed10]] is a very compressed version of 11edo.

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== 11 equal divisions of the octave (11edo proper) ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
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| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
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{{Infobox ET}}
{{ED intro}}
== Theory ==
Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
{{Harmonics in equal|11}}

11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[equal tuning|37ed10]] is a very compressed version of 11edo.

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== 11 equal divisions of the octave (11edo proper) ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
* [https://www.youtube.com/watch?v=4WlTPfRDPCY untitled1], computer

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
      <sha1>q8xwrjmnm09rze6uxqgliic6mz9z4dl</sha1>
    </revision>
    <revision>
      <id>226704</id>
      <parentid>222810</parentid>
      <timestamp>2026-03-23T04:12:04Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Commas */ shows its inaccuracy</comment>
      <origin>226704</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23194" sha1="64i9vzsoazdo8toxbdna5cp45tldpkx" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
{{Harmonics in equal|11}}

11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[144/125]]
| {{Monzo| 4 2 -3 }}
| 244.97
| Trigu
| University comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[equal tuning|37ed10]] is a very compressed version of 11edo.

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== 11 equal divisions of the octave (11edo proper) ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=z0lWcguNsNs ''11 Tone March''] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
* [https://www.youtube.com/watch?v=4WlTPfRDPCY untitled1], computer

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
      <sha1>64i9vzsoazdo8toxbdna5cp45tldpkx</sha1>
    </revision>
    <revision>
      <id>229734</id>
      <parentid>226704</parentid>
      <timestamp>2026-05-06T07:07:03Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* Unequal Derivatives of 11edo */ Bryan Deister's ''11 Tone March'' (2023/2024):  Add short clip, with Lumatone view</comment>
      <origin>229734</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23346" sha1="bvk9okf7x2bz557at6kt10adhb5qau6" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
{{Harmonics in equal|11}}

11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[144/125]]
| {{Monzo| 4 2 -3 }}
| 244.97
| Trigu
| University comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[equal tuning|37ed10]] is a very compressed version of 11edo.

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== 11 equal divisions of the octave (11edo proper) ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* ''11 Tone March'' (2023/2024)
** [https://www.youtube.com/shorts/K2QVvaRUXIQ &lt;nowiki&gt;[short clip]&lt;/nowiki&gt;] (2023, with Lumatone view)
** [https://www.youtube.com/watch?v=z0lWcguNsNs &lt;nowiki&gt;[full version]&lt;/nowiki&gt;] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
* [https://www.youtube.com/watch?v=4WlTPfRDPCY untitled1], computer

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
      <sha1>bvk9okf7x2bz557at6kt10adhb5qau6</sha1>
    </revision>
    <revision>
      <id>229759</id>
      <parentid>229734</parentid>
      <timestamp>2026-05-06T16:12:28Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Notation */ --&gt; Intervals and Notation</comment>
      <origin>229759</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23360" sha1="gp9ywo26dmc13iovkllfwcxq49z30r6" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
{{Harmonics in equal|11}}

11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[144/125]]
| {{Monzo| 4 2 -3 }}
| 244.97
| Trigu
| University comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[equal tuning|37ed10]] is a very compressed version of 11edo.

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== 11 equal divisions of the octave (11edo proper) ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* ''11 Tone March'' (2023/2024)
** [https://www.youtube.com/shorts/K2QVvaRUXIQ &lt;nowiki&gt;[short clip]&lt;/nowiki&gt;] (2023, with Lumatone view)
** [https://www.youtube.com/watch?v=z0lWcguNsNs &lt;nowiki&gt;[full version]&lt;/nowiki&gt;] (2024)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
* [https://www.youtube.com/watch?v=4WlTPfRDPCY untitled1], computer

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
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    </revision>
    <revision>
      <id>229863</id>
      <parentid>229759</parentid>
      <timestamp>2026-05-08T03:13:00Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Unequal Derivatives of 11edo */ Full version has tuning specification</comment>
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      <text bytes="23408" sha1="2n0wv0y6oefbbtsgm4evekzud5sa1ko" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
{{Harmonics in equal|11}}

11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[144/125]]
| {{Monzo| 4 2 -3 }}
| 244.97
| Trigu
| University comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[equal tuning|37ed10]] is a very compressed version of 11edo.

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]


'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.


'''Lumatone'''

There is a [[Lumatone mapping for 11edo]].

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== 11 equal divisions of the octave (11edo proper) ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* ''11 Tone March'' (2023/2024)
** [https://www.youtube.com/shorts/K2QVvaRUXIQ &lt;nowiki&gt;[short clip]&lt;/nowiki&gt;] (2023, with Lumatone view)
** [https://www.youtube.com/watch?v=z0lWcguNsNs &lt;nowiki&gt;[full version]&lt;/nowiki&gt;] (2024, with tuning specification in video description)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
* [https://www.youtube.com/watch?v=4WlTPfRDPCY untitled1], computer

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
      <sha1>2n0wv0y6oefbbtsgm4evekzud5sa1ko</sha1>
    </revision>
    <revision>
      <id>229864</id>
      <parentid>229863</parentid>
      <timestamp>2026-05-08T03:19:19Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Instruments */ Another Lumatone mapping for 11edo is coming real soon</comment>
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| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
{{Harmonics in equal|11}}

11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[144/125]]
| {{Monzo| 4 2 -3 }}
| 244.97
| Trigu
| University comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[equal tuning|37ed10]] is a very compressed version of 11edo.

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

'''Lumatone'''

[[Lumatone mapping for 11edo|Lumatone mappings for 11edo]] are available.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== 11 equal divisions of the octave (11edo proper) ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* ''11 Tone March'' (2023/2024)
** [https://www.youtube.com/shorts/K2QVvaRUXIQ &lt;nowiki&gt;[short clip]&lt;/nowiki&gt;] (2023, with Lumatone view)
** [https://www.youtube.com/watch?v=z0lWcguNsNs &lt;nowiki&gt;[full version]&lt;/nowiki&gt;] (2024, with tuning specification in video description)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
* [https://www.youtube.com/watch?v=4WlTPfRDPCY untitled1], computer

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
      <sha1>9l2gzm0g736lndqf7btk2b6ilmrbtmk</sha1>
    </revision>
    <revision>
      <id>232135</id>
      <parentid>229864</parentid>
      <timestamp>2026-06-10T19:19:02Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* 21st century */ Add Joseph Monzo's ''Monzo, 2026-0608: 11edo, 11/8 time, piano, musescore3'' (2026)</comment>
      <origin>232135</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23569" sha1="j46v8x0kltciyo5597ka38fau3gitag" xml:space="preserve">{{interwiki
| de = 11-EDO
| en = 11edo 
| es = 
| ja = 11平均律
}}
{{Infobox ET}}
{{ED intro}}
== Theory ==
Compared to 12edo, the intervals of 11edo are stretched:

* The "minor second" at 109.09 cents, functions melodically very much like the 100-cent minor second of 12edo.
* The "major second" at 218.18 cents, works in a similar fashion to the 200-cent major second of 12edo, but as a major ninth, it may sound less [[concordant]]. Its inversion, at 981.82 cents, can function as a "bluesy" seventh relative to 12edo's 1000-cent interval, although it is still about 13 cents away from [[7/4]].
* The "minor third" at 327.27 cents, is rather sharp and encroaching upon "[[neutral]] third".
* The "major third" at 436.36 cents, is quite sharp, and closer to the [[supermajor]] third of frequency ratio [[9/7]] than the simpler third of 5/4.
* The "perfect fourth" at 545.45 cents, does not sound like a perfect fourth at all, and passes more easily as the [[11/8]] superfourth than the simpler perfect fourth of 4/3.
{{Harmonics in equal|11}}

11edo does not approximate many small prime harmonics well, only providing good approximations to 7/4 and 11/8. However, 11edo can be treated as a subset of 22edo, and take 22edo's [[6/5]], [[9/7]], and [[16/15]] via direct approximation.

11edo provides the same tuning on the [[k*N subgroups|2*11 subgroup]] 2.9.15.7.11.17 as does 22edo, and on this subgroup it [[tempering out|tempers out]] the same [[comma]]s as 22. Also on this subgroup there is an approximation of the 8:9:11:14:15:16:17 [[chord]] and its subchords. Though the error is rather large, this does provide 11 with a variety of chords approximating [[JI]] chords.

11edo has a good approximation of [[9/7]], hence one natural approach to harmony in 11edo is to generate chords from stacks of this interval. Incidentally, correcting the tuning of 9/7 to just tuning and stacking this interval has the beneficial side effect of also improving the tuning of the 17th harmonic to almost exactly just intonation, with an error of only [[5832/5831|0.3 cents]]. It may therefore be worth considering this JI tuning as an alternative to 11edo.

Being less than twelve, 11edo maps easily to the standard keyboard. The suggested mapping disregards the Ab/G# key, leaving [[Orgone]][7] on the whites. The superfluous Ab can be made a note of [[22edo]], a tuning known as "[[elevenplus]]".

[[File:0-8-16-20 chord.wav|thumb|A 0–8–16–20 chord in 11edo illustrating harmony generated from stacking 9/7 intervals.]]

== Intervals and Notation ==
=== Ups and downs notation ===
11edo can be notated using ups and downs. Conventional notation, including the staff, note names, relative notation, etc. can be used in two ways. The first preserves the ''melodic'' meaning of sharp/flat, major/minor and aug/dim, in that sharp is higher pitched than flat, and major/aug is wider than minor/dim. The disadvantage to this approach is that conventional interval arithmetic no longer works. e.g. M2 + M2 isn't M3, and D + M2 isn't E. Chord names are different because C - E - G isn't P1 - M3 - P5.

The second approach preserves the ''harmonic'' meaning of sharp/flat, major/minor and aug/dim, in that the former is always further fifthwards on the chain of fifths than the latter. Sharp is lower in pitch than flat, and major/aug is narrower than minor/dim. While this approach may seem bizarre at first, interval arithmetic and chord names work as usual. Furthermore, conventional 12edo music can be directly translated to 11edo "on the fly".

The 11edo solfege in the table is derived from [[22edo Solfege|22edo solfege]].

{| class="wikitable center-all right-2 left-4"
! #
! [[Cent]]s
! Solfege
! Approximate Ratios*
! colspan="2" | [[Ups and downs notation|Up/down notation]] &lt;br&gt; with major wider &lt;br&gt; than minor
! colspan="2" | Up/down notation &lt;br&gt; with major narrower &lt;br&gt; than minor
! [[Smitonic]]&lt;br&gt;(3rd-gen)&lt;br&gt;notation 
! [[Tútim Dennsuul Wafiil|TDW]] &lt;br&gt; [[Machinoid|Machine]] &lt;br&gt; notation
! Pseudo-Diatonic Category
!Audio
|-
| 0
| 0.00
| '''do'''
| 1/1
| P1
| A
| P1
| A
| A
| Q, P#
| Unison
|[[File:0-0 unison.mp3|frameless]]
|-
| 1
| 109.09
| '''ra'''
| [[15/14]], [[16/15]], [[17/16]], [[18/17]]
| ^1, m2
| ^A, B
| ^1, M2
| ^A, B
| A#, Bb
| Q#, Rb
| Minor second
|[[File:0-109,09 minor second (11-EDO).mp3|frameless]]
|-
| 2
| 218.18
| '''re'''
| [[8/7]], [[9/8]], [[17/15]]
| ~2, m3
| ^B, Cb
| ~2, M3
| ^B, C#
| B
| R
| Major second
|[[File:0-218,18 major second (11-EDO).mp3|frameless]]
|-
| 3
| 327.27
| '''me'''
| [[6/5]], [[11/9]], [[17/14]]
| M2, ~3
| B#, vC
| m2, ~3
| Bb, vC
| C
| R#, Sb
| Minor third
|[[File:0-327,27 minor third (11-EDO).mp3|frameless]]
|-
| 4
| 436.36
| '''mo'''
| [[9/7]], [[14/11]], [[22/17]]
| M3, v4
| C, vD
| m3, v4
| C, vD
| C#, Db
| S
| Major third/Minor fourth
|[[File:0-436,36 major third (11-EDO).mp3|frameless]]
|-
| 5
| 545.45
| '''fu'''
| [[11/8]], [[15/11]]
| P4, v5
| D, vE
| P4, v5
| D, vE
| D
| S#, Tb
| Major fourth
|[[File:0-545,45 major fourth (11-EDO).mp3|frameless]]
|-
| 6
| 654.55
| '''su'''
| [[16/11]], [[22/15]]
| ^4, P5
| ^D, E
| ^4, P5
| ^D, E
| D#, Eb
| T
| Minor fifth
|[[File:0-654,55 minor fifth (11-EDO).mp3|frameless]]
|-
| 7
| 763.64
| '''lo'''
| [[14/9]], [[11/7]], [[17/11]]
| ^5, m6
| ^E, Fb
| ^5, M6
| ^E, F#
| E
| T#, Ub
| Major fifth/Minor sixth
|[[File:0-763,64 minor sixth (11-EDO).mp3|frameless]]
|-
| 8
| 872.73
| '''la'''
| [[5/3]], [[18/11]], [[28/17]]
| ~6, m7
| vF, Gb
| ~6, M7
| vF, G#
| F
| U
| Major sixth
|[[File:0-872,73 major sixth (11-EDO).mp3|frameless]]
|-
| 9
| 981.82
| '''ta'''
| [[7/4]], [[16/9]], [[30/17]]
| M6, ~7
| F, vG
| m6, ~7
| F, vG
| F#, Gb
| U#, Pb
| Minor seventh
|[[File:0-981,82 minor seventh (11-EDO).mp3|frameless]]
|-
| 10
| 1090.91
| '''ti'''
| [[15/8]], [[17/9]], [[28/15]], [[32/17]]
| M7, v8
| G, vAv
| m7, v8
| G, vAv
| G
| P, Qb
| Major seventh
|[[File:0-1090,91 major seventh (11-EDO).mp3|frameless]]
|-
| 11
| 1200.00
| '''do'''
| 2/1
| P8
| A
| P8
| A
| A
| Q, P#
| Octave
|[[File:0-1200 octave.mp3|frameless]]
|}
&lt;nowiki&gt;* in 2.7.9.11.15.17 subgroup&lt;/nowiki&gt;

The ups and downs notations above are heptatonic systems generated by 5ths (~3/2). Alternative notations include pentatonic 5th-generated, octatonic 5th-generated, nonatonic 5th-generated, heptatonic 3rd-generated, and hexatonic 2nd-generated.

'''&lt;u&gt;Pentatonic 5th-generated:&lt;/u&gt;''' '''D * * E G * * A C * * D'''  ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 5thoid)

D - ^D/Eb - D#/vE - E - G - ^G/Ab - G#/vA - A - C - ^C/Db - C#/vD - D

P1 - ^1/ms3 - A1/~s3 - Ms3 - P4d - ^4d/d5d - A4d/v5d - P5d - ms7 - ~s7/d8d - Ms7/v8d - P8d (s = sub-, d = -oid)

pentatonic genchain of fifths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...

pentatonic genchain of fifths: ...ds3 - ds7 - d4d - d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1 - A5d - As3 - As7... (s = sub-, d = -oid)

'''&lt;u&gt;Octatonic 5th-generated:&lt;/u&gt;''' '''A B * C D E * F G * H A''' ([[Sensoid]] generator = wide 3/2 = 7\11 = perfect 6th)

A - B - B#/Cb - C - D - E - E#/Fb - F - G - G#/Hb - H - A

P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9

octatonic genchain of sixths: ...Db - Ab - Fb - Cb - Hb - E - B - G - D - A - F - C - H - E# - B# - G# - D# - A#...

octatonic genchain of sixths: ...d7 - d4 - d9 - d6 - m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7 - A4 - A1 - A6 - A3...

'''&lt;u&gt;Nonatonic 5th-generated:&lt;/u&gt; A B * C D E F G * H J A''' ([[Joanatonic]] generator = narrow 3/2 = 6\11 = perfect 6th) 

A - B - B#/Cb - C - D - E - F - G - G#/Hb - H - J - A

P1 - m2 - M2/m3 - M3/m4 - M4 - P5 - P6 - m7 - M7/m8 - M8/m9 - M9 - P10

nonotonic genchain of sixths: ...E# - A# - F# - B# - G# - C - H - D - J - E - A - F - B - G - Cb - Hb - Db - Jb - Eb...

nonotonic genchain of sixths: ...M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9...

'''&lt;u&gt;Heptatonic 3rd-generated:&lt;/u&gt; D * E F * G A * B C * D''' ([[Smitonic]] generator = 3\11 = perfect 3rd)

D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D

P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8

genchain of thirds: ...E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb...

genchain of thirds: ...M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6...

'''&lt;u&gt;Hexatonic 2nd-generated:&lt;/u&gt; R * S * T * U * P Q''' '''* R''' ([[Machinoid]] generator = 2\11 = perfect 2nd)

R - R#/Sb - S - S#/Tb - T - T#/Ub - U - U#/Pb - P - Q - Q#/Rb - R

P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - P6 - A6/d7 - P7

genchain of seconds: ... - Qb - Rb - Sb - Tb - Ub - Pb - Q - R - S - T - U - P - Q# - R# - S# - T# - U# - P#...

genchain of seconds: ... - m3 - m4 - m5 - P6 - P1 - P2 - M3 - M4 - M5 - A6 - A1...

===Sagittal notation===
This notation is a subset of the notations for EDOs [[22edo#Sagittal notation|22]], [[44edo#Sagittal notation|44]], and [[66edo#Sagittal notation|66]].
====Evo flavor====

&lt;imagemap&gt;
File:11-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

====Revo flavor====

&lt;imagemap&gt;
File:11-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 368 0 528 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 368 106 [[22-EDO#Sagittal_notation | 22-EDO notation]]
default [[File:11-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=11}}

=== Commas ===
11et [[tempering out|tempers out]] the following [[comma]]s. This assumes val {{val| 11 17 26 31 38 41 }}.

{| class="commatable wikitable center-1 center-2 right-4 center-5"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group=note&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[177147/131072]]
| {{Monzo| -17 11 }}
| 521.50
| sasawa 3rd
| Pythagorean augmented third
|-
| 5
| [[135/128]]
| {{Monzo| -7 3 1 }}
| 92.18
| Layobi
| Major chroma
|-
| 5
| [[144/125]]
| {{Monzo| 4 2 -3 }}
| 244.97
| Trigu
| University comma
|-
| 5
| [[34171875/33554432|(16 digits)]]
| {{Monzo| -25 7 6 }}
| 31.57
| Lala-tribiyo
| [[Ampersand comma]]
|-
| 5
| &lt;abbr title="295578376007080078125/295147905179352825856"&gt;(42 digits)&lt;/abbr&gt;
| {{Monzo| -68 18 17 }}
| 2.52
| Quinla-seyo
| [[Vavoom comma]]
|-
| 7
| &lt;abbr title="854296875/843308032"&gt;(18 digits)&lt;/abbr&gt;
| {{Monzo| -10 7 8 -7 }}
| 22.41
| Lasepru-aquadbiyo
| [[Blackjackisma]]
|-
| 7
| [[1029/1024]]
| {{Monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[16875/16807]]
| {{Monzo| 0 3 4 -5 }}
| 6.99
| Quinru-aquadyo
| Mirkwai comma
|-
| 7
| [[2401/2400]]
| {{Monzo| -5 -1 -2 4 }}
| 0.72
| Bizozogu
| Breedsma
|-
| 11
| [[121/120]]
| {{Monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[65536/65219]]
| {{Monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|}

== Approximation to JI ==
{| class="wikitable"
|-
! Harmonic
! 8
! 
! 9
! 
! 11
! 
! 14
! 
! 16
|-
! JI interval from 1/1
| 1/1 = 0 cents
| 
| 9/8 = 204
| 
| 11/8 = 551
| 
| 7/4 = 969
| 
| 2/1 = 1200
|-
! Nearest 11edo interval
| 0\11 = 0¢
| 
| 2\11 = 218¢
| 
| 5\11 = 545
| 
| 9\11 = 982
| 
| 11\11 = 1200
|-
! Difference
| 0
| 
| +14¢
| 
| -6¢
| 
| +13¢
| 
| 0¢
|-
! JI interval between
| 
| 9:8 = 204¢
| 
| 11:9 = 347
| 
| 14:11 = 418
| 
| 8:7 = 231
| 
|-
! Nearest 11edo interval
| 
| 2\11 = 218¢
| 
| 3\11 = 327
| 
| 4\11 = 436
| 
| 2\11 = 218
| 
|-
! Difference
| 
| +14¢
| 
| -20¢
| 
| +18¢
| 
| -13¢
| 
|}

11edo also may be considered a 2.7.9.11.15.17 subgroup temperament. See diagram:

[[File:11edo_approx_2-7-9-11-15-17_2ndsave.png|alt=11edo_approx_2-7-9-11-15-17_2ndsave.png|11edo_approx_2-7-9-11-15-17_2ndsave.png]]

== Octave stretch or compression ==
11edo has about equally bad sharp and flat mappings of [[prime]]s 3 and 5. The 7 and 13 are quite sharp, but the 11 is a little flat. To use it as a 2.7.11.13 tuning, slight [[octave shrinking]] is advisable. Examples of slightly compressed versions of 11edo include (least to most compressed) [[ed6|28ed6]], [[ed12|39ed12]], [[zpi|30zpi]], [[equal tuning|35ed9]] and [[ed7|31ed7]].

To use its primes 3 or 5, extreme octave shrinking can be used, at the cost of making the octaves sound significantly weaker. [[equal tuning|37ed10]] is a very compressed version of 11edo.

== Scales ==
{{Main|11edo modes}}

=== MOS scales ===
{{Main|List of 11edo MOS scales}}
Although 11edo has one fewer interval in the octave than 12edo, in terms of [[MOS scale|moment-of-symmetry scales]], it offers a great deal more variety. This is because 11 is a prime number, while 12 is composite. Cycles of 2\11 (two degrees of 11edo), 3\11, 4\11 and 5\11 produce scales which do not repeat at the octave until all 11 intervals have been included.

== Instruments ==
'''11edo ukulele'''

[[File:11-edo-ukulele.JPG|alt=11-edo-ukulele.JPG|404x304px|11-edo-ukulele.JPG]]

'''Ensembles'''

In February 2011, [http://oddmusicuc.wordpress.com/ Oddmusic U-C], as part of its Microtonal Design Seminar, generated a 7-piece ensemble for playing music in 11edo. Instrumentation: autotuner, cümbüş, electronic keyboard, kalimba, retrofretted guitar, tuned bottles, udderbot. Recordings forthcoming.

'''Lumatone'''

[[Lumatone mapping for 11edo|Lumatone mappings for 11edo]] are available.

== Introductory Materials ==
* [[File:11edo_1MC.mp3|270px]] 11edo example composition by [[User:Inthar|Inthar]] (first half's in [[4L 3s]], second half is in [[3L 5s]])

== Music ==
{{Catrel|11edo tracks}}
=== 11 equal divisions of the octave (11edo proper) ===
==== Modern renderings ====
; {{W|Arthur Schutt}}
* [https://www.youtube.com/watch?v=GEzxtHILDr8 ''Bluin' The Black Keys''] (1926) – rendered by Francium (2024)

==== 20th century ====
; [[George Secor]]
* [http://xenharmony.wikispaces.com/space/showimage/11edo-improv.mp3 First Piece Ever]{{dead link}} (1970) — apparently the first piece ever written for 11edo.

; [[Bill Sethares]]
* [https://sethares.engr.wisc.edu/mp3s/dabo_girl.html "The Turquoise Dabo Girl"], from [https://sethares.engr.wisc.edu/xentone.html ''Xentonality''] (1997)

==== 21st century ====
; [[Abnormality]]
* [https://www.youtube.com/watch?v=G1rUu9qmXkE ''Scatter Brain''] (2024)

; [[Christopher Bailey]]
* [https://www.youtube.com/playlist?list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3 ''The Stuffed Ones''] (2004) – 4-piece suite ([http://christopherbaileymusic.com/composition-list/ details])
** [https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=1 "Goopy"] · [https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=2 "Ellie"] · [https://www.youtube.com/watch?v=53IiHdXfJwI&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=3 "Ziggy"] · [https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;list=PLby8OiGBluOXODbJpiRhzNftxy5-DV-I3&amp;index=4 "Towelbear"]

; [[Jacob Barton]]
* ''Hyperimprovisations Nuggetwarp'' (2009)
** [https://soundclick.com/share.cfm?id=10267904 "Piece I"] · [https://soundclick.com/share.cfm?id=10267905 "Piece II"] · [https://soundclick.com/share.cfm?id=10267906 "Piece III"]

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/she-is-my-lilac-hued-obsession "She is My Lilac-Hued Obsession"], from [https://cityoftheasleep.bandcamp.com/album/map-of-an-internal-landscape-reissue ''Map of an Internal Landscape''] (2007)

; [[Jason Conklin]]
* ''The City Sleeps, A Madrigal'' (2011) – [http://web.archive.org/web/20201127013549/http://micro.soonlabel.com/gene_ward_smith/Others/Conklin/Conklin-The_City_Sleeps_A_Madrigal.mp3 play] | [https://soundcloud.com/ninly/the-city-sleeps SoundCloud]

; [[E8 Heterotic]]
* [https://youtu.be/9tJHJEZnvFs?si=9n6I3VnejVon_iot ''Olive Flamenco''] (2019)

; [[Francium]]
* "Tostadosto" from ''The Decatonic Album'' (2024) – [https://open.spotify.com/track/27hl1xKswTuaQG0vIjMHhk Spotify] | [https://francium223.bandcamp.com/track/tostadosto Bandcamp] | [https://www.youtube.com/watch?v=28SOTJvT6sw YouTube]
* "Sleep Slope" from ''XenRhythms'' (2024) – [https://open.spotify.com/track/2GnUioPrMlJmMaacE6DK7i Spotify] | [https://francium223.bandcamp.com/track/sleep-slope Bandcamp] | [https://www.youtube.com/watch?v=hM0BAC_YZnQ YouTube]

; [[David Hamill]]
* [http://www.focalchords.com/audio/Cool_My_Head_11EDO.mp3 ''Cool My Head''] (2010)

; [[Andrew Heathwaite]]
* ''Orange Clips on Sausages'' (2004) – [http://web.archive.org/web/20201127012301/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+orangeclipsonsausagesin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=933772 SoundClick]
* ''Blue Gel'' (2004) – [http://web.archive.org/web/20201127012646/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+improvisationin11tet.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songID=834492 SoundClick]
* ''conversation is'' (2010) – [http://web.archive.org/web/20201127012932/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Heathwaite/andrewheathwaite+conversationis.mp3 play] | [https://www.soundclick.com/music/songInfo.cfm?songid=8839070 SoundClick]

; [[Hideya]]
* [https://www.youtube.com/watch?v=PcY3HrwQRRY ''Like Parker 3''] (2019)
* [https://www.youtube.com/watch?v=YawFcH4cXfs ''Like 40s music''] (2022)

; [[Aaron Andrew Hunt]]
* From [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999–2022)
** [https://aaronandrewhunt.bandcamp.com/track/prelude-in-11et "Prelude in 11ET"]
** "Adagio in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/adagio-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/adagio-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}
** "Invention in 11ET" – [https://aaronandrewhunt.bandcamp.com/track/invention-in-11et Bandcamp] | [https://soundcloud.com/uz1kt3k/invention-in-11et?in=uz1kt3k/sets/adagio-invention-in-11et SoundCloud]{{dead link}}

; [[User:Ayceman|Alexandru Ianu]]
* ''Divertimento in 11 tone Orgone'' (2021) – [[:File:Divertimento in 11 tone Orgone.pdf|sheet music]] | [https://youtu.be/8x1f5WFkF4k YouTube] – orgone in 11edo tuning
* ''Sylvian Moon Dance'' (2021) – [[:File:SylvianMoonDance.ogg|audio]] | [[:File:Sylvian_Moon_Dance.pdf|sheet music]] | [https://youtu.be/81uZbsmbet8 YouTube] – orgone in 11edo tuning
* ''Ocean of the Necrophages'' (2021) – orgone in 11edo tuning
** Piano: [[:File:Ocean of the Necrophages (4U UP).ogg|audio]] | [[:File:Ocean of the Necrophages.pdf|sheet music]] | [https://youtu.be/CWU09fXXy1s YouTube]
** Strings: [[:File:Ocean of the Necrophages (strings).ogg|audio]] | [[:File:Ocean of the Necrophages (strings).pdf|sheet music]]

; [[Aaron Krister Johnson]]
* [http://www.akjmusic.com/audio/black_ritual_dirge.mp3 ''Black Ritual Dirge'']{{dead link}}

; [[User:ks26|groundfault]]
* [https://www.youtube.com/watch?v=AEnEYk3X1as ''Ghost Bridge''] (2020)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=bqIDxbc21O8 ''Micropiece in 11edo''] (2020)
* [https://www.youtube.com/watch?v=fwyM3quEzu4 ''Prelude &amp; Fugue in 11edo, in Four Parts, for Recorder, Organ, Cello''] (2022)
* [https://www.youtube.com/watch?v=qNqDuAq6O5k ''George Secor · 11EDO improvisation (1971)''] (2022)

; [[Joseph Monzo]]
* [https://www.youtube.com/shorts/JMrFUKfqfeY ''Monzo, 2026-0608: 11edo, 11/8 time, piano, musescore3''] (2026)

; [[Mundoworld]]
* [https://www.youtube.com/watch?v=69TiqslCgeg ''Fire Memes'' (with Anthony "Pomp" Pompliano)] – Machine[6] in 11edo tuning
* [https://www.youtube.com/watch?v=MfNLxcbVzs8 ''Theory of Creation''] – Machine[6] in 11edo tuning
* "Search Party" from ''No Fun House'' (2025) – [https://open.spotify.com/track/7CMiwDuuRuFpB0skfwT2Ap Spotify] | [https://mundoworld.bandcamp.com/track/search-party Bandcamp] | [https://www.youtube.com/watch?v=hLs6MjuousI YouTube]

; [[User:GlitchyDarkness|No Clue Music]]
* [https://www.youtube.com/watch?v=lPKc1B6YBn4 ''Cursed Star''] (2024)

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=AbWxZ6yh69s "Overcoming"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)

; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=y939ciE9MQY ''33322''] (2024)

; [[X. J. Scott]]
* [https://soundclick.com/share.cfm?id=955383 ''Angkor Wat, September 1066''] (2004)

; [[Sevish]]
* "[[Longwayaway People]]", from ''[[Rhythm and Xen]]'' (2015)
* "[[Make a Dream]]", from ''[[Rhythm and Xen]]'' (2015)

; [[Jon Lyle Smith]]
* [https://archive.org/details/jls_ArchiveVol2/Jaunt_reMix2012.wav ''Jaunt''] (2012) – [http://web.archive.org/web/20201127014902/http://micro.soonlabel.com/jon-lyle-smith/Jaunt.mp3 play] | [https://www.youtube.com/watch?v=HKULte3WhuE YouTube]
* [http://archive.org/download/CounterpointIn11edo/CounterpointIn11edo.mp3 ''Counterpoint in 11EDO'']{{dead link}}

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012602/http://micro.soonlabel.com/11-ET/daily201110-gpo-jeffery-dahmer-cooks.mp3 ''Jeffrey Dahmer Cooks at 11EDO''] (2011)
* [https://web.archive.org/web/20201127015348/http://micro.soonlabel.com/11-ET/20110902_prepared_seagull_metamorphis.mp3 ''The Metamorphosis of Gregor''] (2011)
* ''Eleven Birds'' (2012) – [https://www.chrisvaisvil.com/eleven-birds/ blog] | [http://micro.soonlabel.com/11-ET/20120928-piano-11edo-eleven-birds.mp3 play]
* [https://soundcloud.com/vaisvil/the-execution-of-12-equal ''The Execution of 12 Equal'']{{dead link}}

; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/10.mp3 "10. 11 / octave"], from ''[[Comets Over Flatland]]'' (2007)

; [[Ozan Yarman]]
* [http://www.ozanyarman.com/files/music/Icicle_Caverns.mp3 ''Icicle Caverns''] (2010) ([http://www.ozanyarman.com/files/music/icicle_caverns_score.pdf score])

; [[Yeah Gore]]
* [https://www.youtube.com/watch?v=FL72Z4H1IF8 ''11 TET Hernya''] (2020)
* [https://www.youtube.com/watch?v=dwel2K1Bgds ''YG_A''] (2022)

=== Unequal Derivatives of 11edo ===
; [[Bryan Deister]]
* ''11 Tone March'' (2023/2024)
** [https://www.youtube.com/shorts/K2QVvaRUXIQ &lt;nowiki&gt;[short clip]&lt;/nowiki&gt;] (2023, with Lumatone view)
** [https://www.youtube.com/watch?v=z0lWcguNsNs &lt;nowiki&gt;[full version]&lt;/nowiki&gt;] (2024, with tuning specification in video description)

== Videos ==
* The Stuffed Ones: ''[https://www.youtube.com/watch?v=NU0VvGRelUQ&amp;feature=related Goopy]'', ''[https://www.youtube.com/watch?v=4D9wDl_oxHE&amp;feature=related Ziggy]'', ''[https://www.youtube.com/watch?v=53IiHdXfJwI&amp;feature=related Ellie]'', ''[https://www.youtube.com/watch?v=4sZqpRcB-lk&amp;feature=related Towelbear]'' by [https://www.youtube.com/user/zipzappoozoo zipzappoozoo]

* [https://www.youtube.com/watch?v=AhPjsCoMy-Q 11-equal Improvisation]'', [[Mike Battaglia FAQ|Mike Battaglia]] - youtube
* [https://www.youtube.com/watch?v=4WlTPfRDPCY untitled1], computer

== See also ==
* [[11edo Zine]] — There is an 11edo Zine! As far as we know, 11edo is the first xenharmonic tuning system to have its own zine.

== Notes ==
&lt;references group=note/&gt;

[[Category:Listen]]
{{Todo|add rank 2 temperaments table}}</text>
      <sha1>j46v8x0kltciyo5597ka38fau3gitag</sha1>
    </revision>
  </page>
  <page>
    <title>11edt</title>
    <ns>0</ns>
    <id>91</id>
    <revision>
      <id>91</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>91</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1308" sha1="5wnrto898njeuwnjeykktqivd7dcob9" xml:space="preserve">11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

Tuning in scala format is as follows:

! E:\cakewalk\scales\11_of_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]</text>
      <sha1>5wnrto898njeuwnjeykktqivd7dcob9</sha1>
    </revision>
    <revision>
      <id>4444</id>
      <timestamp>2011-11-28T23:15:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 279979454 - Original comment: **</comment>
      <origin>4444</origin>
      <model>wikitext</model>
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      <text bytes="5246" sha1="teb1quyyac6hiuy4j2t5w740nya6gix" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2011-11-28 23:15:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;279979454&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="10763046"]]

11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

Tuning in scala format is as follows:

! E:\cakewalk\scales\11_of_tritave.scl
!
11 in tritave
!
172.90500
345.81000
518.71500
691.62000
864.52500
1037.43000
1210.33500
1383.24000
1556.14500
1729.05000
3/1




Mozart's [[http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3|sonata #11 in A Major K331 in 11 EDT]] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)
[[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3|Frozen Time Occupies Wall Street]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]] =&amp;gt;[[@http://chrisvaisvil.com/?p=1392| information about the piece]]
[[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3|Molly's Playground]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]] =&amp;gt; [[http://chrisvaisvil.com/?p=1716| information about the piece]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;, with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Tuning in scala format is as follows:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
! E:\cakewalk\scales\11_of_tritave.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
11 in tritave&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
172.90500&amp;lt;br /&amp;gt;
345.81000&amp;lt;br /&amp;gt;
518.71500&amp;lt;br /&amp;gt;
691.62000&amp;lt;br /&amp;gt;
864.52500&amp;lt;br /&amp;gt;
1037.43000&amp;lt;br /&amp;gt;
1210.33500&amp;lt;br /&amp;gt;
1383.24000&amp;lt;br /&amp;gt;
1556.14500&amp;lt;br /&amp;gt;
1729.05000&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Mozart's &amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3" rel="nofollow"&amp;gt;sonata #11 in A Major K331 in 11 EDT&amp;lt;/a&amp;gt; (using a 11 =&amp;amp;gt; 12 key mapping so octaves become tritaves)&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3" rel="nofollow"&amp;gt;Frozen Time Occupies Wall Street&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; =&amp;amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1392" rel="nofollow" target="_blank"&amp;gt; information about the piece&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3" rel="nofollow"&amp;gt;Molly's Playground&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; =&amp;amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1716" rel="nofollow"&amp;gt; information about the piece&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>teb1quyyac6hiuy4j2t5w740nya6gix</sha1>
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        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
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      <comment>**Imported revision 277242062 - Original comment: **</comment>
      <origin>4445</origin>
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      <text bytes="4803" sha1="233mx2kxsy8i9b56rtmgaprsy3stqag" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-11-19 14:52:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;277242062&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="10763046"]]

11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

Tuning in scala format is as follows:

! E:\cakewalk\scales\11_of_tritave.scl
!
11 in tritave
!
172.90500
345.81000
518.71500
691.62000
864.52500
1037.43000
1210.33500
1383.24000
1556.14500
1729.05000
3/1





[[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3|Frozen Time Occupies Wall Street]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]] =&amp;gt;[[@http://chrisvaisvil.com/?p=1392| information about the piece]]
[[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3|Molly's Playground]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]] =&amp;gt; [[http://chrisvaisvil.com/?p=1716| information about the piece]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10763046?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10763046&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;, with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Tuning in scala format is as follows:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
! E:\cakewalk\scales\11_of_tritave.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
11 in tritave&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
172.90500&amp;lt;br /&amp;gt;
345.81000&amp;lt;br /&amp;gt;
518.71500&amp;lt;br /&amp;gt;
691.62000&amp;lt;br /&amp;gt;
864.52500&amp;lt;br /&amp;gt;
1037.43000&amp;lt;br /&amp;gt;
1210.33500&amp;lt;br /&amp;gt;
1383.24000&amp;lt;br /&amp;gt;
1556.14500&amp;lt;br /&amp;gt;
1729.05000&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3" rel="nofollow"&amp;gt;Frozen Time Occupies Wall Street&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; =&amp;amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1392" rel="nofollow" target="_blank"&amp;gt; information about the piece&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3" rel="nofollow"&amp;gt;Molly's Playground&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; =&amp;amp;gt; &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1716" rel="nofollow"&amp;gt; information about the piece&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>233mx2kxsy8i9b56rtmgaprsy3stqag</sha1>
    </revision>
    <revision>
      <id>4446</id>
      <timestamp>2011-10-07T15:06:14Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 262681116 - Original comment: **</comment>
      <origin>4446</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4149" sha1="svxwlg047ae8c5jci42wz4u9dvrq45t" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-10-07 15:06:14 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;262681116&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="10763046"]]

11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

Tuning in scala format is as follows:

! E:\cakewalk\scales\11_of_tritave.scl
!
11 in tritave
!
172.90500
345.81000
518.71500
691.62000
864.52500
1037.43000
1210.33500
1383.24000
1556.14500
1729.05000
3/1





[[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3|Frozen Time Occupies Wall Street]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]] =&amp;gt;[[@http://chrisvaisvil.com/?p=1392| information about the piece]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10763046?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10763046&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;, with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Tuning in scala format is as follows:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
! E:\cakewalk\scales\11_of_tritave.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
11 in tritave&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
172.90500&amp;lt;br /&amp;gt;
345.81000&amp;lt;br /&amp;gt;
518.71500&amp;lt;br /&amp;gt;
691.62000&amp;lt;br /&amp;gt;
864.52500&amp;lt;br /&amp;gt;
1037.43000&amp;lt;br /&amp;gt;
1210.33500&amp;lt;br /&amp;gt;
1383.24000&amp;lt;br /&amp;gt;
1556.14500&amp;lt;br /&amp;gt;
1729.05000&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3" rel="nofollow"&amp;gt;Frozen Time Occupies Wall Street&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; =&amp;amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1392" rel="nofollow" target="_blank"&amp;gt; information about the piece&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>svxwlg047ae8c5jci42wz4u9dvrq45t</sha1>
    </revision>
    <revision>
      <id>4447</id>
      <timestamp>2011-10-07T15:05:16Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 262680646 - Original comment: **</comment>
      <origin>4447</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4133" sha1="i66th95q3bhnkd2liznruj6p4mo4o5x" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-10-07 15:05:16 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;262680646&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="10763046"]]

11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

Tuning in scala format is as follows:

! E:\cakewalk\scales\11_of_tritave.scl
!
11 in tritave
172.90500
345.81000
518.71500
691.62000
864.52500
1037.43000
1210.33500
1383.24000
1556.14500
1729.05000
3/1





[[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3|Frozen Time Occupies Wall Street]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]] =&amp;gt;[[@http://chrisvaisvil.com/?p=1392| information about the piece]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10763046?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10763046&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of &amp;lt;a class="wiki_link" href="/7edo"&amp;gt;7edo&amp;lt;/a&amp;gt;, with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Tuning in scala format is as follows:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
! E:\cakewalk\scales\11_of_tritave.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
11 in tritave&amp;lt;br /&amp;gt;
172.90500&amp;lt;br /&amp;gt;
345.81000&amp;lt;br /&amp;gt;
518.71500&amp;lt;br /&amp;gt;
691.62000&amp;lt;br /&amp;gt;
864.52500&amp;lt;br /&amp;gt;
1037.43000&amp;lt;br /&amp;gt;
1210.33500&amp;lt;br /&amp;gt;
1383.24000&amp;lt;br /&amp;gt;
1556.14500&amp;lt;br /&amp;gt;
1729.05000&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3" rel="nofollow"&amp;gt;Frozen Time Occupies Wall Street&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; =&amp;amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1392" rel="nofollow" target="_blank"&amp;gt; information about the piece&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>i66th95q3bhnkd2liznruj6p4mo4o5x</sha1>
    </revision>
    <revision>
      <id>4448</id>
      <timestamp>2011-10-06T16:49:50Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 262343544 - Original comment: **</comment>
      <origin>4448</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3151" sha1="f0u0k4pamvmlafsn1shkwppzk64x8s6" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2011-10-06 16:49:50 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;262343544&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[media type="custom" key="10763046"]]

Tuning in scala format is as follows:

! E:\cakewalk\scales\11_of_tritave.scl
!
11 in tritave
11
!
172.90500
345.81000
518.71500
691.62000
864.52500
1037.43000
1210.33500
1383.24000
1556.14500
1729.05000
3/1





[[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3|Frozen Time Occupies Wall Street]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]] =&amp;gt;[[@http://chrisvaisvil.com/?p=1392| information about the piece]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10763046?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10763046&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Tuning in scala format is as follows:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
! E:\cakewalk\scales\11_of_tritave.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
11 in tritave&amp;lt;br /&amp;gt;
11&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
172.90500&amp;lt;br /&amp;gt;
345.81000&amp;lt;br /&amp;gt;
518.71500&amp;lt;br /&amp;gt;
691.62000&amp;lt;br /&amp;gt;
864.52500&amp;lt;br /&amp;gt;
1037.43000&amp;lt;br /&amp;gt;
1210.33500&amp;lt;br /&amp;gt;
1383.24000&amp;lt;br /&amp;gt;
1556.14500&amp;lt;br /&amp;gt;
1729.05000&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3" rel="nofollow"&amp;gt;Frozen Time Occupies Wall Street&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt; =&amp;amp;gt;&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=1392" rel="nofollow" target="_blank"&amp;gt; information about the piece&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>f0u0k4pamvmlafsn1shkwppzk64x8s6</sha1>
    </revision>
    <revision>
      <id>33129</id>
      <parentid>91</parentid>
      <timestamp>2018-09-21T07:55:18Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <origin>33129</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1332" sha1="sz3gn3ofv51e7icht6yihs24eg3gg1r" xml:space="preserve">11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

Tuning in scala format is as follows:

! E:\cakewalk\scales\11_of_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>sz3gn3ofv51e7icht6yihs24eg3gg1r</sha1>
    </revision>
    <revision>
      <id>96668</id>
      <parentid>33129</parentid>
      <timestamp>2022-10-05T19:44:32Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
      <origin>96668</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1347" sha1="pwf4kxskrjt490921tug22nv7nci0b1" xml:space="preserve">{{Infobox ET}}
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

Tuning in scala format is as follows:

! E:\cakewalk\scales\11_of_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>pwf4kxskrjt490921tug22nv7nci0b1</sha1>
    </revision>
    <revision>
      <id>103252</id>
      <parentid>96668</parentid>
      <timestamp>2023-02-18T23:02:35Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>103252</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1882" sha1="ppu4dd0rd89ty3eh2sdtrz31fda2wju" xml:space="preserve">{{Infobox ET}}
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

Tuning in scala format is as follows:

! E:\cakewalk\scales\11_of_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1

== Intervals ==

{| class="wikitable center-all"
! #
! Cents
! Hekts
! Approximate ratios
! [[Arcturus]] nonatonic notation
|-
| colspan = "3" | 0
| [[1/1]]
| H
|-
| 1
| 172.9
| 118.1
| [[11/10]], [[10/9]]
| H#, Ib
|-
| 2
| 345.8
| 236.2
| [[11/9]]
| I
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[27/20]]
| J
|-
| 4
| 691.6
| 472.4
| K
|-
| 5
| 864.5
| 590.5
| L
|-
| 6
| 1037.4
| L#, Mb
|-
| 7
| 1210.3
| 826.7
| M
|-
| 8
| 1383.2
| 944.8
| N
|-
| 9
| 1556.1
| 1062.9
| O
|-
| 10
| 1729
| 1181
| P
|-
| 11
| 1902
| 1300
| H
|}

== Pieces ==

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>ppu4dd0rd89ty3eh2sdtrz31fda2wju</sha1>
    </revision>
    <revision>
      <id>103253</id>
      <parentid>103252</parentid>
      <timestamp>2023-02-18T23:06:26Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>103253</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1996" sha1="a4qlcqbjmagy3xmh7lccrdic3f1tyu1" xml:space="preserve">{{Infobox ET}}
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

Tuning in scala format is as follows:

! E:\cakewalk\scales\11_of_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1

== Intervals ==

{| class="wikitable center-all"
! #
! Cents
! Hekts
! Approximate ratios
! [[Arcturus]] nonatonic notation
|-
| colspan = "3" | 0
| [[1/1]]
| H
|-
| 1
| 172.9
| 118.1
| [[11/10]], [[10/9]]
| H#, Ib
|-
| 2
| 345.8
| 236.2
| [[11/9]]
| I
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[27/20]]
| J
|-
| 4
| 691.6
| 472.4
| [[3/2]], [[40/27]]
| K
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[28/17]], [[105/64]]
| L
|-
| 6
| 1037.4
| 708.6
| [[29/16]], [[20/11], [[64/35]]
| L#, Mb
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| M
|-
| 8
| 1383.2
| 944.8
|
| N
|-
| 9
| 1556.1
| 1062.9
| 
| O
|-
| 10
| 1729
| 1181
|
| P
|-
| 11
| 1902
| 1300
|
| H
|}

== Pieces ==

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>a4qlcqbjmagy3xmh7lccrdic3f1tyu1</sha1>
    </revision>
    <revision>
      <id>103254</id>
      <parentid>103253</parentid>
      <timestamp>2023-02-18T23:07:03Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>103254</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2029" sha1="azumni6cm7n1pnak5bsmrxp8ubn2xj3" xml:space="preserve">{{Infobox ET}}
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.


== Intervals ==

{| class="wikitable center-all"
! #
! Cents
! Hekts
! Approximate ratios
! [[Arcturus]] nonatonic notation
|-
| colspan = "3" | 0
| [[1/1]]
| H
|-
| 1
| 172.9
| 118.1
| [[11/10]], [[10/9]]
| H#, Ib
|-
| 2
| 345.8
| 236.2
| [[11/9]]
| I
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[27/20]]
| J
|-
| 4
| 691.6
| 472.4
| [[3/2]], [[40/27]]
| K
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[28/17]], [[105/64]]
| L
|-
| 6
| 1037.4
| 708.6
| [[29/16]], [[20/11], [[64/35]]
| L#, Mb
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| M
|-
| 8
| 1383.2
| 944.8
|
| N
|-
| 9
| 1556.1
| 1062.9
| 
| O
|-
| 10
| 1729
| 1181
|
| P
|-
| 11
| 1902
| 1300
|
| H
|}

== Scala tuning ==
Tuning in scala format is as follows:

&lt;pre&gt;
! E:\cakewalk\scales\11_of_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1
&lt;/pre&gt;

== Pieces ==

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>azumni6cm7n1pnak5bsmrxp8ubn2xj3</sha1>
    </revision>
    <revision>
      <id>103256</id>
      <parentid>103254</parentid>
      <timestamp>2023-02-18T23:07:24Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>103256</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2030" sha1="4ugjemzmiim1yqzr0ahswajddhfzcpg" xml:space="preserve">{{Infobox ET}}
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.


== Intervals ==

{| class="wikitable center-all"
! #
! Cents
! Hekts
! Approximate ratios
! [[Arcturus]] nonatonic notation
|-
| colspan = "3" | 0
| [[1/1]]
| H
|-
| 1
| 172.9
| 118.1
| [[11/10]], [[10/9]]
| H#, Ib
|-
| 2
| 345.8
| 236.2
| [[11/9]]
| I
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[27/20]]
| J
|-
| 4
| 691.6
| 472.4
| [[3/2]], [[40/27]]
| K
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[28/17]], [[105/64]]
| L
|-
| 6
| 1037.4
| 708.6
| [[29/16]], [[20/11]], [[64/35]]
| L#, Mb
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| M
|-
| 8
| 1383.2
| 944.8
|
| N
|-
| 9
| 1556.1
| 1062.9
| 
| O
|-
| 10
| 1729
| 1181
|
| P
|-
| 11
| 1902
| 1300
|
| H
|}

== Scala tuning ==
Tuning in scala format is as follows:

&lt;pre&gt;
! E:\cakewalk\scales\11_of_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1
&lt;/pre&gt;

== Pieces ==

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>4ugjemzmiim1yqzr0ahswajddhfzcpg</sha1>
    </revision>
    <revision>
      <id>103258</id>
      <parentid>103256</parentid>
      <timestamp>2023-02-18T23:14:47Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>103258</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2028" sha1="7y1bdydp3nih673odeeb2vw99udak2g" xml:space="preserve">{{Infobox ET}}
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.


== Intervals ==

{| class="wikitable center-all"
! #
! Cents
! Hekts
! Approximate ratios
! [[Arcturus]] nonatonic notation
|-
| colspan = "3" | 0
| [[1/1]]
| H
|-
| 1
| 172.9
| 118.1
| [[11/10]], [[10/9]]
| H#, Ib
|-
| 2
| 345.8
| 236.2
| [[11/9]]
| I
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[27/20]]
| J
|-
| 4
| 691.6
| 472.4
| [[3/2]], [[40/27]]
| K
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[28/17]], [[105/64]]
| L
|-
| 6
| 1037.4
| 708.6
| [[29/16]], [[20/11]], [[64/35]]
| L#, Mb
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| M
|-
| 8
| 1383.2
| 944.8
|
| N
|-
| 9
| 1556.1
| 1062.9
| 
| O
|-
| 10
| 1729
| 1181
|
| P
|-
| 11
| 1902
| 1300
|
| H
|}

== Scala file ==
Tuning in scala format is as follows:

&lt;pre&gt;
! E:\cakewalk\scales\11_of_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1
&lt;/pre&gt;

== Pieces ==

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>7y1bdydp3nih673odeeb2vw99udak2g</sha1>
    </revision>
    <revision>
      <id>103259</id>
      <parentid>103258</parentid>
      <timestamp>2023-02-18T23:14:53Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>103259</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2027" sha1="cgv7cf2r8u76jswcr4rr7arpmlefv22" xml:space="preserve">{{Infobox ET}}
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

== Intervals ==

{| class="wikitable center-all"
! #
! Cents
! Hekts
! Approximate ratios
! [[Arcturus]] nonatonic notation
|-
| colspan = "3" | 0
| [[1/1]]
| H
|-
| 1
| 172.9
| 118.1
| [[11/10]], [[10/9]]
| H#, Ib
|-
| 2
| 345.8
| 236.2
| [[11/9]]
| I
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[27/20]]
| J
|-
| 4
| 691.6
| 472.4
| [[3/2]], [[40/27]]
| K
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[28/17]], [[105/64]]
| L
|-
| 6
| 1037.4
| 708.6
| [[29/16]], [[20/11]], [[64/35]]
| L#, Mb
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| M
|-
| 8
| 1383.2
| 944.8
|
| N
|-
| 9
| 1556.1
| 1062.9
| 
| O
|-
| 10
| 1729
| 1181
|
| P
|-
| 11
| 1902
| 1300
|
| H
|}

== Scala file ==
Tuning in scala format is as follows:

&lt;pre&gt;
! E:\cakewalk\scales\11_of_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1
&lt;/pre&gt;

== Pieces ==

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>cgv7cf2r8u76jswcr4rr7arpmlefv22</sha1>
    </revision>
    <revision>
      <id>103867</id>
      <parentid>103259</parentid>
      <timestamp>2023-03-01T03:47:25Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <comment>Changing H = 1/1 to J = 1/1 as it seems to be more common</comment>
      <origin>103867</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2037" sha1="pj70706tg0rg9wfxvemolpz84az4lwp" xml:space="preserve">{{Infobox ET}}
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

== Intervals ==

{| class="wikitable center-all"
! #
! Cents
! Hekts
! Approximate ratios
! [[Arcturus]] nonatonic notation (J = 1/1)
|-
| colspan = "3" | 0
| [[1/1]]
| J
|-
| 1
| 172.9
| 118.1
| [[11/10]], [[10/9]]
| J#, Kb
|-
| 2
| 345.8
| 236.2
| [[11/9]]
| K
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[27/20]]
| L
|-
| 4
| 691.6
| 472.4
| [[3/2]], [[40/27]]
| M
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[28/17]], [[105/64]]
| N
|-
| 6
| 1037.4
| 708.6
| [[29/16]], [[20/11]], [[64/35]]
| N#, Ob
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| O
|-
| 8
| 1383.2
| 944.8
|
| P
|-
| 9
| 1556.1
| 1062.9
| 
| Q
|-
| 10
| 1729
| 1181
|
| R
|-
| 11
| 1902
| 1300
|
| J
|}

== Scala file ==
Tuning in scala format is as follows:

&lt;pre&gt;
! E:\cakewalk\scales\11_of_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1
&lt;/pre&gt;

== Pieces ==

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>pj70706tg0rg9wfxvemolpz84az4lwp</sha1>
    </revision>
    <revision>
      <id>132035</id>
      <parentid>103867</parentid>
      <timestamp>2024-01-17T09:30:03Z</timestamp>
      <contributor>
        <username>Akselai</username>
        <id>437</id>
      </contributor>
      <comment>add harmonics in equal</comment>
      <origin>132035</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2090" sha1="ghxu42pbqyn485pilm13hs37saqxi4x" xml:space="preserve">{{Infobox ET}}
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

== Intervals ==

{| class="wikitable center-all"
! #
! Cents
! Hekts
! Approximate ratios
! [[Arcturus]] nonatonic notation (J = 1/1)
|-
| colspan = "3" | 0
| [[1/1]]
| J
|-
| 1
| 172.9
| 118.1
| [[11/10]], [[10/9]]
| J#, Kb
|-
| 2
| 345.8
| 236.2
| [[11/9]]
| K
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[27/20]]
| L
|-
| 4
| 691.6
| 472.4
| [[3/2]], [[40/27]]
| M
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[28/17]], [[105/64]]
| N
|-
| 6
| 1037.4
| 708.6
| [[29/16]], [[20/11]], [[64/35]]
| N#, Ob
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| O
|-
| 8
| 1383.2
| 944.8
|
| P
|-
| 9
| 1556.1
| 1062.9
| 
| Q
|-
| 10
| 1729
| 1181
|
| R
|-
| 11
| 1902
| 1300
|
| J
|}

== Prime harmonics ==
{{Harmonics in equal|11|3|1}}

== Scala file ==
Tuning in scala format is as follows:

&lt;pre&gt;
! E:\cakewalk\scales\11_of_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1
&lt;/pre&gt;

== Pieces ==

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>ghxu42pbqyn485pilm13hs37saqxi4x</sha1>
    </revision>
    <revision>
      <id>132039</id>
      <parentid>132035</parentid>
      <timestamp>2024-01-17T09:39:22Z</timestamp>
      <contributor>
        <username>Akselai</username>
        <id>437</id>
      </contributor>
      <comment>/* Prime harmonics */</comment>
      <origin>132039</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2106" sha1="1zqkatjrggyyq6x22bjdv0dtbpvilu4" xml:space="preserve">{{Infobox ET}}
11edt means the division of 3, the tritave, into 11 equal parts of 175.905 cents each, corresponding to 6.940 edo. It can therefore be seen as a very stretched version of [[7edo|7edo]], with octaves sharpened by ten and a third cents. The octave stretching makes the fifth in better tune, and of course the twelfth is the pure 3/1 tritave.

From a no-two point of view, it tempers out 49/45 and 15625/15309 in the 7-limit and 35/33 and 77/75 in the 11-limit.

== Intervals ==

{| class="wikitable center-all"
! #
! Cents
! Hekts
! Approximate ratios
! [[Arcturus]] nonatonic notation (J = 1/1)
|-
| colspan = "3" | 0
| [[1/1]]
| J
|-
| 1
| 172.9
| 118.1
| [[11/10]], [[10/9]]
| J#, Kb
|-
| 2
| 345.8
| 236.2
| [[11/9]]
| K
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[27/20]]
| L
|-
| 4
| 691.6
| 472.4
| [[3/2]], [[40/27]]
| M
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[28/17]], [[105/64]]
| N
|-
| 6
| 1037.4
| 708.6
| [[29/16]], [[20/11]], [[64/35]]
| N#, Ob
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| O
|-
| 8
| 1383.2
| 944.8
|
| P
|-
| 9
| 1556.1
| 1062.9
| 
| Q
|-
| 10
| 1729
| 1181
|
| R
|-
| 11
| 1902
| 1300
|
| J
|}

== Prime harmonics ==
{{Harmonics in equal|11|3|1|intervals=prime}}

== Scala file ==
Tuning in scala format is as follows:

&lt;pre&gt;
! E:\cakewalk\scales\11_of_tritave.scl

!

11 in tritave

!

172.90500

345.81000

518.71500

691.62000

864.52500

1037.43000

1210.33500

1383.24000

1556.14500

1729.05000

3/1
&lt;/pre&gt;

== Pieces ==

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>1zqkatjrggyyq6x22bjdv0dtbpvilu4</sha1>
    </revision>
    <revision>
      <id>143722</id>
      <parentid>132039</parentid>
      <timestamp>2024-05-20T10:32:03Z</timestamp>
      <contributor>
        <username>Akselai</username>
        <id>437</id>
      </contributor>
      <comment>clean up</comment>
      <origin>143722</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2430" sha1="cnolw0886a2nyhubb1m6rku98633izd" xml:space="preserve">{{Infobox ET}}
{{EDO intro}}

== Theory ==

From a no-two point of view, 11edt has a [[5/3]] major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic [[7/1]], which is off by almost half a step (or about a semitone), which causes it to temper out [[49/45]] in the 7-limit. 11edt is at the extreme end of [[arcturus]] temperament, defined by tempering out [[15625/15309]] in the 3.5.7 subgroup. It gives an equalized interpretation for the [[9L 2s (3/1-equivalent)|sub-arcturus]] [[MOS]] scale.

The 11th harmonic, [[11/1]], only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out [[35/33]] and [[77/75]] in the 11-limit. In the 3.5.11 subgroup, it tempers out [[125/121]].

=== Relation to edos ===
11edt can be seen as a very stretched version of [[7edo]], with octaves sharpened by 10.3 cents. The octave stretching makes the [[3/2]] perfect fifth in better tune, while preserving a just [[3/1]] tritave. 
=== Prime harmonics ===
{{Harmonics in equal|11|3|1}}
{{Harmonics in equal|11|3|1|intervals=prime}}

=== Interval table ===

{| class="wikitable center-all"
! #
! [[Cent]]s
! [[Hekt]]s
! Approximate ratios
! [[Arcturus]] nonatonic notation (J = 1/1)
|-
| colspan = "3" | 0
| [[1/1]]
| J
|-
| 1
| 172.9
| 118.1
| [[11/10]], [[10/9]]
| J#, Kb
|-
| 2
| 345.8
| 236.2
| [[11/9]]
| K
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[27/20]]
| L
|-
| 4
| 691.6
| 472.4
| [[3/2]], [[40/27]]
| M
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[28/17]], [[105/64]]
| N
|-
| 6
| 1037.4
| 708.6
| [[29/16]], [[20/11]], [[64/35]]
| N#, Ob
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| O
|-
| 8
| 1383.2
| 944.8
|
| P
|-
| 9
| 1556.1
| 1062.9
| 
| Q
|-
| 10
| 1729
| 1181
|
| R
|-
| 11
| 1902
| 1300
|
| J
|}

== Pieces ==

Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>cnolw0886a2nyhubb1m6rku98633izd</sha1>
    </revision>
    <revision>
      <id>181326</id>
      <parentid>143722</parentid>
      <timestamp>2025-02-18T16:55:02Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181326</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2429" sha1="ollzl5eh8514ztq315r68tctss0r9qb" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
From a no-two point of view, 11edt has a [[5/3]] major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic [[7/1]], which is off by almost half a step (or about a semitone), which causes it to temper out [[49/45]] in the 7-limit. 11edt is at the extreme end of [[arcturus]] temperament, defined by tempering out [[15625/15309]] in the 3.5.7 subgroup. It gives an equalized interpretation for the [[9L 2s (3/1-equivalent)|sub-arcturus]] [[MOS]] scale.

The 11th harmonic, [[11/1]], only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out [[35/33]] and [[77/75]] in the 11-limit. In the 3.5.11 subgroup, it tempers out [[125/121]].

=== Relation to edos ===
11edt can be seen as a very stretched version of [[7edo]], with octaves sharpened by 10.3 cents. The octave stretching makes the [[3/2]] perfect fifth in better tune, while preserving a just [[3/1]] tritave.

=== Prime harmonics ===
{{Harmonics in equal|11|3|1}}
{{Harmonics in equal|11|3|1|intervals=prime}}

=== Interval table ===
{| class="wikitable center-all"
|-
! #
! [[Cent]]s
! [[Hekt]]s
! Approximate ratios
! [[Arcturus]] nonatonic notation (J = 1/1)
|-
| colspan = "3" | 0
| [[1/1]]
| J
|-
| 1
| 172.9
| 118.1
| [[11/10]], [[10/9]]
| J#, Kb
|-
| 2
| 345.8
| 236.2
| [[11/9]]
| K
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[27/20]]
| L
|-
| 4
| 691.6
| 472.4
| [[3/2]], [[40/27]]
| M
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[28/17]], [[105/64]]
| N
|-
| 6
| 1037.4
| 708.6
| [[29/16]], [[20/11]], [[64/35]]
| N#, Ob
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| O
|-
| 8
| 1383.2
| 944.8
|
| P
|-
| 9
| 1556.1
| 1062.9
| 
| Q
|-
| 10
| 1729
| 1181
|
| R
|-
| 11
| 1902
| 1300
|
| J
|}

== Pieces ==
Mozart's [http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 sonata #11 in A Major K331 in 11 EDT] (using a 11 =&amp;gt; 12 key mapping so octaves become tritaves)

[http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 Frozen Time Occupies Wall Street] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt;[http://chrisvaisvil.com/?p=1392  information about the piece]

[http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 Molly's Playground] by [http://www.chrisvaisvil.com Chris Vaisvil] =&amp;gt; [http://chrisvaisvil.com/?p=1716  information about the piece]
[[category:macrotonal]]</text>
      <sha1>ollzl5eh8514ztq315r68tctss0r9qb</sha1>
    </revision>
    <revision>
      <id>197760</id>
      <parentid>181326</parentid>
      <timestamp>2025-05-26T12:29:47Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>197760</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2707" sha1="i5r4zcact97sb20twredht6a6hif2g0" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
11edt can be seen as a very [[stretched and compressed tuning|stretched]] version of [[7edo]], with octaves sharpened by 10.3 cents. The octave stretching makes the [[3/2]] perfect fifth in better tune, while preserving a just [[3/1]] tritave.

From a no-2 point of view, 11edt has a [[5/3]] major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic [[7/1]], which is off by almost half a step (or about a semitone), which causes it to temper out [[49/45]] in the 7-limit. 11edt is at the extreme end of [[arcturus]] temperament, defined by tempering out [[15625/15309]] in the 3.5.7 subgroup. It gives an equalized interpretation for the [[9L 2s (3/1-equivalent)|sub-arcturus]] [[mos scale]].

The 11th harmonic, [[11/1]], only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out [[35/33]] and [[77/75]] in the 11-limit. In the 3.5.11 subgroup, it tempers out [[125/121]].

=== Harmonics ===
{{Harmonics in equal|11|3|1|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edt (continued)}}

== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
! #
! [[Cent]]s
! [[Hekt]]s
! Approximate ratios
! [[Arcturus]] nonatonic notation (J = 1/1)
|-
| 0
| 0.0
| 0.0
| [[1/1]]
| J
|-
| 1
| 172.9
| 118.1
| [[11/10]], [[10/9]]
| J#, Kb
|-
| 2
| 345.8
| 236.2
| [[11/9]]
| K
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[27/20]]
| L
|-
| 4
| 691.6
| 472.4
| [[3/2]], [[40/27]]
| M
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[28/17]], [[105/64]]
| N
|-
| 6
| 1037.4
| 708.6
| [[29/16]], [[20/11]], [[64/35]]
| N#, Ob
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| O
|-
| 8
| 1383.2
| 944.8
|
| P
|-
| 9
| 1556.1
| 1062.9
| 
| Q
|-
| 10
| 1729.0
| 1181.0
|
| R
|-
| 11
| 1902.0
| 1300.0
|
| J
|}

== Music ==
=== Modern renderings ===
; {{W|Wolfgang Amadeus Mozart}}
* [https://web.archive.org/web/20201127012444/http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 ''Piano Sonata No. 11'' in A major, K. 331] – using a 11 → 12 key mapping so octaves become tritaves

=== 21st century ===
; [[Chris Vaisvil]]
* ''Frozen Time Occupies Wall Street'' (2011) – [https://www.chrisvaisvil.com/frozen-time-occupies-wall-street/ blog] | [https://web.archive.org/web/20220911143825/http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 play] 
* ''Molly's Playground'' (2011) – [https://www.chrisvaisvil.com/mollys-playground/ blog] | [https://web.archive.org/web/20201127013949/http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 play]</text>
      <sha1>i5r4zcact97sb20twredht6a6hif2g0</sha1>
    </revision>
    <revision>
      <id>197761</id>
      <parentid>197760</parentid>
      <timestamp>2025-05-26T12:45:05Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Intervals */ +/-</comment>
      <origin>197761</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2854" sha1="ffzry0m80rfrcbteh8rigog55lkmznl" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
11edt can be seen as a very [[stretched and compressed tuning|stretched]] version of [[7edo]], with octaves sharpened by 10.3 cents. The octave stretching makes the [[3/2]] perfect fifth in better tune, while preserving a just [[3/1]] tritave.

From a no-2 point of view, 11edt has a [[5/3]] major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic [[7/1]], which is off by almost half a step (or about a semitone), which causes it to temper out [[49/45]] in the 7-limit. 11edt is at the extreme end of [[arcturus]] temperament, defined by tempering out [[15625/15309]] in the 3.5.7 subgroup. It gives an equalized interpretation for the [[9L 2s (3/1-equivalent)|sub-arcturus]] [[mos scale]].

The 11th harmonic, [[11/1]], only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out [[35/33]] and [[77/75]] in the 11-limit. In the 3.5.11 subgroup, it tempers out [[125/121]].

=== Harmonics ===
{{Harmonics in equal|11|3|1|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edt (continued)}}

== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
! #
! [[Cent]]s
! [[Hekt]]s
! Approximate ratios*
! [[Arcturus]]&lt;br&gt;enneatonic notation (J = 1/1)
|-
| 0
| 0.0
| 0.0
| [[1/1]]
| J
|-
| 1
| 172.9
| 118.1
| [[10/9]], [[11/10]]
| J#, Kb
|-
| 2
| 345.8
| 236.2
| [[5/4]], [[6/5]], [[11/9]], [[27/22]]
| K
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[15/11]]
| L
|-
| 4
| 691.6
| 472.4
| [[3/2]]
| M
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[18/11]], [[33/20]]
| N
|-
| 6
| 1037.4
| 708.6
| [[9/5]], [[11/6]], [[20/11]]
| N#, Ob
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| O
|-
| 8
| 1383.2
| 944.8
| [[9/4]], [[11/5]]
| P
|-
| 9
| 1556.1
| 1062.9
| [[5/2]], [[12/5]], [[22/9]], [[27/11]]
| Q
|-
| 10
| 1729.0
| 1181.0
| [[8/3]], [[11/4]]
| R
|-
| 11
| 1902.0
| 1300.0
| [[3/1]]
| J
|}
&lt;nowiki/&gt;* As a 2.3.5.11-subgroup temperament

== Music ==
=== Modern renderings ===
; {{W|Wolfgang Amadeus Mozart}}
* [https://web.archive.org/web/20201127012444/http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 ''Piano Sonata No. 11'' in A major, K. 331] – using a 11 → 12 key mapping so octaves become tritaves

=== 21st century ===
; [[Chris Vaisvil]]
* ''Frozen Time Occupies Wall Street'' (2011) – [https://www.chrisvaisvil.com/frozen-time-occupies-wall-street/ blog] | [https://web.archive.org/web/20220911143825/http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 play] 
* ''Molly's Playground'' (2011) – [https://www.chrisvaisvil.com/mollys-playground/ blog] | [https://web.archive.org/web/20201127013949/http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 play]</text>
      <sha1>ffzry0m80rfrcbteh8rigog55lkmznl</sha1>
    </revision>
    <revision>
      <id>197762</id>
      <parentid>197761</parentid>
      <timestamp>2025-05-26T12:46:24Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ +subsets and supersets</comment>
      <origin>197762</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3032" sha1="5e691r91ggnpu5f5p1npn5wbcsm70e6" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
11edt can be seen as a very [[stretched and compressed tuning|stretched]] version of [[7edo]], with octaves sharpened by 10.3 cents. The octave stretching makes the [[3/2]] perfect fifth in better tune, while preserving a just [[3/1]] tritave.

From a no-2 point of view, 11edt has a [[5/3]] major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic [[7/1]], which is off by almost half a step (or about a semitone), which causes it to temper out [[49/45]] in the 7-limit. 11edt is at the extreme end of [[arcturus]] temperament, defined by tempering out [[15625/15309]] in the 3.5.7 subgroup. It gives an equalized interpretation for the [[9L 2s (3/1-equivalent)|sub-arcturus]] [[mos scale]].

The 11th harmonic, [[11/1]], only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out [[35/33]] and [[77/75]] in the 11-limit. In the 3.5.11 subgroup, it tempers out [[125/121]].

=== Harmonics ===
{{Harmonics in equal|11|3|1|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edt (continued)}}

=== Subsets and supersets ===
11edt is the fifth [[prime equal division|prime edt]], following [[7edt]] and before [[13edt]], so it does not contain any nontrivial subset edts.

== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
! #
! [[Cent]]s
! [[Hekt]]s
! Approximate ratios*
! [[Arcturus]]&lt;br&gt;enneatonic notation (J = 1/1)
|-
| 0
| 0.0
| 0.0
| [[1/1]]
| J
|-
| 1
| 172.9
| 118.1
| [[10/9]], [[11/10]]
| J#, Kb
|-
| 2
| 345.8
| 236.2
| [[5/4]], [[6/5]], [[11/9]], [[27/22]]
| K
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[15/11]]
| L
|-
| 4
| 691.6
| 472.4
| [[3/2]]
| M
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[18/11]], [[33/20]]
| N
|-
| 6
| 1037.4
| 708.6
| [[9/5]], [[11/6]], [[20/11]]
| N#, Ob
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| O
|-
| 8
| 1383.2
| 944.8
| [[9/4]], [[11/5]]
| P
|-
| 9
| 1556.1
| 1062.9
| [[5/2]], [[12/5]], [[22/9]], [[27/11]]
| Q
|-
| 10
| 1729.0
| 1181.0
| [[8/3]], [[11/4]]
| R
|-
| 11
| 1902.0
| 1300.0
| [[3/1]]
| J
|}
&lt;nowiki/&gt;* As a 2.3.5.11-subgroup temperament

== Music ==
=== Modern renderings ===
; {{W|Wolfgang Amadeus Mozart}}
* [https://web.archive.org/web/20201127012444/http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 ''Piano Sonata No. 11'' in A major, K. 331] – using a 11 → 12 key mapping so octaves become tritaves

=== 21st century ===
; [[Chris Vaisvil]]
* ''Frozen Time Occupies Wall Street'' (2011) – [https://www.chrisvaisvil.com/frozen-time-occupies-wall-street/ blog] | [https://web.archive.org/web/20220911143825/http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 play] 
* ''Molly's Playground'' (2011) – [https://www.chrisvaisvil.com/mollys-playground/ blog] | [https://web.archive.org/web/20201127013949/http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 play]</text>
      <sha1>5e691r91ggnpu5f5p1npn5wbcsm70e6</sha1>
    </revision>
    <revision>
      <id>197763</id>
      <parentid>197762</parentid>
      <timestamp>2025-05-26T12:48:24Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+see also</comment>
      <origin>197763</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3105" sha1="mb9a6eahw9xeq13eerfslzgiutirokz" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
11edt can be seen as a very [[stretched and compressed tuning|stretched]] version of [[7edo]], with octaves sharpened by 10.3 cents. The octave stretching makes the [[3/2]] perfect fifth in better tune, while preserving a just [[3/1]] tritave.

From a no-2 point of view, 11edt has a [[5/3]] major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic [[7/1]], which is off by almost half a step (or about a semitone), which causes it to temper out [[49/45]] in the 7-limit. 11edt is at the extreme end of [[arcturus]] temperament, defined by tempering out [[15625/15309]] in the 3.5.7 subgroup. It gives an equalized interpretation for the [[9L 2s (3/1-equivalent)|sub-arcturus]] [[mos scale]].

The 11th harmonic, [[11/1]], only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out [[35/33]] and [[77/75]] in the 11-limit. In the 3.5.11 subgroup, it tempers out [[125/121]].

=== Harmonics ===
{{Harmonics in equal|11|3|1|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edt (continued)}}

=== Subsets and supersets ===
11edt is the fifth [[prime equal division|prime edt]], following [[7edt]] and before [[13edt]], so it does not contain any nontrivial subset edts.

== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
! #
! [[Cent]]s
! [[Hekt]]s
! Approximate ratios*
! [[Arcturus]]&lt;br&gt;enneatonic notation (J = 1/1)
|-
| 0
| 0.0
| 0.0
| [[1/1]]
| J
|-
| 1
| 172.9
| 118.1
| [[10/9]], [[11/10]]
| J#, Kb
|-
| 2
| 345.8
| 236.2
| [[5/4]], [[6/5]], [[11/9]], [[27/22]]
| K
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[15/11]]
| L
|-
| 4
| 691.6
| 472.4
| [[3/2]]
| M
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[18/11]], [[33/20]]
| N
|-
| 6
| 1037.4
| 708.6
| [[9/5]], [[11/6]], [[20/11]]
| N#, Ob
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| O
|-
| 8
| 1383.2
| 944.8
| [[9/4]], [[11/5]]
| P
|-
| 9
| 1556.1
| 1062.9
| [[5/2]], [[12/5]], [[22/9]], [[27/11]]
| Q
|-
| 10
| 1729.0
| 1181.0
| [[8/3]], [[11/4]]
| R
|-
| 11
| 1902.0
| 1300.0
| [[3/1]]
| J
|}
&lt;nowiki/&gt;* As a 2.3.5.11-subgroup temperament

== Music ==
=== Modern renderings ===
; {{W|Wolfgang Amadeus Mozart}}
* [https://web.archive.org/web/20201127012444/http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 ''Piano Sonata No. 11'' in A major, K. 331] – using a 11 → 12 key mapping so octaves become tritaves

=== 21st century ===
; [[Chris Vaisvil]]
* ''Frozen Time Occupies Wall Street'' (2011) – [https://www.chrisvaisvil.com/frozen-time-occupies-wall-street/ blog] | [https://web.archive.org/web/20220911143825/http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 play] 
* ''Molly's Playground'' (2011) – [https://www.chrisvaisvil.com/mollys-playground/ blog] | [https://web.archive.org/web/20201127013949/http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 play]

== See also ==
* [[7edo]] – relative edo
* [[18ed6]] – relative ed6</text>
      <sha1>mb9a6eahw9xeq13eerfslzgiutirokz</sha1>
    </revision>
    <revision>
      <id>229655</id>
      <parentid>197763</parentid>
      <timestamp>2026-05-05T00:10:17Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <comment>/* 21st century */ Add ''Microtonal Ambient Music - 11EDT (stretched 7EDO)'' (2020) by The Rasa</comment>
      <origin>229655</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3230" sha1="mihj8uen6lhz4qra6jzh2xfeeyffka9" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
11edt can be seen as a very [[stretched and compressed tuning|stretched]] version of [[7edo]], with octaves sharpened by 10.3 cents. The octave stretching makes the [[3/2]] perfect fifth in better tune, while preserving a just [[3/1]] tritave.

From a no-2 point of view, 11edt has a [[5/3]] major sixth that is 19.8 cents flat. However, 11edt has an extremely inaccurate seventh harmonic [[7/1]], which is off by almost half a step (or about a semitone), which causes it to temper out [[49/45]] in the 7-limit. 11edt is at the extreme end of [[arcturus]] temperament, defined by tempering out [[15625/15309]] in the 3.5.7 subgroup. It gives an equalized interpretation for the [[9L 2s (3/1-equivalent)|sub-arcturus]] [[mos scale]].

The 11th harmonic, [[11/1]], only 1.6 cents flat, is very close to just. By exploiting the badly tuned seventh harmonic, 11edt tempers out [[35/33]] and [[77/75]] in the 11-limit. In the 3.5.11 subgroup, it tempers out [[125/121]].

=== Harmonics ===
{{Harmonics in equal|11|3|1|intervals=integer|columns=11}}
{{Harmonics in equal|11|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 11edt (continued)}}

=== Subsets and supersets ===
11edt is the fifth [[prime equal division|prime edt]], following [[7edt]] and before [[13edt]], so it does not contain any nontrivial subset edts.

== Intervals ==
{| class="wikitable center-1 right-2 right-3"
|-
! #
! [[Cent]]s
! [[Hekt]]s
! Approximate ratios*
! [[Arcturus]]&lt;br&gt;enneatonic notation (J = 1/1)
|-
| 0
| 0.0
| 0.0
| [[1/1]]
| J
|-
| 1
| 172.9
| 118.1
| [[10/9]], [[11/10]]
| J#, Kb
|-
| 2
| 345.8
| 236.2
| [[5/4]], [[6/5]], [[11/9]], [[27/22]]
| K
|-
| 3
| 518.7
| 354.3
| [[4/3]], [[15/11]]
| L
|-
| 4
| 691.6
| 472.4
| [[3/2]]
| M
|-
| 5
| 864.5
| 590.5
| [[5/3]], [[18/11]], [[33/20]]
| N
|-
| 6
| 1037.4
| 708.6
| [[9/5]], [[11/6]], [[20/11]]
| N#, Ob
|-
| 7
| 1210.3
| 826.7
| [[2/1]]
| O
|-
| 8
| 1383.2
| 944.8
| [[9/4]], [[11/5]]
| P
|-
| 9
| 1556.1
| 1062.9
| [[5/2]], [[12/5]], [[22/9]], [[27/11]]
| Q
|-
| 10
| 1729.0
| 1181.0
| [[8/3]], [[11/4]]
| R
|-
| 11
| 1902.0
| 1300.0
| [[3/1]]
| J
|}
&lt;nowiki/&gt;* As a 2.3.5.11-subgroup temperament

== Music ==
=== Modern renderings ===
; {{W|Wolfgang Amadeus Mozart}}
* [https://web.archive.org/web/20201127012444/http://micro.soonlabel.com/6th-comma-meantone/K331-period/k331-walter-piano-11edt.mp3 ''Piano Sonata No. 11'' in A major, K. 331] – using a 11 → 12 key mapping so octaves become tritaves

=== 21st century ===
; [[The Rasa]]
* [https://www.youtube.com/watch?v=cY1tgPbf5Ig ''Microtonal Ambient Music - 11EDT (stretched 7EDO)''] (2020)

; [[Chris Vaisvil]]
* ''Frozen Time Occupies Wall Street'' (2011) – [https://www.chrisvaisvil.com/frozen-time-occupies-wall-street/ blog] | [https://web.archive.org/web/20220911143825/http://micro.soonlabel.com/tritave_in_11/11of_tritave_improv.mp3 play] 
* ''Molly's Playground'' (2011) – [https://www.chrisvaisvil.com/mollys-playground/ blog] | [https://web.archive.org/web/20201127013949/http://micro.soonlabel.com/11edt/daily20111118-3-11of-edt-mollys-playground.mp3 play]

== See also ==
* [[7edo]] – relative edo
* [[18ed6]] – relative ed6</text>
      <sha1>mihj8uen6lhz4qra6jzh2xfeeyffka9</sha1>
    </revision>
  </page>
  <page>
    <title>11L 1s</title>
    <ns>0</ns>
    <id>93</id>
    <revision>
      <id>93</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>93</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3201" sha1="kkkmvaa1oa3wfq0kt4j2wjt8nognemo" xml:space="preserve">&lt;span style="background-color: #ffffff; font-size: 12.8000001907349px;"&gt;This MOS, with its diatonic minor second generator, is the dodecatonic scale of Ripple (9:8=+2 generators) and Passion (6:5=+3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.&lt;/span&gt;

{| class="wikitable"
|-
! colspan="2" | Generator
! | Cents
! | 2g
! | 3g
|-
| | 1/12
| | 
| | 100
| | 200
| | 300
|-
| | 5/59
| | 
| | 101.695
| | 203.39
| | 305.085
|-
| | 
| | 9/106
| | 101.887
| | 203.774
| | 305.66
|-
| | 
| | 13/153
| | 101.961
| | 203.922
| | 305.882
|-
| | 
| | 17/200
| | 102
| | 204
| | 306
|-
| | 4/47
| | 
| | 102.128
| | 204.255
| | 306.383
|-
| | 
| | 7/82
| | 102.439
| | 204.878
| | 307.317
|-
| | 
| | 10/117
| | 102.564
| | 205.128
| | 307.692
|-
| | 
| | 13/152
| | 102.632
| | 205.263
| | 307.895
|-
| | 
| | 16/187
| | 102.674
| | 205.348
| | 308.021
|-
| | 
| | 19/222
| | 102.838
| | 205.676
| | 308.5135
|-
| | 3/35
| | 
| | 102.857
| | 205.714
| | 308.571
|-
| | 
| | 20/233
| | 103.004
| | 206.009
| | 309.013
|-
| | 
| | 17/198
| | 103.03
| | 206.061
| | 309.091
|-
| | 
| | 14/163
| | 103.0675
| | 206.135
| | 309.2025
|-
| | 
| | 
| | 103.123
| | 206.2455
| | 309.368
|-
| | 
| | 11/128
| | 103.125
| | 206.25
| | 309.375
|-
| | 
| | 8/93
| | 103.226
| | 206.452
| | 309.677
|-
| | 
| | 
| | 103.288
| | 206.575
| | 309.863
|-
| | 
| | 5/58
| | 103.448
| | 206.897
| | 310.345
|-
| | 
| | 
| | 103.651
| | 207.301
| | 310.952
|-
| | 
| | 7/81
| | 103.703
| | 207.407
| | 311.111
|-
| | 
| | 9/104
| | 103.846
| | 207.692
| | 311.5385
|-
| | 
| | 11/127
| | 103.937
| | 207.874
| | 311.811
|-
| | 
| | 13/150
| | 104
| | 208
| | 312
|-
| | 2/23
| | 
| | 104.348
| | 208.969
| | 313.0435
|-
| | 
| | 11/126
| | 104.762
| | 209.524
| | 314.286
|-
| | 
| | 9/103
| | 104.854
| | 209.709
| | 314.563
|-
| | 
| | 7/80
| | 105
| | 210
| | 315
|-
| | 
| | 5/57
| | 105.263
| | 210.526
| | 315.7895
|-
| | 
| | 
| | 105.43
| | 210.86
| | 316.29
|-
| | 
| | 8/91
| | 105.4945
| | 210.989
| | 316.4835
|-
| | 
| | 
| | 105.561
| | 211.121
| | 316.682
|-
| | 
| | 11/125
| | 105.6
| | 211.2
| | 316.8
|-
| | 3/34
| | 
| | 105.822
| | 211.765
| | 317.647
|-
| | 
| | 28/317
| | 105.994
| | 211.987
| | 317.981
|-
| | 
| | 25/283
| | 106.007
| | 212.014
| | 318.021
|-
| | 
| | 
| | 106.023
| | 212.046
| | 318.069
|-
| | 
| | 22/249
| | 106.024
| | 212.048
| | 318.072
|-
| | 
| | 19/215
| | 106.0465
| | 212.093
| | 318.1395
|-
| | 
| | 16/181
| | 106.077
| | 212.155
| | 318.232
|-
| | 
| | 13/147
| | 106.122
| | 212.245
| | 318.367
|-
| | 
| | 10/113
| | 106.195
| | 212.389
| | 318.581
|-
| | 
| | 7/79
| | 106.329
| | 212.658
| | 318.987
|-
| | 
| | 11/124
| | 106.452
| | 212.903
| | 319.355
|-
| | 4/45
| | 
| | 106.667
| | 213.333
| | 320
|-
| | 5/56
| | 
| | 107.143
| | 214.286
| | 321.429
|-
| | 1/11
| | 
| | 109.091
| | 218.182
| | 327.273
|}</text>
      <sha1>kkkmvaa1oa3wfq0kt4j2wjt8nognemo</sha1>
    </revision>
    <revision>
      <id>4449</id>
      <timestamp>2015-11-11T14:18:27Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 566074907 - Original comment: **</comment>
      <origin>4449</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18111" sha1="ogt5b6oulzwyx73jbti9378173wgwaw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-11 14:18:27 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566074907&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="background-color: #ffffff; font-size: 12.8000001907349px;"&amp;gt;This MOS, with its diatonic minor second generator, is the dodecatonic scale of Ripple (9:8=+2 generators) and Passion (6:5=+3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.&amp;lt;/span&amp;gt;
||||~ Generator ||~ Cents ||~ 2g ||~ 3g ||
|| 1/12 ||   || 100 || 200 || 300 ||
|| 5/59 ||   || 101.695 || 203.39 || 305.085 ||
||   || 9/106 || 101.887 || 203.774 || 305.66 ||
||   || 13/153 || 101.961 || 203.922 || 305.882 ||
||   || 17/200 || 102 || 204 || 306 ||
|| 4/47 ||   || 102.128 || 204.255 || 306.383 ||
||   || 7/82 || 102.439 || 204.878 || 307.317 ||
||   || 10/117 || 102.564 || 205.128 || 307.692 ||
||   || 13/152 || 102.632 || 205.263 || 307.895 ||
||   || 16/187 || 102.674 || 205.348 || 308.021 ||
||   || 19/222 || 102.838 || 205.676 || 308.5135 ||
|| 3/35 ||   || 102.857 || 205.714 || 308.571 ||
||   || 20/233 || 103.004 || 206.009 || 309.013 ||
||   || 17/198 || 103.03 || 206.061 || 309.091 ||
||   || 14/163 || 103.0675 || 206.135 || 309.2025 ||
||   ||   || 103.123 || 206.2455 || 309.368 ||
||   || 11/128 || 103.125 || 206.25 || 309.375 ||
||   || 8/93 || 103.226 || 206.452 || 309.677 ||
||   ||   || 103.288 || 206.575 || 309.863 ||
||   || 5/58 || 103.448 || 206.897 || 310.345 ||
||   ||   || 103.651 || 207.301 || 310.952 ||
||   || 7/81 || 103.703 || 207.407 || 311.111 ||
||   || 9/104 || 103.846 || 207.692 || 311.5385 ||
||   || 11/127 || 103.937 || 207.874 || 311.811 ||
||   || 13/150 || 104 || 208 || 312 ||
|| 2/23 ||   || 104.348 || 208.969 || 313.0435 ||
||   || 11/126 || 104.762 || 209.524 || 314.286 ||
||   || 9/103 || 104.854 || 209.709 || 314.563 ||
||   || 7/80 || 105 || 210 || 315 ||
||   || 5/57 || 105.263 || 210.526 || 315.7895 ||
||   ||   || 105.43 || 210.86 || 316.29 ||
||   || 8/91 || 105.4945 || 210.989 || 316.4835 ||
||   ||   || 105.561 || 211.121 || 316.682 ||
||   || 11/125 || 105.6 || 211.2 || 316.8 ||
|| 3/34 ||   || 105.822 || 211.765 || 317.647 ||
||   || 28/317 || 105.994 || 211.987 || 317.981 ||
||   || 25/283 || 106.007 || 212.014 || 318.021 ||
||   ||   || 106.023 || 212.046 || 318.069 ||
||   || 22/249 || 106.024 || 212.048 || 318.072 ||
||   || 19/215 || 106.0465 || 212.093 || 318.1395 ||
||   || 16/181 || 106.077 || 212.155 || 318.232 ||
||   || 13/147 || 106.122 || 212.245 || 318.367 ||
||   || 10/113 || 106.195 || 212.389 || 318.581 ||
||   || 7/79 || 106.329 || 212.658 || 318.987 ||
||   || 11/124 || 106.452 || 212.903 || 319.355 ||
|| 4/45 ||   || 106.667 || 213.333 || 320 ||
|| 5/56 ||   || 107.143 || 214.286 || 321.429 ||
|| 1/11 ||   || 109.091 || 218.182 || 327.273 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 1s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="background-color: #ffffff; font-size: 12.8000001907349px;"&amp;gt;This MOS, with its diatonic minor second generator, is the dodecatonic scale of Ripple (9:8=+2 generators) and Passion (6:5=+3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;2g&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;3g&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;300&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.695&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;203.39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;305.085&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/106&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.887&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;203.774&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;305.66&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/153&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.961&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;203.922&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;305.882&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;306&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;204.255&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;306.383&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.439&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;204.878&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;307.317&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.564&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205.128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;307.692&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/152&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.632&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205.263&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;307.895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.674&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205.348&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308.021&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/222&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.838&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205.676&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308.5135&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.857&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.004&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206.009&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309.013&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/198&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.03&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206.061&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309.091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.0675&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206.135&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309.2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.123&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206.2455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309.368&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309.375&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.226&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206.452&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309.677&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206.575&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309.863&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206.897&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;310.345&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.651&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;207.301&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;310.952&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.703&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;207.407&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;311.111&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/104&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.846&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;207.692&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;311.5385&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.937&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;207.874&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;311.811&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/150&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;208&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;312&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.348&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;208.969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;313.0435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/126&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.762&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;209.524&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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      <sha1>ogt5b6oulzwyx73jbti9378173wgwaw</sha1>
    </revision>
    <revision>
      <id>4450</id>
      <timestamp>2015-05-23T17:50:53Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 551978048 - Original comment: **</comment>
      <origin>4450</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17507" sha1="moictq7fackj3avl3wfq54kqnaduq2s" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-23 17:50:53 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;551978048&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="background-color: #ffffff; font-size: 12.8000001907349px;"&amp;gt;This MOS, with its diatonic minor second generator, is the dodecatonic scale of Ripple (9:8=+2 generators) and Passion (6:5=+3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.&amp;lt;/span&amp;gt;
||||~ Generator ||~ Cents ||~ 2g ||~ 3g ||
|| 1/12 ||   || 100 || 200 || 300 ||
|| 5/59 ||   || 101 41/59 || 203 23/59 || 305 5/59 ||
||   || 9/106 || 101 47/53 || 203 41/53 || 305 35/53 ||
||   || 13/153 || 101 49/51 || 203 47/51 || 305 15/17 ||
||   || 17/200 || 102 || 204 || 306 ||
|| 4/47 ||   || 102 6/47 || 204 12/47 || 306 18/47 ||
||   || 7/82 || 102 18/41 || 204 36/41 || 307 13/41 ||
||   || 10/117 || 102 22/39 || 205 5/39 || 307 9/13 ||
||   || 13/152 || 102 12/19 || 205 5/19 || 307 17/19 ||
||   || 16/187 || 102 126/187 || 205 65/187 || 308 4/187 ||
||   || 19/222 || 102 31/37 || 205 25/37 || 308 19/37 ||
|| 3/35 ||   || 102 6/7 || 205 5/7 || 308 4/7 ||
||   || 20/233 || 103 1/233 || 206 2/233 || 309 3/233 ||
||   || 17/198 || 103 1/33 || 206 2/33 || 309 1/11 ||
||   || 14/163 || 103 11/163 || 206 22/163 || 309 33/163 ||
||   || 11/128 || 103.125 || 206.25 || 309.375 ||
||   || 8/93 || 103 7/31 || 206 14/31 || 309 21/31 ||
||   || 5/58 || 103 13/29 || 206 26/29 || 310 10/29 ||
||   || 7/81 || 103 19/27 || 206 11/27 || 311 1/9 ||
||   || 9/104 || 103 11/13 || 207 9/13 || 311 7/13 ||
||   || 11/127 || 103 119/127 || 207 111/127 || 311 123/127 ||
||   || 13/150 || 104 || 208 || 312 ||
|| 2/23 ||   || 104 8/23 || 208 16/23 || 313 1/23 ||
||   || 11/126 || 104 16/21 || 209 11/21 || 314 2/7 ||
||   || 9/103 || 104 88/103 || 209 73/103 || 314 58/103 ||
||   || 7/80 || 105 || 210 || 315 ||
||   || 5/57 || 105 5/19 || 210 10/19 || 315 15/19 ||
||   || 8/91 || 105 45/91 || 210 90/91 || 316 44/91 ||
||   ||   || 105.560585 || 211.121169 || 316.681754 ||
||   || 11/125 || 105.6 || 211.2 || 316.8 ||
|| 3/34 ||   || 105 15/17 || 211 13/17 || 317 11/17 ||
||   || 28/317 || 105 315/317 || 211 &amp;lt;span style="line-height: 1.5;"&amp;gt;313/317&amp;lt;/span&amp;gt; || 317 311/317 ||
||   || 25/283 || 106 2/283 || 212 4/283 || 318 6/283 ||
||   ||   || 106.022897 || 212.045793 || 318.06869 ||
||   || 22/249 || 106 2/83 || 212 4/83 || 318 6/83 ||
||   || 19/215 || 106 2/43 || 212 4/43 || 318 6/43 ||
||   || 16/181 || 106 14/181 || 212 28/181 || 318 42/181 ||
||   || 13/147 || 106 6&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;/49&amp;lt;/span&amp;gt; || 212 12/49 || 318 18/49 ||
||   || 10/113 || 106 22/113 || 212 44/113 || 318 66/113 ||
||   || 7/79 || 106 26/79 || 212 52/79 || 318 78/79 ||
||   || 11/124 || 106 14/31 || 212 28/31 || 319 11/31 ||
|| 4/45 ||   || 106 2/3 || 213 1/3 || 320 ||
|| 5/56 ||   || 107 1/7 || 214 2/7 || 321 3/7 ||
|| 1/11 ||   || 109 1/11 || 218 2/11 || 327 3/11 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 1s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="background-color: #ffffff; font-size: 12.8000001907349px;"&amp;gt;This MOS, with its diatonic minor second generator, is the dodecatonic scale of Ripple (9:8=+2 generators) and Passion (6:5=+3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;2g&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;3g&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;300&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;5/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101 41/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;203 23/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;305 35/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/153&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101 49/51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;203 47/51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;305 15/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;306&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 6/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;204 12/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;306 18/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 18/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;204 36/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;307 13/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 22/39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 5/39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;307 9/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/152&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 12/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 5/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;307 17/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 126/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 65/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308 4/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/222&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 31/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 25/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308 19/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 6/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 5/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308 4/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 1/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 2/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309 3/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/198&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 1/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 2/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309 1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 11/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 22/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309 33/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309.375&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 7/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 14/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309 21/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 13/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 26/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;310 10/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 19/27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 11/27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;311 1/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/104&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 11/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;207 9/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;311 7/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 119/127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;207 111/127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;311 123/127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/150&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;208&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;312&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104 8/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;208 16/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;313 1/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/126&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104 16/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;209 11/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;314 2/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104 88/103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;209 73/103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;314 58/103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;210&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;315&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105 5/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;210 10/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;315 15/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105 45/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;210 90/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;316 44/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.560585&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;211.121169&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;316.681754&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;211.2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;316.8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105 15/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;211 13/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;317 11/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28/317&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105 315/317&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;211 &amp;lt;span style="line-height: 1.5;"&amp;gt;313/317&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;317 311/317&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25/283&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 2/283&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 4/283&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 6/283&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106.022897&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212.045793&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318.06869&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22/249&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 2/83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 4/83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 6/83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/215&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 2/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 4/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 6/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/181&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 14/181&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 28/181&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 42/181&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/147&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 6&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;/49&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 12/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 18/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 22/113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 44/113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 66/113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 26/79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 52/79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 78/79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/124&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 14/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 28/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;319 11/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;213 1/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;320&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107 1/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;214 2/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;321 3/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109 1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218 2/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327 3/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>moictq7fackj3avl3wfq54kqnaduq2s</sha1>
    </revision>
    <revision>
      <id>4451</id>
      <timestamp>2015-02-12T11:33:38Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540763866 - Original comment: **</comment>
      <origin>4451</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17409" sha1="c5k9mbewnvjvukpoynb87rik6f42d2v" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-12 11:33:38 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540763866&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="background-color: #ffffff; font-size: 12.8000001907349px;"&amp;gt;This MOS, with its diatonic minor second generator, is the dodecatonic scale of Ripple (9:8=+2 generators) and Passion (6:5=+3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.&amp;lt;/span&amp;gt;
||||~ Generator ||~ Cents ||~ 2g ||~ 3g ||
|| 1/12 ||   || 100 || 200 || 300 ||
|| 5/59 ||   || 101 41/59 || 203 23/59 || 305 5/59 ||
||   || 9/106 || 101 47/53 || 203 41/53 || 305 35/53 ||
||   || 13/153 || 101 49/51 || 203 47/51 || 305 15/17 ||
||   || 17/200 || 102 || 204 || 306 ||
|| 4/47 ||   || 102 6/47 || 204 12/47 || 306 18/47 ||
||   || 7/82 || 102 18/41 || 204 36/41 || 307 13/41 ||
||   || 10/117 || 102 22/39 || 205 5/39 || 307 9/13 ||
||   || 13/152 || 102 12/19 || 205 5/19 || 307 17/19 ||
||   || 16/187 || 102 126/187 || 205 65/187 || 308 4/187 ||
||   || 19/222 || 102 31/37 || 205 25/37 || 308 19/37 ||
|| 3/35 ||   || 102 6/7 || 205 5/7 || 308 4/7 ||
||   || 20/233 || 103 1/233 || 206 2/233 || 309 3/233 ||
||   || 17/198 || 103 1/33 || 206 2/33 || 309 1/11 ||
||   || 14/163 || 103 11/163 || 206 22/163 || 309 33/163 ||
||   || 11/128 || 103.125 || 206.25 || 309.375 ||
||   || 8/93 || 103 7/31 || 206 14/31 || 309 21/31 ||
||   || 5/58 || 103 13/29 || 206 26/29 || 310 10/29 ||
||   || 7/81 || 103 19/27 || 206 11/27 || 311 1/9 ||
||   || 9/104 || 103 11/13 || 207 9/13 || 311 7/13 ||
||   || 11/127 || 103 119/127 || 207 111/127 || 311 123/127 ||
||   || 13/150 || 104 || 208 || 312 ||
|| 2/23 ||   || 104 8/23 || 208 16/23 || 313 1/23 ||
||   || 11/126 || 104 16/21 || 209 11/21 || 314 2/7 ||
||   || 9/103 || 104 88/103 || 209 73/103 || 314 58/103 ||
||   || 7/80 || 105 || 210 || 315 ||
||   || 5/57 || 105 5/19 || 210 10/19 || 315 15/19 ||
||   || 8/91 || 105 45/91 || 210 90/91 || 316 44/91 ||
||   || 11/125 || 105.6 || 211.2 || 316.8 ||
||   ||   || 105.853446 || 211.7068925 || 317.560339 ||
|| 3/34 ||   || 105 15/17 || 211 13/17 || 317 11/17 ||
||   ||   || 105.911009 || 211.822019 || 317.733028 ||
||   || 28/317 || 105 315/317 || 211 313/317 || 317 311/317 ||
||   || 25/283 || 106 2/283 || 212 4/283 || 318 6/283 ||
||   || 22/249 || 106 2/83 || 212 4/83 || 318 6/83 ||
||   || 19/215 || 106 2/43 || 212 4/43 || 318 6/43 ||
||   || 16/181 || 106 14/181 || 212 28/181 || 318 42/181 ||
||   || 13/147 || 106 6&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;/49&amp;lt;/span&amp;gt; || 212 12/49 || 318 18/49 ||
||   || 10/113 || 106 22/113 || 212 44/113 || 318 66/113 ||
||   || 7/79 || 106 26/79 || 212 52/79 || 318 78/79 ||
||   || 11/124 || 106 14/31 || 212 28/31 || 319 11/31 ||
|| 4/45 ||   || 106 2/3 || 213 1/3 || 320 ||
|| 5/56 ||   || 107 1/7 || 214 2/7 || 321 3/7 ||
|| 1/11 ||   || 109 1/11 || 218 2/11 || 327 3/11 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 1s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="background-color: #ffffff; font-size: 12.8000001907349px;"&amp;gt;This MOS, with its diatonic minor second generator, is the dodecatonic scale of Ripple (9:8=+2 generators) and Passion (6:5=+3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;2g&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;3g&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;300&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101 41/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;203 23/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;305 5/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/106&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101 47/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;203 41/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;305 35/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/153&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101 49/51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;203 47/51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;305 15/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;306&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 6/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;204 12/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;306 18/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 18/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;204 36/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;307 13/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 22/39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 5/39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;307 9/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/152&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 12/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 5/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;307 17/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 126/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 65/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308 4/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/222&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 31/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 25/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308 19/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 6/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 5/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308 4/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 1/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 2/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309 3/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/198&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 1/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 2/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309 1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 11/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 22/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309 33/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309.375&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 7/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 14/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309 21/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/58&amp;lt;br /&amp;gt;
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      <sha1>c5k9mbewnvjvukpoynb87rik6f42d2v</sha1>
    </revision>
    <revision>
      <id>4452</id>
      <timestamp>2014-12-01T20:01:41Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 533423956 - Original comment: **</comment>
      <origin>4452</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16745" sha1="72hxa13ozkpp6uhdtaqdj965gilkw4l" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2014-12-01 20:01:41 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;533423956&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="background-color: #ffffff; font-size: 12.8000001907349px;"&amp;gt;This MOS, with its diatonic minor second generator, is the dodecatonic scale of Ripple (9:8=+2 generators) and Passion (6:5=+3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.&amp;lt;/span&amp;gt;
||||~ Generator ||~ Cents ||~ 2g ||~ 3g ||
|| 1/12 ||   || 100 || 200 || 300 ||
|| 5/59 ||   || 101 41/59 || 203 23/59 || 305 5/59 ||
||   || 9/106 || 101 47/53 || 203 41/53 || 305 35/53 ||
||   || 13/153 || 101 49/51 || 203 47/51 || 305 15/17 ||
||   || 17/200 || 102 || 204 || 306 ||
|| 4/47 ||   || 102 6/47 || 204 12/47 || 306 18/47 ||
||   || 7/82 || 102 18/41 || 204 36/41 || 307 13/41 ||
||   || 10/117 || 102 22/39 || 205 5/39 || 307 9/13 ||
||   || 13/152 || 102 12/19 || 205 5/19 || 307 17/19 ||
||   || 16/187 || 102 126/187 || 205 65/187 || 308 4/187 ||
||   || 19/222 || 102 31/37 || 205 25/37 || 308 19/37 ||
|| 3/35 ||   || 102 6/7 || 205 5/7 || 308 4/7 ||
||   || 20/233 || 103 1/233 || 206 2/233 || 309 3/233 ||
||   || 17/198 || 103 1/33 || 206 2/33 || 309 1/11 ||
||   || 14/163 || 103 11/163 || 206 22/163 || 309 33/163 ||
||   || 11/128 || 103.125 || 206.25 || 309.375 ||
||   || 8/93 || 103 7/31 || 206 14/31 || 309 21/31 ||
||   || 5/58 || 103 13/29 || 206 26/29 || 310 10/29 ||
||   || 7/81 || 103 19/27 || 206 11/27 || 311 1/9 ||
||   || 9/104 || 103 11/13 || 207 9/13 || 311 7/13 ||
||   || 11/127 || 103 119/127 || 207 111/127 || 311 123/127 ||
||   || 13/150 || 104 || 208 || 312 ||
|| 2/23 ||   || 104 8/23 || 208 16/23 || 313 1/23 ||
||   || 11/126 || 104 16/21 || 209 11/21 || 314 2/7 ||
||   || 9/103 || 104 88/103 || 209 73/103 || 314 58/103 ||
||   || 7/80 || 105 || 210 || 315 ||
||   || 5/57 || 105 5/19 || 210 10/19 || 315 15/19 ||
||   || 8/91 || 105 45/91 || 210 90/91 || 316 44/91 ||
||   || 11/125 || 105.6 || 211.2 || 316.8 ||
|| 3/34 ||   || 105 15/17 || 211 13/17 || 317 11/17 ||
||   || 28/317 || 105 315/317 || 211 313/317 || 317 311/317 ||
||   || 25/283 || 106 2/283 || 212 4/283 || 318 6/283 ||
||   || 22/249 || 106 2/83 || 212 4/83 || 318 6/83 ||
||   || 19/215 || 106 2/43 || 212 4/43 || 318 6/43 ||
||   || 16/181 || 106 14/181 || 212 28/181 || 318 42/181 ||
||   || 13/147 || 106 6&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;/49&amp;lt;/span&amp;gt; || 212 12/49 || 318 18/49 ||
||   || 10/113 || 106 22/113 || 212 44/113 || 318 66/113 ||
||   || 7/79 || 106 26/79 || 212 52/79 || 318 78/79 ||
||   || 11/124 || 106 14/31 || 212 28/31 || 319 11/31 ||
|| 4/45 ||   || 106 2/3 || 213 1/3 || 320 ||
|| 5/56 ||   || 107 1/7 || 214 2/7 || 321 3/7 ||
|| 1/11 ||   || 109 1/11 || 218 2/11 || 327 3/11 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 1s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="background-color: #ffffff; font-size: 12.8000001907349px;"&amp;gt;This MOS, with its diatonic minor second generator, is the dodecatonic scale of Ripple (9:8=+2 generators) and Passion (6:5=+3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th colspan="2"&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Cents&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;2g&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;3g&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;300&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101 41/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;203 23/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;305 5/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/106&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101 47/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;203 41/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;305 35/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/153&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101 49/51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;203 47/51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;305 15/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/200&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;204&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;306&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 6/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;204 12/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;306 18/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 18/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;204 36/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;307 13/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/117&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 22/39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 5/39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;307 9/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/152&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 12/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 5/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;307 17/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 126/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 65/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308 4/187&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/222&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 31/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 25/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308 19/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 6/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;205 5/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;308 4/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 1/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 2/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309 3/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/198&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 1/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 2/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309 1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 11/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 22/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309 33/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309.375&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 7/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 14/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;309 21/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 13/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 26/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;310 10/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 19/27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;206 11/27&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;311 1/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/104&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 11/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;207 9/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;311 7/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 119/127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;207 111/127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;311 123/127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/150&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;208&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;312&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104 8/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;208 16/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;313 1/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/126&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104 16/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;209 11/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;314 2/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104 88/103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;209 73/103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;314 58/103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;210&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;315&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105 5/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;210 10/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;315 15/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105 45/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;210 90/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;316 44/91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;211.2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;316.8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105 15/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;211 13/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;317 11/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28/317&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105 315/317&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;211 313/317&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;317 311/317&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25/283&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 2/283&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 4/283&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 6/283&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22/249&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 2/83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 4/83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 6/83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/215&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 2/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 4/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 6/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/181&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 14/181&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 28/181&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 42/181&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/147&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 6&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;/49&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 12/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 18/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 22/113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 44/113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 66/113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 26/79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 52/79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;318 78/79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/124&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 14/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;212 28/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;319 11/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/45&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;213 1/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;320&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107 1/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;214 2/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;321 3/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109 1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;218 2/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327 3/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>72hxa13ozkpp6uhdtaqdj965gilkw4l</sha1>
    </revision>
    <revision>
      <id>63329</id>
      <parentid>93</parentid>
      <timestamp>2021-03-17T02:39:04Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>-bad formatting; +categories</comment>
      <origin>63329</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3202" sha1="bjxo0tnmlhpbxp3rz8o5o636xxe6ab0" xml:space="preserve">This MOS, with its diatonic minor second generator, is the dodecatonic scale of Ripple (9:8=+2 generators) and Passion (6:5=+3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.

{| class="wikitable"
|-
! colspan="2" | Generator
! | Cents
! | 2g
! | 3g
|-
| | 1/12
| | 
| | 100
| | 200
| | 300
|-
| | 5/59
| | 
| | 101.695
| | 203.39
| | 305.085
|-
| | 
| | 9/106
| | 101.887
| | 203.774
| | 305.66
|-
| | 
| | 13/153
| | 101.961
| | 203.922
| | 305.882
|-
| | 
| | 17/200
| | 102
| | 204
| | 306
|-
| | 4/47
| | 
| | 102.128
| | 204.255
| | 306.383
|-
| | 
| | 7/82
| | 102.439
| | 204.878
| | 307.317
|-
| | 
| | 10/117
| | 102.564
| | 205.128
| | 307.692
|-
| | 
| | 13/152
| | 102.632
| | 205.263
| | 307.895
|-
| | 
| | 16/187
| | 102.674
| | 205.348
| | 308.021
|-
| | 
| | 19/222
| | 102.838
| | 205.676
| | 308.5135
|-
| | 3/35
| | 
| | 102.857
| | 205.714
| | 308.571
|-
| | 
| | 20/233
| | 103.004
| | 206.009
| | 309.013
|-
| | 
| | 17/198
| | 103.03
| | 206.061
| | 309.091
|-
| | 
| | 14/163
| | 103.0675
| | 206.135
| | 309.2025
|-
| | 
| | 
| | 103.123
| | 206.2455
| | 309.368
|-
| | 
| | 11/128
| | 103.125
| | 206.25
| | 309.375
|-
| | 
| | 8/93
| | 103.226
| | 206.452
| | 309.677
|-
| | 
| | 
| | 103.288
| | 206.575
| | 309.863
|-
| | 
| | 5/58
| | 103.448
| | 206.897
| | 310.345
|-
| | 
| | 
| | 103.651
| | 207.301
| | 310.952
|-
| | 
| | 7/81
| | 103.703
| | 207.407
| | 311.111
|-
| | 
| | 9/104
| | 103.846
| | 207.692
| | 311.5385
|-
| | 
| | 11/127
| | 103.937
| | 207.874
| | 311.811
|-
| | 
| | 13/150
| | 104
| | 208
| | 312
|-
| | 2/23
| | 
| | 104.348
| | 208.969
| | 313.0435
|-
| | 
| | 11/126
| | 104.762
| | 209.524
| | 314.286
|-
| | 
| | 9/103
| | 104.854
| | 209.709
| | 314.563
|-
| | 
| | 7/80
| | 105
| | 210
| | 315
|-
| | 
| | 5/57
| | 105.263
| | 210.526
| | 315.7895
|-
| | 
| | 
| | 105.43
| | 210.86
| | 316.29
|-
| | 
| | 8/91
| | 105.4945
| | 210.989
| | 316.4835
|-
| | 
| | 
| | 105.561
| | 211.121
| | 316.682
|-
| | 
| | 11/125
| | 105.6
| | 211.2
| | 316.8
|-
| | 3/34
| | 
| | 105.822
| | 211.765
| | 317.647
|-
| | 
| | 28/317
| | 105.994
| | 211.987
| | 317.981
|-
| | 
| | 25/283
| | 106.007
| | 212.014
| | 318.021
|-
| | 
| | 
| | 106.023
| | 212.046
| | 318.069
|-
| | 
| | 22/249
| | 106.024
| | 212.048
| | 318.072
|-
| | 
| | 19/215
| | 106.0465
| | 212.093
| | 318.1395
|-
| | 
| | 16/181
| | 106.077
| | 212.155
| | 318.232
|-
| | 
| | 13/147
| | 106.122
| | 212.245
| | 318.367
|-
| | 
| | 10/113
| | 106.195
| | 212.389
| | 318.581
|-
| | 
| | 7/79
| | 106.329
| | 212.658
| | 318.987
|-
| | 
| | 11/124
| | 106.452
| | 212.903
| | 319.355
|-
| | 4/45
| | 
| | 106.667
| | 213.333
| | 320
|-
| | 5/56
| | 
| | 107.143
| | 214.286
| | 321.429
|-
| | 1/11
| | 
| | 109.091
| | 218.182
| | 327.273
|}

[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]</text>
      <sha1>bjxo0tnmlhpbxp3rz8o5o636xxe6ab0</sha1>
    </revision>
    <revision>
      <id>63330</id>
      <parentid>63329</parentid>
      <timestamp>2021-03-17T02:40:52Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Table formatting; +links</comment>
      <origin>63330</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2728" sha1="1p94dqmgaqxygt0mft5r3ltyf3tyh7e" xml:space="preserve">This MOS, with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators) and [[passion]] (6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.

{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
|-
| 1/12
| 
| 100
| 200
| 300
|-
| 5/59
| 
| 101.695
| 203.39
| 305.085
|-
| 
| 9/106
| 101.887
| 203.774
| 305.66
|-
| 
| 13/153
| 101.961
| 203.922
| 305.882
|-
| 
| 17/200
| 102
| 204
| 306
|-
| 4/47
| 
| 102.128
| 204.255
| 306.383
|-
| 
| 7/82
| 102.439
| 204.878
| 307.317
|-
| 
| 10/117
| 102.564
| 205.128
| 307.692
|-
| 
| 13/152
| 102.632
| 205.263
| 307.895
|-
| 
| 16/187
| 102.674
| 205.348
| 308.021
|-
| 
| 19/222
| 102.838
| 205.676
| 308.5135
|-
| 3/35
| 
| 102.857
| 205.714
| 308.571
|-
| 
| 20/233
| 103.004
| 206.009
| 309.013
|-
| 
| 17/198
| 103.03
| 206.061
| 309.091
|-
| 
| 14/163
| 103.0675
| 206.135
| 309.2025
|-
| 
| 
| 103.123
| 206.2455
| 309.368
|-
| 
| 11/128
| 103.125
| 206.25
| 309.375
|-
| 
| 8/93
| 103.226
| 206.452
| 309.677
|-
| 
| 
| 103.288
| 206.575
| 309.863
|-
| 
| 5/58
| 103.448
| 206.897
| 310.345
|-
| 
| 
| 103.651
| 207.301
| 310.952
|-
| 
| 7/81
| 103.703
| 207.407
| 311.111
|-
| 
| 9/104
| 103.846
| 207.692
| 311.5385
|-
| 
| 11/127
| 103.937
| 207.874
| 311.811
|-
| 
| 13/150
| 104
| 208
| 312
|-
| 2/23
| 
| 104.348
| 208.969
| 313.0435
|-
| 
| 11/126
| 104.762
| 209.524
| 314.286
|-
| 
| 9/103
| 104.854
| 209.709
| 314.563
|-
| 
| 7/80
| 105
| 210
| 315
|-
| 
| 5/57
| 105.263
| 210.526
| 315.7895
|-
| 
| 
| 105.43
| 210.86
| 316.29
|-
| 
| 8/91
| 105.4945
| 210.989
| 316.4835
|-
| 
| 
| 105.561
| 211.121
| 316.682
|-
| 
| 11/125
| 105.6
| 211.2
| 316.8
|-
| 3/34
| 
| 105.822
| 211.765
| 317.647
|-
| 
| 28/317
| 105.994
| 211.987
| 317.981
|-
| 
| 25/283
| 106.007
| 212.014
| 318.021
|-
| 
| 
| 106.023
| 212.046
| 318.069
|-
| 
| 22/249
| 106.024
| 212.048
| 318.072
|-
| 
| 19/215
| 106.0465
| 212.093
| 318.1395
|-
| 
| 16/181
| 106.077
| 212.155
| 318.232
|-
| 
| 13/147
| 106.122
| 212.245
| 318.367
|-
| 
| 10/113
| 106.195
| 212.389
| 318.581
|-
| 
| 7/79
| 106.329
| 212.658
| 318.987
|-
| 
| 11/124
| 106.452
| 212.903
| 319.355
|-
| 4/45
| 
| 106.667
| 213.333
| 320
|-
| 5/56
| 
| 107.143
| 214.286
| 321.429
|-
| 1/11
| 
| 109.091
| 218.182
| 327.273
|}

[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]</text>
      <sha1>1p94dqmgaqxygt0mft5r3ltyf3tyh7e</sha1>
    </revision>
    <revision>
      <id>68557</id>
      <parentid>63330</parentid>
      <timestamp>2021-04-24T19:44:20Z</timestamp>
      <contributor>
        <username>Keenan Pepper</username>
        <id>25</id>
      </contributor>
      <minor/>
      <comment>Clean up categories</comment>
      <origin>68557</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2704" sha1="77viug4yrsqrtmzgbm79xaj0d6vcpqy" xml:space="preserve">This MOS, with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators) and [[passion]] (6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.

{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
|-
| 1/12
| 
| 100
| 200
| 300
|-
| 5/59
| 
| 101.695
| 203.39
| 305.085
|-
| 
| 9/106
| 101.887
| 203.774
| 305.66
|-
| 
| 13/153
| 101.961
| 203.922
| 305.882
|-
| 
| 17/200
| 102
| 204
| 306
|-
| 4/47
| 
| 102.128
| 204.255
| 306.383
|-
| 
| 7/82
| 102.439
| 204.878
| 307.317
|-
| 
| 10/117
| 102.564
| 205.128
| 307.692
|-
| 
| 13/152
| 102.632
| 205.263
| 307.895
|-
| 
| 16/187
| 102.674
| 205.348
| 308.021
|-
| 
| 19/222
| 102.838
| 205.676
| 308.5135
|-
| 3/35
| 
| 102.857
| 205.714
| 308.571
|-
| 
| 20/233
| 103.004
| 206.009
| 309.013
|-
| 
| 17/198
| 103.03
| 206.061
| 309.091
|-
| 
| 14/163
| 103.0675
| 206.135
| 309.2025
|-
| 
| 
| 103.123
| 206.2455
| 309.368
|-
| 
| 11/128
| 103.125
| 206.25
| 309.375
|-
| 
| 8/93
| 103.226
| 206.452
| 309.677
|-
| 
| 
| 103.288
| 206.575
| 309.863
|-
| 
| 5/58
| 103.448
| 206.897
| 310.345
|-
| 
| 
| 103.651
| 207.301
| 310.952
|-
| 
| 7/81
| 103.703
| 207.407
| 311.111
|-
| 
| 9/104
| 103.846
| 207.692
| 311.5385
|-
| 
| 11/127
| 103.937
| 207.874
| 311.811
|-
| 
| 13/150
| 104
| 208
| 312
|-
| 2/23
| 
| 104.348
| 208.969
| 313.0435
|-
| 
| 11/126
| 104.762
| 209.524
| 314.286
|-
| 
| 9/103
| 104.854
| 209.709
| 314.563
|-
| 
| 7/80
| 105
| 210
| 315
|-
| 
| 5/57
| 105.263
| 210.526
| 315.7895
|-
| 
| 
| 105.43
| 210.86
| 316.29
|-
| 
| 8/91
| 105.4945
| 210.989
| 316.4835
|-
| 
| 
| 105.561
| 211.121
| 316.682
|-
| 
| 11/125
| 105.6
| 211.2
| 316.8
|-
| 3/34
| 
| 105.822
| 211.765
| 317.647
|-
| 
| 28/317
| 105.994
| 211.987
| 317.981
|-
| 
| 25/283
| 106.007
| 212.014
| 318.021
|-
| 
| 
| 106.023
| 212.046
| 318.069
|-
| 
| 22/249
| 106.024
| 212.048
| 318.072
|-
| 
| 19/215
| 106.0465
| 212.093
| 318.1395
|-
| 
| 16/181
| 106.077
| 212.155
| 318.232
|-
| 
| 13/147
| 106.122
| 212.245
| 318.367
|-
| 
| 10/113
| 106.195
| 212.389
| 318.581
|-
| 
| 7/79
| 106.329
| 212.658
| 318.987
|-
| 
| 11/124
| 106.452
| 212.903
| 319.355
|-
| 4/45
| 
| 106.667
| 213.333
| 320
|-
| 5/56
| 
| 107.143
| 214.286
| 321.429
|-
| 1/11
| 
| 109.091
| 218.182
| 327.273
|}

[[Category:Scales]]
[[Category:Abstract MOS patterns]]</text>
      <sha1>77viug4yrsqrtmzgbm79xaj0d6vcpqy</sha1>
    </revision>
    <revision>
      <id>87062</id>
      <parentid>68557</parentid>
      <timestamp>2022-02-25T14:54:41Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87062</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2684" sha1="6wd7oreeifl4ranhyzie58lefja9jt6" xml:space="preserve">This MOS, with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators) and [[passion]] (6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.

{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
|-
| 1/12
| 
| 100
| 200
| 300
|-
| 5/59
| 
| 101.695
| 203.39
| 305.085
|-
| 
| 9/106
| 101.887
| 203.774
| 305.66
|-
| 
| 13/153
| 101.961
| 203.922
| 305.882
|-
| 
| 17/200
| 102
| 204
| 306
|-
| 4/47
| 
| 102.128
| 204.255
| 306.383
|-
| 
| 7/82
| 102.439
| 204.878
| 307.317
|-
| 
| 10/117
| 102.564
| 205.128
| 307.692
|-
| 
| 13/152
| 102.632
| 205.263
| 307.895
|-
| 
| 16/187
| 102.674
| 205.348
| 308.021
|-
| 
| 19/222
| 102.838
| 205.676
| 308.5135
|-
| 3/35
| 
| 102.857
| 205.714
| 308.571
|-
| 
| 20/233
| 103.004
| 206.009
| 309.013
|-
| 
| 17/198
| 103.03
| 206.061
| 309.091
|-
| 
| 14/163
| 103.0675
| 206.135
| 309.2025
|-
| 
| 
| 103.123
| 206.2455
| 309.368
|-
| 
| 11/128
| 103.125
| 206.25
| 309.375
|-
| 
| 8/93
| 103.226
| 206.452
| 309.677
|-
| 
| 
| 103.288
| 206.575
| 309.863
|-
| 
| 5/58
| 103.448
| 206.897
| 310.345
|-
| 
| 
| 103.651
| 207.301
| 310.952
|-
| 
| 7/81
| 103.703
| 207.407
| 311.111
|-
| 
| 9/104
| 103.846
| 207.692
| 311.5385
|-
| 
| 11/127
| 103.937
| 207.874
| 311.811
|-
| 
| 13/150
| 104
| 208
| 312
|-
| 2/23
| 
| 104.348
| 208.969
| 313.0435
|-
| 
| 11/126
| 104.762
| 209.524
| 314.286
|-
| 
| 9/103
| 104.854
| 209.709
| 314.563
|-
| 
| 7/80
| 105
| 210
| 315
|-
| 
| 5/57
| 105.263
| 210.526
| 315.7895
|-
| 
| 
| 105.43
| 210.86
| 316.29
|-
| 
| 8/91
| 105.4945
| 210.989
| 316.4835
|-
| 
| 
| 105.561
| 211.121
| 316.682
|-
| 
| 11/125
| 105.6
| 211.2
| 316.8
|-
| 3/34
| 
| 105.822
| 211.765
| 317.647
|-
| 
| 28/317
| 105.994
| 211.987
| 317.981
|-
| 
| 25/283
| 106.007
| 212.014
| 318.021
|-
| 
| 
| 106.023
| 212.046
| 318.069
|-
| 
| 22/249
| 106.024
| 212.048
| 318.072
|-
| 
| 19/215
| 106.0465
| 212.093
| 318.1395
|-
| 
| 16/181
| 106.077
| 212.155
| 318.232
|-
| 
| 13/147
| 106.122
| 212.245
| 318.367
|-
| 
| 10/113
| 106.195
| 212.389
| 318.581
|-
| 
| 7/79
| 106.329
| 212.658
| 318.987
|-
| 
| 11/124
| 106.452
| 212.903
| 319.355
|-
| 4/45
| 
| 106.667
| 213.333
| 320
|-
| 5/56
| 
| 107.143
| 214.286
| 321.429
|-
| 1/11
| 
| 109.091
| 218.182
| 327.273
|}

[[Category:Abstract MOS patterns]]</text>
      <sha1>6wd7oreeifl4ranhyzie58lefja9jt6</sha1>
    </revision>
    <revision>
      <id>93940</id>
      <parentid>87062</parentid>
      <timestamp>2022-08-05T03:16:43Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Abstract MOS patterns]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>93940</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2648" sha1="0qfflx7frtyd1vmodspdxh1csgkh89x" xml:space="preserve">This MOS, with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators) and [[passion]] (6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.

{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
|-
| 1/12
| 
| 100
| 200
| 300
|-
| 5/59
| 
| 101.695
| 203.39
| 305.085
|-
| 
| 9/106
| 101.887
| 203.774
| 305.66
|-
| 
| 13/153
| 101.961
| 203.922
| 305.882
|-
| 
| 17/200
| 102
| 204
| 306
|-
| 4/47
| 
| 102.128
| 204.255
| 306.383
|-
| 
| 7/82
| 102.439
| 204.878
| 307.317
|-
| 
| 10/117
| 102.564
| 205.128
| 307.692
|-
| 
| 13/152
| 102.632
| 205.263
| 307.895
|-
| 
| 16/187
| 102.674
| 205.348
| 308.021
|-
| 
| 19/222
| 102.838
| 205.676
| 308.5135
|-
| 3/35
| 
| 102.857
| 205.714
| 308.571
|-
| 
| 20/233
| 103.004
| 206.009
| 309.013
|-
| 
| 17/198
| 103.03
| 206.061
| 309.091
|-
| 
| 14/163
| 103.0675
| 206.135
| 309.2025
|-
| 
| 
| 103.123
| 206.2455
| 309.368
|-
| 
| 11/128
| 103.125
| 206.25
| 309.375
|-
| 
| 8/93
| 103.226
| 206.452
| 309.677
|-
| 
| 
| 103.288
| 206.575
| 309.863
|-
| 
| 5/58
| 103.448
| 206.897
| 310.345
|-
| 
| 
| 103.651
| 207.301
| 310.952
|-
| 
| 7/81
| 103.703
| 207.407
| 311.111
|-
| 
| 9/104
| 103.846
| 207.692
| 311.5385
|-
| 
| 11/127
| 103.937
| 207.874
| 311.811
|-
| 
| 13/150
| 104
| 208
| 312
|-
| 2/23
| 
| 104.348
| 208.969
| 313.0435
|-
| 
| 11/126
| 104.762
| 209.524
| 314.286
|-
| 
| 9/103
| 104.854
| 209.709
| 314.563
|-
| 
| 7/80
| 105
| 210
| 315
|-
| 
| 5/57
| 105.263
| 210.526
| 315.7895
|-
| 
| 
| 105.43
| 210.86
| 316.29
|-
| 
| 8/91
| 105.4945
| 210.989
| 316.4835
|-
| 
| 
| 105.561
| 211.121
| 316.682
|-
| 
| 11/125
| 105.6
| 211.2
| 316.8
|-
| 3/34
| 
| 105.822
| 211.765
| 317.647
|-
| 
| 28/317
| 105.994
| 211.987
| 317.981
|-
| 
| 25/283
| 106.007
| 212.014
| 318.021
|-
| 
| 
| 106.023
| 212.046
| 318.069
|-
| 
| 22/249
| 106.024
| 212.048
| 318.072
|-
| 
| 19/215
| 106.0465
| 212.093
| 318.1395
|-
| 
| 16/181
| 106.077
| 212.155
| 318.232
|-
| 
| 13/147
| 106.122
| 212.245
| 318.367
|-
| 
| 10/113
| 106.195
| 212.389
| 318.581
|-
| 
| 7/79
| 106.329
| 212.658
| 318.987
|-
| 
| 11/124
| 106.452
| 212.903
| 319.355
|-
| 4/45
| 
| 106.667
| 213.333
| 320
|-
| 5/56
| 
| 107.143
| 214.286
| 321.429
|-
| 1/11
| 
| 109.091
| 218.182
| 327.273
|}</text>
      <sha1>0qfflx7frtyd1vmodspdxh1csgkh89x</sha1>
    </revision>
    <revision>
      <id>94028</id>
      <parentid>93940</parentid>
      <timestamp>2022-08-05T03:26:34Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox, name still missing, misc. edits</comment>
      <origin>94028</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2823" sha1="l5yb4mho7x4kuugr9syykzssb4he4x1" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Paucitonic = 1
| Pattern = LLLLLLLLLLLs
}}
The '''11L 1s''' [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators) and [[passion]] (6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
|-
| 1/12
| 
| 100
| 200
| 300
|-
| 5/59
| 
| 101.695
| 203.39
| 305.085
|-
| 
| 9/106
| 101.887
| 203.774
| 305.66
|-
| 
| 13/153
| 101.961
| 203.922
| 305.882
|-
| 
| 17/200
| 102
| 204
| 306
|-
| 4/47
| 
| 102.128
| 204.255
| 306.383
|-
| 
| 7/82
| 102.439
| 204.878
| 307.317
|-
| 
| 10/117
| 102.564
| 205.128
| 307.692
|-
| 
| 13/152
| 102.632
| 205.263
| 307.895
|-
| 
| 16/187
| 102.674
| 205.348
| 308.021
|-
| 
| 19/222
| 102.838
| 205.676
| 308.5135
|-
| 3/35
| 
| 102.857
| 205.714
| 308.571
|-
| 
| 20/233
| 103.004
| 206.009
| 309.013
|-
| 
| 17/198
| 103.03
| 206.061
| 309.091
|-
| 
| 14/163
| 103.0675
| 206.135
| 309.2025
|-
| 
| 
| 103.123
| 206.2455
| 309.368
|-
| 
| 11/128
| 103.125
| 206.25
| 309.375
|-
| 
| 8/93
| 103.226
| 206.452
| 309.677
|-
| 
| 
| 103.288
| 206.575
| 309.863
|-
| 
| 5/58
| 103.448
| 206.897
| 310.345
|-
| 
| 
| 103.651
| 207.301
| 310.952
|-
| 
| 7/81
| 103.703
| 207.407
| 311.111
|-
| 
| 9/104
| 103.846
| 207.692
| 311.5385
|-
| 
| 11/127
| 103.937
| 207.874
| 311.811
|-
| 
| 13/150
| 104
| 208
| 312
|-
| 2/23
| 
| 104.348
| 208.969
| 313.0435
|-
| 
| 11/126
| 104.762
| 209.524
| 314.286
|-
| 
| 9/103
| 104.854
| 209.709
| 314.563
|-
| 
| 7/80
| 105
| 210
| 315
|-
| 
| 5/57
| 105.263
| 210.526
| 315.7895
|-
| 
| 
| 105.43
| 210.86
| 316.29
|-
| 
| 8/91
| 105.4945
| 210.989
| 316.4835
|-
| 
| 
| 105.561
| 211.121
| 316.682
|-
| 
| 11/125
| 105.6
| 211.2
| 316.8
|-
| 3/34
| 
| 105.822
| 211.765
| 317.647
|-
| 
| 28/317
| 105.994
| 211.987
| 317.981
|-
| 
| 25/283
| 106.007
| 212.014
| 318.021
|-
| 
| 
| 106.023
| 212.046
| 318.069
|-
| 
| 22/249
| 106.024
| 212.048
| 318.072
|-
| 
| 19/215
| 106.0465
| 212.093
| 318.1395
|-
| 
| 16/181
| 106.077
| 212.155
| 318.232
|-
| 
| 13/147
| 106.122
| 212.245
| 318.367
|-
| 
| 10/113
| 106.195
| 212.389
| 318.581
|-
| 
| 7/79
| 106.329
| 212.658
| 318.987
|-
| 
| 11/124
| 106.452
| 212.903
| 319.355
|-
| 4/45
| 
| 106.667
| 213.333
| 320
|-
| 5/56
| 
| 107.143
| 214.286
| 321.429
|-
| 1/11
| 
| 109.091
| 218.182
| 327.273
|}</text>
      <sha1>l5yb4mho7x4kuugr9syykzssb4he4x1</sha1>
    </revision>
    <revision>
      <id>94198</id>
      <parentid>94028</parentid>
      <timestamp>2022-08-14T06:10:51Z</timestamp>
      <contributor>
        <username>Frostburn</username>
        <id>326</id>
      </contributor>
      <comment>Add name.</comment>
      <origin>94198</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2831" sha1="tjab8v0bjw80thhdcyoj356f5esx9m3" xml:space="preserve">{{Infobox MOS
| Name = ripploid
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Paucitonic = 1
| Pattern = LLLLLLLLLLLs
}}
The '''11L 1s''' [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators) and [[passion]] (6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2/23edo, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
|-
| 1/12
| 
| 100
| 200
| 300
|-
| 5/59
| 
| 101.695
| 203.39
| 305.085
|-
| 
| 9/106
| 101.887
| 203.774
| 305.66
|-
| 
| 13/153
| 101.961
| 203.922
| 305.882
|-
| 
| 17/200
| 102
| 204
| 306
|-
| 4/47
| 
| 102.128
| 204.255
| 306.383
|-
| 
| 7/82
| 102.439
| 204.878
| 307.317
|-
| 
| 10/117
| 102.564
| 205.128
| 307.692
|-
| 
| 13/152
| 102.632
| 205.263
| 307.895
|-
| 
| 16/187
| 102.674
| 205.348
| 308.021
|-
| 
| 19/222
| 102.838
| 205.676
| 308.5135
|-
| 3/35
| 
| 102.857
| 205.714
| 308.571
|-
| 
| 20/233
| 103.004
| 206.009
| 309.013
|-
| 
| 17/198
| 103.03
| 206.061
| 309.091
|-
| 
| 14/163
| 103.0675
| 206.135
| 309.2025
|-
| 
| 
| 103.123
| 206.2455
| 309.368
|-
| 
| 11/128
| 103.125
| 206.25
| 309.375
|-
| 
| 8/93
| 103.226
| 206.452
| 309.677
|-
| 
| 
| 103.288
| 206.575
| 309.863
|-
| 
| 5/58
| 103.448
| 206.897
| 310.345
|-
| 
| 
| 103.651
| 207.301
| 310.952
|-
| 
| 7/81
| 103.703
| 207.407
| 311.111
|-
| 
| 9/104
| 103.846
| 207.692
| 311.5385
|-
| 
| 11/127
| 103.937
| 207.874
| 311.811
|-
| 
| 13/150
| 104
| 208
| 312
|-
| 2/23
| 
| 104.348
| 208.969
| 313.0435
|-
| 
| 11/126
| 104.762
| 209.524
| 314.286
|-
| 
| 9/103
| 104.854
| 209.709
| 314.563
|-
| 
| 7/80
| 105
| 210
| 315
|-
| 
| 5/57
| 105.263
| 210.526
| 315.7895
|-
| 
| 
| 105.43
| 210.86
| 316.29
|-
| 
| 8/91
| 105.4945
| 210.989
| 316.4835
|-
| 
| 
| 105.561
| 211.121
| 316.682
|-
| 
| 11/125
| 105.6
| 211.2
| 316.8
|-
| 3/34
| 
| 105.822
| 211.765
| 317.647
|-
| 
| 28/317
| 105.994
| 211.987
| 317.981
|-
| 
| 25/283
| 106.007
| 212.014
| 318.021
|-
| 
| 
| 106.023
| 212.046
| 318.069
|-
| 
| 22/249
| 106.024
| 212.048
| 318.072
|-
| 
| 19/215
| 106.0465
| 212.093
| 318.1395
|-
| 
| 16/181
| 106.077
| 212.155
| 318.232
|-
| 
| 13/147
| 106.122
| 212.245
| 318.367
|-
| 
| 10/113
| 106.195
| 212.389
| 318.581
|-
| 
| 7/79
| 106.329
| 212.658
| 318.987
|-
| 
| 11/124
| 106.452
| 212.903
| 319.355
|-
| 4/45
| 
| 106.667
| 213.333
| 320
|-
| 5/56
| 
| 107.143
| 214.286
| 321.429
|-
| 1/11
| 
| 109.091
| 218.182
| 327.273
|}</text>
      <sha1>tjab8v0bjw80thhdcyoj356f5esx9m3</sha1>
    </revision>
    <revision>
      <id>94360</id>
      <parentid>94198</parentid>
      <timestamp>2022-08-15T18:32:40Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Merged comments from 11L 1s-15L 1s, backslashes</comment>
      <origin>94360</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2995" sha1="cfd0me32n9o7827ppnxdn36no4lszby" xml:space="preserve">{{Infobox MOS
| Name = ripploid
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Paucitonic = 1
| Pattern = LLLLLLLLLLLs
}}
The '''11L 1s''' [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators) and [[passion]] (6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
! Comments
|-
| 1\12
| 
| 100
| 200
| 300
| Ripple
|-
| 5\59
| 
| 101.695
| 203.39
| 305.085
| 
|-
| 
| 9\106
| 101.887
| 203.774
| 305.66
| 
|-
| 
| 13\153
| 101.961
| 203.922
| 305.882
| 
|-
| 
| 17\200
| 102
| 204
| 306
| 
|-
| 4\47
| 
| 102.128
| 204.255
| 306.383
| 
|-
| 
| 7\82
| 102.439
| 204.878
| 307.317
| 
|-
| 
| 10\117
| 102.564
| 205.128
| 307.692
| 
|-
| 
| 13\152
| 102.632
| 205.263
| 307.895
| 
|-
| 
| 16\187
| 102.674
| 205.348
| 308.021
| 
|-
| 
| 19\222
| 102.838
| 205.676
| 308.5135
| 
|-
| 3\35
| 
| 102.857
| 205.714
| 308.571
| 
|-
| 
| 20\233
| 103.004
| 206.009
| 309.013
| 
|-
| 
| 17\198
| 103.03
| 206.061
| 309.091
| 
|-
| 
| 14\163
| 103.0675
| 206.135
| 309.2025
| 
|-
| 
| 
| 103.123
| 206.2455
| 309.368
| 
|-
| 
| 11\128
| 103.125
| 206.25
| 309.375
| 
|-
| 
| 8\93
| 103.226
| 206.452
| 309.677
| 
|-
| 
| 
| 103.288
| 206.575
| 309.863
| 
|-
| 
| 5\58
| 103.448
| 206.897
| 310.345
| 
|-
| 
| 
| 103.651
| 207.301
| 310.952
| 
|-
| 
| 7\81
| 103.703
| 207.407
| 311.111
| 
|-
| 
| 9\104
| 103.846
| 207.692
| 311.5385
| 
|-
| 
| 11\127
| 103.937
| 207.874
| 311.811
| 
|-
| 
| 13\150
| 104
| 208
| 312
| 
|-
| 2\23
| 
| 104.348
| 208.969
| 313.0435
| Ripple
|-
| 
| 11\126
| 104.762
| 209.524
| 314.286
| 
|-
| 
| 9\103
| 104.854
| 209.709
| 314.563
| 
|-
| 
| 7\80
| 105
| 210
| 315
| 
|-
| 
| 5\57
| 105.263
| 210.526
| 315.7895
| 
|-
| 
| 
| 105.43
| 210.86
| 316.29
| 
|-
| 
| 8\91
| 105.4945
| 210.989
| 316.4835
| 
|-
| 
| 
| 105.561
| 211.121
| 316.682
| 
|-
| 
| 11\125
| 105.6
| 211.2
| 316.8
| 
|-
| 3\34
| 
| 105.822
| 211.765
| 317.647
| 
|-
| 
| 28\317
| 105.994
| 211.987
| 317.981
| 
|-
| 
| 25\283
| 106.007
| 212.014
| 318.021
| 
|-
| 
| 
| 106.023
| 212.046
| 318.069
| 
|-
| 
| 22\249
| 106.024
| 212.048
| 318.072
| 
|-
| 
| 19\215
| 106.0465
| 212.093
| 318.1395
| 
|-
| 
| 16\181
| 106.077
| 212.155
| 318.232
| 
|-
| 
| 13\147
| 106.122
| 212.245
| 318.367
| 
|-
| 
| 10\113
| 106.195
| 212.389
| 318.581
| 
|-
| 
| 7\79
| 106.329
| 212.658
| 318.987
| 
|-
| 
| 11\124
| 106.452
| 212.903
| 319.355
| 
|-
| 4\45
| 
| 106.667
| 213.333
| 320
| 
|-
| 5\56
| 
| 107.143
| 214.286
| 321.429
| 
|-
| 1\11
| 
| 109.091
| 218.182
| 327.273
| 
|}</text>
      <sha1>cfd0me32n9o7827ppnxdn36no4lszby</sha1>
    </revision>
    <revision>
      <id>94362</id>
      <parentid>94360</parentid>
      <timestamp>2022-08-15T19:14:36Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Unify precision (except where no edo size was given), fixed some errors</comment>
      <origin>94362</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3014" sha1="dazjxrvc40sjlapee80lgik0xr9uwvr" xml:space="preserve">{{Infobox MOS
| Name = ripploid
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Paucitonic = 1
| Pattern = LLLLLLLLLLLs
}}
The '''11L 1s''' [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators) and [[passion]] (6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
! Comments
|-
| 1\12
| 
| 100
| 200
| 300
| Ripple
|-
| 5\59
| 
| 101.695
| 203.390
| 305.085
| 
|-
| 
| 9\106
| 101.887
| 203.774
| 305.660
| 
|-
| 
| 13\153
| 101.961
| 203.922
| 305.882
| 
|-
| 
| 17\200
| 102
| 204
| 306
| 
|-
| 4\47
| 
| 102.128
| 204.255
| 306.383
| 
|-
| 
| 7\82
| 102.439
| 204.878
| 307.317
| 
|-
| 
| 10\117
| 102.564
| 205.128
| 307.692
| 
|-
| 
| 13\152
| 102.632
| 205.263
| 307.895
| 
|-
| 
| 16\187
| 102.674
| 205.348
| 308.021
| 
|-
| 
| 19\222
| 102.703
| 205.405
| 308.108
| 
|-
| 3\35
| 
| 102.857
| 205.714
| 308.571
| 
|-
| 
| 20\233
| 103.004
| 206.009
| 309.013
| 
|-
| 
| 17\198
| 103.030
| 206.061
| 309.091
| 
|-
| 
| 14\163
| 103.067
| 206.135
| 309.202
| 
|-
| 
| 
| 103.123
| 206.2455
| 309.368
| 
|-
| 
| 11\128
| 103.125
| 206.25
| 309.375
| 
|-
| 
| 8\93
| 103.226
| 206.452
| 309.677
| 
|-
| 
| 
| 103.288
| 206.575
| 309.863
| 
|-
| 
| 5\58
| 103.448
| 206.897
| 310.345
| 
|-
| 
| 
| 103.651
| 207.301
| 310.952
| 
|-
| 
| 7\81
| 103.703
| 207.407
| 311.111
| 
|-
| 
| 9\104
| 103.846
| 207.692
| 311.538
| 
|-
| 
| 11\127
| 103.937
| 207.874
| 311.811
| 
|-
| 
| 13\150
| 104
| 208
| 312
| 
|-
| 2\23
| 
| 104.348
| 208.969
| 313.043
| Ripple
|-
| 
| 11\126
| 104.762
| 209.524
| 314.286
| 
|-
| 
| 9\103
| 104.854
| 209.709
| 314.563
| 
|-
| 
| 7\80
| 105
| 210
| 315
| 
|-
| 
| 5\57
| 105.263
| 210.526
| 315.789
| 
|-
| 
| 
| 105.43
| 210.86
| 316.29
| 
|-
| 
| 8\91
| 105.495
| 210.989
| 316.484
| 
|-
| 
| 
| 105.561
| 211.121
| 316.682
| 
|-
| 
| 11\125
| 105.6
| 211.2
| 316.8
| 
|-
| 3\34
| 
| 105.822
| 211.765
| 317.647
| 
|-
| 
| 28\317
| 105.994
| 211.987
| 317.981
| 
|-
| 
| 25\283
| 106.007
| 212.014
| 318.021
| 
|-
| 
| 
| 106.023
| 212.046
| 318.069
| 
|-
| 
| 22\249
| 106.024
| 212.048
| 318.072
| 
|-
| 
| 19\215
| 106.047
| 212.093
| 318.140
| 
|-
| 
| 16\181
| 106.077
| 212.155
| 318.232
| 
|-
| 
| 13\147
| 106.122
| 212.245
| 318.367
| 
|-
| 
| 10\113
| 106.195
| 212.389
| 318.581
| 
|-
| 
| 7\79
| 106.329
| 212.658
| 318.987
| 
|-
| 
| 11\124
| 106.452
| 212.903
| 319.355
| 
|-
| 4\45
| 
| 106.667
| 213.333
| 320
| 
|-
| 5\56
| 
| 107.143
| 214.286
| 321.429
| 
|-
| 1\11
| 
| 109.091
| 218.182
| 327.273
| 
|}

{{todo|unify precision}}</text>
      <sha1>dazjxrvc40sjlapee80lgik0xr9uwvr</sha1>
    </revision>
    <revision>
      <id>94506</id>
      <parentid>94362</parentid>
      <timestamp>2022-08-17T09:19:00Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94506</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3005" sha1="rj0x5bukyu6ounvr6z2kc7ydtzg4wu8" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
The '''11L 1s''' [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators) and [[passion]] (6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
! Comments
|-
| 1\12
| 
| 100
| 200
| 300
| Ripple
|-
| 5\59
| 
| 101.695
| 203.390
| 305.085
| 
|-
| 
| 9\106
| 101.887
| 203.774
| 305.660
| 
|-
| 
| 13\153
| 101.961
| 203.922
| 305.882
| 
|-
| 
| 17\200
| 102
| 204
| 306
| 
|-
| 4\47
| 
| 102.128
| 204.255
| 306.383
| 
|-
| 
| 7\82
| 102.439
| 204.878
| 307.317
| 
|-
| 
| 10\117
| 102.564
| 205.128
| 307.692
| 
|-
| 
| 13\152
| 102.632
| 205.263
| 307.895
| 
|-
| 
| 16\187
| 102.674
| 205.348
| 308.021
| 
|-
| 
| 19\222
| 102.703
| 205.405
| 308.108
| 
|-
| 3\35
| 
| 102.857
| 205.714
| 308.571
| 
|-
| 
| 20\233
| 103.004
| 206.009
| 309.013
| 
|-
| 
| 17\198
| 103.030
| 206.061
| 309.091
| 
|-
| 
| 14\163
| 103.067
| 206.135
| 309.202
| 
|-
| 
| 
| 103.123
| 206.2455
| 309.368
| 
|-
| 
| 11\128
| 103.125
| 206.25
| 309.375
| 
|-
| 
| 8\93
| 103.226
| 206.452
| 309.677
| 
|-
| 
| 
| 103.288
| 206.575
| 309.863
| 
|-
| 
| 5\58
| 103.448
| 206.897
| 310.345
| 
|-
| 
| 
| 103.651
| 207.301
| 310.952
| 
|-
| 
| 7\81
| 103.703
| 207.407
| 311.111
| 
|-
| 
| 9\104
| 103.846
| 207.692
| 311.538
| 
|-
| 
| 11\127
| 103.937
| 207.874
| 311.811
| 
|-
| 
| 13\150
| 104
| 208
| 312
| 
|-
| 2\23
| 
| 104.348
| 208.969
| 313.043
| Ripple
|-
| 
| 11\126
| 104.762
| 209.524
| 314.286
| 
|-
| 
| 9\103
| 104.854
| 209.709
| 314.563
| 
|-
| 
| 7\80
| 105
| 210
| 315
| 
|-
| 
| 5\57
| 105.263
| 210.526
| 315.789
| 
|-
| 
| 
| 105.43
| 210.86
| 316.29
| 
|-
| 
| 8\91
| 105.495
| 210.989
| 316.484
| 
|-
| 
| 
| 105.561
| 211.121
| 316.682
| 
|-
| 
| 11\125
| 105.6
| 211.2
| 316.8
| 
|-
| 3\34
| 
| 105.822
| 211.765
| 317.647
| 
|-
| 
| 28\317
| 105.994
| 211.987
| 317.981
| 
|-
| 
| 25\283
| 106.007
| 212.014
| 318.021
| 
|-
| 
| 
| 106.023
| 212.046
| 318.069
| 
|-
| 
| 22\249
| 106.024
| 212.048
| 318.072
| 
|-
| 
| 19\215
| 106.047
| 212.093
| 318.140
| 
|-
| 
| 16\181
| 106.077
| 212.155
| 318.232
| 
|-
| 
| 13\147
| 106.122
| 212.245
| 318.367
| 
|-
| 
| 10\113
| 106.195
| 212.389
| 318.581
| 
|-
| 
| 7\79
| 106.329
| 212.658
| 318.987
| 
|-
| 
| 11\124
| 106.452
| 212.903
| 319.355
| 
|-
| 4\45
| 
| 106.667
| 213.333
| 320
| 
|-
| 5\56
| 
| 107.143
| 214.286
| 321.429
| 
|-
| 1\11
| 
| 109.091
| 218.182
| 327.273
| 
|}

{{todo|unify precision}}</text>
      <sha1>rj0x5bukyu6ounvr6z2kc7ydtzg4wu8</sha1>
    </revision>
    <revision>
      <id>101023</id>
      <parentid>94506</parentid>
      <timestamp>2023-01-06T16:13:28Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Copying from [[Category:Abstract MOS patterns]] to [[Category:12-tone scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>101023</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3034" sha1="9npv774olmse9x33trr0isl41lj7y0i" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
The '''11L 1s''' [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators) and [[passion]] (6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
! Comments
|-
| 1\12
| 
| 100
| 200
| 300
| Ripple
|-
| 5\59
| 
| 101.695
| 203.390
| 305.085
| 
|-
| 
| 9\106
| 101.887
| 203.774
| 305.660
| 
|-
| 
| 13\153
| 101.961
| 203.922
| 305.882
| 
|-
| 
| 17\200
| 102
| 204
| 306
| 
|-
| 4\47
| 
| 102.128
| 204.255
| 306.383
| 
|-
| 
| 7\82
| 102.439
| 204.878
| 307.317
| 
|-
| 
| 10\117
| 102.564
| 205.128
| 307.692
| 
|-
| 
| 13\152
| 102.632
| 205.263
| 307.895
| 
|-
| 
| 16\187
| 102.674
| 205.348
| 308.021
| 
|-
| 
| 19\222
| 102.703
| 205.405
| 308.108
| 
|-
| 3\35
| 
| 102.857
| 205.714
| 308.571
| 
|-
| 
| 20\233
| 103.004
| 206.009
| 309.013
| 
|-
| 
| 17\198
| 103.030
| 206.061
| 309.091
| 
|-
| 
| 14\163
| 103.067
| 206.135
| 309.202
| 
|-
| 
| 
| 103.123
| 206.2455
| 309.368
| 
|-
| 
| 11\128
| 103.125
| 206.25
| 309.375
| 
|-
| 
| 8\93
| 103.226
| 206.452
| 309.677
| 
|-
| 
| 
| 103.288
| 206.575
| 309.863
| 
|-
| 
| 5\58
| 103.448
| 206.897
| 310.345
| 
|-
| 
| 
| 103.651
| 207.301
| 310.952
| 
|-
| 
| 7\81
| 103.703
| 207.407
| 311.111
| 
|-
| 
| 9\104
| 103.846
| 207.692
| 311.538
| 
|-
| 
| 11\127
| 103.937
| 207.874
| 311.811
| 
|-
| 
| 13\150
| 104
| 208
| 312
| 
|-
| 2\23
| 
| 104.348
| 208.969
| 313.043
| Ripple
|-
| 
| 11\126
| 104.762
| 209.524
| 314.286
| 
|-
| 
| 9\103
| 104.854
| 209.709
| 314.563
| 
|-
| 
| 7\80
| 105
| 210
| 315
| 
|-
| 
| 5\57
| 105.263
| 210.526
| 315.789
| 
|-
| 
| 
| 105.43
| 210.86
| 316.29
| 
|-
| 
| 8\91
| 105.495
| 210.989
| 316.484
| 
|-
| 
| 
| 105.561
| 211.121
| 316.682
| 
|-
| 
| 11\125
| 105.6
| 211.2
| 316.8
| 
|-
| 3\34
| 
| 105.822
| 211.765
| 317.647
| 
|-
| 
| 28\317
| 105.994
| 211.987
| 317.981
| 
|-
| 
| 25\283
| 106.007
| 212.014
| 318.021
| 
|-
| 
| 
| 106.023
| 212.046
| 318.069
| 
|-
| 
| 22\249
| 106.024
| 212.048
| 318.072
| 
|-
| 
| 19\215
| 106.047
| 212.093
| 318.140
| 
|-
| 
| 16\181
| 106.077
| 212.155
| 318.232
| 
|-
| 
| 13\147
| 106.122
| 212.245
| 318.367
| 
|-
| 
| 10\113
| 106.195
| 212.389
| 318.581
| 
|-
| 
| 7\79
| 106.329
| 212.658
| 318.987
| 
|-
| 
| 11\124
| 106.452
| 212.903
| 319.355
| 
|-
| 4\45
| 
| 106.667
| 213.333
| 320
| 
|-
| 5\56
| 
| 107.143
| 214.286
| 321.429
| 
|-
| 1\11
| 
| 109.091
| 218.182
| 327.273
| 
|}

{{todo|unify precision}}

[[Category:12-tone scales]]</text>
      <sha1>9npv774olmse9x33trr0isl41lj7y0i</sha1>
    </revision>
    <revision>
      <id>102472</id>
      <parentid>101023</parentid>
      <timestamp>2023-02-06T05:21:09Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Scale tree */</comment>
      <origin>102472</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2698" sha1="f3hyzbsmj8mgcggpmxwsexedotvblya" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
The '''11L 1s''' [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators) and [[passion]] (6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
! Comments
|-
| 1\12
| 
| 100
| 200
| 300
| Ripple
|-
| 5\59
| 
| 101.695
| 203.390
| 305.085
| 
|-
| 
| 9\106
| 101.887
| 203.774
| 305.660
| 
|-
| 
| 13\153
| 101.961
| 203.922
| 305.882
| 
|-
| 
| 17\200
| 102
| 204
| 306
| 
|-
| 4\47
| 
| 102.128
| 204.255
| 306.383
| 
|-
| 
| 7\82
| 102.439
| 204.878
| 307.317
| 
|-
| 
| 10\117
| 102.564
| 205.128
| 307.692
| 
|-
| 
| 13\152
| 102.632
| 205.263
| 307.895
| 
|-
| 
| 16\187
| 102.674
| 205.348
| 308.021
| 
|-
| 
| 19\222
| 102.703
| 205.405
| 308.108
| 
|-
| 3\35
| 
| 102.857
| 205.714
| 308.571
| 
|-
| 
| 20\233
| 103.004
| 206.009
| 309.013
| 
|-
| 
| 17\198
| 103.030
| 206.061
| 309.091
| 
|-
| 
| 14\163
| 103.067
| 206.135
| 309.202
| 
|-
| 
| 
| 103.123
| 206.2455
| 309.368
| 
|-
| 
| 11\128
| 103.125
| 206.25
| 309.375
| 
|-
| 
| 8\93
| 103.226
| 206.452
| 309.677
| 
|-
| 
| 
| 103.288
| 206.575
| 309.863
| 
|-
| 
| 5\58
| 103.448
| 206.897
| 310.345
| 
|-
| 
| 
| 103.651
| 207.301
| 310.952
| 
|-
| 
| 7\81
| 103.703
| 207.407
| 311.111
| 
|-
| 
| 9\104
| 103.846
| 207.692
| 311.538
| 
|-
| 
| 11\127
| 103.937
| 207.874
| 311.811
| 
|-
| 
| 13\150
| 104
| 208
| 312
| 
|-
| 2\23
| 
| 104.348
| 208.969
| 313.043
| Ripple
|-
| 
| 11\126
| 104.762
| 209.524
| 314.286
| 
|-
| 
| 9\103
| 104.854
| 209.709
| 314.563
| 
|-
| 
| 7\80
| 105
| 210
| 315
| 
|-
| 
| 5\57
| 105.263
| 210.526
| 315.789
| 
|-
| 
| 
| 105.43
| 210.86
| 316.29
| 
|-
| 
| 8\91
| 105.495
| 210.989
| 316.484
| 
|-
| 
| 
| 105.561
| 211.121
| 316.682
| 
|-
| 
| 11\125
| 105.6
| 211.2
| 316.8
| 
|-
| 3\34
| 
| 105.822
| 211.765
| 317.647
| 
|-
| 
| 
| 106.023
| 212.046
| 318.069
| 
|-
| 
| 7\79
| 106.329
| 212.658
| 318.987
| 
|-
| 
| 11\124
| 106.452
| 212.903
| 319.355
| 
|-
| 4\45
| 
| 106.667
| 213.333
| 320
| 
|-
| 5\56
| 
| 107.143
| 214.286
| 321.429
| 
|-
| 1\11
| 
| 109.091
| 218.182
| 327.273
| 
|}

{{todo|unify precision}}

[[Category:12-tone scales]]</text>
      <sha1>f3hyzbsmj8mgcggpmxwsexedotvblya</sha1>
    </revision>
    <revision>
      <id>102474</id>
      <parentid>102472</parentid>
      <timestamp>2023-02-06T05:35:48Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>102474</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2697" sha1="ano3z0sy8m4pucetpl7y4r13kd8u6yo" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
The '''11L 1s''' [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators-6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
! Comments
|-
| 1\12
| 
| 100
| 200
| 300
| Ripple
|-
| 5\59
| 
| 101.695
| 203.390
| 305.085
| 
|-
| 
| 9\106
| 101.887
| 203.774
| 305.660
| 
|-
| 
| 13\153
| 101.961
| 203.922
| 305.882
| 
|-
| 
| 17\200
| 102
| 204
| 306
| 
|-
| 4\47
| 
| 102.128
| 204.255
| 306.383
| 
|-
| 
| 7\82
| 102.439
| 204.878
| 307.317
| 
|-
| 
| 10\117
| 102.564
| 205.128
| 307.692
| 
|-
| 
| 13\152
| 102.632
| 205.263
| 307.895
| 
|-
| 
| 16\187
| 102.674
| 205.348
| 308.021
| 
|-
| 
| 19\222
| 102.703
| 205.405
| 308.108
| 
|-
| 3\35
| 
| 102.857
| 205.714
| 308.571
| 
|-
| 
| 20\233
| 103.004
| 206.009
| 309.013
| 
|-
| 
| 17\198
| 103.030
| 206.061
| 309.091
| 
|-
| 
| 14\163
| 103.067
| 206.135
| 309.202
| 
|-
| 
| 
| 103.123
| [[Tel:206.2455|206.2455]]
| 309.368
| 
|-
| 
| 11\128
| 103.125
| 206.25
| 309.375
| 
|-
| 
| 8\93
| 103.226
| 206.452
| 309.677
| 
|-
| 
| 
| 103.288
| 206.575
| 309.863
| 
|-
| 
| 5\58
| 103.448
| 206.897
| 310.345
| 
|-
| 
| 
| 103.651
| 207.301
| 310.952
| 
|-
| 
| 7\81
| 103.703
| 207.407
| 311.111
| 
|-
| 
| 9\104
| 103.846
| 207.692
| 311.538
| 
|-
| 
| 11\127
| 103.937
| 207.874
| 311.811
| 
|-
| 
| 13\150
| 104
| 208
| 312
| 
|-
| 2\23
| 
| 104.348
| 208.969
| 313.043
| Ripple
|-
| 
| 11\126
| 104.762
| 209.524
| 314.286
| 
|-
| 
| 9\103
| 104.854
| 209.709
| 314.563
| 
|-
| 
| 7\80
| 105
| 210
| 315
| 
|-
| 
| 5\57
| 105.263
| 210.526
| 315.789
| 
|-
| 
| 
| 105.43
| 210.86
| 316.29
| 
|-
| 
| 8\91
| 105.495
| 210.989
| 316.484
| 
|-
| 
| 
| 105.561
| 211.121
| 316.682
| 
|-
| 
| 11\125
| 105.6
| 211.2
| 316.8
| 
|-
| 3\34
| 
| 105.822
| 211.765
| 317.647
| 
|-
| 
| 
| 106.023
| 212.046
| 318.069
| 
|-
| 
| 7\79
| 106.329
| 212.658
| 318.987
| 
|-
| 
| 11\124
| 106.452
| 212.903
| 319.355
| 
|-
| 4\45
| 
| 106.667
| 213.333
| 320
| 
|-
| 5\56
| 
| 107.143
| 214.286
| 321.429
| 
|-
| 1\11
| 
| 109.091
| 218.182
| 327.273
| 
|}

{{todo|unify precision}}

[[Category:12-tone scales]]</text>
      <sha1>ano3z0sy8m4pucetpl7y4r13kd8u6yo</sha1>
    </revision>
    <revision>
      <id>102507</id>
      <parentid>102474</parentid>
      <timestamp>2023-02-07T04:02:51Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Fix telephone link</comment>
      <origin>102507</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2680" sha1="6zriutn0m9u8w2ov9dapg6efe2aw0vn" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
The '''11L 1s''' [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators-6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
! Comments
|-
| 1\12
| 
| 100
| 200
| 300
| Ripple
|-
| 5\59
| 
| 101.695
| 203.390
| 305.085
| 
|-
| 
| 9\106
| 101.887
| 203.774
| 305.660
| 
|-
| 
| 13\153
| 101.961
| 203.922
| 305.882
| 
|-
| 
| 17\200
| 102
| 204
| 306
| 
|-
| 4\47
| 
| 102.128
| 204.255
| 306.383
| 
|-
| 
| 7\82
| 102.439
| 204.878
| 307.317
| 
|-
| 
| 10\117
| 102.564
| 205.128
| 307.692
| 
|-
| 
| 13\152
| 102.632
| 205.263
| 307.895
| 
|-
| 
| 16\187
| 102.674
| 205.348
| 308.021
| 
|-
| 
| 19\222
| 102.703
| 205.405
| 308.108
| 
|-
| 3\35
| 
| 102.857
| 205.714
| 308.571
| 
|-
| 
| 20\233
| 103.004
| 206.009
| 309.013
| 
|-
| 
| 17\198
| 103.030
| 206.061
| 309.091
| 
|-
| 
| 14\163
| 103.067
| 206.135
| 309.202
| 
|-
| 
| 
| 103.123
| 206.2455
| 309.368
| 
|-
| 
| 11\128
| 103.125
| 206.25
| 309.375
| 
|-
| 
| 8\93
| 103.226
| 206.452
| 309.677
| 
|-
| 
| 
| 103.288
| 206.575
| 309.863
| 
|-
| 
| 5\58
| 103.448
| 206.897
| 310.345
| 
|-
| 
| 
| 103.651
| 207.301
| 310.952
| 
|-
| 
| 7\81
| 103.703
| 207.407
| 311.111
| 
|-
| 
| 9\104
| 103.846
| 207.692
| 311.538
| 
|-
| 
| 11\127
| 103.937
| 207.874
| 311.811
| 
|-
| 
| 13\150
| 104
| 208
| 312
| 
|-
| 2\23
| 
| 104.348
| 208.969
| 313.043
| Ripple
|-
| 
| 11\126
| 104.762
| 209.524
| 314.286
| 
|-
| 
| 9\103
| 104.854
| 209.709
| 314.563
| 
|-
| 
| 7\80
| 105
| 210
| 315
| 
|-
| 
| 5\57
| 105.263
| 210.526
| 315.789
| 
|-
| 
| 
| 105.43
| 210.86
| 316.29
| 
|-
| 
| 8\91
| 105.495
| 210.989
| 316.484
| 
|-
| 
| 
| 105.561
| 211.121
| 316.682
| 
|-
| 
| 11\125
| 105.6
| 211.2
| 316.8
| 
|-
| 3\34
| 
| 105.822
| 211.765
| 317.647
| 
|-
| 
| 
| 106.023
| 212.046
| 318.069
| 
|-
| 
| 7\79
| 106.329
| 212.658
| 318.987
| 
|-
| 
| 11\124
| 106.452
| 212.903
| 319.355
| 
|-
| 4\45
| 
| 106.667
| 213.333
| 320
| 
|-
| 5\56
| 
| 107.143
| 214.286
| 321.429
| 
|-
| 1\11
| 
| 109.091
| 218.182
| 327.273
| 
|}

{{todo|unify precision}}

[[Category:12-tone scales]]</text>
      <sha1>6zriutn0m9u8w2ov9dapg6efe2aw0vn</sha1>
    </revision>
    <revision>
      <id>118925</id>
      <parentid>102507</parentid>
      <timestamp>2023-07-07T01:40:38Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Unified precision</comment>
      <origin>118925</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2716" sha1="0wdn9oyvx6t10wy68kzgjuailma0vrw" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
The '''11L 1s''' [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators-6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
! Comments
|-
| 1\12
| 
| 100.000
| 200.000
| 300.000
| Ripple
|-
| 5\59
| 
| 101.695
| 203.390
| 305.085
| 
|-
| 
| 9\106
| 101.887
| 203.774
| 305.660
| 
|-
| 
| 13\153
| 101.961
| 203.922
| 305.882
| 
|-
| 
| 17\200
| 102.000
| 204.000
| 306.000
| 
|-
| 4\47
| 
| 102.128
| 204.255
| 306.383
| 
|-
| 
| 7\82
| 102.439
| 204.878
| 307.317
| 
|-
| 
| 10\117
| 102.564
| 205.128
| 307.692
| 
|-
| 
| 13\152
| 102.632
| 205.263
| 307.895
| 
|-
| 
| 16\187
| 102.674
| 205.348
| 308.021
| 
|-
| 
| 19\222
| 102.703
| 205.405
| 308.108
| 
|-
| 3\35
| 
| 102.857
| 205.714
| 308.571
| 
|-
| 
| 20\233
| 103.004
| 206.009
| 309.013
| 
|-
| 
| 17\198
| 103.030
| 206.061
| 309.091
| 
|-
| 
| 14\163
| 103.067
| 206.135
| 309.202
| 
|-
| 
| 
| 103.123
| 206.2455
| 309.368
| 
|-
| 
| 11\128
| 103.125
| 206.250
| 309.375
| 
|-
| 
| 8\93
| 103.226
| 206.452
| 309.677
| 
|-
| 
| 
| 103.288
| 206.575
| 309.863
| 
|-
| 
| 5\58
| 103.448
| 206.897
| 310.345
| 
|-
| 
| 
| 103.651
| 207.301
| 310.952
| 
|-
| 
| 7\81
| 103.703
| 207.407
| 311.111
| 
|-
| 
| 9\104
| 103.846
| 207.692
| 311.538
| 
|-
| 
| 11\127
| 103.937
| 207.874
| 311.811
| 
|-
| 
| 13\150
| 104.000
| 208.000
| 312.000
| 
|-
| 2\23
| 
| 104.348
| 208.969
| 313.043
| Ripple
|-
| 
| 11\126
| 104.762
| 209.524
| 314.286
| 
|-
| 
| 9\103
| 104.854
| 209.709
| 314.563
| 
|-
| 
| 7\80
| 105.000
| 210.000
| 315.000
| 
|-
| 
| 5\57
| 105.263
| 210.526
| 315.789
| 
|-
| 
| 
| 105.430
| 210.860
| 316.290
| 
|-
| 
| 8\91
| 105.495
| 210.989
| 316.484
| 
|-
| 
| 
| 105.561
| 211.121
| 316.682
| 
|-
| 
| 11\125
| 105.600
| 211.200
| 316.800
| 
|-
| 3\34
| 
| 105.822
| 211.765
| 317.647
| 
|-
| 
| 
| 106.023
| 212.046
| 318.069
| 
|-
| 
| 7\79
| 106.329
| 212.658
| 318.987
| 
|-
| 
| 11\124
| 106.452
| 212.903
| 319.355
| 
|-
| 4\45
| 
| 106.667
| 213.333
| 320.000
| 
|-
| 5\56
| 
| 107.143
| 214.286
| 321.429
| 
|-
| 1\11
| 
| 109.091
| 218.182
| 327.273
| 
|}

[[Category:12-tone scales]]</text>
      <sha1>0wdn9oyvx6t10wy68kzgjuailma0vrw</sha1>
    </revision>
    <revision>
      <id>118926</id>
      <parentid>118925</parentid>
      <timestamp>2023-07-07T01:40:55Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Scale tree */</comment>
      <origin>118926</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2715" sha1="kctw4sc27fr2ako14v5dpx5qh2jvuxa" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
The '''11L 1s''' [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators-6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
! Comments
|-
| 1\12
| 
| 100.000
| 200.000
| 300.000
| Ripple
|-
| 5\59
| 
| 101.695
| 203.390
| 305.085
| 
|-
| 
| 9\106
| 101.887
| 203.774
| 305.660
| 
|-
| 
| 13\153
| 101.961
| 203.922
| 305.882
| 
|-
| 
| 17\200
| 102.000
| 204.000
| 306.000
| 
|-
| 4\47
| 
| 102.128
| 204.255
| 306.383
| 
|-
| 
| 7\82
| 102.439
| 204.878
| 307.317
| 
|-
| 
| 10\117
| 102.564
| 205.128
| 307.692
| 
|-
| 
| 13\152
| 102.632
| 205.263
| 307.895
| 
|-
| 
| 16\187
| 102.674
| 205.348
| 308.021
| 
|-
| 
| 19\222
| 102.703
| 205.405
| 308.108
| 
|-
| 3\35
| 
| 102.857
| 205.714
| 308.571
| 
|-
| 
| 20\233
| 103.004
| 206.009
| 309.013
| 
|-
| 
| 17\198
| 103.030
| 206.061
| 309.091
| 
|-
| 
| 14\163
| 103.067
| 206.135
| 309.202
| 
|-
| 
| 
| 103.123
| 206.246
| 309.368
| 
|-
| 
| 11\128
| 103.125
| 206.250
| 309.375
| 
|-
| 
| 8\93
| 103.226
| 206.452
| 309.677
| 
|-
| 
| 
| 103.288
| 206.575
| 309.863
| 
|-
| 
| 5\58
| 103.448
| 206.897
| 310.345
| 
|-
| 
| 
| 103.651
| 207.301
| 310.952
| 
|-
| 
| 7\81
| 103.703
| 207.407
| 311.111
| 
|-
| 
| 9\104
| 103.846
| 207.692
| 311.538
| 
|-
| 
| 11\127
| 103.937
| 207.874
| 311.811
| 
|-
| 
| 13\150
| 104.000
| 208.000
| 312.000
| 
|-
| 2\23
| 
| 104.348
| 208.969
| 313.043
| Ripple
|-
| 
| 11\126
| 104.762
| 209.524
| 314.286
| 
|-
| 
| 9\103
| 104.854
| 209.709
| 314.563
| 
|-
| 
| 7\80
| 105.000
| 210.000
| 315.000
| 
|-
| 
| 5\57
| 105.263
| 210.526
| 315.789
| 
|-
| 
| 
| 105.430
| 210.860
| 316.290
| 
|-
| 
| 8\91
| 105.495
| 210.989
| 316.484
| 
|-
| 
| 
| 105.561
| 211.121
| 316.682
| 
|-
| 
| 11\125
| 105.600
| 211.200
| 316.800
| 
|-
| 3\34
| 
| 105.822
| 211.765
| 317.647
| 
|-
| 
| 
| 106.023
| 212.046
| 318.069
| 
|-
| 
| 7\79
| 106.329
| 212.658
| 318.987
| 
|-
| 
| 11\124
| 106.452
| 212.903
| 319.355
| 
|-
| 4\45
| 
| 106.667
| 213.333
| 320.000
| 
|-
| 5\56
| 
| 107.143
| 214.286
| 321.429
| 
|-
| 1\11
| 
| 109.091
| 218.182
| 327.273
| 
|}

[[Category:12-tone scales]]</text>
      <sha1>kctw4sc27fr2ako14v5dpx5qh2jvuxa</sha1>
    </revision>
    <revision>
      <id>133693</id>
      <parentid>118926</parentid>
      <timestamp>2024-01-30T08:18:51Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>+intro +modes</comment>
      <origin>133693</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2748" sha1="7j0fwm63g374k4l5ctpjd4uf2n3oqwe" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
{{MOS intro}}
The 11L 1s [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators-6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.
== Modes ==
{{MOS modes}}
== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
! Comments
|-
| 1\12
| 
| 100.000
| 200.000
| 300.000
| Ripple
|-
| 5\59
| 
| 101.695
| 203.390
| 305.085
| 
|-
| 
| 9\106
| 101.887
| 203.774
| 305.660
| 
|-
| 
| 13\153
| 101.961
| 203.922
| 305.882
| 
|-
| 
| 17\200
| 102.000
| 204.000
| 306.000
| 
|-
| 4\47
| 
| 102.128
| 204.255
| 306.383
| 
|-
| 
| 7\82
| 102.439
| 204.878
| 307.317
| 
|-
| 
| 10\117
| 102.564
| 205.128
| 307.692
| 
|-
| 
| 13\152
| 102.632
| 205.263
| 307.895
| 
|-
| 
| 16\187
| 102.674
| 205.348
| 308.021
| 
|-
| 
| 19\222
| 102.703
| 205.405
| 308.108
| 
|-
| 3\35
| 
| 102.857
| 205.714
| 308.571
| 
|-
| 
| 20\233
| 103.004
| 206.009
| 309.013
| 
|-
| 
| 17\198
| 103.030
| 206.061
| 309.091
| 
|-
| 
| 14\163
| 103.067
| 206.135
| 309.202
| 
|-
| 
| 
| 103.123
| 206.246
| 309.368
| 
|-
| 
| 11\128
| 103.125
| 206.250
| 309.375
| 
|-
| 
| 8\93
| 103.226
| 206.452
| 309.677
| 
|-
| 
| 
| 103.288
| 206.575
| 309.863
| 
|-
| 
| 5\58
| 103.448
| 206.897
| 310.345
| 
|-
| 
| 
| 103.651
| 207.301
| 310.952
| 
|-
| 
| 7\81
| 103.703
| 207.407
| 311.111
| 
|-
| 
| 9\104
| 103.846
| 207.692
| 311.538
| 
|-
| 
| 11\127
| 103.937
| 207.874
| 311.811
| 
|-
| 
| 13\150
| 104.000
| 208.000
| 312.000
| 
|-
| 2\23
| 
| 104.348
| 208.969
| 313.043
| Ripple
|-
| 
| 11\126
| 104.762
| 209.524
| 314.286
| 
|-
| 
| 9\103
| 104.854
| 209.709
| 314.563
| 
|-
| 
| 7\80
| 105.000
| 210.000
| 315.000
| 
|-
| 
| 5\57
| 105.263
| 210.526
| 315.789
| 
|-
| 
| 
| 105.430
| 210.860
| 316.290
| 
|-
| 
| 8\91
| 105.495
| 210.989
| 316.484
| 
|-
| 
| 
| 105.561
| 211.121
| 316.682
| 
|-
| 
| 11\125
| 105.600
| 211.200
| 316.800
| 
|-
| 3\34
| 
| 105.822
| 211.765
| 317.647
| 
|-
| 
| 
| 106.023
| 212.046
| 318.069
| 
|-
| 
| 7\79
| 106.329
| 212.658
| 318.987
| 
|-
| 
| 11\124
| 106.452
| 212.903
| 319.355
| 
|-
| 4\45
| 
| 106.667
| 213.333
| 320.000
| 
|-
| 5\56
| 
| 107.143
| 214.286
| 321.429
| 
|-
| 1\11
| 
| 109.091
| 218.182
| 327.273
| 
|}

[[Category:12-tone scales]]</text>
      <sha1>7j0fwm63g374k4l5ctpjd4uf2n3oqwe</sha1>
    </revision>
    <revision>
      <id>133694</id>
      <parentid>133693</parentid>
      <timestamp>2024-01-30T08:19:46Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Scale tree */  + link to ripple</comment>
      <origin>133694</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2752" sha1="s8ihkwr47nvzpy19ksubqakyxpeuzsd" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
{{MOS intro}}
The 11L 1s [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators-6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.
== Modes ==
{{MOS modes}}
== Scale tree ==
{| class="wikitable"
|-
! colspan="2" | Generator
! Cents
! 2g
! 3g
! Comments
|-
| 1\12
| 
| 100.000
| 200.000
| 300.000
| [[Ripple]]
|-
| 5\59
| 
| 101.695
| 203.390
| 305.085
| 
|-
| 
| 9\106
| 101.887
| 203.774
| 305.660
| 
|-
| 
| 13\153
| 101.961
| 203.922
| 305.882
| 
|-
| 
| 17\200
| 102.000
| 204.000
| 306.000
| 
|-
| 4\47
| 
| 102.128
| 204.255
| 306.383
| 
|-
| 
| 7\82
| 102.439
| 204.878
| 307.317
| 
|-
| 
| 10\117
| 102.564
| 205.128
| 307.692
| 
|-
| 
| 13\152
| 102.632
| 205.263
| 307.895
| 
|-
| 
| 16\187
| 102.674
| 205.348
| 308.021
| 
|-
| 
| 19\222
| 102.703
| 205.405
| 308.108
| 
|-
| 3\35
| 
| 102.857
| 205.714
| 308.571
| 
|-
| 
| 20\233
| 103.004
| 206.009
| 309.013
| 
|-
| 
| 17\198
| 103.030
| 206.061
| 309.091
| 
|-
| 
| 14\163
| 103.067
| 206.135
| 309.202
| 
|-
| 
| 
| 103.123
| 206.246
| 309.368
| 
|-
| 
| 11\128
| 103.125
| 206.250
| 309.375
| 
|-
| 
| 8\93
| 103.226
| 206.452
| 309.677
| 
|-
| 
| 
| 103.288
| 206.575
| 309.863
| 
|-
| 
| 5\58
| 103.448
| 206.897
| 310.345
| 
|-
| 
| 
| 103.651
| 207.301
| 310.952
| 
|-
| 
| 7\81
| 103.703
| 207.407
| 311.111
| 
|-
| 
| 9\104
| 103.846
| 207.692
| 311.538
| 
|-
| 
| 11\127
| 103.937
| 207.874
| 311.811
| 
|-
| 
| 13\150
| 104.000
| 208.000
| 312.000
| 
|-
| 2\23
| 
| 104.348
| 208.969
| 313.043
| Ripple
|-
| 
| 11\126
| 104.762
| 209.524
| 314.286
| 
|-
| 
| 9\103
| 104.854
| 209.709
| 314.563
| 
|-
| 
| 7\80
| 105.000
| 210.000
| 315.000
| 
|-
| 
| 5\57
| 105.263
| 210.526
| 315.789
| 
|-
| 
| 
| 105.430
| 210.860
| 316.290
| 
|-
| 
| 8\91
| 105.495
| 210.989
| 316.484
| 
|-
| 
| 
| 105.561
| 211.121
| 316.682
| 
|-
| 
| 11\125
| 105.600
| 211.200
| 316.800
| 
|-
| 3\34
| 
| 105.822
| 211.765
| 317.647
| 
|-
| 
| 
| 106.023
| 212.046
| 318.069
| 
|-
| 
| 7\79
| 106.329
| 212.658
| 318.987
| 
|-
| 
| 11\124
| 106.452
| 212.903
| 319.355
| 
|-
| 4\45
| 
| 106.667
| 213.333
| 320.000
| 
|-
| 5\56
| 
| 107.143
| 214.286
| 321.429
| 
|-
| 1\11
| 
| 109.091
| 218.182
| 327.273
| 
|}

[[Category:12-tone scales]]</text>
      <sha1>s8ihkwr47nvzpy19ksubqakyxpeuzsd</sha1>
    </revision>
    <revision>
      <id>136110</id>
      <parentid>133694</parentid>
      <timestamp>2024-02-19T00:26:23Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>/* Scale tree */ Replaced scale tree, where ripple falls in scale tree may need clarification</comment>
      <origin>136110</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="838" sha1="8kfa4v9u9nkc2rm77f528gh4sew1kvb" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
{{MOS intro}}
The 11L 1s [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators-6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.
== Modes ==
{{MOS modes}}
== Scale tree ==
{{Scale tree|Comments=1/1: [[Ripple]] (for generators approaching this value)}}
[[Category:12-tone scales]]</text>
      <sha1>8kfa4v9u9nkc2rm77f528gh4sew1kvb</sha1>
    </revision>
    <revision>
      <id>136220</id>
      <parentid>136110</parentid>
      <timestamp>2024-02-19T06:41:27Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <minor/>
      <comment>/* Scale tree */ Added cent value for ripple</comment>
      <origin>136220</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="810" sha1="gb9ypjfuk0f7tfqtv4pjahjdy17k79r" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
{{MOS intro}}
The 11L 1s [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators-6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.
== Modes ==
{{MOS modes}}
== Scale tree ==
{{Scale tree|Comments=1/1: [[Ripple]] (100.838¢)}}
[[Category:12-tone scales]]</text>
      <sha1>gb9ypjfuk0f7tfqtv4pjahjdy17k79r</sha1>
    </revision>
    <revision>
      <id>170571</id>
      <parentid>136220</parentid>
      <timestamp>2024-12-15T06:56:57Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>== Intervals == {{MOS intervals}}</comment>
      <origin>170571</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="847" sha1="c4qtrm1f42umys4cdg3x0jvzwqyyigw" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
{{MOS intro}}
The 11L 1s [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators-6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Intervals ==
{{MOS intervals}}

== Modes ==
{{MOS modes}}

== Scale tree ==
{{Scale tree|Comments=1/1: [[Ripple]] (100.838¢)}}
[[Category:12-tone scales]]</text>
      <sha1>c4qtrm1f42umys4cdg3x0jvzwqyyigw</sha1>
    </revision>
    <revision>
      <id>170572</id>
      <parentid>170571</parentid>
      <timestamp>2024-12-15T06:57:25Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>170572</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="847" sha1="2lxoxw2mzhyxluhi608q5f2jit2pygy" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
{{MOS intro}}
The 11L 1s [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators-6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree|Comments=1/1: [[Ripple]] (100.838¢)}}
[[Category:12-tone scales]]</text>
      <sha1>2lxoxw2mzhyxluhi608q5f2jit2pygy</sha1>
    </revision>
    <revision>
      <id>183729</id>
      <parentid>170572</parentid>
      <timestamp>2025-02-28T16:38:20Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183729</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="852" sha1="r3yqwrzjg36pbc21kqytq2jc7iprkde" xml:space="preserve">{{Infobox MOS
| Name = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 1
| Equalized = 1
| Collapsed = 1
| Pattern = LLLLLLLLLLLs
}}
{{MOS intro}}
The 11L 1s [[MOS scale]], with its diatonic minor second generator, is the dodecatonic scale of [[ripple]] (9:8 = +2 generators-6:5 = +3 generators) temperaments. However, it feels markedly more hendecatonic to listeners used to the equally divided chromatic scale because of its unbroken chain of eleven equal steps–never mind that the 12th degree is actually less than the octave. This hendecatonic feel is especially pronounced above 2\23, where the octave feels like a comma shift of the 12th degree rather than an independent note.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 1/1 = [[Ripple]] (100.838¢)
}}

[[Category:12-tone scales]]</text>
      <sha1>r3yqwrzjg36pbc21kqytq2jc7iprkde</sha1>
    </revision>
  </page>
  <page>
    <title>11L 2s</title>
    <ns>0</ns>
    <id>94</id>
    <revision>
      <id>94</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>94</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="751" sha1="frr80hsoe4uzp15ouihpkgsmgo5s82e" xml:space="preserve">This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.

{| class="wikitable"
|-
| | 5\11
| | 
| | 545.5455
|-
| | 26\57
| | 
| | 547.368
|-
| | 21\46
| | 
| | 547.826
|-
| | 
| | 
| | 548.439
|-
| | 16\35
| | 
| | 548.571
|-
| | 
| | 
| | 548.874
|-
| | 
| | 43\94
| | 548.936
|-
| | 
| | 
| | 548.997
|-
| | 
| | 27\59
| | 549.1525
|-
| | 11\24
| | 
| | 550
|-
| | 
| | 
| | 550.636
|-
| | 
| | 28\61
| | 550.82
|-
| | 
| | 
| | 550.965
|-
| | 
| | 45\98
| | 551.02
|-
| | 
| | 
| | 551.113
|-
| | 17\37
| | 
| | 551.351
|-
| | 23\50
| | 
| | 552
|-
| | 29\63
| | 
| | 552.381
|-
| | 6\13
| | 
| | 553.846
|}</text>
      <sha1>frr80hsoe4uzp15ouihpkgsmgo5s82e</sha1>
    </revision>
    <revision>
      <id>4456</id>
      <timestamp>2015-11-13T15:30:34Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 566387013 - Original comment: **</comment>
      <origin>4456</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5405" sha1="821nr79nh3lu47koaeg2y1c6bpbr40s" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-13 15:30:34 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566387013&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.
|| 5\11 ||   || 545.5455 ||
|| 26\57 ||   || 547.368 ||
|| 21\46 ||   || 547.826 ||
||   ||   || 548.439 ||
|| 16\35 ||   || 548.571 ||
||   ||   || 548.874 ||
||   || 43\94 || 548.936 ||
||   ||   || 548.997 ||
||   || 27\59 || 549.1525 ||
|| 11\24 ||   || 550 ||
||   ||   || 550.636 ||
||   || 28\61 || 550.82 ||
||   ||   || 550.965 ||
||   || 45\98 || 551.02 ||
||   ||   || 551.113 ||
|| 17\37 ||   || 551.351 ||
|| 23\50 ||   || 552 ||
|| 29\63 ||   || 552.381 ||
|| 6\13 ||   || 553.846 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.5455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26\57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.368&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21\46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.826&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.439&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16\35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.874&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43\94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.936&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.997&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27\59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;549.1525&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11\24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;550&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;550.636&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28\61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;550.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;550.965&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;45\98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17\37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.351&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;23\50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;29\63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552.381&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6\13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;553.846&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>821nr79nh3lu47koaeg2y1c6bpbr40s</sha1>
    </revision>
    <revision>
      <id>4457</id>
      <timestamp>2015-10-19T17:41:03Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 563036079 - Original comment: **</comment>
      <origin>4457</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4825" sha1="o767jknz0gygp7fimxzleyjnmbng4lv" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-10-19 17:41:03 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;563036079&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.
|| 5\11 ||   || 545.5455 ||
|| 26\57 ||   || 547.368 ||
|| 21\46 ||   || 547.826 ||
||   ||   || 548.439 ||
|| 16\35 ||   || 548.571 ||
||   ||   || 548.874 ||
||   || 43\94 || 548.936 ||
||   ||   || 548.997 ||
||   || 27\59 || 549.1525 ||
|| 11\24 ||   || 550 ||
||   || 28\61 || 550.82 ||
||   ||   || 550.965 ||
|| 17\37 ||   || 551.351 ||
|| 23\50 ||   || 552 ||
|| 29\63 ||   || 552.381 ||
|| 6\13 ||   || 553.846 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.5455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26\57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.368&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21\46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.826&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.439&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16\35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.874&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43\94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.936&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.997&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27\59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;549.1525&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11\24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;550&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28\61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;550.82&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;550.965&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17\37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.351&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;23\50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;29\63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552.381&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6\13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;553.846&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>o767jknz0gygp7fimxzleyjnmbng4lv</sha1>
    </revision>
    <revision>
      <id>4458</id>
      <timestamp>2015-05-23T22:40:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 551986292 - Original comment: **</comment>
      <origin>4458</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4045" sha1="03ajrbwhbsnkxalbnor77rntw67hdbm" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-23 22:40:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;551986292&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.
|| 5\11 ||   || 545.5455 ||
|| 26\57 ||   || 547.368 ||
|| 21\46 ||   || 547.826 ||
||   ||   || 548.439 ||
|| 16\35 ||   || 548.571 ||
||   ||   || 548.874 ||
||   || 27\59 || 549.1525 ||
|| 11\24 ||   || 550 ||
|| 17\37 ||   || 551.351 ||
|| 23\50 ||   || 552 ||
|| 29\63 ||   || 552.381 ||
|| 6\13 ||   || 553.846 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.5455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26\57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.368&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21\46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.826&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.439&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16\35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.874&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27\59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;549.1525&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11\24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;550&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17\37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.351&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;23\50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;29\63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552.381&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6\13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;553.846&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>03ajrbwhbsnkxalbnor77rntw67hdbm</sha1>
    </revision>
    <revision>
      <id>4459</id>
      <timestamp>2015-02-14T19:58:33Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540988936 - Original comment: **</comment>
      <origin>4459</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4045" sha1="2ufyrz1qnzmxtjkjyjrrteoyee2cel2" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-14 19:58:33 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540988936&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.
|| 5\11 ||   || 545.5455 ||
|| 26\57 ||   || 547.368 ||
|| 21\46 ||   || 547.826 ||
||   ||   || 548.544 ||
|| 16\35 ||   || 548.571 ||
||   ||   || 548.599 ||
||   || 27\59 || 549.1525 ||
|| 11\24 ||   || 550 ||
|| 17\37 ||   || 551.351 ||
|| 23\50 ||   || 552 ||
|| 29\63 ||   || 552.381 ||
|| 6\13 ||   || 553.846 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;545.5455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;26\57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.368&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;21\46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;547.826&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.544&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16\35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;548.599&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27\59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;549.1525&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11\24&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;550&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;17\37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;551.351&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;23\50&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;29\63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;552.381&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6\13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;553.846&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>2ufyrz1qnzmxtjkjyjrrteoyee2cel2</sha1>
    </revision>
    <revision>
      <id>4460</id>
      <timestamp>2015-02-14T19:30:24Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540988138 - Original comment: **</comment>
      <origin>4460</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1615" sha1="jcgxoo6244ts8832i7x66rvrbop0inu" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-14 19:30:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540988138&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 2s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>jcgxoo6244ts8832i7x66rvrbop0inu</sha1>
    </revision>
    <revision>
      <id>39202</id>
      <parentid>94</parentid>
      <timestamp>2018-12-30T04:36:46Z</timestamp>
      <contributor>
        <username>Spt3125</username>
        <id>16</id>
      </contributor>
      <comment>new table</comment>
      <origin>39202</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2024" sha1="lln9djamzug4mcdwadh7eb7niyo1axt" xml:space="preserve">This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.

{| class="wikitable"
! colspan=6 | generator
! | L
! | s
! | L/s
! | gen (cents)
! | comment
|-
| | 5\11
| |
| |
| |
| |
| |
| | 1
| | 0
| |
| | 545.455
| |
|-
| |
| |
| |
| |
| | &lt;sup&gt;&amp;rarr;&lt;/sup&gt;41\90
| |
| | 8
| | 1
| | 8.000
| | 546.667
| |
|-
| |
| |
| |
| |
| | &lt;sup&gt;&amp;rarr;&lt;/sup&gt;36\79
| |
| | 7
| | 1
| | 7.000
| | 546.835
| |
|-
| |
| |
| |
| |
| | &lt;sup&gt;&amp;rarr;&lt;/sup&gt;31\68
| |
| | 6
| | 1
| | 6.000
| | 547.059
| |
|-
| |
| |
| |
| |
| | 26\57
| |
| | 5
| | 1
| | 5.000
| | 547.368
| |
|-
| |
| |
| |
| | 21\46
| |
| |
| | 4
| | 1
| | 4.000
| | 547.826
| |
|-
| |
| |
| |
| |
| | 37\81
| |
| | 7
| | 2
| | 3.500
| | 548.148
| |
|-
| |
| |
| | 16\35
| |
| |
| |
| | 3
| | 1
| | 3.000
| | 548.571
| |
|-
| |
| |
| |
| |
| | 43\94
| |
| | 8
| | 3
| | 2.667
| | 548.936
| |
|-
| |
| |
| |
| | 27\59
| |
| |
| | 5
| | 2
| | 2.500
| | 549.153
| |
|-
| |
| |
| |
| |
| | 38\83
| |
| | 7
| | 3
| | 2.333
| | 549.398
| |
|-
| |
| | 11\24
| |
| |
| |
| |
| | 2
| | 1
| | 2.000
| | 550.000
| |
|-
| |
| |
| |
| |
| | 39\85
| |
| | 7
| | 4
| | 1.750
| | 550.588
| |
|-
| |
| |
| |
| | 28\61
| |
| |
| | 5
| | 3
| | 1.667
| | 550.820
| |
|-
| |
| |
| |
| |
| |
| | (5&amp;phi;+1)/(11&amp;phi;+2)
| | &amp;phi;
| | 1
| | 1.618
| | 550.965
| |
|-
| |
| |
| |
| |
| | 45\98
| |
| | 8
| | 5
| | 1.600
| | 551.020
| |
|-
| |
| |
| | 17\37
| |
| |
| |
| | 3
| | 2
| | 1.500
| | 551.351
| |
|-
| |
| |
| |
| |
| | 40\87
| |
| | 7
| | 5
| | 1.400
| | 551.724
| |
|-
| |
| |
| |
| | 23\50
| |
| |
| | 4
| | 3
| | 1.333
| | 552.000
| |
|-
| |
| |
| |
| |
| | 29\63
| |
| | 5
| | 4
| | 1.250
| | 552.381
| |
|-
| |
| |
| |
| |
| | &lt;sup&gt;&amp;rarr;&lt;/sup&gt;35\76
| |
| | 6
| | 5
| | 1.200
| | 552.632
| |
|-
| |
| |
| |
| |
| | &lt;sup&gt;&amp;rarr;&lt;/sup&gt;41\89
| |
| | 7
| | 6
| | 1.167
| | 552.809
| |
|-
| | 6\13
| |
| |
| |
| |
| |
| | 1
| | 1
| | 1.000
| | 553.846
| |
|}</text>
      <sha1>lln9djamzug4mcdwadh7eb7niyo1axt</sha1>
    </revision>
    <revision>
      <id>39920</id>
      <parentid>39202</parentid>
      <timestamp>2019-02-01T23:28:41Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>39920</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2077" sha1="l6286dyvv6bl1j7bhu4yo4ok5pngmur" xml:space="preserve">This is the mega chromatic scale of Heinz temperament. Its generator of 5\11edo to 6\13edo hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.

{| class="wikitable"
! colspan="8" | generator
! | L
! | s
! | L/s
! | gen (cents)
! | comment
|-
| | 5\11
| |
| |
| |
| |
| |
|
|
| | 1
| | 0
| |
| | 545.455
| |
|-
| |
| |
| |
| |
| |
| |
|
|41\90
| | 8
| | 1
| | 8.000
| | 546.667
| |
|-
| |
| |
| |
| |
| |
| |
|36\79
|
| | 7
| | 1
| | 7.000
| | 546.835
| |
|-
| |
| |
| |
| |
| |
| |31\68
|
|
| | 6
| | 1
| | 6.000
| | 547.059
| |
|-
| |
| |
| |
| |
| | 26\57
| |
|
|
| | 5
| | 1
| | 5.000
| | 547.368
| |
|-
| |
| |
| |
| | 21\46
| |
| |
|
|
| | 4
| | 1
| | 4.000
| | 547.826
| |
|-
| |
| |
| |
| |
| | 37\81
| |
|
|
| | 7
| | 2
| | 3.500
| | 548.148
| |
|-
| |
| |
| | 16\35
| |
| |
| |
|
|
| | 3
| | 1
| | 3.000
| | 548.571
| |
|-
| |
| |
| |
| |
| | 43\94
| |
|
|
| | 8
| | 3
| | 2.667
| | 548.936
| |
|-
| |
| |
| |
| | 27\59
| |
| |
|
|
| | 5
| | 2
| | 2.500
| | 549.153
| |
|-
| |
| |
| |
| |
| | 38\83
| |
|
|
| | 7
| | 3
| | 2.333
| | 549.398
| |
|-
| |
| | 11\24
| |
| |
| |
| |
|
|
| | 2
| | 1
| | 2.000
| | 550.000
| |
|-
| |
| |
| |
| |
| | 39\85
| |
|
|
| | 7
| | 4
| | 1.750
| | 550.588
| |
|-
| |
| |
| |
| | 28\61
| |
| |
|
|
| | 5
| | 3
| | 1.667
| | 550.820
| |
|-
| |
| |
| |
| |
| |
| | (5&amp;phi;+1)/(11&amp;phi;+2)
|
|
| | &amp;phi;
| | 1
| | 1.618
| | 550.965
| |
|-
| |
| |
| |
| |
| | 45\98
| |
|
|
| | 8
| | 5
| | 1.600
| | 551.020
| |
|-
| |
| |
| | 17\37
| |
| |
| |
|
|
| | 3
| | 2
| | 1.500
| | 551.351
| |
|-
| |
| |
| |
| |
| | 40\87
| |
|
|
| | 7
| | 5
| | 1.400
| | 551.724
| |
|-
| |
| |
| |
| | 23\50
| |
| |
|
|
| | 4
| | 3
| | 1.333
| | 552.000
| |
|-
| |
| |
| |
| |
| | 29\63
| |
|
|
| | 5
| | 4
| | 1.250
| | 552.381
| |
|-
| |
| |
| |
| |
| |
| |35\76
|
|
| | 6
| | 5
| | 1.200
| | 552.632
| |
|-
| |
| |
| |
| |
| |
| |
|41\89
|
| | 7
| | 6
| | 1.167
| | 552.809
| |
|-
|
|
|
|
|
|
|
|47\102
|8
|7
|1.125
|552.941
|
|-
| | 6\13
| |
| |
| |
| |
| |
|
|
| | 1
| | 1
| | 1.000
| | 553.846
| |
|}</text>
      <sha1>l6286dyvv6bl1j7bhu4yo4ok5pngmur</sha1>
    </revision>
    <revision>
      <id>94372</id>
      <parentid>39920</parentid>
      <timestamp>2022-08-15T20:21:23Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox, name still missing, misc. edits</comment>
      <origin>94372</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1805" sha1="cmj5733vy41pgekrjfyyadoxgt7sfds" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 6
| Paucitonic = 5
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.

== Scale tree ==
{| class="wikitable"
! colspan="8" | generator
! L
! s
! L/s
! gen (cents)
! comment
|-
| 5\11
|
|
|
|
|
|
|
| 1
| 0
|
| 545.455
|
|-
|
|
|
|
|
|
|
| 41\90
| 8
| 1
| 8.000
| 546.667
|
|-
|
|
|
|
|
|
| 36\79
|
| 7
| 1
| 7.000
| 546.835
|
|-
|
|
|
|
|
| 31\68
|
|
| 6
| 1
| 6.000
| 547.059
|
|-
|
|
|
|
| 26\57
|
|
|
| 5
| 1
| 5.000
| 547.368
|
|-
|
|
|
| 21\46
|
|
|
|
| 4
| 1
| 4.000
| 547.826
| Heinz is around here
|-
|
|
|
|
| 37\81
|
|
|
| 7
| 2
| 3.500
| 548.148
|
|-
|
|
| 16\35
|
|
|
|
|
| 3
| 1
| 3.000
| 548.571
|
|-
|
|
|
|
| 43\94
|
|
|
| 8
| 3
| 2.667
| 548.936
|
|-
|
|
|
| 27\59
|
|
|
|
| 5
| 2
| 2.500
| 549.153
|
|-
|
|
|
|
| 38\83
|
|
|
| 7
| 3
| 2.333
| 549.398
|
|-
|
| 11\24
|
|
|
|
|
|
| 2
| 1
| 2.000
| 550.000
|
|-
|
|
|
|
| 39\85
|
|
|
| 7
| 4
| 1.750
| 550.588
|
|-
|
|
|
| 28\61
|
|
|
|
| 5
| 3
| 1.667
| 550.820
|
|-
|
|
|
|
|
| (5&amp;phi;+1)/(11&amp;phi;+2)
|
|
| &amp;phi;
| 1
| 1.618
| 550.965
|
|-
|
|
|
|
| 45\98
|
|
|
| 8
| 5
| 1.600
| 551.020
|
|-
|
|
| 17\37
|
|
|
|
|
| 3
| 2
| 1.500
| 551.351
|
|-
|
|
|
|
| 40\87
|
|
|
| 7
| 5
| 1.400
| 551.724
|
|-
|
|
|
| 23\50
|
|
|
|
| 4
| 3
| 1.333
| 552.000
|
|-
|
|
|
|
| 29\63
|
|
|
| 5
| 4
| 1.250
| 552.381
|
|-
|
|
|
|
|
| 35\76
|
|
| 6
| 5
| 1.200
| 552.632
|
|-
|
|
|
|
|
|
| 41\89
|
| 7
| 6
| 1.167
| 552.809
|
|-
|
|
|
|
|
|
|
| 47\102
| 8
| 7
| 1.125
| 552.941
|
|-
| 6\13
|
|
|
|
|
|
|
| 1
| 1
| 1.000
| 553.846
|
|}</text>
      <sha1>cmj5733vy41pgekrjfyyadoxgt7sfds</sha1>
    </revision>
    <revision>
      <id>94508</id>
      <parentid>94372</parentid>
      <timestamp>2022-08-17T09:20:10Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94508</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1804" sha1="jndl3a6zzx34tdfht3s48gio2p6zjdp" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 6
| Collapsed = 5
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.

== Scale tree ==
{| class="wikitable"
! colspan="8" | generator
! L
! s
! L/s
! gen (cents)
! comment
|-
| 5\11
|
|
|
|
|
|
|
| 1
| 0
|
| 545.455
|
|-
|
|
|
|
|
|
|
| 41\90
| 8
| 1
| 8.000
| 546.667
|
|-
|
|
|
|
|
|
| 36\79
|
| 7
| 1
| 7.000
| 546.835
|
|-
|
|
|
|
|
| 31\68
|
|
| 6
| 1
| 6.000
| 547.059
|
|-
|
|
|
|
| 26\57
|
|
|
| 5
| 1
| 5.000
| 547.368
|
|-
|
|
|
| 21\46
|
|
|
|
| 4
| 1
| 4.000
| 547.826
| Heinz is around here
|-
|
|
|
|
| 37\81
|
|
|
| 7
| 2
| 3.500
| 548.148
|
|-
|
|
| 16\35
|
|
|
|
|
| 3
| 1
| 3.000
| 548.571
|
|-
|
|
|
|
| 43\94
|
|
|
| 8
| 3
| 2.667
| 548.936
|
|-
|
|
|
| 27\59
|
|
|
|
| 5
| 2
| 2.500
| 549.153
|
|-
|
|
|
|
| 38\83
|
|
|
| 7
| 3
| 2.333
| 549.398
|
|-
|
| 11\24
|
|
|
|
|
|
| 2
| 1
| 2.000
| 550.000
|
|-
|
|
|
|
| 39\85
|
|
|
| 7
| 4
| 1.750
| 550.588
|
|-
|
|
|
| 28\61
|
|
|
|
| 5
| 3
| 1.667
| 550.820
|
|-
|
|
|
|
|
| (5&amp;phi;+1)/(11&amp;phi;+2)
|
|
| &amp;phi;
| 1
| 1.618
| 550.965
|
|-
|
|
|
|
| 45\98
|
|
|
| 8
| 5
| 1.600
| 551.020
|
|-
|
|
| 17\37
|
|
|
|
|
| 3
| 2
| 1.500
| 551.351
|
|-
|
|
|
|
| 40\87
|
|
|
| 7
| 5
| 1.400
| 551.724
|
|-
|
|
|
| 23\50
|
|
|
|
| 4
| 3
| 1.333
| 552.000
|
|-
|
|
|
|
| 29\63
|
|
|
| 5
| 4
| 1.250
| 552.381
|
|-
|
|
|
|
|
| 35\76
|
|
| 6
| 5
| 1.200
| 552.632
|
|-
|
|
|
|
|
|
| 41\89
|
| 7
| 6
| 1.167
| 552.809
|
|-
|
|
|
|
|
|
|
| 47\102
| 8
| 7
| 1.125
| 552.941
|
|-
| 6\13
|
|
|
|
|
|
|
| 1
| 1
| 1.000
| 553.846
|
|}</text>
      <sha1>jndl3a6zzx34tdfht3s48gio2p6zjdp</sha1>
    </revision>
    <revision>
      <id>100976</id>
      <parentid>94508</parentid>
      <timestamp>2023-01-06T15:44:02Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Fix equalized/collapsed in infobox</comment>
      <origin>100976</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1804" sha1="pbp4ltu85l6c74gysyyrv39j8yabeia" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 7
| Collapsed = 6
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.

== Scale tree ==
{| class="wikitable"
! colspan="8" | generator
! L
! s
! L/s
! gen (cents)
! comment
|-
| 5\11
|
|
|
|
|
|
|
| 1
| 0
|
| 545.455
|
|-
|
|
|
|
|
|
|
| 41\90
| 8
| 1
| 8.000
| 546.667
|
|-
|
|
|
|
|
|
| 36\79
|
| 7
| 1
| 7.000
| 546.835
|
|-
|
|
|
|
|
| 31\68
|
|
| 6
| 1
| 6.000
| 547.059
|
|-
|
|
|
|
| 26\57
|
|
|
| 5
| 1
| 5.000
| 547.368
|
|-
|
|
|
| 21\46
|
|
|
|
| 4
| 1
| 4.000
| 547.826
| Heinz is around here
|-
|
|
|
|
| 37\81
|
|
|
| 7
| 2
| 3.500
| 548.148
|
|-
|
|
| 16\35
|
|
|
|
|
| 3
| 1
| 3.000
| 548.571
|
|-
|
|
|
|
| 43\94
|
|
|
| 8
| 3
| 2.667
| 548.936
|
|-
|
|
|
| 27\59
|
|
|
|
| 5
| 2
| 2.500
| 549.153
|
|-
|
|
|
|
| 38\83
|
|
|
| 7
| 3
| 2.333
| 549.398
|
|-
|
| 11\24
|
|
|
|
|
|
| 2
| 1
| 2.000
| 550.000
|
|-
|
|
|
|
| 39\85
|
|
|
| 7
| 4
| 1.750
| 550.588
|
|-
|
|
|
| 28\61
|
|
|
|
| 5
| 3
| 1.667
| 550.820
|
|-
|
|
|
|
|
| (5&amp;phi;+1)/(11&amp;phi;+2)
|
|
| &amp;phi;
| 1
| 1.618
| 550.965
|
|-
|
|
|
|
| 45\98
|
|
|
| 8
| 5
| 1.600
| 551.020
|
|-
|
|
| 17\37
|
|
|
|
|
| 3
| 2
| 1.500
| 551.351
|
|-
|
|
|
|
| 40\87
|
|
|
| 7
| 5
| 1.400
| 551.724
|
|-
|
|
|
| 23\50
|
|
|
|
| 4
| 3
| 1.333
| 552.000
|
|-
|
|
|
|
| 29\63
|
|
|
| 5
| 4
| 1.250
| 552.381
|
|-
|
|
|
|
|
| 35\76
|
|
| 6
| 5
| 1.200
| 552.632
|
|-
|
|
|
|
|
|
| 41\89
|
| 7
| 6
| 1.167
| 552.809
|
|-
|
|
|
|
|
|
|
| 47\102
| 8
| 7
| 1.125
| 552.941
|
|-
| 6\13
|
|
|
|
|
|
|
| 1
| 1
| 1.000
| 553.846
|
|}</text>
      <sha1>pbp4ltu85l6c74gysyyrv39j8yabeia</sha1>
    </revision>
    <revision>
      <id>101028</id>
      <parentid>100976</parentid>
      <timestamp>2023-01-06T16:14:29Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Copying from [[Category:Abstract MOS patterns]] to [[Category:13-tone scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>101028</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1833" sha1="eir52nbb5fgm0bw8lq28mhinlx8id2p" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 7
| Collapsed = 6
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.

== Scale tree ==
{| class="wikitable"
! colspan="8" | generator
! L
! s
! L/s
! gen (cents)
! comment
|-
| 5\11
|
|
|
|
|
|
|
| 1
| 0
|
| 545.455
|
|-
|
|
|
|
|
|
|
| 41\90
| 8
| 1
| 8.000
| 546.667
|
|-
|
|
|
|
|
|
| 36\79
|
| 7
| 1
| 7.000
| 546.835
|
|-
|
|
|
|
|
| 31\68
|
|
| 6
| 1
| 6.000
| 547.059
|
|-
|
|
|
|
| 26\57
|
|
|
| 5
| 1
| 5.000
| 547.368
|
|-
|
|
|
| 21\46
|
|
|
|
| 4
| 1
| 4.000
| 547.826
| Heinz is around here
|-
|
|
|
|
| 37\81
|
|
|
| 7
| 2
| 3.500
| 548.148
|
|-
|
|
| 16\35
|
|
|
|
|
| 3
| 1
| 3.000
| 548.571
|
|-
|
|
|
|
| 43\94
|
|
|
| 8
| 3
| 2.667
| 548.936
|
|-
|
|
|
| 27\59
|
|
|
|
| 5
| 2
| 2.500
| 549.153
|
|-
|
|
|
|
| 38\83
|
|
|
| 7
| 3
| 2.333
| 549.398
|
|-
|
| 11\24
|
|
|
|
|
|
| 2
| 1
| 2.000
| 550.000
|
|-
|
|
|
|
| 39\85
|
|
|
| 7
| 4
| 1.750
| 550.588
|
|-
|
|
|
| 28\61
|
|
|
|
| 5
| 3
| 1.667
| 550.820
|
|-
|
|
|
|
|
| (5&amp;phi;+1)/(11&amp;phi;+2)
|
|
| &amp;phi;
| 1
| 1.618
| 550.965
|
|-
|
|
|
|
| 45\98
|
|
|
| 8
| 5
| 1.600
| 551.020
|
|-
|
|
| 17\37
|
|
|
|
|
| 3
| 2
| 1.500
| 551.351
|
|-
|
|
|
|
| 40\87
|
|
|
| 7
| 5
| 1.400
| 551.724
|
|-
|
|
|
| 23\50
|
|
|
|
| 4
| 3
| 1.333
| 552.000
|
|-
|
|
|
|
| 29\63
|
|
|
| 5
| 4
| 1.250
| 552.381
|
|-
|
|
|
|
|
| 35\76
|
|
| 6
| 5
| 1.200
| 552.632
|
|-
|
|
|
|
|
|
| 41\89
|
| 7
| 6
| 1.167
| 552.809
|
|-
|
|
|
|
|
|
|
| 47\102
| 8
| 7
| 1.125
| 552.941
|
|-
| 6\13
|
|
|
|
|
|
|
| 1
| 1
| 1.000
| 553.846
|
|}

[[Category:13-tone scales]]</text>
      <sha1>eir52nbb5fgm0bw8lq28mhinlx8id2p</sha1>
    </revision>
    <revision>
      <id>102469</id>
      <parentid>101028</parentid>
      <timestamp>2023-02-05T15:43:44Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>this fact is the most prominent about 11L 2s, why it isn't at the top of the page</comment>
      <origin>102469</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2032" sha1="eq6yhy2uxqxi7pbphln90h2hyix1313" xml:space="preserve">{{Infobox MOS
| Other names =Wyschnegradsky's&lt;br&gt; diatonicized chromatic 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 7
| Collapsed = 6
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is most notable for being used by [[Ivan Wyschnegradsky]] and having a name "diatonicized chromatic scale".

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.

== Scale tree ==
{| class="wikitable"
! colspan="8" | generator
! L
! s
! L/s
! gen (cents)
! comment
|-
| 5\11
|
|
|
|
|
|
|
| 1
| 0
|
| 545.455
|
|-
|
|
|
|
|
|
|
| 41\90
| 8
| 1
| 8.000
| 546.667
|
|-
|
|
|
|
|
|
| 36\79
|
| 7
| 1
| 7.000
| 546.835
|
|-
|
|
|
|
|
| 31\68
|
|
| 6
| 1
| 6.000
| 547.059
|
|-
|
|
|
|
| 26\57
|
|
|
| 5
| 1
| 5.000
| 547.368
|
|-
|
|
|
| 21\46
|
|
|
|
| 4
| 1
| 4.000
| 547.826
| Heinz is around here
|-
|
|
|
|
| 37\81
|
|
|
| 7
| 2
| 3.500
| 548.148
|
|-
|
|
| 16\35
|
|
|
|
|
| 3
| 1
| 3.000
| 548.571
|
|-
|
|
|
|
| 43\94
|
|
|
| 8
| 3
| 2.667
| 548.936
|
|-
|
|
|
| 27\59
|
|
|
|
| 5
| 2
| 2.500
| 549.153
|
|-
|
|
|
|
| 38\83
|
|
|
| 7
| 3
| 2.333
| 549.398
|
|-
|
| 11\24
|
|
|
|
|
|
| 2
| 1
| 2.000
| 550.000
|
|-
|
|
|
|
| 39\85
|
|
|
| 7
| 4
| 1.750
| 550.588
|
|-
|
|
|
| 28\61
|
|
|
|
| 5
| 3
| 1.667
| 550.820
|
|-
|
|
|
|
|
| (5&amp;phi;+1)/(11&amp;phi;+2)
|
|
| &amp;phi;
| 1
| 1.618
| 550.965
|
|-
|
|
|
|
| 45\98
|
|
|
| 8
| 5
| 1.600
| 551.020
|
|-
|
|
| 17\37
|
|
|
|
|
| 3
| 2
| 1.500
| 551.351
|
|-
|
|
|
|
| 40\87
|
|
|
| 7
| 5
| 1.400
| 551.724
|
|-
|
|
|
| 23\50
|
|
|
|
| 4
| 3
| 1.333
| 552.000
|
|-
|
|
|
|
| 29\63
|
|
|
| 5
| 4
| 1.250
| 552.381
|
|-
|
|
|
|
|
| 35\76
|
|
| 6
| 5
| 1.200
| 552.632
|
|-
|
|
|
|
|
|
| 41\89
|
| 7
| 6
| 1.167
| 552.809
|
|-
|
|
|
|
|
|
|
| 47\102
| 8
| 7
| 1.125
| 552.941
|
|-
| 6\13
|
|
|
|
|
|
|
| 1
| 1
| 1.000
| 553.846
|
|}

[[Category:13-tone scales]]</text>
      <sha1>eq6yhy2uxqxi7pbphln90h2hyix1313</sha1>
    </revision>
    <revision>
      <id>103329</id>
      <parentid>102469</parentid>
      <timestamp>2023-02-20T01:03:57Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>103329</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2120" sha1="t5x1wuszeatj8rg5p78n36713v30r5h" xml:space="preserve">{{Infobox MOS
| Other names =hendecoid, Wyschnegradsky's&lt;br&gt; diatonicized chromatic 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 7
| Collapsed = 6
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is most notable for being used by [[Ivan Wyschnegradsky]] and having a name "diatonicized chromatic scale". The more concise name for the scale, proposed by Eliora, is '''hendecoid'''.

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale.

== Scale tree ==
{| class="wikitable"
! colspan="8" | generator
! L
! s
! L/s
! gen (cents)
! comment
|-
| 5\11
|
|
|
|
|
|
|
| 1
| 0
|
| 545.455
|
|-
|
|
|
|
|
|
|
| 41\90
| 8
| 1
| 8.000
| 546.667
|
|-
|
|
|
|
|
|
| 36\79
|
| 7
| 1
| 7.000
| 546.835
|
|-
|
|
|
|
|
| 31\68
|
|
| 6
| 1
| 6.000
| 547.059
|
|-
|
|
|
|
| 26\57
|
|
|
| 5
| 1
| 5.000
| 547.368
|
|-
|
|
|
| 21\46
|
|
|
|
| 4
| 1
| 4.000
| 547.826
| Heinz is around here
|-
|
|
|
|
| 37\81
|
|
|
| 7
| 2
| 3.500
| 548.148
|
|-
|
|
| 16\35
|
|
|
|
|
| 3
| 1
| 3.000
| 548.571
|
|-
|
|
|
|
| 43\94
|
|
|
| 8
| 3
| 2.667
| 548.936
|
|-
|
|
|
| 27\59
|
|
|
|
| 5
| 2
| 2.500
| 549.153
|
|-
|
|
|
|
| 38\83
|
|
|
| 7
| 3
| 2.333
| 549.398
|
|-
|
| 11\24
|
|
|
|
|
|
| 2
| 1
| 2.000
| 550.000
|
|-
|
|
|
|
| 39\85
|
|
|
| 7
| 4
| 1.750
| 550.588
|
|-
|
|
|
| 28\61
|
|
|
|
| 5
| 3
| 1.667
| 550.820
|
|-
|
|
|
|
|
| (5&amp;phi;+1)/(11&amp;phi;+2)
|
|
| &amp;phi;
| 1
| 1.618
| 550.965
|
|-
|
|
|
|
| 45\98
|
|
|
| 8
| 5
| 1.600
| 551.020
|
|-
|
|
| 17\37
|
|
|
|
|
| 3
| 2
| 1.500
| 551.351
|
|-
|
|
|
|
| 40\87
|
|
|
| 7
| 5
| 1.400
| 551.724
|
|-
|
|
|
| 23\50
|
|
|
|
| 4
| 3
| 1.333
| 552.000
|
|-
|
|
|
|
| 29\63
|
|
|
| 5
| 4
| 1.250
| 552.381
|
|-
|
|
|
|
|
| 35\76
|
|
| 6
| 5
| 1.200
| 552.632
|
|-
|
|
|
|
|
|
| 41\89
|
| 7
| 6
| 1.167
| 552.809
|
|-
|
|
|
|
|
|
|
| 47\102
| 8
| 7
| 1.125
| 552.941
|
|-
| 6\13
|
|
|
|
|
|
|
| 1
| 1
| 1.000
| 553.846
|
|}

[[Category:13-tone scales]]</text>
      <sha1>t5x1wuszeatj8rg5p78n36713v30r5h</sha1>
    </revision>
    <revision>
      <id>107213</id>
      <parentid>103329</parentid>
      <timestamp>2023-04-10T11:55:18Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>107213</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2202" sha1="0ryullh78hby7yxw1fx71liy5lcp9ko" xml:space="preserve">{{Infobox MOS
| Other names =hendecoid, Wyschnegradsky's&lt;br&gt; diatonicized chromatic 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 7
| Collapsed = 6
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is most notable for being used by [[Ivan Wyschnegradsky]] and having a name "diatonicized chromatic scale". The more concise name for the scale, proposed by Eliora, is '''hendecoid'''.

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale - the strong relationship to the number 11 is the reason for the name "hendecoid".

== Scale tree ==
{| class="wikitable"
! colspan="8" | generator
! L
! s
! L/s
! gen (cents)
! comment
|-
| 5\11
|
|
|
|
|
|
|
| 1
| 0
|
| 545.455
|
|-
|
|
|
|
|
|
|
| 41\90
| 8
| 1
| 8.000
| 546.667
|
|-
|
|
|
|
|
|
| 36\79
|
| 7
| 1
| 7.000
| 546.835
|
|-
|
|
|
|
|
| 31\68
|
|
| 6
| 1
| 6.000
| 547.059
|
|-
|
|
|
|
| 26\57
|
|
|
| 5
| 1
| 5.000
| 547.368
|
|-
|
|
|
| 21\46
|
|
|
|
| 4
| 1
| 4.000
| 547.826
| Heinz is around here
|-
|
|
|
|
| 37\81
|
|
|
| 7
| 2
| 3.500
| 548.148
|
|-
|
|
| 16\35
|
|
|
|
|
| 3
| 1
| 3.000
| 548.571
|
|-
|
|
|
|
| 43\94
|
|
|
| 8
| 3
| 2.667
| 548.936
|
|-
|
|
|
| 27\59
|
|
|
|
| 5
| 2
| 2.500
| 549.153
|
|-
|
|
|
|
| 38\83
|
|
|
| 7
| 3
| 2.333
| 549.398
|
|-
|
| 11\24
|
|
|
|
|
|
| 2
| 1
| 2.000
| 550.000
|
|-
|
|
|
|
| 39\85
|
|
|
| 7
| 4
| 1.750
| 550.588
|
|-
|
|
|
| 28\61
|
|
|
|
| 5
| 3
| 1.667
| 550.820
|
|-
|
|
|
|
|
| (5&amp;phi;+1)/(11&amp;phi;+2)
|
|
| &amp;phi;
| 1
| 1.618
| 550.965
|
|-
|
|
|
|
| 45\98
|
|
|
| 8
| 5
| 1.600
| 551.020
|
|-
|
|
| 17\37
|
|
|
|
|
| 3
| 2
| 1.500
| 551.351
|
|-
|
|
|
|
| 40\87
|
|
|
| 7
| 5
| 1.400
| 551.724
|
|-
|
|
|
| 23\50
|
|
|
|
| 4
| 3
| 1.333
| 552.000
|
|-
|
|
|
|
| 29\63
|
|
|
| 5
| 4
| 1.250
| 552.381
|
|-
|
|
|
|
|
| 35\76
|
|
| 6
| 5
| 1.200
| 552.632
|
|-
|
|
|
|
|
|
| 41\89
|
| 7
| 6
| 1.167
| 552.809
|
|-
|
|
|
|
|
|
|
| 47\102
| 8
| 7
| 1.125
| 552.941
|
|-
| 6\13
|
|
|
|
|
|
|
| 1
| 1
| 1.000
| 553.846
|
|}

[[Category:13-tone scales]]</text>
      <sha1>0ryullh78hby7yxw1fx71liy5lcp9ko</sha1>
    </revision>
    <revision>
      <id>107214</id>
      <parentid>107213</parentid>
      <timestamp>2023-04-10T12:00:09Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Scale tree */ remove phi as it's in the way of the scale tree</comment>
      <origin>107214</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2128" sha1="9gqdebhxqyq7yu78zbgbc5c78cu5jwz" xml:space="preserve">{{Infobox MOS
| Other names =hendecoid, Wyschnegradsky's&lt;br&gt; diatonicized chromatic 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 7
| Collapsed = 6
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is most notable for being used by [[Ivan Wyschnegradsky]] and having a name "diatonicized chromatic scale". The more concise name for the scale, proposed by Eliora, is '''hendecoid'''.

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale - the strong relationship to the number 11 is the reason for the name "hendecoid".

== Scale tree ==
{| class="wikitable"
! colspan="8" | generator
! L
! s
! L/s
! gen (cents)
! comment
|-
| 5\11
|
|
|
|
|
|
|
| 1
| 0
|
| 545.455
|
|-
|
|
|
|
|
|
|
| 41\90
| 8
| 1
| 8.000
| 546.667
|
|-
|
|
|
|
|
|
| 36\79
|
| 7
| 1
| 7.000
| 546.835
|
|-
|
|
|
|
|
| 31\68
|
|
| 6
| 1
| 6.000
| 547.059
|
|-
|
|
|
|
| 26\57
|
|
|
| 5
| 1
| 5.000
| 547.368
|
|-
|
|
|
| 21\46
|
|
|
|
| 4
| 1
| 4.000
| 547.826
| Heinz is around here
|-
|
|
|
|
| 37\81
|
|
|
| 7
| 2
| 3.500
| 548.148
|
|-
|
|
| 16\35
|
|
|
|
|
| 3
| 1
| 3.000
| 548.571
|
|-
|
|
|
|
| 43\94
|
|
|
| 8
| 3
| 2.667
| 548.936
|
|-
|
|
|
| 27\59
|
|
|
|
| 5
| 2
| 2.500
| 549.153
|
|-
|
|
|
|
| 38\83
|
|
|
| 7
| 3
| 2.333
| 549.398
|
|-
|
| 11\24
|
|
|
|
|
|
| 2
| 1
| 2.000
| 550.000
|
|-
|
|
|
|
| 39\85
|
|
|
| 7
| 4
| 1.750
| 550.588
|
|-
|
|
|
| 28\61
|
|
|
|
| 5
| 3
| 1.667
| 550.820
|
|-
|
|
|
|
| 45\98
|
|
|
| 8
| 5
| 1.600
| 551.020
|
|-
|
|
| 17\37
|
|
|
|
|
| 3
| 2
| 1.500
| 551.351
|
|-
|
|
|
|
| 40\87
|
|
|
| 7
| 5
| 1.400
| 551.724
|
|-
|
|
|
| 23\50
|
|
|
|
| 4
| 3
| 1.333
| 552.000
|
|-
|
|
|
|
| 29\63
|
|
|
| 5
| 4
| 1.250
| 552.381
|
|-
|
|
|
|
|
| 35\76
|
|
| 6
| 5
| 1.200
| 552.632
|
|-
|
|
|
|
|
|
| 41\89
|
| 7
| 6
| 1.167
| 552.809
|
|-
|
|
|
|
|
|
|
| 47\102
| 8
| 7
| 1.125
| 552.941
|
|-
| 6\13
|
|
|
|
|
|
|
| 1
| 1
| 1.000
| 553.846
|
|}

[[Category:13-tone scales]]</text>
      <sha1>9gqdebhxqyq7yu78zbgbc5c78cu5jwz</sha1>
    </revision>
    <revision>
      <id>107215</id>
      <parentid>107214</parentid>
      <timestamp>2023-04-10T12:42:11Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Scale tree */ improve the scale tree</comment>
      <origin>107215</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2495" sha1="jjfxjvst5r0kfhpt0qvj3qntc5a4ni8" xml:space="preserve">{{Infobox MOS
| Other names =hendecoid, Wyschnegradsky's&lt;br&gt; diatonicized chromatic 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 7
| Collapsed = 6
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is most notable for being used by [[Ivan Wyschnegradsky]] and having a name "diatonicized chromatic scale". The more concise name for the scale, proposed by Eliora, is '''hendecoid'''. Another possible name for this mos in TAMNAMS is '''p-enhar balzano''', being one of four enharmonic scales of [[2L 7s]].

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale - the strong relationship to the number 11 is the reason for the name "hendecoid".

== Modes ==

* 12|0 LLLLLLsLLLLLs
* 11|1 LLLLLsLLLLLLs
* 10|2 LLLLLsLLLLLsL
* 9|3 LLLLsLLLLLLsL

...

== Scale tree ==
{| class="wikitable center-all"
! colspan="6" |Generator
!L
!s
!L/s
!Comments
|-
|7\13
|
|
|
|
|
|1
|1
|1.000
|
|-
| || || || || ||41\76||6||5||1.200
|
|-
| || || || ||34\63|| ||5||4||1.250
|
|-
| || || || || ||61\113||9||7||1.286
|
|-
| || || ||27\50|| || ||4||3||1.333
|
|-
| || || || || ||74\137||11||8||1.375
|
|-
| || || || ||47\87|| ||7||5||1.400
|
|-
| || || || || ||67\124||10||7||1.428
|
|-
| || ||20\37|| || || ||3||2||1.500
|
|-
| || || || || ||73\135||11||7||1.571
|
|-
| || || || ||53\98|| ||8||5||1.600
|
|-
| || || || || ||86\159||13||8||1.625
|
|-
| || || ||33\61|| || ||5||3||1.667
|Freivald / emka is around here
|-
| || || || || ||79\146||12||7||1.714
|
|-
| || || || ||46\85|| ||7||4||1.750
|
|-
| || || || || ||71\109||9||5||1.800
|
|-
| ||13\24|| || || || ||2||1||2.000
|Basic hendecoid
|-
| || || || || ||70\107||9||4||2.250
|
|-
| || || || ||45\83|| ||7||3||2.333
|
|-
| || || || || ||77\142||12||5||2.400
|
|-
| || || ||32\59|| || ||5||2||2.500
|
|-
| || || || || ||83\152||13||5||2.600
|
|-
| || || || ||51\94|| ||8||3||2.667
|
|-
| || || || || ||70\129||11||4||2.750
|
|-
| || ||19\35|| || || ||3||1||3.000
|
|-
| || || || || ||63\116||10||3||3.333
|
|-
| || || || ||44\81|| ||7||2||3.500
|
|-
| || || || || ||69\127||11||3||3.667
|
|-
| || || ||25\46|| || ||4||1||4.000
|
|-
| || || || || ||56\103||9||2||4.500
|
|-
| || || || ||31\57|| ||5||1||5.000
|
|-
| || || || || ||37\68||6||1||6.000
|
|-
|6\11|| || || || || ||1||0||→ inf
|
|}
[[Category:13-tone scales]]</text>
      <sha1>jjfxjvst5r0kfhpt0qvj3qntc5a4ni8</sha1>
    </revision>
    <revision>
      <id>107216</id>
      <parentid>107215</parentid>
      <timestamp>2023-04-10T12:48:21Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>add modes</comment>
      <origin>107216</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2726" sha1="5uzeunosgj9p6m85oxppggetww75j7v" xml:space="preserve">{{Infobox MOS
| Other names =hendecoid, Wyschnegradsky's&lt;br&gt; diatonicized chromatic 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 7
| Collapsed = 6
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is most notable for being used by [[Ivan Wyschnegradsky]] and having a name "diatonicized chromatic scale". The more concise name for the scale, proposed by Eliora, is '''hendecoid'''. Another possible name for this mos in TAMNAMS is '''p-enhar balzano''', being one of four enharmonic scales of [[2L 7s]].

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale - the strong relationship to the number 11 is the reason for the name "hendecoid".

== Modes ==

* 12|0 LLLLLLsLLLLLs
* 11|1 LLLLLsLLLLLLs
* 10|2 LLLLLsLLLLLsL
* 9|3 LLLLsLLLLLLsL

* 8|4 LLLLsLLLLLsLL
* 7|5 LLLsLLLLLLsLL
* 6|6 LLLsLLLLLsLLL
* 5|7 LLsLLLLLLsLLL
* 4|8 LLsLLLLLsLLLL
* 3|9 LsLLLLLLsLLLL
* 2|10 LsLLLLLsLLLLL
* 1|11 sLLLLLLsLLLLL

== Scale tree ==
{| class="wikitable center-all"
! colspan="6" |Generator
!L
!s
!L/s
!Comments
|-
|7\13
|
|
|
|
|
|1
|1
|1.000
|
|-
| || || || || ||41\76||6||5||1.200
|
|-
| || || || ||34\63|| ||5||4||1.250
|
|-
| || || || || ||61\113||9||7||1.286
|
|-
| || || ||27\50|| || ||4||3||1.333
|
|-
| || || || || ||74\137||11||8||1.375
|
|-
| || || || ||47\87|| ||7||5||1.400
|
|-
| || || || || ||67\124||10||7||1.428
|
|-
| || ||20\37|| || || ||3||2||1.500
|
|-
| || || || || ||73\135||11||7||1.571
|
|-
| || || || ||53\98|| ||8||5||1.600
|
|-
| || || || || ||86\159||13||8||1.625
|
|-
| || || ||33\61|| || ||5||3||1.667
|[[Freivald]] / [[emka]] is around here
|-
| || || || || ||79\146||12||7||1.714
|
|-
| || || || ||46\85|| ||7||4||1.750
|
|-
| || || || || ||71\109||9||5||1.800
|
|-
| ||13\24|| || || || ||2||1||2.000
|Basic hendecoid,
Wyschnegradsky's diatonicized chromatic
|-
| || || || || ||70\107||9||4||2.250
|
|-
| || || || ||45\83|| ||7||3||2.333
|
|-
| || || || || ||77\142||12||5||2.400
|
|-
| || || ||32\59|| || ||5||2||2.500
|
|-
| || || || || ||83\152||13||5||2.600
|
|-
| || || || ||51\94|| ||8||3||2.667
|
|-
| || || || || ||70\129||11||4||2.750
|
|-
| || ||19\35|| || || ||3||1||3.000
|
|-
| || || || || ||63\116||10||3||3.333
|
|-
| || || || ||44\81|| ||7||2||3.500
|
|-
| || || || || ||69\127||11||3||3.667
|
|-
| || || ||25\46|| || ||4||1||4.000
|[[Heinz]] is around here
|-
| || || || || ||56\103||9||2||4.500
|
|-
| || || || ||31\57|| ||5||1||5.000
|
|-
| || || || || ||37\68||6||1||6.000
|
|-
|6\11|| || || || || ||1||0||→ inf
|
|}
[[Category:13-tone scales]]</text>
      <sha1>5uzeunosgj9p6m85oxppggetww75j7v</sha1>
    </revision>
    <revision>
      <id>107217</id>
      <parentid>107216</parentid>
      <timestamp>2023-04-10T12:50:48Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>107217</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2801" sha1="bt85msjl6zzb714rtz1rcelvy5vr7ky" xml:space="preserve">{{Infobox MOS
| Other names =hendecoid, Wyschnegradsky's&lt;br&gt; diatonicized chromatic 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 7
| Collapsed = 6
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is most notable for being used by [[Ivan Wyschnegradsky]] and having a name "diatonicized chromatic scale". The more concise name for the scale, proposed by Eliora, is '''hendecoid'''. Another possible name for this mos in TAMNAMS is '''p-enhar balzano''', being one of four enharmonic scales of [[2L 7s]].

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale - the strong relationship to the number 11 is the reason for the name "hendecoid". If just 11/8 is used as generator, the step ratio is around 1.962389572...

== Modes ==

* 12|0 LLLLLLsLLLLLs
* 11|1 LLLLLsLLLLLLs
* 10|2 LLLLLsLLLLLsL
* 9|3 LLLLsLLLLLLsL

* 8|4 LLLLsLLLLLsLL
* 7|5 LLLsLLLLLLsLL
* 6|6 LLLsLLLLLsLLL
* 5|7 LLsLLLLLLsLLL
* 4|8 LLsLLLLLsLLLL
* 3|9 LsLLLLLLsLLLL
* 2|10 LsLLLLLsLLLLL
* 1|11 sLLLLLLsLLLLL

== Scale tree ==
{| class="wikitable center-all"
! colspan="6" |Generator
!L
!s
!L/s
!Comments
|-
|7\13
|
|
|
|
|
|1
|1
|1.000
|
|-
| || || || || ||41\76||6||5||1.200
|
|-
| || || || ||34\63|| ||5||4||1.250
|
|-
| || || || || ||61\113||9||7||1.286
|
|-
| || || ||27\50|| || ||4||3||1.333
|
|-
| || || || || ||74\137||11||8||1.375
|
|-
| || || || ||47\87|| ||7||5||1.400
|
|-
| || || || || ||67\124||10||7||1.428
|
|-
| || ||20\37|| || || ||3||2||1.500
|
|-
| || || || || ||73\135||11||7||1.571
|
|-
| || || || ||53\98|| ||8||5||1.600
|
|-
| || || || || ||86\159||13||8||1.625
|
|-
| || || ||33\61|| || ||5||3||1.667
|[[Freivald]] / [[emka]] is around here
|-
| || || || || ||79\146||12||7||1.714
|
|-
| || || || ||46\85|| ||7||4||1.750
|
|-
| || || || || ||71\109||9||5||1.800
|
|-
| ||13\24|| || || || ||2||1||2.000
|Basic hendecoid,
Wyschnegradsky's diatonicized chromatic
|-
| || || || || ||70\107||9||4||2.250
|
|-
| || || || ||45\83|| ||7||3||2.333
|
|-
| || || || || ||77\142||12||5||2.400
|
|-
| || || ||32\59|| || ||5||2||2.500
|
|-
| || || || || ||83\152||13||5||2.600
|
|-
| || || || ||51\94|| ||8||3||2.667
|
|-
| || || || || ||70\129||11||4||2.750
|
|-
| || ||19\35|| || || ||3||1||3.000
|
|-
| || || || || ||63\116||10||3||3.333
|
|-
| || || || ||44\81|| ||7||2||3.500
|
|-
| || || || || ||69\127||11||3||3.667
|
|-
| || || ||25\46|| || ||4||1||4.000
|[[Heinz]] is around here
|-
| || || || || ||56\103||9||2||4.500
|
|-
| || || || ||31\57|| ||5||1||5.000
|
|-
| || || || || ||37\68||6||1||6.000
|
|-
|6\11|| || || || || ||1||0||→ inf
|
|}
[[Category:13-tone scales]]</text>
      <sha1>bt85msjl6zzb714rtz1rcelvy5vr7ky</sha1>
    </revision>
    <revision>
      <id>107218</id>
      <parentid>107217</parentid>
      <timestamp>2023-04-10T12:53:59Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>107218</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2793" sha1="k6wb3gtqkurlon3x53dph1tdb5kkozg" xml:space="preserve">{{Infobox MOS
| Other names =hendecoid, Wyschnegradsky's&lt;br&gt; diatonicized chromatic 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 7
| Collapsed = 6
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is most notable for being used by [[Ivan Wyschnegradsky]] and having a name "diatonicized chromatic scale". The more concise name for the scale, proposed by Eliora, is '''hendecoid'''. Another possible name for this mos in TAMNAMS is '''p-enhar balzano''', being one of four enharmonic scales of [[2L 7s]].

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale - the strong relationship to the number 11 is the reason for the name "hendecoid". If just 11/8 is used as generator, the step ratio is around 1.509.

== Modes ==

* 12|0 LLLLLLsLLLLLs
* 11|1 LLLLLsLLLLLLs
* 10|2 LLLLLsLLLLLsL
* 9|3 LLLLsLLLLLLsL

* 8|4 LLLLsLLLLLsLL
* 7|5 LLLsLLLLLLsLL
* 6|6 LLLsLLLLLsLLL
* 5|7 LLsLLLLLLsLLL
* 4|8 LLsLLLLLsLLLL
* 3|9 LsLLLLLLsLLLL
* 2|10 LsLLLLLsLLLLL
* 1|11 sLLLLLLsLLLLL

== Scale tree ==
{| class="wikitable center-all"
! colspan="6" |Generator
!L
!s
!L/s
!Comments
|-
|7\13
|
|
|
|
|
|1
|1
|1.000
|
|-
| || || || || ||41\76||6||5||1.200
|
|-
| || || || ||34\63|| ||5||4||1.250
|
|-
| || || || || ||61\113||9||7||1.286
|
|-
| || || ||27\50|| || ||4||3||1.333
|
|-
| || || || || ||74\137||11||8||1.375
|
|-
| || || || ||47\87|| ||7||5||1.400
|
|-
| || || || || ||67\124||10||7||1.428
|
|-
| || ||20\37|| || || ||3||2||1.500
|
|-
| || || || || ||73\135||11||7||1.571
|
|-
| || || || ||53\98|| ||8||5||1.600
|
|-
| || || || || ||86\159||13||8||1.625
|
|-
| || || ||33\61|| || ||5||3||1.667
|[[Freivald]] / [[emka]] is around here
|-
| || || || || ||79\146||12||7||1.714
|
|-
| || || || ||46\85|| ||7||4||1.750
|
|-
| || || || || ||71\109||9||5||1.800
|
|-
| ||13\24|| || || || ||2||1||2.000
|Basic hendecoid,
Wyschnegradsky's diatonicized chromatic
|-
| || || || || ||70\107||9||4||2.250
|
|-
| || || || ||45\83|| ||7||3||2.333
|
|-
| || || || || ||77\142||12||5||2.400
|
|-
| || || ||32\59|| || ||5||2||2.500
|
|-
| || || || || ||83\152||13||5||2.600
|
|-
| || || || ||51\94|| ||8||3||2.667
|
|-
| || || || || ||70\129||11||4||2.750
|
|-
| || ||19\35|| || || ||3||1||3.000
|
|-
| || || || || ||63\116||10||3||3.333
|
|-
| || || || ||44\81|| ||7||2||3.500
|
|-
| || || || || ||69\127||11||3||3.667
|
|-
| || || ||25\46|| || ||4||1||4.000
|[[Heinz]] is around here
|-
| || || || || ||56\103||9||2||4.500
|
|-
| || || || ||31\57|| ||5||1||5.000
|
|-
| || || || || ||37\68||6||1||6.000
|
|-
|6\11|| || || || || ||1||0||→ inf
|
|}
[[Category:13-tone scales]]</text>
      <sha1>k6wb3gtqkurlon3x53dph1tdb5kkozg</sha1>
    </revision>
    <revision>
      <id>114655</id>
      <parentid>107218</parentid>
      <timestamp>2023-05-25T22:04:17Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Modes */</comment>
      <origin>114655</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2561" sha1="t7pojwdotzb3jtk3e93gvlmxr2hp0mq" xml:space="preserve">{{Infobox MOS
| Other names =hendecoid, Wyschnegradsky's&lt;br&gt; diatonicized chromatic 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 7
| Collapsed = 6
| Pattern = LLLLLLsLLLLLs
}}
The '''11L 2s''' [[MOS scale]] is most notable for being used by [[Ivan Wyschnegradsky]] and having a name "diatonicized chromatic scale". The more concise name for the scale, proposed by Eliora, is '''hendecoid'''. Another possible name for this mos in TAMNAMS is '''p-enhar balzano''', being one of four enharmonic scales of [[2L 7s]].

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale - the strong relationship to the number 11 is the reason for the name "hendecoid". If just 11/8 is used as generator, the step ratio is around 1.509.

== Modes ==

{{MOS modes}}

== Scale tree ==
{| class="wikitable center-all"
! colspan="6" |Generator
!L
!s
!L/s
!Comments
|-
|7\13
|
|
|
|
|
|1
|1
|1.000
|
|-
| || || || || ||41\76||6||5||1.200
|
|-
| || || || ||34\63|| ||5||4||1.250
|
|-
| || || || || ||61\113||9||7||1.286
|
|-
| || || ||27\50|| || ||4||3||1.333
|
|-
| || || || || ||74\137||11||8||1.375
|
|-
| || || || ||47\87|| ||7||5||1.400
|
|-
| || || || || ||67\124||10||7||1.428
|
|-
| || ||20\37|| || || ||3||2||1.500
|
|-
| || || || || ||73\135||11||7||1.571
|
|-
| || || || ||53\98|| ||8||5||1.600
|
|-
| || || || || ||86\159||13||8||1.625
|
|-
| || || ||33\61|| || ||5||3||1.667
|[[Freivald]] / [[emka]] is around here
|-
| || || || || ||79\146||12||7||1.714
|
|-
| || || || ||46\85|| ||7||4||1.750
|
|-
| || || || || ||71\109||9||5||1.800
|
|-
| ||13\24|| || || || ||2||1||2.000
|Basic hendecoid,
Wyschnegradsky's diatonicized chromatic
|-
| || || || || ||70\107||9||4||2.250
|
|-
| || || || ||45\83|| ||7||3||2.333
|
|-
| || || || || ||77\142||12||5||2.400
|
|-
| || || ||32\59|| || ||5||2||2.500
|
|-
| || || || || ||83\152||13||5||2.600
|
|-
| || || || ||51\94|| ||8||3||2.667
|
|-
| || || || || ||70\129||11||4||2.750
|
|-
| || ||19\35|| || || ||3||1||3.000
|
|-
| || || || || ||63\116||10||3||3.333
|
|-
| || || || ||44\81|| ||7||2||3.500
|
|-
| || || || || ||69\127||11||3||3.667
|
|-
| || || ||25\46|| || ||4||1||4.000
|[[Heinz]] is around here
|-
| || || || || ||56\103||9||2||4.500
|
|-
| || || || ||31\57|| ||5||1||5.000
|
|-
| || || || || ||37\68||6||1||6.000
|
|-
|6\11|| || || || || ||1||0||→ inf
|
|}
[[Category:13-tone scales]]</text>
      <sha1>t7pojwdotzb3jtk3e93gvlmxr2hp0mq</sha1>
    </revision>
    <revision>
      <id>136149</id>
      <parentid>114655</parentid>
      <timestamp>2024-02-19T01:29:38Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Rewrote intro</comment>
      <origin>136149</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2594" sha1="eogppe7ev3udj7w34aqtok39ey2ie67" xml:space="preserve">{{MOS intro|Other Names=hendecoid; Wyschnegradsky's diatonicized chromatic scale}}{{Infobox MOS
| Other names =hendecoid, Wyschnegradsky's&lt;br&gt; diatonicized chromatic 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 2
| Equalized = 7
| Collapsed = 6
| Pattern = LLLLLLsLLLLLs
}}
This scale is most notable for being used by [[Ivan Wyschnegradsky]], bearing the name '''diatonicized chromatic scale'''. Eliora has proposed the name '''hendecoid''' for its strong relationship to the number 11, as it's an 11+-limit scale and has generators that are close to 11/8. Frostburn has proposed the name '''p-enhar balzano''', as a grandchild scale of 2L 7s.

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale. If just 11/8 is used as generator, the step ratio is around 1.509.

== Modes ==

{{MOS modes}}

== Scale tree ==
{| class="wikitable center-all"
! colspan="6" |Generator
!L
!s
!L/s
!Comments
|-
|7\13
|
|
|
|
|
|1
|1
|1.000
|
|-
| || || || || ||41\76||6||5||1.200
|
|-
| || || || ||34\63|| ||5||4||1.250
|
|-
| || || || || ||61\113||9||7||1.286
|
|-
| || || ||27\50|| || ||4||3||1.333
|
|-
| || || || || ||74\137||11||8||1.375
|
|-
| || || || ||47\87|| ||7||5||1.400
|
|-
| || || || || ||67\124||10||7||1.428
|
|-
| || ||20\37|| || || ||3||2||1.500
|
|-
| || || || || ||73\135||11||7||1.571
|
|-
| || || || ||53\98|| ||8||5||1.600
|
|-
| || || || || ||86\159||13||8||1.625
|
|-
| || || ||33\61|| || ||5||3||1.667
|[[Freivald]] / [[emka]] is around here
|-
| || || || || ||79\146||12||7||1.714
|
|-
| || || || ||46\85|| ||7||4||1.750
|
|-
| || || || || ||71\109||9||5||1.800
|
|-
| ||13\24|| || || || ||2||1||2.000
|Basic hendecoid,
Wyschnegradsky's diatonicized chromatic
|-
| || || || || ||70\107||9||4||2.250
|
|-
| || || || ||45\83|| ||7||3||2.333
|
|-
| || || || || ||77\142||12||5||2.400
|
|-
| || || ||32\59|| || ||5||2||2.500
|
|-
| || || || || ||83\152||13||5||2.600
|
|-
| || || || ||51\94|| ||8||3||2.667
|
|-
| || || || || ||70\129||11||4||2.750
|
|-
| || ||19\35|| || || ||3||1||3.000
|
|-
| || || || || ||63\116||10||3||3.333
|
|-
| || || || ||44\81|| ||7||2||3.500
|
|-
| || || || || ||69\127||11||3||3.667
|
|-
| || || ||25\46|| || ||4||1||4.000
|[[Heinz]] is around here
|-
| || || || || ||56\103||9||2||4.500
|
|-
| || || || ||31\57|| ||5||1||5.000
|
|-
| || || || || ||37\68||6||1||6.000
|
|-
|6\11|| || || || || ||1||0||→ inf
|
|}
[[Category:13-tone scales]]</text>
      <sha1>eogppe7ev3udj7w34aqtok39ey2ie67</sha1>
    </revision>
    <revision>
      <id>136150</id>
      <parentid>136149</parentid>
      <timestamp>2024-02-19T01:30:10Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <minor/>
      <comment>Rearranged templates</comment>
      <origin>136150</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2414" sha1="5vvymfufomlb0abbrl93ri9cjyt21qx" xml:space="preserve">{{Infobox MOS}}
{{MOS intro|Other Names=hendecoid; Wyschnegradsky's diatonicized chromatic scale}}
This scale is most notable for being used by [[Ivan Wyschnegradsky]], bearing the name '''diatonicized chromatic scale'''. Eliora has proposed the name '''hendecoid''' for its strong relationship to the number 11, as it's an 11+-limit scale and has generators that are close to 11/8. Frostburn has proposed the name '''p-enhar balzano''', as a grandchild scale of 2L 7s.

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale. If just 11/8 is used as generator, the step ratio is around 1.509.

== Modes ==

{{MOS modes}}

== Scale tree ==
{| class="wikitable center-all"
! colspan="6" |Generator
!L
!s
!L/s
!Comments
|-
|7\13
|
|
|
|
|
|1
|1
|1.000
|
|-
| || || || || ||41\76||6||5||1.200
|
|-
| || || || ||34\63|| ||5||4||1.250
|
|-
| || || || || ||61\113||9||7||1.286
|
|-
| || || ||27\50|| || ||4||3||1.333
|
|-
| || || || || ||74\137||11||8||1.375
|
|-
| || || || ||47\87|| ||7||5||1.400
|
|-
| || || || || ||67\124||10||7||1.428
|
|-
| || ||20\37|| || || ||3||2||1.500
|
|-
| || || || || ||73\135||11||7||1.571
|
|-
| || || || ||53\98|| ||8||5||1.600
|
|-
| || || || || ||86\159||13||8||1.625
|
|-
| || || ||33\61|| || ||5||3||1.667
|[[Freivald]] / [[emka]] is around here
|-
| || || || || ||79\146||12||7||1.714
|
|-
| || || || ||46\85|| ||7||4||1.750
|
|-
| || || || || ||71\109||9||5||1.800
|
|-
| ||13\24|| || || || ||2||1||2.000
|Basic hendecoid,
Wyschnegradsky's diatonicized chromatic
|-
| || || || || ||70\107||9||4||2.250
|
|-
| || || || ||45\83|| ||7||3||2.333
|
|-
| || || || || ||77\142||12||5||2.400
|
|-
| || || ||32\59|| || ||5||2||2.500
|
|-
| || || || || ||83\152||13||5||2.600
|
|-
| || || || ||51\94|| ||8||3||2.667
|
|-
| || || || || ||70\129||11||4||2.750
|
|-
| || ||19\35|| || || ||3||1||3.000
|
|-
| || || || || ||63\116||10||3||3.333
|
|-
| || || || ||44\81|| ||7||2||3.500
|
|-
| || || || || ||69\127||11||3||3.667
|
|-
| || || ||25\46|| || ||4||1||4.000
|[[Heinz]] is around here
|-
| || || || || ||56\103||9||2||4.500
|
|-
| || || || ||31\57|| ||5||1||5.000
|
|-
| || || || || ||37\68||6||1||6.000
|
|-
|6\11|| || || || || ||1||0||→ inf
|
|}
[[Category:13-tone scales]]</text>
      <sha1>5vvymfufomlb0abbrl93ri9cjyt21qx</sha1>
    </revision>
    <revision>
      <id>137489</id>
      <parentid>136150</parentid>
      <timestamp>2024-02-29T04:53:49Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>/* Scale tree */ Replaced scale tree, copied entries to template</comment>
      <origin>137489</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="957" sha1="me6ovy8lzgiqiaums84aey0qr8306jn" xml:space="preserve">{{Infobox MOS}}
{{MOS intro|Other Names=hendecoid; Wyschnegradsky's diatonicized chromatic scale}}
This scale is most notable for being used by [[Ivan Wyschnegradsky]], bearing the name '''diatonicized chromatic scale'''. Eliora has proposed the name '''hendecoid''' for its strong relationship to the number 11, as it's an 11+-limit scale and has generators that are close to 11/8. Frostburn has proposed the name '''p-enhar balzano''', as a grandchild scale of 2L 7s.

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale. If just 11/8 is used as generator, the step ratio is around 1.509.

== Modes ==

{{MOS modes}}

== Scale tree ==
{{Scale tree|Comments=4/1: [[Heinz]];
7/5: ↕ [[Emka]];
8/5: ↕ [[Freivald]]}}
[[Category:13-tone scales]]</text>
      <sha1>me6ovy8lzgiqiaums84aey0qr8306jn</sha1>
    </revision>
    <revision>
      <id>170573</id>
      <parentid>137489</parentid>
      <timestamp>2024-12-15T06:57:54Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>== Intervals == {{MOS intervals}}</comment>
      <origin>170573</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="991" sha1="naugnagvs0o2yjq0eish9du14604rm8" xml:space="preserve">{{Infobox MOS}}
{{MOS intro|Other Names=hendecoid; Wyschnegradsky's diatonicized chromatic scale}}
This scale is most notable for being used by [[Ivan Wyschnegradsky]], bearing the name '''diatonicized chromatic scale'''. Eliora has proposed the name '''hendecoid''' for its strong relationship to the number 11, as it's an 11+-limit scale and has generators that are close to 11/8. Frostburn has proposed the name '''p-enhar balzano''', as a grandchild scale of 2L 7s.

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale. If just 11/8 is used as generator, the step ratio is around 1.509.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree|Comments=4/1: [[Heinz]];
7/5: ↕ [[Emka]];
8/5: ↕ [[Freivald]]}}
[[Category:13-tone scales]]</text>
      <sha1>naugnagvs0o2yjq0eish9du14604rm8</sha1>
    </revision>
    <revision>
      <id>183730</id>
      <parentid>170573</parentid>
      <timestamp>2025-02-28T16:41:05Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183730</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1000" sha1="epmrxql0k3kfm44qeolhm7w8ifubi69" xml:space="preserve">{{Infobox MOS}}
{{MOS intro|Other Names=hendecoid; Wyschnegradsky's diatonicized chromatic scale}}
This scale is most notable for being used by [[Ivan Wyschnegradsky]], bearing the name '''diatonicized chromatic scale'''. Eliora has proposed the name '''hendecoid''' for its strong relationship to the number 11, as it's an 11+-limit scale and has generators that are close to 11/8. Frostburn has proposed the name '''p-enhar balzano''', as a grandchild scale of 2L 7s.

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale. If just 11/8 is used as generator, the step ratio is around 1.509.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 4/1 = [[Heinz]]
| 7/5 = ↕ [[Emka]]
| 8/5 = ↕ [[Freivald]]
}}

[[Category:13-tone scales]]</text>
      <sha1>epmrxql0k3kfm44qeolhm7w8ifubi69</sha1>
    </revision>
    <revision>
      <id>190251</id>
      <parentid>183730</parentid>
      <timestamp>2025-04-08T14:05:18Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>190251</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1004" sha1="0ntt17bmo440fg2h1mqhhhk8h12mln6" xml:space="preserve">{{Infobox MOS}}
{{MOS intro|Other Names=hendecoid; Wyschnegradsky's diatonicized chromatic scale}}
This scale is most notable for being used by [[Ivan Wyschnegradsky]], bearing the name '''diatonicized chromatic scale'''. Eliora has proposed the name '''hendecoid''' for its strong relationship to the number 11, as it's an 11+-limit scale and has generators that are close to [[11/8]]. Frostburn has proposed the name '''p-enhar balzano''', as a grandchild scale of 2L 7s.

From a regular temperament theory perspective, is notable for correponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale. If just 11/8 is used as generator, the step ratio is around 1.509.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 4/1 = [[Heinz]]
| 7/5 = ↕ [[Emka]]
| 8/5 = ↕ [[Freivald]]
}}

[[Category:13-tone scales]]</text>
      <sha1>0ntt17bmo440fg2h1mqhhhk8h12mln6</sha1>
    </revision>
    <revision>
      <id>194752</id>
      <parentid>190251</parentid>
      <timestamp>2025-04-29T19:14:25Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>Fix a couple of typos; reorder comments in MOS tuning spectrum table to make it more intuitive for somebody else editing this page (should not affect normal page display)</comment>
      <origin>194752</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1016" sha1="60umy8pea6urw8u7rt7atliy4074hrv" xml:space="preserve">{{Infobox MOS}}
{{MOS intro|Other Names=hendecoid; Wyschnegradsky's diatonicized chromatic scale}}
This scale is most notable for being used by [[Ivan Wyschnegradsky]], bearing the name '''diatonicized chromatic scale'''. Eliora has proposed the name '''hendecoid''' for its strong relationship to the number 11, as it's an 11+-limit scale and has generators that are close to [[11/8]]. Frostburn has proposed the name '''p-enhar balzano''', as a grandchild scale of 2L 7s.

From a regular temperament theory perspective, this scale is notable for corresponding to the mega chromatic scale of [[Heinz]] temperament. Its generator of 5\11 to 6\13 hits so close to 11/8 as to be able to be called nothing but that interval, making it an 11+-limit scale. If just 11/8 is used as generator, the step ratio is around 1.509.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 7/5 = ↕ [[Emka]]
| 8/5 = ↕ [[Freivald]]
| 4/1 = [[Heinz]]
}}

[[Category:13-tone scales]]</text>
      <sha1>60umy8pea6urw8u7rt7atliy4074hrv</sha1>
    </revision>
  </page>
  <page>
    <title>11L 3s</title>
    <ns>0</ns>
    <id>95</id>
    <revision>
      <id>95</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>95</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8911" sha1="7j31osau7subze8fzezhnuc5dp8eklj" xml:space="preserve">=&lt;span style="color: #800061; font-size: 103%;"&gt;The Ketradektriatoh Scale&lt;/span&gt;=

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

&lt;u&gt;'''ED2s that contains this scale:'''&lt;/u&gt;

'''2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)'''

'''3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)'''

'''3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)'''

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]

4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]

4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]

'''5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ'''

'''5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]''' '''φ'''

5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]

6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]

7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]

7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]

7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]

'''8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ'''

'''8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]''' '''φ'''

8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]

9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]

9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]

9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]

10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]

10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]

10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''' 11 11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''' 11 11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''': [[124edo|124]]

11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]

11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]

11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]

11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]

11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]

11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]

11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]

12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]

12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]

12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]

13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]

13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]

13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]

'''13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ'''

13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]

13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]

'''13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]''' '''φ'''

13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]

13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]

13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]

13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]

14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]

14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]

14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]

14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]

14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]

15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]

15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]

15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]

15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]

15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]

15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]

15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]

16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]

16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]

16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]

16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]

16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]

16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]

16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]

17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]

17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]

17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]

'''17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)'''

'''17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]'''

'''17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]'''

'''17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]'''

'''17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]'''

'''17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]'''

'''17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]'''

'''17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)'''

17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]

17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]

17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]

17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:

{| class="wikitable"
|-
| | 4\[[11edo|11]]
| | 
| | 
| | 
| | 
| | 
| | 
| | 436.364
| | 109.091
| | 0
| style="text-align:center;" | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 29\[[80edo|80]]
| | 435
| | 105
| | 15
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 25\[[69edo|69]]
| | 
| | 434.783
| | 104.348
| | 17.391
| | 
|-
| | 
| | 
| | 
| | 
| | 21\[[58edo|58]]
| | 
| | 
| | 434.483
| | 103.448
| | 20.69
| | 
|-
| | 
| | 
| | 
| | 17\[[47edo|47]]
| | 
| | 
| | 
| | 434.043
| | 102.128
| | 25.532
| | 
|-
| | 
| | 
| | 
| | 
| | 30\[[83edo|83]]
| | 
| | 
| | 433.735
| | 101.208
| | 28.916
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 73\[[202edo|202]]
| | 433.663
| | 100.990
| | 29.703
| | Since here are the optimal range Lufsur mode (?)
|-
| | 
| | 
| | 
| | 
| | 
| | 43\[[119edo|119]]
| | 
| | 433.613
| | 100.840
| | 30.252
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 
| | 433.459
| | 100.377
| | 31.95
| | 
|-
| | 
| | 
| | 13\[[36edo|36]]
| | 
| | 
| | 
| | 
| | 433.333
| | 100
| | 33.333
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 
| | 433.048
| | 99.144
| | 36.473
| | 
|-
| | 
| | 
| | 
| | 
| | 35\97
| | 
| | 
| | 432.99
| | 98.969
| | 37.113
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 
| | 432.933
| | 98.799
| | 37.738
| | 
|-
| | 
| | 
| | 
| | 22\[[61edo|61]]
| | 
| | 
| | 
| | 432.787
| | 98.361
| | 39.344
| | 
|-
| | 
| | 9\[[25edo|25]]
| | 
| | 
| | 
| | 
| | 
| | 432
| | 96
| | 48
| style="text-align:center;" | Boundary of propriety;

generators smaller than this are proper
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 
| | 431.417
| | 94.25
| | 54.4155
| | 
|-
| | 
| | 
| | 
| | 23\[[64edo|64]]
| | 
| | 
| | 
| | 431.25
| | 93.75
| | 56.25
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 
| | 431.1185
| | 93.355
| | 57.697
| | 
|-
| | 
| | 
| | 
| | 
| | 37\103
| | 
| | 
| | 431.068
| | 93.204
| | 58.25
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 
| | 430.984
| | 92.952
| | 58.175
| | 
|-
| | 
| | 
| | 14\[[39edo|39]]
| | 
| | 
| | 
| | 
| | 430.769
| | 92.308
| | 61.538
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 47\[[131edo|131]]
| | 
| | 430.534
| | 91.603
| | 64.122
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 80\[[223edo|223]]
| | 430.493
| | 91.480
| | 64.575
| | Until here are the optimal range Fuslur mode (?)
|-
| | 
| | 
| | 
| | 
| | 33\[[92edo|92]]
| | 
| | 
| | 430.435
| | 91.304
| | 65.217
| | 
|-
| | 
| | 
| | 
| | 19\[[53edo|53]]
| | 
| | 
| | 
| | 430.189
| | 90.566
| | 67.925
| | 
|-
| | 
| | 
| | 
| | 
| | 24\[[67edo|67]]
| | 
| | 
| | 429.851
| | 89.552
| | 71.642
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 29\[[81edo|81]]
| | 
| | 429.63
| | 88.889
| | 74.074
| | 
|-
| | 
| | 
| | 
| | 
| | 
| | 
| | 34\[[95edo|95]]
| | 429.474
| | 88.421
| | 75.7895
| | 
|-
| | 5\[[14edo|14]]
| | 
| | 
| | 
| | 
| | 
| | 
| | 428.571
| | 85.714
| | 85.714
| style="text-align:center;" | 
|}</text>
      <sha1>7j31osau7subze8fzezhnuc5dp8eklj</sha1>
    </revision>
    <revision>
      <id>4461</id>
      <timestamp>2015-11-14T14:35:13Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 566463619 - Original comment: **</comment>
      <origin>4461</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="38764" sha1="ljrahx1hrf40myh7igegnkar0yrgthw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-14 14:35:13 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566463619&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]** **φ**
5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]
6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]** **φ**
8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]
9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]
9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]
10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]
10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]
10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;**: [[124edo|124]]
11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]
11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]
11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]
12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]
12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]
12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]
13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]
13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]
13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]
**13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ**
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
**13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]** **φ**
13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]
13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]
13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]
13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]
14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]
14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]
14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]
14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]
14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]
15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]
15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]
15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]
15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]
15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]
15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]
15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]
16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]
16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]
16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]
16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]
16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]
16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]
16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]
17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]
17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]
17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]
17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]
17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]
17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   || 436.364 || 109.091 || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   || 434.783 || 104.348 || 17.391 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   || 434.483 || 103.448 || 20.69 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   || 434.043 || 102.128 || 25.532 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   || 433.735 || 101.208 || 28.916 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] || 433.663 || 100.990 || 29.703 || Since here are the optimal range Lufsur mode (?) ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   || 433.613 || 100.840 || 30.252 ||   ||
||   ||   ||   ||   ||   ||   ||   || 433.459 || 100.377 || 31.95 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   || 433.333 || 100 || 33.333 ||   ||
||   ||   ||   ||   ||   ||   ||   || 433.048 || 99.144 || 36.473 ||   ||
||   ||   ||   ||   || 35\97 ||   ||   || 432.99 || 98.969 || 37.113 ||   ||
||   ||   ||   ||   ||   ||   ||   || 432.933 || 98.799 || 37.738 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   || 432.787 || 98.361 || 39.344 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   ||   || 431.417 || 94.25 || 54.4155 ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   ||   || 431.1185 || 93.355 || 57.697 ||   ||
||   ||   ||   ||   || 37\103 ||   ||   || 431.068 || 93.204 || 58.25 ||   ||
||   ||   ||   ||   ||   ||   ||   || 430.984 || 92.952 || 58.175 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   || 430.769 || 92.308 || 61.538 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   || 430.534 || 91.603 || 64.122 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] || 430.493 || 91.480 || 64.575 || Until here are the optimal range Fuslur mode (?) ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   || 430.435 || 91.304 || 65.217 ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   || 430.189 || 90.566 || 67.925 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   || 429.851 || 89.552 || 71.642 ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   || 429.63 || 88.889 || 74.074 ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] || 429.474 || 88.421 || 75.7895 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   || 428.571 || 85.714 || 85.714 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 1 5 5 5 5 1 5 5 5 5 1: &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 4 5 5 5 5 4 5 5 5 5 4: &amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: &amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: &amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: &amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: &amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
8 8 8 1 8 8 8 8 1 8 8 8 8 1: &amp;lt;a class="wiki_link" href="/91edo"&amp;gt;91&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
8 8 8 7 8 8 8 8 7 8 8 8 8 7: &amp;lt;a class="wiki_link" href="/109edo"&amp;gt;109&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: &amp;lt;a class="wiki_link" href="/102edo"&amp;gt;102&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: &amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: &amp;lt;a class="wiki_link" href="/120edo"&amp;gt;120&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: &amp;lt;a class="wiki_link" href="/123edo"&amp;gt;123&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1:&amp;lt;a class="wiki_link" href="/113edo"&amp;gt;113&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: &amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: &amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: &amp;lt;a class="wiki_link" href="/137edo"&amp;gt;137&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt;: &amp;lt;a class="wiki_link" href="/124edo"&amp;gt;124&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: &amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: &amp;lt;a class="wiki_link" href="/145edo"&amp;gt;145&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :&amp;lt;a class="wiki_link" href="/148edo"&amp;gt;148&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: &amp;lt;a class="wiki_link" href="/151edo"&amp;gt;151&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: &amp;lt;a class="wiki_link" href="/135edo"&amp;gt;135&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: &amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: &amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: &amp;lt;a class="wiki_link" href="/165edo"&amp;gt;165&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: &amp;lt;a class="wiki_link" href="/146edo"&amp;gt;146&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: &amp;lt;a class="wiki_link" href="/149edo"&amp;gt;149&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: &amp;lt;a class="wiki_link" href="/155edo"&amp;gt;155&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: &amp;lt;a class="wiki_link" href="/170edo"&amp;gt;170&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: &amp;lt;a class="wiki_link" href="/173edo"&amp;gt;173&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: &amp;lt;a class="wiki_link" href="/176edo"&amp;gt;176&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
14 14 14 1 14 14 14 14 1 14 14 14 14 1: &amp;lt;a class="wiki_link" href="/157edo"&amp;gt;157&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 3 14 14 14 14 3 14 14 14 14 3: &amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 5 14 14 14 14 5 14 14 14 14 5: &amp;lt;a class="wiki_link" href="/169edo"&amp;gt;169&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 9 14 14 14 14 9 14 14 14 14 9: &amp;lt;a class="wiki_link" href="/181edo"&amp;gt;181&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 11 14 14 14 14 11 14 14 14 14 11: &amp;lt;a class="wiki_link" href="/187edo"&amp;gt;187&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 13 14 14 14 14 13 14 14 14 14 13: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
15 15 15 1 15 15 15 15 1 15 15 15 15 1: &amp;lt;a class="wiki_link" href="/168edo"&amp;gt;168&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 2 15 15 15 15 2 15 15 15 15 2: &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 4 15 15 15 15 4 15 15 15 15 4: &amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 7 15 15 15 15 7 15 15 15 15 7: &amp;lt;a class="wiki_link" href="/186edo"&amp;gt;186&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 8 15 15 15 15 8 15 15 15 15 8: &amp;lt;a class="wiki_link" href="/189edo"&amp;gt;189&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 11 15 15 15 15 11 15 15 15 15 11: &amp;lt;a class="wiki_link" href="/198edo"&amp;gt;198&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 13 15 15 15 15 13 15 15 15 15 13: &amp;lt;a class="wiki_link" href="/204edo"&amp;gt;204&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 14 15 15 15 15 14 15 15 15 15 14: &amp;lt;a class="wiki_link" href="/207edo"&amp;gt;207&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
16 16 16 1 16 16 16 16 1 16 16 16 16 1: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 3 16 16 16 16 3 16 16 16 16 3: &amp;lt;a class="wiki_link" href="/185edo"&amp;gt;185&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 5 16 16 16 16 5 16 16 16 16 5: &amp;lt;a class="wiki_link" href="/191edo"&amp;gt;191&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 7 16 16 16 16 7 16 16 16 16 7: &amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 9 16 16 16 16 9 16 16 16 16 9: &amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 11 16 16 16 16 11 16 16 16 16 11: &amp;lt;a class="wiki_link" href="/209edo"&amp;gt;209&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 13 16 16 16 16 13 16 16 16 16 13: &amp;lt;a class="wiki_link" href="/215edo"&amp;gt;215&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 15 16 16 16 16 15 16 16 16 16 15: &amp;lt;a class="wiki_link" href="/221edo"&amp;gt;221&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
17 17 17 1 17 17 17 17 1 17 17 17 17 1: &amp;lt;a class="wiki_link" href="/190edo"&amp;gt;190&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 2 17 17 17 17 2 17 17 17 17 2: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 3 17 17 17 17 3 17 17 17 17 3: &amp;lt;a class="wiki_link" href="/196edo"&amp;gt;196&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 4 17 17 17 17 4 17 17 17 17 4: &amp;lt;a class="wiki_link" href="/199edo"&amp;gt;199&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: &amp;lt;a class="wiki_link" href="/205edo"&amp;gt;205&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: &amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: &amp;lt;a class="wiki_link" href="/211edo"&amp;gt;211&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: &amp;lt;a class="wiki_link" href="/214edo"&amp;gt;214&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: &amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: &amp;lt;a class="wiki_link" href="/220edo"&amp;gt;220&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 13 17 17 17 17 13 17 17 17 17 13: &amp;lt;a class="wiki_link" href="/226edo"&amp;gt;226&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 14 17 17 17 17 14 17 17 17 17 14: &amp;lt;a class="wiki_link" href="/229edo"&amp;gt;229&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 15 17 17 17 17 15 17 17 17 17 15: &amp;lt;a class="wiki_link" href="/232edo"&amp;gt;232&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 16 17 17 17 17 16 17 17 17 17 16: &amp;lt;a class="wiki_link" href="/235edo"&amp;gt;235&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.364&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.348&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.483&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17\&amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.043&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.532&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30\&amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.208&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.916&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.663&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.990&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29.703&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Since here are the optimal range Lufsur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43\&amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.613&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.459&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.377&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;31.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;98.969&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.068&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;93.204&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.603&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.122&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;430.493&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.575&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;71.642&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.889&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;74.074&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34\&amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.474&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.421&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;75.7895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\&amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;428.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ljrahx1hrf40myh7igegnkar0yrgthw</sha1>
    </revision>
    <revision>
      <id>4462</id>
      <timestamp>2015-10-24T16:12:05Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 563749003 - Original comment: **</comment>
      <origin>4462</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="36964" sha1="53wux5dcpykb1kp31e7a09x7iofa40p" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-10-24 16:12:05 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;563749003&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]** **φ**
5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]
6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]** **φ**
8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]
9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]
9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]
10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]
10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]
10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;**: [[124edo|124]]
11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]
11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]
11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]
12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]
12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]
12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]
13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]
13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]
13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]
**13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ**
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
**13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]** **φ**
13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]
13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]
13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]
13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]
14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]
14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]
14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]
14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]
14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]
15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]
15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]
15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]
15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]
15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]
15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]
15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]
16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]
16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]
16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]
16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]
16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]
16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]
16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]
17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]
17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]
17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]
17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]
17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]
17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   || 436.364 || 109.091 || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   || 434.783 || 104.348 || 17.391 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   || 434.483 || 103.448 || 20.69 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   || 434.043 || 102.128 || 25.532 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   || 433.735 || 101.208 || 28.916 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] || 433.663 || 100.990 || 29.703 || Since here are the optimal range Lufsur mode (?) ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   || 433.613 || 100.840 || 30.252 ||   ||
||   ||   ||   ||   ||   ||   ||   || 433.459 || 100.377 || 31.95 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   || 433.333 || 100 || 33.333 ||   ||
||   ||   ||   ||   ||   ||   ||   || 433.048 || 99.144 || 36.473 ||   ||
||   ||   ||   ||   || 35\97 ||   ||   || 432.99 || 98.969 || 37.113 ||   ||
||   ||   ||   ||   ||   ||   ||   || 432.933 || 98.799 || 37.738 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   || 432.787 || 98.361 || 39.344 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   ||   || 431.1185 || 93.355 || 57.697 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   || 430.769 || 92.308 || 61.538 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   || 430.534 || 91.603 || 64.122 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] || 430.493 || 91.480 || 64.575 || Until here are the optimal range Fuslur mode (?) ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   || 430.435 || 91.304 || 65.217 ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   || 430.189 || 90.566 || 67.925 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   || 429.851 || 89.552 || 71.642 ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   || 429.63 || 88.889 || 74.074 ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] || 429.474 || 88.421 || 75.7895 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   || 428.571 || 85.714 || 85.714 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 1 5 5 5 5 1 5 5 5 5 1: &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 4 5 5 5 5 4 5 5 5 5 4: &amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: &amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: &amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: &amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: &amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
8 8 8 1 8 8 8 8 1 8 8 8 8 1: &amp;lt;a class="wiki_link" href="/91edo"&amp;gt;91&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
8 8 8 7 8 8 8 8 7 8 8 8 8 7: &amp;lt;a class="wiki_link" href="/109edo"&amp;gt;109&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: &amp;lt;a class="wiki_link" href="/102edo"&amp;gt;102&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: &amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: &amp;lt;a class="wiki_link" href="/120edo"&amp;gt;120&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: &amp;lt;a class="wiki_link" href="/123edo"&amp;gt;123&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1:&amp;lt;a class="wiki_link" href="/113edo"&amp;gt;113&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: &amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: &amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: &amp;lt;a class="wiki_link" href="/137edo"&amp;gt;137&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt;: &amp;lt;a class="wiki_link" href="/124edo"&amp;gt;124&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: &amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: &amp;lt;a class="wiki_link" href="/145edo"&amp;gt;145&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :&amp;lt;a class="wiki_link" href="/148edo"&amp;gt;148&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: &amp;lt;a class="wiki_link" href="/151edo"&amp;gt;151&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: &amp;lt;a class="wiki_link" href="/135edo"&amp;gt;135&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: &amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: &amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: &amp;lt;a class="wiki_link" href="/165edo"&amp;gt;165&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: &amp;lt;a class="wiki_link" href="/146edo"&amp;gt;146&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: &amp;lt;a class="wiki_link" href="/149edo"&amp;gt;149&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: &amp;lt;a class="wiki_link" href="/155edo"&amp;gt;155&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: &amp;lt;a class="wiki_link" href="/170edo"&amp;gt;170&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: &amp;lt;a class="wiki_link" href="/173edo"&amp;gt;173&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: &amp;lt;a class="wiki_link" href="/176edo"&amp;gt;176&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
14 14 14 1 14 14 14 14 1 14 14 14 14 1: &amp;lt;a class="wiki_link" href="/157edo"&amp;gt;157&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 3 14 14 14 14 3 14 14 14 14 3: &amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 5 14 14 14 14 5 14 14 14 14 5: &amp;lt;a class="wiki_link" href="/169edo"&amp;gt;169&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 9 14 14 14 14 9 14 14 14 14 9: &amp;lt;a class="wiki_link" href="/181edo"&amp;gt;181&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 11 14 14 14 14 11 14 14 14 14 11: &amp;lt;a class="wiki_link" href="/187edo"&amp;gt;187&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 13 14 14 14 14 13 14 14 14 14 13: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
15 15 15 1 15 15 15 15 1 15 15 15 15 1: &amp;lt;a class="wiki_link" href="/168edo"&amp;gt;168&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 2 15 15 15 15 2 15 15 15 15 2: &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 4 15 15 15 15 4 15 15 15 15 4: &amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 7 15 15 15 15 7 15 15 15 15 7: &amp;lt;a class="wiki_link" href="/186edo"&amp;gt;186&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 8 15 15 15 15 8 15 15 15 15 8: &amp;lt;a class="wiki_link" href="/189edo"&amp;gt;189&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 11 15 15 15 15 11 15 15 15 15 11: &amp;lt;a class="wiki_link" href="/198edo"&amp;gt;198&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 13 15 15 15 15 13 15 15 15 15 13: &amp;lt;a class="wiki_link" href="/204edo"&amp;gt;204&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 14 15 15 15 15 14 15 15 15 15 14: &amp;lt;a class="wiki_link" href="/207edo"&amp;gt;207&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
16 16 16 1 16 16 16 16 1 16 16 16 16 1: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 3 16 16 16 16 3 16 16 16 16 3: &amp;lt;a class="wiki_link" href="/185edo"&amp;gt;185&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 5 16 16 16 16 5 16 16 16 16 5: &amp;lt;a class="wiki_link" href="/191edo"&amp;gt;191&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 7 16 16 16 16 7 16 16 16 16 7: &amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 9 16 16 16 16 9 16 16 16 16 9: &amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 11 16 16 16 16 11 16 16 16 16 11: &amp;lt;a class="wiki_link" href="/209edo"&amp;gt;209&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 13 16 16 16 16 13 16 16 16 16 13: &amp;lt;a class="wiki_link" href="/215edo"&amp;gt;215&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 15 16 16 16 16 15 16 16 16 16 15: &amp;lt;a class="wiki_link" href="/221edo"&amp;gt;221&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
17 17 17 1 17 17 17 17 1 17 17 17 17 1: &amp;lt;a class="wiki_link" href="/190edo"&amp;gt;190&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 2 17 17 17 17 2 17 17 17 17 2: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 3 17 17 17 17 3 17 17 17 17 3: &amp;lt;a class="wiki_link" href="/196edo"&amp;gt;196&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 4 17 17 17 17 4 17 17 17 17 4: &amp;lt;a class="wiki_link" href="/199edo"&amp;gt;199&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: &amp;lt;a class="wiki_link" href="/205edo"&amp;gt;205&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: &amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: &amp;lt;a class="wiki_link" href="/211edo"&amp;gt;211&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: &amp;lt;a class="wiki_link" href="/214edo"&amp;gt;214&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: &amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: &amp;lt;a class="wiki_link" href="/220edo"&amp;gt;220&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 13 17 17 17 17 13 17 17 17 17 13: &amp;lt;a class="wiki_link" href="/226edo"&amp;gt;226&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 14 17 17 17 17 14 17 17 17 17 14: &amp;lt;a class="wiki_link" href="/229edo"&amp;gt;229&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 15 17 17 17 17 15 17 17 17 17 15: &amp;lt;a class="wiki_link" href="/232edo"&amp;gt;232&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 16 17 17 17 17 16 17 17 17 17 16: &amp;lt;a class="wiki_link" href="/235edo"&amp;gt;235&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.364&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.348&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.483&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17\&amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.043&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.532&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30\&amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.208&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.916&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.663&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.990&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29.703&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Since here are the optimal range Lufsur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43\&amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.613&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.459&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.377&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;31.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13\&amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.048&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;99.144&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;36.473&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;35\97&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;37.113&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.933&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.799&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;37.738&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22\&amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.787&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.361&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;39.344&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23\&amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.1185&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.355&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;57.697&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14\&amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.769&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92.308&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;61.538&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;47\&amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.534&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.603&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.122&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;80\&amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.493&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.575&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Until here are the optimal range Fuslur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33\&amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.304&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;65.217&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19\&amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.189&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;90.566&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;67.925&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;24\&amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.851&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;89.552&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71.642&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.889&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;74.074&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34\&amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.474&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.421&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;75.7895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\&amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;428.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>53wux5dcpykb1kp31e7a09x7iofa40p</sha1>
    </revision>
    <revision>
      <id>4463</id>
      <timestamp>2015-05-23T23:00:24Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 551986736 - Original comment: **</comment>
      <origin>4463</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="35164" sha1="s6pranrbebgqcs5s40au25tpvs0vn70" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-23 23:00:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;551986736&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]** **φ**
5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]
6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]** **φ**
8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]
9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]
9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]
10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]
10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]
10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;**: [[124edo|124]]
11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]
11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]
11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]
12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]
12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]
12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]
13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]
13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]
13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]
**13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ**
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
**13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]** **φ**
13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]
13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]
13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]
13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]
14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]
14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]
14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]
14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]
14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]
15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]
15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]
15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]
15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]
15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]
15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]
15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]
16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]
16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]
16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]
16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]
16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]
16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]
16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]
17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]
17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]
17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]
17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]
17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]
17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   || 436.364 || 109.091 || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   || 434.783 || 104.348 || 17.391 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   || 434.483 || 103.448 || 20.69 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   || 434.043 || 102.128 || 25.532 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   || 433.735 || 101.208 || 28.916 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] || 433.663 || 100.990 || 29.703 || Since here are the optimal range Lufsur mode (?) ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   || 433.613 || 100.840 || 30.252 ||   ||
||   ||   ||   ||   ||   ||   ||   || 433.459 || 100.377 || 31.95 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   || 433.333 || 100 || 33.333 ||   ||
||   ||   ||   ||   ||   ||   ||   || 433.048 || 99.144 || 36.473 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   || 432.787 || 98.361 || 39.344 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   || 430.769 || 92.308 || 61.538 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   || 430.534 || 91.603 || 64.122 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] || 430.493 || 91.480 || 64.575 || Until here are the optimal range Fuslur mode (?) ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   || 430.435 || 91.304 || 65.217 ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   || 430.189 || 90.566 || 67.925 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   || 429.851 || 89.552 || 71.642 ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   || 429.63 || 88.889 || 74.074 ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] || 429.474 || 88.421 || 75.7895 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   || 428.571 || 85.714 || 85.714 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 1 5 5 5 5 1 5 5 5 5 1: &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 4 5 5 5 5 4 5 5 5 5 4: &amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: &amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: &amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: &amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: &amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
8 8 8 1 8 8 8 8 1 8 8 8 8 1: &amp;lt;a class="wiki_link" href="/91edo"&amp;gt;91&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
8 8 8 7 8 8 8 8 7 8 8 8 8 7: &amp;lt;a class="wiki_link" href="/109edo"&amp;gt;109&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: &amp;lt;a class="wiki_link" href="/102edo"&amp;gt;102&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: &amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: &amp;lt;a class="wiki_link" href="/120edo"&amp;gt;120&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: &amp;lt;a class="wiki_link" href="/123edo"&amp;gt;123&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1:&amp;lt;a class="wiki_link" href="/113edo"&amp;gt;113&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: &amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: &amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: &amp;lt;a class="wiki_link" href="/137edo"&amp;gt;137&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt;: &amp;lt;a class="wiki_link" href="/124edo"&amp;gt;124&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: &amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: &amp;lt;a class="wiki_link" href="/145edo"&amp;gt;145&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :&amp;lt;a class="wiki_link" href="/148edo"&amp;gt;148&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: &amp;lt;a class="wiki_link" href="/151edo"&amp;gt;151&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: &amp;lt;a class="wiki_link" href="/135edo"&amp;gt;135&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: &amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: &amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: &amp;lt;a class="wiki_link" href="/165edo"&amp;gt;165&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: &amp;lt;a class="wiki_link" href="/146edo"&amp;gt;146&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: &amp;lt;a class="wiki_link" href="/149edo"&amp;gt;149&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: &amp;lt;a class="wiki_link" href="/155edo"&amp;gt;155&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: &amp;lt;a class="wiki_link" href="/170edo"&amp;gt;170&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: &amp;lt;a class="wiki_link" href="/173edo"&amp;gt;173&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: &amp;lt;a class="wiki_link" href="/176edo"&amp;gt;176&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
14 14 14 1 14 14 14 14 1 14 14 14 14 1: &amp;lt;a class="wiki_link" href="/157edo"&amp;gt;157&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 3 14 14 14 14 3 14 14 14 14 3: &amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 5 14 14 14 14 5 14 14 14 14 5: &amp;lt;a class="wiki_link" href="/169edo"&amp;gt;169&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 9 14 14 14 14 9 14 14 14 14 9: &amp;lt;a class="wiki_link" href="/181edo"&amp;gt;181&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 11 14 14 14 14 11 14 14 14 14 11: &amp;lt;a class="wiki_link" href="/187edo"&amp;gt;187&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 13 14 14 14 14 13 14 14 14 14 13: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
15 15 15 1 15 15 15 15 1 15 15 15 15 1: &amp;lt;a class="wiki_link" href="/168edo"&amp;gt;168&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 2 15 15 15 15 2 15 15 15 15 2: &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 4 15 15 15 15 4 15 15 15 15 4: &amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 7 15 15 15 15 7 15 15 15 15 7: &amp;lt;a class="wiki_link" href="/186edo"&amp;gt;186&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 8 15 15 15 15 8 15 15 15 15 8: &amp;lt;a class="wiki_link" href="/189edo"&amp;gt;189&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 11 15 15 15 15 11 15 15 15 15 11: &amp;lt;a class="wiki_link" href="/198edo"&amp;gt;198&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 13 15 15 15 15 13 15 15 15 15 13: &amp;lt;a class="wiki_link" href="/204edo"&amp;gt;204&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 14 15 15 15 15 14 15 15 15 15 14: &amp;lt;a class="wiki_link" href="/207edo"&amp;gt;207&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
16 16 16 1 16 16 16 16 1 16 16 16 16 1: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 3 16 16 16 16 3 16 16 16 16 3: &amp;lt;a class="wiki_link" href="/185edo"&amp;gt;185&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 5 16 16 16 16 5 16 16 16 16 5: &amp;lt;a class="wiki_link" href="/191edo"&amp;gt;191&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 7 16 16 16 16 7 16 16 16 16 7: &amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 9 16 16 16 16 9 16 16 16 16 9: &amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 11 16 16 16 16 11 16 16 16 16 11: &amp;lt;a class="wiki_link" href="/209edo"&amp;gt;209&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 13 16 16 16 16 13 16 16 16 16 13: &amp;lt;a class="wiki_link" href="/215edo"&amp;gt;215&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 15 16 16 16 16 15 16 16 16 16 15: &amp;lt;a class="wiki_link" href="/221edo"&amp;gt;221&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
17 17 17 1 17 17 17 17 1 17 17 17 17 1: &amp;lt;a class="wiki_link" href="/190edo"&amp;gt;190&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 2 17 17 17 17 2 17 17 17 17 2: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 3 17 17 17 17 3 17 17 17 17 3: &amp;lt;a class="wiki_link" href="/196edo"&amp;gt;196&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 4 17 17 17 17 4 17 17 17 17 4: &amp;lt;a class="wiki_link" href="/199edo"&amp;gt;199&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: &amp;lt;a class="wiki_link" href="/205edo"&amp;gt;205&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: &amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: &amp;lt;a class="wiki_link" href="/211edo"&amp;gt;211&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: &amp;lt;a class="wiki_link" href="/214edo"&amp;gt;214&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: &amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: &amp;lt;a class="wiki_link" href="/220edo"&amp;gt;220&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 13 17 17 17 17 13 17 17 17 17 13: &amp;lt;a class="wiki_link" href="/226edo"&amp;gt;226&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 14 17 17 17 17 14 17 17 17 17 14: &amp;lt;a class="wiki_link" href="/229edo"&amp;gt;229&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 15 17 17 17 17 15 17 17 17 17 15: &amp;lt;a class="wiki_link" href="/232edo"&amp;gt;232&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 16 17 17 17 17 16 17 17 17 17 16: &amp;lt;a class="wiki_link" href="/235edo"&amp;gt;235&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.364&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.348&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.483&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17\&amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.043&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.532&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30\&amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.208&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.916&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.663&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.990&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29.703&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Since here are the optimal range Lufsur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43\&amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.613&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.459&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.377&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;31.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13\&amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.048&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;99.144&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;36.473&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22\&amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.787&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.361&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;39.344&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\&amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23\&amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14\&amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.769&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92.308&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;61.538&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;47\&amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.534&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.603&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.122&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;80\&amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.493&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.575&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Until here are the optimal range Fuslur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33\&amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.304&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;65.217&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19\&amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.189&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;90.566&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;67.925&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;24\&amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.851&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;89.552&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71.642&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.889&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;74.074&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34\&amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.474&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.421&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;75.7895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\&amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;428.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>s6pranrbebgqcs5s40au25tpvs0vn70</sha1>
    </revision>
    <revision>
      <id>4464</id>
      <timestamp>2015-02-12T14:10:18Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540792018 - Original comment: **</comment>
      <origin>4464</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="35166" sha1="8efeelsjkgqv023lvsmxdvt7lvxghi4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-12 14:10:18 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540792018&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]** **φ**
5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]
6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]** **φ**
8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]
9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]
9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]
10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]
10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]
10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;**: [[124edo|124]]
11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]
11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]
11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]
12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]
12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]
12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]
13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]
13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]
13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]
**13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ**
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
**13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]** **φ**
13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]
13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]
13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]
13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]
14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]
14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]
14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]
14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]
14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]
15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]
15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]
15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]
15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]
15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]
15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]
15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]
16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]
16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]
16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]
16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]
16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]
16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]
16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]
17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]
17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]
17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]
17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]
17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]
17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   || 436.364 || 109.091 || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   || 434.783 || 104.348 || 17.391 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   || 434.483 || 103.448 || 20.69 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   || 434.043 || 102.128 || 25.532 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   || 433.735 || 101.208 || 28.916 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] || 433.663 || 100.990 || 29.703 || Since here are the optimal range Lufsur mode (?) ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   || 433.613 || 100.840 || 30.252 ||   ||
||   ||   ||   ||   ||   ||   ||   || 433.359 || 100.077 || 33.052 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   || 433.333 || 100 || 33.333 ||   ||
||   ||   ||   ||   ||   ||   ||   || 433.308 || 99.923 || 33.617 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   || 432.787 || 98.361 || 39.344 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   || 430.769 || 92.308 || 61.538 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   || 430.534 || 91.603 || 64.122 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] || 430.493 || 91.480 || 64.575 || Until here are the optimal range Fuslur mode (?) ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   || 430.435 || 91.304 || 65.217 ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   || 430.189 || 90.566 || 67.925 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   || 429.851 || 89.552 || 71.642 ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   || 429.63 || 88.889 || 74.074 ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] || 429.474 || 88.421 || 75.7895 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   || 428.571 || 85.714 || 85.714 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 1 5 5 5 5 1 5 5 5 5 1: &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 4 5 5 5 5 4 5 5 5 5 4: &amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: &amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: &amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: &amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: &amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
8 8 8 1 8 8 8 8 1 8 8 8 8 1: &amp;lt;a class="wiki_link" href="/91edo"&amp;gt;91&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
8 8 8 7 8 8 8 8 7 8 8 8 8 7: &amp;lt;a class="wiki_link" href="/109edo"&amp;gt;109&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: &amp;lt;a class="wiki_link" href="/102edo"&amp;gt;102&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: &amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: &amp;lt;a class="wiki_link" href="/120edo"&amp;gt;120&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: &amp;lt;a class="wiki_link" href="/123edo"&amp;gt;123&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1:&amp;lt;a class="wiki_link" href="/113edo"&amp;gt;113&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: &amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: &amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: &amp;lt;a class="wiki_link" href="/137edo"&amp;gt;137&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt;: &amp;lt;a class="wiki_link" href="/124edo"&amp;gt;124&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: &amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: &amp;lt;a class="wiki_link" href="/145edo"&amp;gt;145&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :&amp;lt;a class="wiki_link" href="/148edo"&amp;gt;148&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: &amp;lt;a class="wiki_link" href="/151edo"&amp;gt;151&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: &amp;lt;a class="wiki_link" href="/135edo"&amp;gt;135&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: &amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: &amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: &amp;lt;a class="wiki_link" href="/165edo"&amp;gt;165&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: &amp;lt;a class="wiki_link" href="/146edo"&amp;gt;146&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: &amp;lt;a class="wiki_link" href="/149edo"&amp;gt;149&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: &amp;lt;a class="wiki_link" href="/155edo"&amp;gt;155&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: &amp;lt;a class="wiki_link" href="/170edo"&amp;gt;170&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: &amp;lt;a class="wiki_link" href="/173edo"&amp;gt;173&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: &amp;lt;a class="wiki_link" href="/176edo"&amp;gt;176&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
14 14 14 1 14 14 14 14 1 14 14 14 14 1: &amp;lt;a class="wiki_link" href="/157edo"&amp;gt;157&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 3 14 14 14 14 3 14 14 14 14 3: &amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 5 14 14 14 14 5 14 14 14 14 5: &amp;lt;a class="wiki_link" href="/169edo"&amp;gt;169&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 9 14 14 14 14 9 14 14 14 14 9: &amp;lt;a class="wiki_link" href="/181edo"&amp;gt;181&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 11 14 14 14 14 11 14 14 14 14 11: &amp;lt;a class="wiki_link" href="/187edo"&amp;gt;187&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 13 14 14 14 14 13 14 14 14 14 13: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
15 15 15 1 15 15 15 15 1 15 15 15 15 1: &amp;lt;a class="wiki_link" href="/168edo"&amp;gt;168&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 2 15 15 15 15 2 15 15 15 15 2: &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 4 15 15 15 15 4 15 15 15 15 4: &amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 7 15 15 15 15 7 15 15 15 15 7: &amp;lt;a class="wiki_link" href="/186edo"&amp;gt;186&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 8 15 15 15 15 8 15 15 15 15 8: &amp;lt;a class="wiki_link" href="/189edo"&amp;gt;189&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 11 15 15 15 15 11 15 15 15 15 11: &amp;lt;a class="wiki_link" href="/198edo"&amp;gt;198&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 13 15 15 15 15 13 15 15 15 15 13: &amp;lt;a class="wiki_link" href="/204edo"&amp;gt;204&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 14 15 15 15 15 14 15 15 15 15 14: &amp;lt;a class="wiki_link" href="/207edo"&amp;gt;207&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
16 16 16 1 16 16 16 16 1 16 16 16 16 1: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 3 16 16 16 16 3 16 16 16 16 3: &amp;lt;a class="wiki_link" href="/185edo"&amp;gt;185&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 5 16 16 16 16 5 16 16 16 16 5: &amp;lt;a class="wiki_link" href="/191edo"&amp;gt;191&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 7 16 16 16 16 7 16 16 16 16 7: &amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 9 16 16 16 16 9 16 16 16 16 9: &amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 11 16 16 16 16 11 16 16 16 16 11: &amp;lt;a class="wiki_link" href="/209edo"&amp;gt;209&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 13 16 16 16 16 13 16 16 16 16 13: &amp;lt;a class="wiki_link" href="/215edo"&amp;gt;215&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 15 16 16 16 16 15 16 16 16 16 15: &amp;lt;a class="wiki_link" href="/221edo"&amp;gt;221&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
17 17 17 1 17 17 17 17 1 17 17 17 17 1: &amp;lt;a class="wiki_link" href="/190edo"&amp;gt;190&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 2 17 17 17 17 2 17 17 17 17 2: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 3 17 17 17 17 3 17 17 17 17 3: &amp;lt;a class="wiki_link" href="/196edo"&amp;gt;196&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 4 17 17 17 17 4 17 17 17 17 4: &amp;lt;a class="wiki_link" href="/199edo"&amp;gt;199&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: &amp;lt;a class="wiki_link" href="/205edo"&amp;gt;205&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: &amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: &amp;lt;a class="wiki_link" href="/211edo"&amp;gt;211&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: &amp;lt;a class="wiki_link" href="/214edo"&amp;gt;214&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: &amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: &amp;lt;a class="wiki_link" href="/220edo"&amp;gt;220&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 13 17 17 17 17 13 17 17 17 17 13: &amp;lt;a class="wiki_link" href="/226edo"&amp;gt;226&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 14 17 17 17 17 14 17 17 17 17 14: &amp;lt;a class="wiki_link" href="/229edo"&amp;gt;229&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 15 17 17 17 17 15 17 17 17 17 15: &amp;lt;a class="wiki_link" href="/232edo"&amp;gt;232&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 16 17 17 17 17 16 17 17 17 17 16: &amp;lt;a class="wiki_link" href="/235edo"&amp;gt;235&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.364&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.348&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.483&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17\&amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.043&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.532&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30\&amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.208&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.916&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.663&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.990&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29.703&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Since here are the optimal range Lufsur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43\&amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.613&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.359&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.077&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33.052&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13\&amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.308&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;99.923&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33.617&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22\&amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.787&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.361&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;39.344&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\&amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23\&amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14\&amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.769&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92.308&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;61.538&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;47\&amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.534&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.603&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.122&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;80\&amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.493&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.575&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Until here are the optimal range Fuslur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;33\&amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;430.435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.304&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;65.217&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19\&amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.189&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;90.566&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;67.925&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;24\&amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.851&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;89.552&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71.642&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.889&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;74.074&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34\&amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.474&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.421&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;75.7895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\&amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;428.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>8efeelsjkgqv023lvsmxdvt7lvxghi4</sha1>
    </revision>
    <revision>
      <id>4465</id>
      <timestamp>2012-12-16T02:24:42Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 392655808 - Original comment: **</comment>
      <origin>4465</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="33976" sha1="lcyecdxfm6534ukoi1zgncdpi7mncpd" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-12-16 02:24:42 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;392655808&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]** **φ**
5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]
6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]** **φ**
8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]
9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]
9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]
10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]
10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]
10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;**: [[124edo|124]]
11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]
11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]
11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]
12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]
12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]
12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]
13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]
13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]
13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]
**13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ**
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
**13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]** **φ**
13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]
13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]
13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]
13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]
14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]
14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]
14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]
14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]
14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]
15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]
15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]
15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]
15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]
15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]
15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]
15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]
16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]
16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]
16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]
16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]
16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]
16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]
16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]
17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]
17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]
17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]
17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]
17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]
17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   || 436.364 || 109.091 || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   || 434.783 || 104.348 || 17.391 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   || 434.483 || 103.448 || 20.69 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   || 434.043 || 102.128 || 25.532 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   || 433.735 || 101.208 || 28.916 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] || 433.663 || 100.990 || 29.703 || Since here are the optimal range Lufsur mode (?) ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   || 433.613 || 100.840 || 30.252 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   || 433.333 || 100 || 33.333 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   || 432.787 || 98.361 || 39.344 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   || 430.769 || 92.308 || 61.538 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   || 430.534 || 91.603 || 64.122 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] || 430.493 || 91.480 || 64.575 || Until here are the optimal range Fuslur mode (?) ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   || 430.435 || 91.304 || 65.217 ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   || 430.189 || 90.566 || 67.925 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   || 429.851 || 89.552 || 71.642 ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   || 429.63 || 88.889 || 74.074 ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] || 429.474 || 88.421 || 75.7895 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   || 428.571 || 85.714 || 85.714 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 1 5 5 5 5 1 5 5 5 5 1: &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 4 5 5 5 5 4 5 5 5 5 4: &amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: &amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: &amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: &amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: &amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
8 8 8 1 8 8 8 8 1 8 8 8 8 1: &amp;lt;a class="wiki_link" href="/91edo"&amp;gt;91&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
8 8 8 7 8 8 8 8 7 8 8 8 8 7: &amp;lt;a class="wiki_link" href="/109edo"&amp;gt;109&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: &amp;lt;a class="wiki_link" href="/102edo"&amp;gt;102&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: &amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: &amp;lt;a class="wiki_link" href="/120edo"&amp;gt;120&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: &amp;lt;a class="wiki_link" href="/123edo"&amp;gt;123&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1:&amp;lt;a class="wiki_link" href="/113edo"&amp;gt;113&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: &amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: &amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: &amp;lt;a class="wiki_link" href="/137edo"&amp;gt;137&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt;: &amp;lt;a class="wiki_link" href="/124edo"&amp;gt;124&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: &amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: &amp;lt;a class="wiki_link" href="/145edo"&amp;gt;145&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :&amp;lt;a class="wiki_link" href="/148edo"&amp;gt;148&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: &amp;lt;a class="wiki_link" href="/151edo"&amp;gt;151&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: &amp;lt;a class="wiki_link" href="/135edo"&amp;gt;135&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: &amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: &amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: &amp;lt;a class="wiki_link" href="/165edo"&amp;gt;165&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: &amp;lt;a class="wiki_link" href="/146edo"&amp;gt;146&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: &amp;lt;a class="wiki_link" href="/149edo"&amp;gt;149&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: &amp;lt;a class="wiki_link" href="/155edo"&amp;gt;155&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: &amp;lt;a class="wiki_link" href="/170edo"&amp;gt;170&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: &amp;lt;a class="wiki_link" href="/173edo"&amp;gt;173&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: &amp;lt;a class="wiki_link" href="/176edo"&amp;gt;176&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
14 14 14 1 14 14 14 14 1 14 14 14 14 1: &amp;lt;a class="wiki_link" href="/157edo"&amp;gt;157&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 3 14 14 14 14 3 14 14 14 14 3: &amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 5 14 14 14 14 5 14 14 14 14 5: &amp;lt;a class="wiki_link" href="/169edo"&amp;gt;169&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 9 14 14 14 14 9 14 14 14 14 9: &amp;lt;a class="wiki_link" href="/181edo"&amp;gt;181&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 11 14 14 14 14 11 14 14 14 14 11: &amp;lt;a class="wiki_link" href="/187edo"&amp;gt;187&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 13 14 14 14 14 13 14 14 14 14 13: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
15 15 15 1 15 15 15 15 1 15 15 15 15 1: &amp;lt;a class="wiki_link" href="/168edo"&amp;gt;168&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 2 15 15 15 15 2 15 15 15 15 2: &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 4 15 15 15 15 4 15 15 15 15 4: &amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 7 15 15 15 15 7 15 15 15 15 7: &amp;lt;a class="wiki_link" href="/186edo"&amp;gt;186&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 8 15 15 15 15 8 15 15 15 15 8: &amp;lt;a class="wiki_link" href="/189edo"&amp;gt;189&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 11 15 15 15 15 11 15 15 15 15 11: &amp;lt;a class="wiki_link" href="/198edo"&amp;gt;198&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 13 15 15 15 15 13 15 15 15 15 13: &amp;lt;a class="wiki_link" href="/204edo"&amp;gt;204&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 14 15 15 15 15 14 15 15 15 15 14: &amp;lt;a class="wiki_link" href="/207edo"&amp;gt;207&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
16 16 16 1 16 16 16 16 1 16 16 16 16 1: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 3 16 16 16 16 3 16 16 16 16 3: &amp;lt;a class="wiki_link" href="/185edo"&amp;gt;185&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 5 16 16 16 16 5 16 16 16 16 5: &amp;lt;a class="wiki_link" href="/191edo"&amp;gt;191&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 7 16 16 16 16 7 16 16 16 16 7: &amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 9 16 16 16 16 9 16 16 16 16 9: &amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 11 16 16 16 16 11 16 16 16 16 11: &amp;lt;a class="wiki_link" href="/209edo"&amp;gt;209&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 13 16 16 16 16 13 16 16 16 16 13: &amp;lt;a class="wiki_link" href="/215edo"&amp;gt;215&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 15 16 16 16 16 15 16 16 16 16 15: &amp;lt;a class="wiki_link" href="/221edo"&amp;gt;221&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
17 17 17 1 17 17 17 17 1 17 17 17 17 1: &amp;lt;a class="wiki_link" href="/190edo"&amp;gt;190&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 2 17 17 17 17 2 17 17 17 17 2: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 3 17 17 17 17 3 17 17 17 17 3: &amp;lt;a class="wiki_link" href="/196edo"&amp;gt;196&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 4 17 17 17 17 4 17 17 17 17 4: &amp;lt;a class="wiki_link" href="/199edo"&amp;gt;199&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: &amp;lt;a class="wiki_link" href="/205edo"&amp;gt;205&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: &amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: &amp;lt;a class="wiki_link" href="/211edo"&amp;gt;211&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: &amp;lt;a class="wiki_link" href="/214edo"&amp;gt;214&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: &amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: &amp;lt;a class="wiki_link" href="/220edo"&amp;gt;220&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 13 17 17 17 17 13 17 17 17 17 13: &amp;lt;a class="wiki_link" href="/226edo"&amp;gt;226&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 14 17 17 17 17 14 17 17 17 17 14: &amp;lt;a class="wiki_link" href="/229edo"&amp;gt;229&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 15 17 17 17 17 15 17 17 17 17 15: &amp;lt;a class="wiki_link" href="/232edo"&amp;gt;232&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 16 17 17 17 17 16 17 17 17 17 16: &amp;lt;a class="wiki_link" href="/235edo"&amp;gt;235&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.364&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.348&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.483&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17\&amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.043&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.532&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30\&amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.208&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.916&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.663&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.990&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29.703&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Since here are the optimal range Lufsur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43\&amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.613&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13\&amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22\&amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.787&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.361&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;39.344&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\&amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23\&amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14\&amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.769&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92.308&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;61.538&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;47\&amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.534&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.603&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.122&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;80\&amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.493&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.575&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Until here are the optimal range Fuslur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33\&amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.304&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;65.217&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19\&amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.189&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;90.566&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;67.925&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;24\&amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.851&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;89.552&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71.642&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.889&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;74.074&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34\&amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.474&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.421&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;75.7895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\&amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;428.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>lcyecdxfm6534ukoi1zgncdpi7mncpd</sha1>
    </revision>
    <revision>
      <id>4466</id>
      <timestamp>2012-12-03T23:27:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 388812104 - Original comment: **</comment>
      <origin>4466</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="33726" sha1="i0ooqnp832bhuj72fe4mb0dly7o7gs4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-12-03 23:27:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;388812104&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]** **φ**
5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]
6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]** **φ**
8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]
9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]
9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]
10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]
10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]
10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;**: [[124edo|124]]
11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]
11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]
11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]
12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]
12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]
12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]
13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]
13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]
13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]
**13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ**
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
**13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]** **φ**
13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]
13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]
13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]
13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]
14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]
14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]
14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]
14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]
14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]
15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]
15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]
15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]
15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]
15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]
15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]
15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]
16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]
16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]
16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]
16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]
16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]
16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]
16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]
17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]
17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]
17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]
17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]
17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]
17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   || 436.364 || 109.091 || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\80 || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\69 ||   || 434.783 || 104.348 || 17.391 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   || 434.483 || 103.448 || 20.69 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   || 434.043 || 102.128 || 25.532 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   || 433.735 || 101.208 || 28.916 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] || 433.663 || 100.990 || 29.703 || Since here are the optimal range Lufsur mode (?) ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   || 433.613 || 100.840 || 30.252 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   || 433.333 || 100 || 33.333 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   || 432.787 || 98.361 || 39.344 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   || 430.769 || 92.308 || 61.538 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   || 430.534 || 91.603 || 64.122 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] || 430.493 || 91.480 || 64.575 || Until here are the optimal range Fuslur mode (?) ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   || 430.435 || 91.304 || 65.217 ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   || 430.189 || 90.566 || 67.925 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   || 429.851 || 89.552 || 71.642 ||   ||
||   ||   ||   ||   ||   || 29\81 ||   || 429.63 || 88.889 || 74.074 ||   ||
||   ||   ||   ||   ||   ||   || 34\95 || 429.474 || 88.421 || 75.7895 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   || 428.571 || 85.714 || 85.714 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 1 5 5 5 5 1 5 5 5 5 1: &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 4 5 5 5 5 4 5 5 5 5 4: &amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: &amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: &amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: &amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: &amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
8 8 8 1 8 8 8 8 1 8 8 8 8 1: &amp;lt;a class="wiki_link" href="/91edo"&amp;gt;91&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
8 8 8 7 8 8 8 8 7 8 8 8 8 7: &amp;lt;a class="wiki_link" href="/109edo"&amp;gt;109&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: &amp;lt;a class="wiki_link" href="/102edo"&amp;gt;102&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: &amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: &amp;lt;a class="wiki_link" href="/120edo"&amp;gt;120&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: &amp;lt;a class="wiki_link" href="/123edo"&amp;gt;123&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1:&amp;lt;a class="wiki_link" href="/113edo"&amp;gt;113&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: &amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: &amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: &amp;lt;a class="wiki_link" href="/137edo"&amp;gt;137&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt;: &amp;lt;a class="wiki_link" href="/124edo"&amp;gt;124&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: &amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: &amp;lt;a class="wiki_link" href="/145edo"&amp;gt;145&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :&amp;lt;a class="wiki_link" href="/148edo"&amp;gt;148&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: &amp;lt;a class="wiki_link" href="/151edo"&amp;gt;151&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: &amp;lt;a class="wiki_link" href="/135edo"&amp;gt;135&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: &amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: &amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: &amp;lt;a class="wiki_link" href="/165edo"&amp;gt;165&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: &amp;lt;a class="wiki_link" href="/146edo"&amp;gt;146&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: &amp;lt;a class="wiki_link" href="/149edo"&amp;gt;149&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: &amp;lt;a class="wiki_link" href="/155edo"&amp;gt;155&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: &amp;lt;a class="wiki_link" href="/170edo"&amp;gt;170&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: &amp;lt;a class="wiki_link" href="/173edo"&amp;gt;173&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: &amp;lt;a class="wiki_link" href="/176edo"&amp;gt;176&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
14 14 14 1 14 14 14 14 1 14 14 14 14 1: &amp;lt;a class="wiki_link" href="/157edo"&amp;gt;157&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 3 14 14 14 14 3 14 14 14 14 3: &amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 5 14 14 14 14 5 14 14 14 14 5: &amp;lt;a class="wiki_link" href="/169edo"&amp;gt;169&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 9 14 14 14 14 9 14 14 14 14 9: &amp;lt;a class="wiki_link" href="/181edo"&amp;gt;181&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 11 14 14 14 14 11 14 14 14 14 11: &amp;lt;a class="wiki_link" href="/187edo"&amp;gt;187&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 13 14 14 14 14 13 14 14 14 14 13: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
15 15 15 1 15 15 15 15 1 15 15 15 15 1: &amp;lt;a class="wiki_link" href="/168edo"&amp;gt;168&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 2 15 15 15 15 2 15 15 15 15 2: &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 4 15 15 15 15 4 15 15 15 15 4: &amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 7 15 15 15 15 7 15 15 15 15 7: &amp;lt;a class="wiki_link" href="/186edo"&amp;gt;186&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 8 15 15 15 15 8 15 15 15 15 8: &amp;lt;a class="wiki_link" href="/189edo"&amp;gt;189&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 11 15 15 15 15 11 15 15 15 15 11: &amp;lt;a class="wiki_link" href="/198edo"&amp;gt;198&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 13 15 15 15 15 13 15 15 15 15 13: &amp;lt;a class="wiki_link" href="/204edo"&amp;gt;204&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 14 15 15 15 15 14 15 15 15 15 14: &amp;lt;a class="wiki_link" href="/207edo"&amp;gt;207&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
16 16 16 1 16 16 16 16 1 16 16 16 16 1: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 3 16 16 16 16 3 16 16 16 16 3: &amp;lt;a class="wiki_link" href="/185edo"&amp;gt;185&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 5 16 16 16 16 5 16 16 16 16 5: &amp;lt;a class="wiki_link" href="/191edo"&amp;gt;191&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 7 16 16 16 16 7 16 16 16 16 7: &amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 9 16 16 16 16 9 16 16 16 16 9: &amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 11 16 16 16 16 11 16 16 16 16 11: &amp;lt;a class="wiki_link" href="/209edo"&amp;gt;209&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 13 16 16 16 16 13 16 16 16 16 13: &amp;lt;a class="wiki_link" href="/215edo"&amp;gt;215&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 15 16 16 16 16 15 16 16 16 16 15: &amp;lt;a class="wiki_link" href="/221edo"&amp;gt;221&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
17 17 17 1 17 17 17 17 1 17 17 17 17 1: &amp;lt;a class="wiki_link" href="/190edo"&amp;gt;190&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 2 17 17 17 17 2 17 17 17 17 2: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 3 17 17 17 17 3 17 17 17 17 3: &amp;lt;a class="wiki_link" href="/196edo"&amp;gt;196&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 4 17 17 17 17 4 17 17 17 17 4: &amp;lt;a class="wiki_link" href="/199edo"&amp;gt;199&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: &amp;lt;a class="wiki_link" href="/205edo"&amp;gt;205&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: &amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: &amp;lt;a class="wiki_link" href="/211edo"&amp;gt;211&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: &amp;lt;a class="wiki_link" href="/214edo"&amp;gt;214&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: &amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: &amp;lt;a class="wiki_link" href="/220edo"&amp;gt;220&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 13 17 17 17 17 13 17 17 17 17 13: &amp;lt;a class="wiki_link" href="/226edo"&amp;gt;226&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 14 17 17 17 17 14 17 17 17 17 14: &amp;lt;a class="wiki_link" href="/229edo"&amp;gt;229&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 15 17 17 17 17 15 17 17 17 17 15: &amp;lt;a class="wiki_link" href="/232edo"&amp;gt;232&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 16 17 17 17 17 16 17 17 17 17 16: &amp;lt;a class="wiki_link" href="/235edo"&amp;gt;235&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.364&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.348&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.483&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17\&amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.043&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.532&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30\&amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.208&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.916&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.663&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.990&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29.703&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Since here are the optimal range Lufsur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43\&amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.613&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13\&amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22\&amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.787&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.361&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;39.344&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\&amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23\&amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14\&amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.769&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92.308&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;61.538&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;47\&amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.534&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.603&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.122&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;80\&amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.493&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.575&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Until here are the optimal range Fuslur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33\&amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.304&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;65.217&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19\&amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.189&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;90.566&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;67.925&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;24\&amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.851&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;89.552&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71.642&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;29\81&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.889&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;74.074&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34\95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.474&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.421&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;75.7895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\&amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;428.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>i0ooqnp832bhuj72fe4mb0dly7o7gs4</sha1>
    </revision>
    <revision>
      <id>4467</id>
      <timestamp>2012-12-01T01:02:46Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 387921300 - Original comment: **</comment>
      <origin>4467</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="31328" sha1="fbjr6y1cjixcufw8i7mq30reg3w5qrf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-12-01 01:02:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;387921300&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]** **φ**
5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]
6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]** **φ**
8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]
9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]
9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]
10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]
10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]
10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;**: [[124edo|124]]
11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]
11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]
11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]
12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]
12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]
12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]
13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]
13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]
13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]
**13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ**
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
**13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]** **φ**
13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]
13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]
13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]
13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]
14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]
14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]
14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]
14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]
14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]
15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]
15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]
15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]
15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]
15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]
15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]
15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]
16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]
16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]
16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]
16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]
16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]
16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]
16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]
17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]
17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]
17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]
17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]
17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]
17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   || 436.364 || 109.091 || 0 ||=   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   || 434.483 || 103.448 || 20.69 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   || 434.043 || 102.128 || 25.532 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   || 433.735 || 101.208 || 28.916 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] || 433.663 || 100.990 || 29.703 || Since here are the optimal range Lufsur mode (?) ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   || 433.613 || 100.840 || 30.252 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   || 433.333 || 100 || 33.333 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   || 432.787 || 98.361 || 39.344 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   || 430.769 || 92.308 || 61.538 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   || 430.534 || 91.603 || 64.122 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] || 430.493 || 91.480 || 64.575 || Until here are the optimal range Fuslur mode (?) ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   || 430.435 || 91.304 || 65.217 ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   || 430.189 || 90.566 || 67.925 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   || 429.851 || 89.552 || 71.642 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   || 428.571 || 85.714 || 85.714 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 1 5 5 5 5 1 5 5 5 5 1: &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 4 5 5 5 5 4 5 5 5 5 4: &amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: &amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: &amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: &amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: &amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
8 8 8 1 8 8 8 8 1 8 8 8 8 1: &amp;lt;a class="wiki_link" href="/91edo"&amp;gt;91&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
8 8 8 7 8 8 8 8 7 8 8 8 8 7: &amp;lt;a class="wiki_link" href="/109edo"&amp;gt;109&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: &amp;lt;a class="wiki_link" href="/102edo"&amp;gt;102&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: &amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: &amp;lt;a class="wiki_link" href="/120edo"&amp;gt;120&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: &amp;lt;a class="wiki_link" href="/123edo"&amp;gt;123&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1:&amp;lt;a class="wiki_link" href="/113edo"&amp;gt;113&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: &amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: &amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: &amp;lt;a class="wiki_link" href="/137edo"&amp;gt;137&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt;: &amp;lt;a class="wiki_link" href="/124edo"&amp;gt;124&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: &amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: &amp;lt;a class="wiki_link" href="/145edo"&amp;gt;145&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :&amp;lt;a class="wiki_link" href="/148edo"&amp;gt;148&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: &amp;lt;a class="wiki_link" href="/151edo"&amp;gt;151&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: &amp;lt;a class="wiki_link" href="/135edo"&amp;gt;135&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: &amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: &amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: &amp;lt;a class="wiki_link" href="/165edo"&amp;gt;165&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: &amp;lt;a class="wiki_link" href="/146edo"&amp;gt;146&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: &amp;lt;a class="wiki_link" href="/149edo"&amp;gt;149&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: &amp;lt;a class="wiki_link" href="/155edo"&amp;gt;155&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: &amp;lt;a class="wiki_link" href="/170edo"&amp;gt;170&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: &amp;lt;a class="wiki_link" href="/173edo"&amp;gt;173&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: &amp;lt;a class="wiki_link" href="/176edo"&amp;gt;176&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
14 14 14 1 14 14 14 14 1 14 14 14 14 1: &amp;lt;a class="wiki_link" href="/157edo"&amp;gt;157&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 3 14 14 14 14 3 14 14 14 14 3: &amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 5 14 14 14 14 5 14 14 14 14 5: &amp;lt;a class="wiki_link" href="/169edo"&amp;gt;169&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 9 14 14 14 14 9 14 14 14 14 9: &amp;lt;a class="wiki_link" href="/181edo"&amp;gt;181&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 11 14 14 14 14 11 14 14 14 14 11: &amp;lt;a class="wiki_link" href="/187edo"&amp;gt;187&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 13 14 14 14 14 13 14 14 14 14 13: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
15 15 15 1 15 15 15 15 1 15 15 15 15 1: &amp;lt;a class="wiki_link" href="/168edo"&amp;gt;168&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 2 15 15 15 15 2 15 15 15 15 2: &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 4 15 15 15 15 4 15 15 15 15 4: &amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 7 15 15 15 15 7 15 15 15 15 7: &amp;lt;a class="wiki_link" href="/186edo"&amp;gt;186&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 8 15 15 15 15 8 15 15 15 15 8: &amp;lt;a class="wiki_link" href="/189edo"&amp;gt;189&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 11 15 15 15 15 11 15 15 15 15 11: &amp;lt;a class="wiki_link" href="/198edo"&amp;gt;198&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 13 15 15 15 15 13 15 15 15 15 13: &amp;lt;a class="wiki_link" href="/204edo"&amp;gt;204&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 14 15 15 15 15 14 15 15 15 15 14: &amp;lt;a class="wiki_link" href="/207edo"&amp;gt;207&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
16 16 16 1 16 16 16 16 1 16 16 16 16 1: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 3 16 16 16 16 3 16 16 16 16 3: &amp;lt;a class="wiki_link" href="/185edo"&amp;gt;185&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 5 16 16 16 16 5 16 16 16 16 5: &amp;lt;a class="wiki_link" href="/191edo"&amp;gt;191&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 7 16 16 16 16 7 16 16 16 16 7: &amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 9 16 16 16 16 9 16 16 16 16 9: &amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 11 16 16 16 16 11 16 16 16 16 11: &amp;lt;a class="wiki_link" href="/209edo"&amp;gt;209&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 13 16 16 16 16 13 16 16 16 16 13: &amp;lt;a class="wiki_link" href="/215edo"&amp;gt;215&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 15 16 16 16 16 15 16 16 16 16 15: &amp;lt;a class="wiki_link" href="/221edo"&amp;gt;221&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
17 17 17 1 17 17 17 17 1 17 17 17 17 1: &amp;lt;a class="wiki_link" href="/190edo"&amp;gt;190&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 2 17 17 17 17 2 17 17 17 17 2: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 3 17 17 17 17 3 17 17 17 17 3: &amp;lt;a class="wiki_link" href="/196edo"&amp;gt;196&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 4 17 17 17 17 4 17 17 17 17 4: &amp;lt;a class="wiki_link" href="/199edo"&amp;gt;199&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: &amp;lt;a class="wiki_link" href="/205edo"&amp;gt;205&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: &amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: &amp;lt;a class="wiki_link" href="/211edo"&amp;gt;211&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: &amp;lt;a class="wiki_link" href="/214edo"&amp;gt;214&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: &amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: &amp;lt;a class="wiki_link" href="/220edo"&amp;gt;220&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 13 17 17 17 17 13 17 17 17 17 13: &amp;lt;a class="wiki_link" href="/226edo"&amp;gt;226&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 14 17 17 17 17 14 17 17 17 17 14: &amp;lt;a class="wiki_link" href="/229edo"&amp;gt;229&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 15 17 17 17 17 15 17 17 17 17 15: &amp;lt;a class="wiki_link" href="/232edo"&amp;gt;232&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 16 17 17 17 17 16 17 17 17 17 16: &amp;lt;a class="wiki_link" href="/235edo"&amp;gt;235&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.364&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.483&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17\&amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.043&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.532&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30\&amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.208&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.916&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.663&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.990&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29.703&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Since here are the optimal range Lufsur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43\&amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.613&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13\&amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22\&amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.787&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.361&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;39.344&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\&amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23\&amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14\&amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.769&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92.308&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;61.538&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;47\&amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.534&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.603&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.122&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;80\&amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.493&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.575&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Until here are the optimal range Fuslur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33\&amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.304&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;65.217&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19\&amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.189&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;90.566&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;67.925&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;24\&amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.851&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;89.552&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71.642&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\&amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;428.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>fbjr6y1cjixcufw8i7mq30reg3w5qrf</sha1>
    </revision>
    <revision>
      <id>4468</id>
      <timestamp>2012-10-05T02:11:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;keenanpepper</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 370449306 - Original comment: **</comment>
      <origin>4468</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="30659" sha1="isba2h0o0anuk740shfkuvip5prvmom" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on &lt;tt&gt;2012-10-05 02:11:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;370449306&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]** **φ**
5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]
6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]** **φ**
8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]
9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]
9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]
10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]
10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]
10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;**: [[124edo|124]]
11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]
11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]
11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]
12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]
12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]
12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]
13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]
13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]
13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]
**13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ**
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
**13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]** **φ**
13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]
13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]
13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]
13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]
14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]
14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]
14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]
14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]
14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]
15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]
15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]
15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]
15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]
15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]
15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]
15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]
16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]
16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]
16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]
16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]
16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]
16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]
16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]
17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]
17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]
17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]
17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]
17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]
17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   || 436.364 || 109.091 || 0 ||=   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   || 434.043 || 102.128 || 25.532 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   || 433.735 || 101.208 || 28.916 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] || 433.663 || 100.990 || 29.703 || Since here are the optimal range Lufsur mode (?) ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   || 433.613 || 100.840 || 30.252 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   || 433.333 || 100 || 33.333 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   || 432.787 || 98.361 || 39.344 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   || 430.769 || 92.308 || 61.538 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   || 430.534 || 91.603 || 64.122 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] || 430.493 || 91.480 || 64.575 || Until here are the optimal range Fuslur mode (?) ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   || 430.435 || 91.304 || 65.217 ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   || 430.189 || 90.566 || 67.925 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   || 429.851 || 89.552 || 71.642 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   || 428.571 || 85.714 || 85.714 ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 1 5 5 5 5 1 5 5 5 5 1: &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 4 5 5 5 5 4 5 5 5 5 4: &amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: &amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: &amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: &amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: &amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
8 8 8 1 8 8 8 8 1 8 8 8 8 1: &amp;lt;a class="wiki_link" href="/91edo"&amp;gt;91&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
8 8 8 7 8 8 8 8 7 8 8 8 8 7: &amp;lt;a class="wiki_link" href="/109edo"&amp;gt;109&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: &amp;lt;a class="wiki_link" href="/102edo"&amp;gt;102&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: &amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: &amp;lt;a class="wiki_link" href="/120edo"&amp;gt;120&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: &amp;lt;a class="wiki_link" href="/123edo"&amp;gt;123&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1:&amp;lt;a class="wiki_link" href="/113edo"&amp;gt;113&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: &amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: &amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: &amp;lt;a class="wiki_link" href="/137edo"&amp;gt;137&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt;: &amp;lt;a class="wiki_link" href="/124edo"&amp;gt;124&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: &amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: &amp;lt;a class="wiki_link" href="/145edo"&amp;gt;145&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :&amp;lt;a class="wiki_link" href="/148edo"&amp;gt;148&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: &amp;lt;a class="wiki_link" href="/151edo"&amp;gt;151&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: &amp;lt;a class="wiki_link" href="/135edo"&amp;gt;135&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: &amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: &amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: &amp;lt;a class="wiki_link" href="/165edo"&amp;gt;165&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: &amp;lt;a class="wiki_link" href="/146edo"&amp;gt;146&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: &amp;lt;a class="wiki_link" href="/149edo"&amp;gt;149&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: &amp;lt;a class="wiki_link" href="/155edo"&amp;gt;155&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: &amp;lt;a class="wiki_link" href="/170edo"&amp;gt;170&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: &amp;lt;a class="wiki_link" href="/173edo"&amp;gt;173&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: &amp;lt;a class="wiki_link" href="/176edo"&amp;gt;176&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
14 14 14 1 14 14 14 14 1 14 14 14 14 1: &amp;lt;a class="wiki_link" href="/157edo"&amp;gt;157&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 3 14 14 14 14 3 14 14 14 14 3: &amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 5 14 14 14 14 5 14 14 14 14 5: &amp;lt;a class="wiki_link" href="/169edo"&amp;gt;169&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 9 14 14 14 14 9 14 14 14 14 9: &amp;lt;a class="wiki_link" href="/181edo"&amp;gt;181&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 11 14 14 14 14 11 14 14 14 14 11: &amp;lt;a class="wiki_link" href="/187edo"&amp;gt;187&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 13 14 14 14 14 13 14 14 14 14 13: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
15 15 15 1 15 15 15 15 1 15 15 15 15 1: &amp;lt;a class="wiki_link" href="/168edo"&amp;gt;168&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 2 15 15 15 15 2 15 15 15 15 2: &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 4 15 15 15 15 4 15 15 15 15 4: &amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 7 15 15 15 15 7 15 15 15 15 7: &amp;lt;a class="wiki_link" href="/186edo"&amp;gt;186&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 8 15 15 15 15 8 15 15 15 15 8: &amp;lt;a class="wiki_link" href="/189edo"&amp;gt;189&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 11 15 15 15 15 11 15 15 15 15 11: &amp;lt;a class="wiki_link" href="/198edo"&amp;gt;198&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 13 15 15 15 15 13 15 15 15 15 13: &amp;lt;a class="wiki_link" href="/204edo"&amp;gt;204&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 14 15 15 15 15 14 15 15 15 15 14: &amp;lt;a class="wiki_link" href="/207edo"&amp;gt;207&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
16 16 16 1 16 16 16 16 1 16 16 16 16 1: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 3 16 16 16 16 3 16 16 16 16 3: &amp;lt;a class="wiki_link" href="/185edo"&amp;gt;185&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 5 16 16 16 16 5 16 16 16 16 5: &amp;lt;a class="wiki_link" href="/191edo"&amp;gt;191&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 7 16 16 16 16 7 16 16 16 16 7: &amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 9 16 16 16 16 9 16 16 16 16 9: &amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 11 16 16 16 16 11 16 16 16 16 11: &amp;lt;a class="wiki_link" href="/209edo"&amp;gt;209&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 13 16 16 16 16 13 16 16 16 16 13: &amp;lt;a class="wiki_link" href="/215edo"&amp;gt;215&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 15 16 16 16 16 15 16 16 16 16 15: &amp;lt;a class="wiki_link" href="/221edo"&amp;gt;221&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
17 17 17 1 17 17 17 17 1 17 17 17 17 1: &amp;lt;a class="wiki_link" href="/190edo"&amp;gt;190&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 2 17 17 17 17 2 17 17 17 17 2: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 3 17 17 17 17 3 17 17 17 17 3: &amp;lt;a class="wiki_link" href="/196edo"&amp;gt;196&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 4 17 17 17 17 4 17 17 17 17 4: &amp;lt;a class="wiki_link" href="/199edo"&amp;gt;199&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: &amp;lt;a class="wiki_link" href="/205edo"&amp;gt;205&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: &amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: &amp;lt;a class="wiki_link" href="/211edo"&amp;gt;211&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: &amp;lt;a class="wiki_link" href="/214edo"&amp;gt;214&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: &amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: &amp;lt;a class="wiki_link" href="/220edo"&amp;gt;220&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 13 17 17 17 17 13 17 17 17 17 13: &amp;lt;a class="wiki_link" href="/226edo"&amp;gt;226&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 14 17 17 17 17 14 17 17 17 17 14: &amp;lt;a class="wiki_link" href="/229edo"&amp;gt;229&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 15 17 17 17 17 15 17 17 17 17 15: &amp;lt;a class="wiki_link" href="/232edo"&amp;gt;232&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 16 17 17 17 17 16 17 17 17 17 16: &amp;lt;a class="wiki_link" href="/235edo"&amp;gt;235&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.364&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17\&amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.043&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.532&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30\&amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.208&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.916&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.663&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.990&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29.703&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Since here are the optimal range Lufsur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43\&amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.613&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.840&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30.252&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13\&amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33.333&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22\&amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.787&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.361&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;39.344&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9\&amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23\&amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14\&amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.769&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92.308&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;61.538&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;47\&amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.534&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.603&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.122&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;80\&amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.493&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.480&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64.575&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Until here are the optimal range Fuslur mode (?)&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33\&amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;91.304&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;65.217&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19\&amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;430.189&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;90.566&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;67.925&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;24\&amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.851&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;89.552&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71.642&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\&amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;428.571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>isba2h0o0anuk740shfkuvip5prvmom</sha1>
    </revision>
    <revision>
      <id>4469</id>
      <timestamp>2012-06-27T00:50:13Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 348421978 - Original comment: **</comment>
      <origin>4469</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="111141" sha1="iaxq9ohmtxql4he4b1afaws1wt2potq" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-06-27 00:50:13 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;348421978&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]** **φ**
5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]
6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]** **φ**
8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]
9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]
9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]
10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]
10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]
10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;**: [[124edo|124]]
11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]
11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]
11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]
12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]
12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]
12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]
13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]
13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]
13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]
**13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ**
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
**13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]** **φ**
13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]
13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]
13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]
13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]
14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]
14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]
14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]
14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]
14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]
15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]
15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]
15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]
15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]
15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]
15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]
15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]
16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]
16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]
16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]
16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]
16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]
16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]
16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]
17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]
17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]
17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]
17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]
17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]
17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   ||   || 436.__36__ || 109.__09__ || 0 ||=   ||
||   ||   ||   ||   ||   ||   ||   || 33\91 || 435.16484 || 105.494505 || 13.18681 ||   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] ||   || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   ||   ||   || 54\149 || 434.89933 || 104.69799 || 16.10738 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   ||   || 434.78261 || 104.34783 || 17.391 ||   ||
||   ||   ||   ||   ||   ||   ||   || 71\196 || 434.69388 || 104.08163 || 18.36735 ||   ||
||   ||   ||   ||   ||   ||   || 46\[[127edo|127]] ||   || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   ||   ||   ||   || 67\185 || 434.__594__ || 103.__783__ || 19.__459__ ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   ||   || 434.48276 || 103.448275 || 20.68966 ||   ||
||   ||   ||   ||   ||   ||   ||   || 80\221 || 434.38914 || 103.16742 || 21.71946 ||   ||
||   ||   ||   ||   ||   ||   || 59\[[163edo|163]] ||   || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   ||   ||   || 97\268 || 434.32836 || 102.985075 || 22.38806 ||   ||
||   ||   ||   ||   ||   || 38\[[105edo|105]] ||   ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   ||   || 93\257 || 434.241245 || 102.723735 || 23.3463 ||   ||
||   ||   ||   ||   ||   ||   || 55\[[152edo|152]] ||   || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   ||   ||   ||   ||   || 72\199 || 434.17085 || 102.51256 || 24.1206 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   ||   || 81\224 || 433.92857 || 101.78571 || 26.78571 ||   ||
||   ||   ||   ||   ||   ||   || 64\[[177edo|177]] ||   || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   ||   ||   || 111\307 || 433.87622 || 101.62866 || 27.36156 ||   ||
||   ||   ||   ||   ||   || 47\[[130edo|130]] ||   ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   ||   || 124\343 || 433.81924 || 101.45773 || 27.98834 ||   ||
||   ||   ||   ||   ||   ||   || 77\[[213edo|213]] ||   || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   ||   ||   ||   || 107\296 || 433.__783__ || 101.__351__ || 28.__378__ ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   ||   || 103\285 || 433.68421 || 101.05263 || 29.47368 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] ||   || 433.__6633__ || 100.__9900__ || 29.__7030__ || Since here are the optimal range Lufsur mode ||
||   ||   ||   ||   ||   ||   ||   || 116\321 || 433.64486 || 100.93458 || 29.96054 ||   ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   ||   || 99\274 || 433.57664 || 100.72993 || 30.65693 ||   ||
||   ||   ||   ||   ||   ||   || 56\[[155edo|155]] ||   || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   ||   ||   ||   ||   ||   || 69\191 || 433.50785 || 100.52356 || 31.41361 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   ||   || 433.__3__ || 100 || 33.__3__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 74\205 || 433.__17073__ || 99.__51219__ || 35.__12195__ ||   ||
||   ||   ||   ||   ||   ||   || 61\[[169edo|169]] ||   || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   ||   ||   || 109\302 || 433.11258 || 99.33775 || 35.76159 ||   ||
||   ||   ||   ||   ||   || 48\[[133edo|133]] ||   ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   ||   || 131\363 || 433.05785 || 99.17355 || 36.__36__ ||   ||
||   ||   ||   ||   ||   ||   || 83\[[230edo|230]] ||   || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   ||   ||   ||   || 118\327 || 433.02752 || 99.08257 || 36.69725 ||   ||
||   ||   ||   ||   || 35\[[97edo|97]] ||   ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   ||   || 127\352 || 432.95__45__ || 98.8__63__ || 37.5 ||   ||
||   ||   ||   ||   ||   ||   || 92\[[255edo|255]] ||   || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   ||   ||   || 149\413 || 432.92978 || 98.78935 || 37.7724 ||   ||
||   ||   ||   ||   ||   || 57\[[158edo|158]] ||   ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   ||   || 136\377 || 432.89125 || 98.67374 || 38.19629 ||   ||
||   ||   ||   ||   ||   ||   || 79\[[219edo|219]] ||   || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   ||   ||   ||   ||   || 101\280 || 432.85714 || 98.57143 || 38.57143 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   ||   || 97\269 || 432.713755 || 98.14126 || 40.1487 ||   ||
||   ||   ||   ||   ||   ||   || 75\[[208edo|208]] ||   || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   ||   ||   || 128\355 || 432.67606 || 98.02817 || 40.56338 ||   ||
||   ||   ||   ||   ||   || 53\[[147edo|147]] ||   ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   ||   || 137\380 || 432.63158 || 97.89474 || 41.05263 ||   ||
||   ||   ||   ||   ||   ||   || 84\[[233edo|233]] ||   || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   ||   ||   ||   || 115\319 || 432.60188 || 97.80564 || 41.37931 ||   ||
||   ||   ||   ||   || 31\[[86edo|86]] ||   ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   ||   || 102\283 || 432.50883 || 97.5265 || 42.40283 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[197edo|197]] ||   || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   ||   ||   || 111\308 || 432.46753 || 97.4026 || 42.85714 ||   ||
||   ||   ||   ||   ||   || 40\[[111edo|111]] ||   ||   || 432.__432__ || 97.__297__ || 43.__135__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 89\247 || 432.38866 || 97.16599 || 43.7247 ||   ||
||   ||   ||   ||   ||   ||   || 49\[[136edo|136]] ||   || 432.35294 || 97.05882 || 44.11765 ||   ||
||   ||   ||   ||   ||   ||   ||   || 58\[[161edo|161]] || 432.29814 || 96.89441 || 44.72051 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   ||   || 59\[[164edo|164]] || 431.__70731__ || 95.__12195__ || 51.__21951__ ||   ||
||   ||   ||   ||   ||   ||   || 50\[[139edo|139]] ||   || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   ||   ||   || 91\253 || 431.62055 || 94.86166 || 52.17391 ||   ||
||   ||   ||   ||   ||   || 41\[[114edo|114]] ||   ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   ||   || 114\317 || 431.54574 || 94.63722 || 52.996845 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[203edo|203]] ||   || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   ||   ||   ||   || 105\292 || 431.50685 || 94.52055 || 53.42466 ||   ||
||   ||   ||   ||   || 32\[[89edo|89]] ||   ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   ||   || 119\331 || 431.41664 || 94.29582 || 54.380665 ||   ||
||   ||   ||   ||   ||   ||   || 87\[[242edo|242]] ||   || 431.40496 || 94.21488 || 54.__54__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 142\395 || 431.392405 || 94.177215 || 54.68354 ||   ||
||   ||   ||   ||   ||   || 55\[[153edo|153]] ||   ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   ||   || 133\370 || 431.__351__ || 94.__054__ || 55.__135__ ||   ||
||   ||   ||   ||   ||   ||   || 78\[[217edo|217]] ||   || 431.33641 || 94.00922 || 55.29955 ||   ||
||   ||   ||   ||   ||   ||   ||   || 101\281 || 431.31673 || 93.95018 || 55.51601 ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   ||   || 106\295 || 431.18644 || 93.55932 || 56.94915 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[231edo|231]] ||   || 431.16833 || 93.50649 || 57.14286 ||   ||
||   ||   ||   ||   ||   ||   ||   || 143\398 || 431.15578 || 93.46734 || 57.28643 ||   ||
||   ||   ||   ||   ||   || 60\[[167edo|167]] ||   ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   ||   || 157\437 || 431.12128 || 93.36384 || 57.6659 ||   ||
||   ||   ||   ||   ||   ||   || 97\[[270edo|270]] ||   || 431.__1__ || 93.__3__ || 57.__8__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 134\373 || 431.0992 || 93.29759 || 57.90885 ||   ||
||   ||   ||   ||   || 37\[[103edo|103]] ||   ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   ||   || 125\348 || 431.03448 || 93.10345 || 58.60269 ||   ||
||   ||   ||   ||   ||   ||   || 88\[[245edo|245]] ||   || 431.02041 || 93.06122 || 58.77551 ||   ||
||   ||   ||   ||   ||   ||   ||   || 139\387 || 431.00775 || 93.02326 || 58.91473 ||   ||
||   ||   ||   ||   ||   || 51\[[142edo|142]] ||   ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   ||   || 116\323 || 430.95975 || 92.87926 || 59.44272 ||   ||
||   ||   ||   ||   ||   ||   || 65\[[181edo|181]] ||   || 430.93923 || 92.81768 || 59.66852 ||   ||
||   ||   ||   ||   ||   ||   ||   || 79\220 || 430.__90__ || 92.__72__ || 60 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   ||   || 75\209 || 430.62201 || 91.86603 || 63.157895 ||   ||
||   ||   ||   ||   ||   ||   || 61\[[170edo|170]] ||   || 430.58824 || 91.76471 || 62.54942 ||   ||
||   ||   ||   ||   ||   ||   ||   || 108\301 || 430.56478 || 91.69435 || 63.787375 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   ||   || 127\354 || 430.508475 || 91.52542 || 64.40678 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] ||   || 430.49327 || 91.47982 || 64.57505 || Until here are the optimal range Fuslur mode ||
||   ||   ||   ||   ||   ||   ||   || 113\315 || 430.47619 || 91.42857 || 64.761905 ||   ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   ||   || 118\329 || 430.39514 || 91.18541 || 65.653495 ||   ||
||   ||   ||   ||   ||   ||   || 85\[[237edo|237]] ||   || 430.37975 || 91.13924 || 65.82309 ||   ||
||   ||   ||   ||   ||   ||   ||   || 137\382 || 430.36649 || 91.09948 || 65.96859 ||   ||
||   ||   ||   ||   ||   || 52\[[145edo|145]] ||   ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   ||   || 123\343 || 430.3207 || 90.9621 || 66.4723 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[198edo|198]] ||   || 430.__30__ || 90.__90__ || 66.__67__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 90\251 || 430.27888 || 90.83665 || 66.93227 ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   ||   || 81\226 || 430.0885 || 90.26549 || 69.02655 ||   ||
||   ||   ||   ||   ||   ||   || 62\[[173edo|173]] ||   || 430.0578 || 90.17341 || 69.36416 ||   ||
||   ||   ||   ||   ||   ||   ||   || 105\293 || 430.03413 || 90.10239 || 69.62457 ||   ||
||   ||   ||   ||   ||   || 43\[[120edo|120]] ||   ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   ||   || 110\307 || 429.96743 || 89.90228 || 70.35831 ||   ||
||   ||   ||   ||   ||   ||   || 67\[[187edo|187]] ||   || 429.94652 || 89.83957 || 70.58824 ||   ||
||   ||   ||   ||   ||   ||   ||   || 91\254 || 429.92126 || 89.76378 || 70.86614 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   ||   || 77\215 || 429.76744 || 89.30233 || 72.55814 ||   ||
||   ||   ||   ||   ||   ||   || 53\[[148edo|148]] ||   || 429.__729__ || 89.__189__ || 72.__973__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 82\229 || 429.69432 || 89.08297 || 73.362445 ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   ||   || 429.__629__ || 88.__8__ || 74.__074__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 63\176 || 429.__54__ || 88.__63__ || 75 ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] ||   || 429.47368 || 88.42105 || 75.78947 ||   ||
||   ||   ||   ||   ||   ||   ||   || 39\109 || 429.3578 || 88.07339 || 77.06422 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   ||   || 428.57143 || 85.__714285__ || 85.__714285__ ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 1 5 5 5 5 1 5 5 5 5 1: &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 4 5 5 5 5 4 5 5 5 5 4: &amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: &amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: &amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: &amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: &amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
8 8 8 1 8 8 8 8 1 8 8 8 8 1: &amp;lt;a class="wiki_link" href="/91edo"&amp;gt;91&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
8 8 8 7 8 8 8 8 7 8 8 8 8 7: &amp;lt;a class="wiki_link" href="/109edo"&amp;gt;109&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: &amp;lt;a class="wiki_link" href="/102edo"&amp;gt;102&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: &amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: &amp;lt;a class="wiki_link" href="/120edo"&amp;gt;120&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: &amp;lt;a class="wiki_link" href="/123edo"&amp;gt;123&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1:&amp;lt;a class="wiki_link" href="/113edo"&amp;gt;113&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: &amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: &amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: &amp;lt;a class="wiki_link" href="/137edo"&amp;gt;137&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt;: &amp;lt;a class="wiki_link" href="/124edo"&amp;gt;124&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: &amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: &amp;lt;a class="wiki_link" href="/145edo"&amp;gt;145&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :&amp;lt;a class="wiki_link" href="/148edo"&amp;gt;148&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: &amp;lt;a class="wiki_link" href="/151edo"&amp;gt;151&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: &amp;lt;a class="wiki_link" href="/135edo"&amp;gt;135&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: &amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: &amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: &amp;lt;a class="wiki_link" href="/165edo"&amp;gt;165&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: &amp;lt;a class="wiki_link" href="/146edo"&amp;gt;146&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: &amp;lt;a class="wiki_link" href="/149edo"&amp;gt;149&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: &amp;lt;a class="wiki_link" href="/155edo"&amp;gt;155&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt; Split-φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt; &amp;lt;strong&amp;gt;φ&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: &amp;lt;a class="wiki_link" href="/170edo"&amp;gt;170&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: &amp;lt;a class="wiki_link" href="/173edo"&amp;gt;173&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: &amp;lt;a class="wiki_link" href="/176edo"&amp;gt;176&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
14 14 14 1 14 14 14 14 1 14 14 14 14 1: &amp;lt;a class="wiki_link" href="/157edo"&amp;gt;157&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 3 14 14 14 14 3 14 14 14 14 3: &amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 5 14 14 14 14 5 14 14 14 14 5: &amp;lt;a class="wiki_link" href="/169edo"&amp;gt;169&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 9 14 14 14 14 9 14 14 14 14 9: &amp;lt;a class="wiki_link" href="/181edo"&amp;gt;181&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 11 14 14 14 14 11 14 14 14 14 11: &amp;lt;a class="wiki_link" href="/187edo"&amp;gt;187&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 13 14 14 14 14 13 14 14 14 14 13: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
15 15 15 1 15 15 15 15 1 15 15 15 15 1: &amp;lt;a class="wiki_link" href="/168edo"&amp;gt;168&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 2 15 15 15 15 2 15 15 15 15 2: &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 4 15 15 15 15 4 15 15 15 15 4: &amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 7 15 15 15 15 7 15 15 15 15 7: &amp;lt;a class="wiki_link" href="/186edo"&amp;gt;186&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 8 15 15 15 15 8 15 15 15 15 8: &amp;lt;a class="wiki_link" href="/189edo"&amp;gt;189&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 11 15 15 15 15 11 15 15 15 15 11: &amp;lt;a class="wiki_link" href="/198edo"&amp;gt;198&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 13 15 15 15 15 13 15 15 15 15 13: &amp;lt;a class="wiki_link" href="/204edo"&amp;gt;204&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 14 15 15 15 15 14 15 15 15 15 14: &amp;lt;a class="wiki_link" href="/207edo"&amp;gt;207&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
16 16 16 1 16 16 16 16 1 16 16 16 16 1: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 3 16 16 16 16 3 16 16 16 16 3: &amp;lt;a class="wiki_link" href="/185edo"&amp;gt;185&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 5 16 16 16 16 5 16 16 16 16 5: &amp;lt;a class="wiki_link" href="/191edo"&amp;gt;191&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 7 16 16 16 16 7 16 16 16 16 7: &amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 9 16 16 16 16 9 16 16 16 16 9: &amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 11 16 16 16 16 11 16 16 16 16 11: &amp;lt;a class="wiki_link" href="/209edo"&amp;gt;209&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 13 16 16 16 16 13 16 16 16 16 13: &amp;lt;a class="wiki_link" href="/215edo"&amp;gt;215&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 15 16 16 16 16 15 16 16 16 16 15: &amp;lt;a class="wiki_link" href="/221edo"&amp;gt;221&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
17 17 17 1 17 17 17 17 1 17 17 17 17 1: &amp;lt;a class="wiki_link" href="/190edo"&amp;gt;190&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 2 17 17 17 17 2 17 17 17 17 2: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 3 17 17 17 17 3 17 17 17 17 3: &amp;lt;a class="wiki_link" href="/196edo"&amp;gt;196&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 4 17 17 17 17 4 17 17 17 17 4: &amp;lt;a class="wiki_link" href="/199edo"&amp;gt;199&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: &amp;lt;a class="wiki_link" href="/205edo"&amp;gt;205&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: &amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: &amp;lt;a class="wiki_link" href="/211edo"&amp;gt;211&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: &amp;lt;a class="wiki_link" href="/214edo"&amp;gt;214&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: &amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: &amp;lt;a class="wiki_link" href="/220edo"&amp;gt;220&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 13 17 17 17 17 13 17 17 17 17 13: &amp;lt;a class="wiki_link" href="/226edo"&amp;gt;226&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 14 17 17 17 17 14 17 17 17 17 14: &amp;lt;a class="wiki_link" href="/229edo"&amp;gt;229&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 15 17 17 17 17 15 17 17 17 17 15: &amp;lt;a class="wiki_link" href="/232edo"&amp;gt;232&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 16 17 17 17 17 16 17 17 17 17 16: &amp;lt;a class="wiki_link" href="/235edo"&amp;gt;235&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.&amp;lt;u&amp;gt;36&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.&amp;lt;u&amp;gt;09&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33\91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435.16484&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.494505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13.18681&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;54\149&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.89933&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.69799&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16.10738&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71\196&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.69388&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.08163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.36735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46\&amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.64567&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.93701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.89764&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;67\185&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.&amp;lt;u&amp;gt;594&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.&amp;lt;u&amp;gt;783&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19.&amp;lt;u&amp;gt;459&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448275&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20.68966&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;80\221&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.38914&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.16742&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21.71946&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;59\&amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.35583&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.06748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.0859&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97\268&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.32836&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.985075&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.38806&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;38\&amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.28571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93\257&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.241245&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.723735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23.3463&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;53\&amp;lt;a class="wiki_link" href="/148edo"&amp;gt;148&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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      <sha1>iaxq9ohmtxql4he4b1afaws1wt2potq</sha1>
    </revision>
    <revision>
      <id>4470</id>
      <timestamp>2012-06-27T00:44:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 348420226 - Original comment: **</comment>
      <origin>4470</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="111078" sha1="2p84emm2m3rl2z7z5iplb8w8jtdmsk7" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-06-27 00:44:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;348420226&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] (Split-Phi)**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]] (Phi)**
5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]
6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] (Split-Phi)**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]] (Phi)**
8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]
9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]
9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]
10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]
10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]
10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;** 11 11 11 11 **&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;**: [[124edo|124]]
11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]
11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]
11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]
12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]
12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]
12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]
13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]
13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]
13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]
**13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] (Split-Phi)**
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
**13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]] (Phi)**
13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]
13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]
13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]
13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]
14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]
14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]
14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]
14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]
14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]
15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]
15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]
15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]
15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]
15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]
15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]
15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]
16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]
16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]
16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]
16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]
16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]
16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]
16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]
17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]
17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]
17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]
17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]
17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]
17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   ||   || 436.__36__ || 109.__09__ || 0 ||=   ||
||   ||   ||   ||   ||   ||   ||   || 33\91 || 435.16484 || 105.494505 || 13.18681 ||   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] ||   || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   ||   ||   || 54\149 || 434.89933 || 104.69799 || 16.10738 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   ||   || 434.78261 || 104.34783 || 17.391 ||   ||
||   ||   ||   ||   ||   ||   ||   || 71\196 || 434.69388 || 104.08163 || 18.36735 ||   ||
||   ||   ||   ||   ||   ||   || 46\[[127edo|127]] ||   || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   ||   ||   ||   || 67\185 || 434.__594__ || 103.__783__ || 19.__459__ ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   ||   || 434.48276 || 103.448275 || 20.68966 ||   ||
||   ||   ||   ||   ||   ||   ||   || 80\221 || 434.38914 || 103.16742 || 21.71946 ||   ||
||   ||   ||   ||   ||   ||   || 59\[[163edo|163]] ||   || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   ||   ||   || 97\268 || 434.32836 || 102.985075 || 22.38806 ||   ||
||   ||   ||   ||   ||   || 38\[[105edo|105]] ||   ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   ||   || 93\257 || 434.241245 || 102.723735 || 23.3463 ||   ||
||   ||   ||   ||   ||   ||   || 55\[[152edo|152]] ||   || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   ||   ||   ||   ||   || 72\199 || 434.17085 || 102.51256 || 24.1206 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   ||   || 81\224 || 433.92857 || 101.78571 || 26.78571 ||   ||
||   ||   ||   ||   ||   ||   || 64\[[177edo|177]] ||   || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   ||   ||   || 111\307 || 433.87622 || 101.62866 || 27.36156 ||   ||
||   ||   ||   ||   ||   || 47\[[130edo|130]] ||   ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   ||   || 124\343 || 433.81924 || 101.45773 || 27.98834 ||   ||
||   ||   ||   ||   ||   ||   || 77\[[213edo|213]] ||   || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   ||   ||   ||   || 107\296 || 433.__783__ || 101.__351__ || 28.__378__ ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   ||   || 103\285 || 433.68421 || 101.05263 || 29.47368 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] ||   || 433.__6633__ || 100.__9900__ || 29.__7030__ || Since here are the optimal range Lufsur mode ||
||   ||   ||   ||   ||   ||   ||   || 116\321 || 433.64486 || 100.93458 || 29.96054 ||   ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   ||   || 99\274 || 433.57664 || 100.72993 || 30.65693 ||   ||
||   ||   ||   ||   ||   ||   || 56\[[155edo|155]] ||   || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   ||   ||   ||   ||   ||   || 69\191 || 433.50785 || 100.52356 || 31.41361 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   ||   || 433.__3__ || 100 || 33.__3__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 74\205 || 433.__17073__ || 99.__51219__ || 35.__12195__ ||   ||
||   ||   ||   ||   ||   ||   || 61\[[169edo|169]] ||   || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   ||   ||   || 109\302 || 433.11258 || 99.33775 || 35.76159 ||   ||
||   ||   ||   ||   ||   || 48\[[133edo|133]] ||   ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   ||   || 131\363 || 433.05785 || 99.17355 || 36.__36__ ||   ||
||   ||   ||   ||   ||   ||   || 83\[[230edo|230]] ||   || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   ||   ||   ||   || 118\327 || 433.02752 || 99.08257 || 36.69725 ||   ||
||   ||   ||   ||   || 35\[[97edo|97]] ||   ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   ||   || 127\352 || 432.95__45__ || 98.8__63__ || 37.5 ||   ||
||   ||   ||   ||   ||   ||   || 92\[[255edo|255]] ||   || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   ||   ||   || 149\413 || 432.92978 || 98.78935 || 37.7724 ||   ||
||   ||   ||   ||   ||   || 57\[[158edo|158]] ||   ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   ||   || 136\377 || 432.89125 || 98.67374 || 38.19629 ||   ||
||   ||   ||   ||   ||   ||   || 79\[[219edo|219]] ||   || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   ||   ||   ||   ||   || 101\280 || 432.85714 || 98.57143 || 38.57143 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   ||   || 97\269 || 432.713755 || 98.14126 || 40.1487 ||   ||
||   ||   ||   ||   ||   ||   || 75\[[208edo|208]] ||   || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   ||   ||   || 128\355 || 432.67606 || 98.02817 || 40.56338 ||   ||
||   ||   ||   ||   ||   || 53\[[147edo|147]] ||   ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   ||   || 137\380 || 432.63158 || 97.89474 || 41.05263 ||   ||
||   ||   ||   ||   ||   ||   || 84\[[233edo|233]] ||   || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   ||   ||   ||   || 115\319 || 432.60188 || 97.80564 || 41.37931 ||   ||
||   ||   ||   ||   || 31\[[86edo|86]] ||   ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   ||   || 102\283 || 432.50883 || 97.5265 || 42.40283 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[197edo|197]] ||   || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   ||   ||   || 111\308 || 432.46753 || 97.4026 || 42.85714 ||   ||
||   ||   ||   ||   ||   || 40\[[111edo|111]] ||   ||   || 432.__432__ || 97.__297__ || 43.__135__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 89\247 || 432.38866 || 97.16599 || 43.7247 ||   ||
||   ||   ||   ||   ||   ||   || 49\[[136edo|136]] ||   || 432.35294 || 97.05882 || 44.11765 ||   ||
||   ||   ||   ||   ||   ||   ||   || 58\[[161edo|161]] || 432.29814 || 96.89441 || 44.72051 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   ||   || 59\[[164edo|164]] || 431.__70731__ || 95.__12195__ || 51.__21951__ ||   ||
||   ||   ||   ||   ||   ||   || 50\[[139edo|139]] ||   || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   ||   ||   || 91\253 || 431.62055 || 94.86166 || 52.17391 ||   ||
||   ||   ||   ||   ||   || 41\[[114edo|114]] ||   ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   ||   || 114\317 || 431.54574 || 94.63722 || 52.996845 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[203edo|203]] ||   || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   ||   ||   ||   || 105\292 || 431.50685 || 94.52055 || 53.42466 ||   ||
||   ||   ||   ||   || 32\[[89edo|89]] ||   ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   ||   || 119\331 || 431.41664 || 94.29582 || 54.380665 ||   ||
||   ||   ||   ||   ||   ||   || 87\[[242edo|242]] ||   || 431.40496 || 94.21488 || 54.__54__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 142\395 || 431.392405 || 94.177215 || 54.68354 ||   ||
||   ||   ||   ||   ||   || 55\[[153edo|153]] ||   ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   ||   || 133\370 || 431.__351__ || 94.__054__ || 55.__135__ ||   ||
||   ||   ||   ||   ||   ||   || 78\[[217edo|217]] ||   || 431.33641 || 94.00922 || 55.29955 ||   ||
||   ||   ||   ||   ||   ||   ||   || 101\281 || 431.31673 || 93.95018 || 55.51601 ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   ||   || 106\295 || 431.18644 || 93.55932 || 56.94915 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[231edo|231]] ||   || 431.16833 || 93.50649 || 57.14286 ||   ||
||   ||   ||   ||   ||   ||   ||   || 143\398 || 431.15578 || 93.46734 || 57.28643 ||   ||
||   ||   ||   ||   ||   || 60\[[167edo|167]] ||   ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   ||   || 157\437 || 431.12128 || 93.36384 || 57.6659 ||   ||
||   ||   ||   ||   ||   ||   || 97\[[270edo|270]] ||   || 431.__1__ || 93.__3__ || 57.__8__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 134\373 || 431.0992 || 93.29759 || 57.90885 ||   ||
||   ||   ||   ||   || 37\[[103edo|103]] ||   ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   ||   || 125\348 || 431.03448 || 93.10345 || 58.60269 ||   ||
||   ||   ||   ||   ||   ||   || 88\[[245edo|245]] ||   || 431.02041 || 93.06122 || 58.77551 ||   ||
||   ||   ||   ||   ||   ||   ||   || 139\387 || 431.00775 || 93.02326 || 58.91473 ||   ||
||   ||   ||   ||   ||   || 51\[[142edo|142]] ||   ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   ||   || 116\323 || 430.95975 || 92.87926 || 59.44272 ||   ||
||   ||   ||   ||   ||   ||   || 65\[[181edo|181]] ||   || 430.93923 || 92.81768 || 59.66852 ||   ||
||   ||   ||   ||   ||   ||   ||   || 79\220 || 430.__90__ || 92.__72__ || 60 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   ||   || 75\209 || 430.62201 || 91.86603 || 63.157895 ||   ||
||   ||   ||   ||   ||   ||   || 61\[[170edo|170]] ||   || 430.58824 || 91.76471 || 62.54942 ||   ||
||   ||   ||   ||   ||   ||   ||   || 108\301 || 430.56478 || 91.69435 || 63.787375 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   ||   || 127\354 || 430.508475 || 91.52542 || 64.40678 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] ||   || 430.49327 || 91.47982 || 64.57505 || Until here are the optimal range Fuslur mode ||
||   ||   ||   ||   ||   ||   ||   || 113\315 || 430.47619 || 91.42857 || 64.761905 ||   ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   ||   || 118\329 || 430.39514 || 91.18541 || 65.653495 ||   ||
||   ||   ||   ||   ||   ||   || 85\[[237edo|237]] ||   || 430.37975 || 91.13924 || 65.82309 ||   ||
||   ||   ||   ||   ||   ||   ||   || 137\382 || 430.36649 || 91.09948 || 65.96859 ||   ||
||   ||   ||   ||   ||   || 52\[[145edo|145]] ||   ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   ||   || 123\343 || 430.3207 || 90.9621 || 66.4723 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[198edo|198]] ||   || 430.__30__ || 90.__90__ || 66.__67__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 90\251 || 430.27888 || 90.83665 || 66.93227 ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   ||   || 81\226 || 430.0885 || 90.26549 || 69.02655 ||   ||
||   ||   ||   ||   ||   ||   || 62\[[173edo|173]] ||   || 430.0578 || 90.17341 || 69.36416 ||   ||
||   ||   ||   ||   ||   ||   ||   || 105\293 || 430.03413 || 90.10239 || 69.62457 ||   ||
||   ||   ||   ||   ||   || 43\[[120edo|120]] ||   ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   ||   || 110\307 || 429.96743 || 89.90228 || 70.35831 ||   ||
||   ||   ||   ||   ||   ||   || 67\[[187edo|187]] ||   || 429.94652 || 89.83957 || 70.58824 ||   ||
||   ||   ||   ||   ||   ||   ||   || 91\254 || 429.92126 || 89.76378 || 70.86614 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   ||   || 77\215 || 429.76744 || 89.30233 || 72.55814 ||   ||
||   ||   ||   ||   ||   ||   || 53\[[148edo|148]] ||   || 429.__729__ || 89.__189__ || 72.__973__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 82\229 || 429.69432 || 89.08297 || 73.362445 ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   ||   || 429.__629__ || 88.__8__ || 74.__074__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 63\176 || 429.__54__ || 88.__63__ || 75 ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] ||   || 429.47368 || 88.42105 || 75.78947 ||   ||
||   ||   ||   ||   ||   ||   ||   || 39\109 || 429.3578 || 88.07339 || 77.06422 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   ||   || 428.57143 || 85.__714285__ || 85.__714285__ ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 1 5 5 5 5 1 5 5 5 5 1: &amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt; (Split-Phi)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt; (Phi)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 4 5 5 5 5 4 5 5 5 5 4: &amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: &amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: &amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: &amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: &amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
8 8 8 1 8 8 8 8 1 8 8 8 8 1: &amp;lt;a class="wiki_link" href="/91edo"&amp;gt;91&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt; (Split-Phi)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt; (Phi)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
8 8 8 7 8 8 8 8 7 8 8 8 8 7: &amp;lt;a class="wiki_link" href="/109edo"&amp;gt;109&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: &amp;lt;a class="wiki_link" href="/102edo"&amp;gt;102&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: &amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: &amp;lt;a class="wiki_link" href="/120edo"&amp;gt;120&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: &amp;lt;a class="wiki_link" href="/123edo"&amp;gt;123&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1:&amp;lt;a class="wiki_link" href="/113edo"&amp;gt;113&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: &amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: &amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: &amp;lt;a class="wiki_link" href="/137edo"&amp;gt;137&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt; 11 11 11 11 &amp;lt;strong&amp;gt;&amp;lt;span style="color: #006209;"&amp;gt;1&amp;lt;/span&amp;gt;&amp;lt;/strong&amp;gt;: &amp;lt;a class="wiki_link" href="/124edo"&amp;gt;124&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: &amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: &amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: &amp;lt;a class="wiki_link" href="/145edo"&amp;gt;145&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :&amp;lt;a class="wiki_link" href="/148edo"&amp;gt;148&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: &amp;lt;a class="wiki_link" href="/151edo"&amp;gt;151&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: &amp;lt;a class="wiki_link" href="/135edo"&amp;gt;135&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: &amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: &amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: &amp;lt;a class="wiki_link" href="/165edo"&amp;gt;165&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: &amp;lt;a class="wiki_link" href="/146edo"&amp;gt;146&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: &amp;lt;a class="wiki_link" href="/149edo"&amp;gt;149&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: &amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: &amp;lt;a class="wiki_link" href="/155edo"&amp;gt;155&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt; (Split-Phi)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt; (Phi)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: &amp;lt;a class="wiki_link" href="/170edo"&amp;gt;170&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: &amp;lt;a class="wiki_link" href="/173edo"&amp;gt;173&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: &amp;lt;a class="wiki_link" href="/176edo"&amp;gt;176&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
14 14 14 1 14 14 14 14 1 14 14 14 14 1: &amp;lt;a class="wiki_link" href="/157edo"&amp;gt;157&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 3 14 14 14 14 3 14 14 14 14 3: &amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 5 14 14 14 14 5 14 14 14 14 5: &amp;lt;a class="wiki_link" href="/169edo"&amp;gt;169&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 9 14 14 14 14 9 14 14 14 14 9: &amp;lt;a class="wiki_link" href="/181edo"&amp;gt;181&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 11 14 14 14 14 11 14 14 14 14 11: &amp;lt;a class="wiki_link" href="/187edo"&amp;gt;187&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
14 14 14 13 14 14 14 14 13 14 14 14 14 13: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
15 15 15 1 15 15 15 15 1 15 15 15 15 1: &amp;lt;a class="wiki_link" href="/168edo"&amp;gt;168&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 2 15 15 15 15 2 15 15 15 15 2: &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 4 15 15 15 15 4 15 15 15 15 4: &amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 7 15 15 15 15 7 15 15 15 15 7: &amp;lt;a class="wiki_link" href="/186edo"&amp;gt;186&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 8 15 15 15 15 8 15 15 15 15 8: &amp;lt;a class="wiki_link" href="/189edo"&amp;gt;189&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 11 15 15 15 15 11 15 15 15 15 11: &amp;lt;a class="wiki_link" href="/198edo"&amp;gt;198&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 13 15 15 15 15 13 15 15 15 15 13: &amp;lt;a class="wiki_link" href="/204edo"&amp;gt;204&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
15 15 15 14 15 15 15 15 14 15 15 15 15 14: &amp;lt;a class="wiki_link" href="/207edo"&amp;gt;207&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
16 16 16 1 16 16 16 16 1 16 16 16 16 1: &amp;lt;a class="wiki_link" href="/179edo"&amp;gt;179&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 3 16 16 16 16 3 16 16 16 16 3: &amp;lt;a class="wiki_link" href="/185edo"&amp;gt;185&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 5 16 16 16 16 5 16 16 16 16 5: &amp;lt;a class="wiki_link" href="/191edo"&amp;gt;191&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 7 16 16 16 16 7 16 16 16 16 7: &amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 9 16 16 16 16 9 16 16 16 16 9: &amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 11 16 16 16 16 11 16 16 16 16 11: &amp;lt;a class="wiki_link" href="/209edo"&amp;gt;209&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 13 16 16 16 16 13 16 16 16 16 13: &amp;lt;a class="wiki_link" href="/215edo"&amp;gt;215&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
16 16 16 15 16 16 16 16 15 16 16 16 16 15: &amp;lt;a class="wiki_link" href="/221edo"&amp;gt;221&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
17 17 17 1 17 17 17 17 1 17 17 17 17 1: &amp;lt;a class="wiki_link" href="/190edo"&amp;gt;190&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 2 17 17 17 17 2 17 17 17 17 2: &amp;lt;a class="wiki_link" href="/193edo"&amp;gt;193&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 3 17 17 17 17 3 17 17 17 17 3: &amp;lt;a class="wiki_link" href="/196edo"&amp;gt;196&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 4 17 17 17 17 4 17 17 17 17 4: &amp;lt;a class="wiki_link" href="/199edo"&amp;gt;199&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: &amp;lt;a class="wiki_link" href="/205edo"&amp;gt;205&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: &amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: &amp;lt;a class="wiki_link" href="/211edo"&amp;gt;211&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: &amp;lt;a class="wiki_link" href="/214edo"&amp;gt;214&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: &amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: &amp;lt;a class="wiki_link" href="/220edo"&amp;gt;220&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 13 17 17 17 17 13 17 17 17 17 13: &amp;lt;a class="wiki_link" href="/226edo"&amp;gt;226&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 14 17 17 17 17 14 17 17 17 17 14: &amp;lt;a class="wiki_link" href="/229edo"&amp;gt;229&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 15 17 17 17 17 15 17 17 17 17 15: &amp;lt;a class="wiki_link" href="/232edo"&amp;gt;232&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
17 17 17 16 17 17 17 17 16 17 17 17 17 16: &amp;lt;a class="wiki_link" href="/235edo"&amp;gt;235&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.&amp;lt;u&amp;gt;36&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.&amp;lt;u&amp;gt;09&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33\91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435.16484&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.494505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13.18681&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;54\149&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.89933&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.69799&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16.10738&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71\196&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.69388&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.08163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.36735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46\&amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.64567&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.93701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.89764&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;67\185&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.&amp;lt;u&amp;gt;594&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.&amp;lt;u&amp;gt;783&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19.&amp;lt;u&amp;gt;459&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
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      <sha1>2p84emm2m3rl2z7z5iplb8w8jtdmsk7</sha1>
    </revision>
    <revision>
      <id>4471</id>
      <timestamp>2012-06-22T11:25:19Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 347272142 - Original comment: **</comment>
      <origin>4471</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="106509" sha1="c79dmg71fto1660kuu10x0kh64ymjlm" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-06-22 11:25:19 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;347272142&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

**5 5 5 1 5 5 5 5 1 5 5 5 5 1: 58**
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]**
**5 5 5 4 5 5 5 5 4 5 5 5 5 4: 67**

6 6 6 1 6 6 6 6 1 6 6 6 6 1: 69
6 6 6 5 6 6 6 6 5 6 6 6 6 5: 81

7 7 7 1 7 7 7 7 1 7 7 7 7 1: 80
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: 95

**8 8 8 1 8 8 8 8 1 8 8 8 8 1: 91**
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]]**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]**
**8 8 8 7 8 8 8 8 7 8 8 8 8 7: 109**

9 9 9 1 9 9 9 9 1 9 9 9 9 1: 102
9 9 9 2 9 9 9 9 2 9 9 9 9 2: 105
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: 120
9 9 9 8 9 9 9 9 8 9 9 9 9 8: 123

10 10 10 1 10 10 10 10 1 10 10 10 10 1: 113
10 10 10 3 10 10 10 10 3 10 10 10 10 3: 119
10 10 10 7 10 10 10 10 7 10 10 10 10 7: 131
10 10 10 9 10 10 10 10 9 10 10 10 10 9: 137

11 11 11 1 11 11 11 11 1 11 11 11 11 1: 124
11 11 11 2 11 11 11 11 2 11 11 11 11 2: 127
11 11 11 3 11 11 11 11 3 11 11 11 11 3: 130
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: 145
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :148
11 11 11 10 11 11 11 11 10 11 11 11 11 10: 151

12 12 12 1 12 12 12 12 1 12 12 12 12 1: 135
12 12 12 5 12 12 12 12 5 12 12 12 12 5: 147
12 12 12 7 12 12 12 12 7 12 12 12 12 7: 153
12 12 12 11 12 12 12 12 11 12 12 12 12 11: 165

13 13 13 1 13 13 13 13 1 13 13 13 13 1: 146
13 13 13 2 13 13 13 13 2 13 13 13 13 2: 149
13 13 13 3 13 13 13 13 3 13 13 13 13 3: 152
13 13 13 4 13 13 13 13 4 13 13 13 13 4: 155
13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]
13 13 13 9 13 13 13 13 9 13 13 13 13 9: 170
13 13 13 10 13 13 13 13 10 13 13 13 13 10: 173
13 13 13 11 13 13 13 13 11 13 13 13 13 11: 176
13 13 13 12 13 13 13 13 12 13 13 13 13 12: 179

14 14 14 1 14 14 14 14 1 14 14 14 14 1: 157
14 14 14 3 14 14 14 14 3 14 14 14 14 3: 163
14 14 14 5 14 14 14 14 5 14 14 14 14 5: 169
14 14 14 9 14 14 14 14 9 14 14 14 14 9: 181
14 14 14 11 14 14 14 14 11 14 14 14 14 11: 187
14 14 14 13 14 14 14 14 13 14 14 14 14 13: 193

15 15 15 1 15 15 15 15 1 15 15 15 15 1: 168
15 15 15 2 15 15 15 15 2 15 15 15 15 2: 171
15 15 15 4 15 15 15 15 4 15 15 15 15 4: 177
15 15 15 7 15 15 15 15 7 15 15 15 15 7: 186
15 15 15 8 15 15 15 15 8 15 15 15 15 8: 189
15 15 15 11 15 15 15 15 11 15 15 15 15 11: 198
15 15 15 13 15 15 15 15 13 15 15 15 15 13: 204
15 15 15 14 15 15 15 15 14 15 15 15 15 14: 207

16 16 16 1 16 16 16 16 1 16 16 16 16 1: 179
16 16 16 3 16 16 16 16 3 16 16 16 16 3: 185
16 16 16 5 16 16 16 16 5 16 16 16 16 5: 191
16 16 16 7 16 16 16 16 7 16 16 16 16 7: 197
16 16 16 9 16 16 16 16 9 16 16 16 16 9: 203
16 16 16 11 16 16 16 16 11 16 16 16 16 11: 209
16 16 16 13 16 16 16 16 13 16 16 16 16 13: 215
16 16 16 15 16 16 16 16 15 16 16 16 16 15: 221 

**17 17 17 1 17 17 17 17 1 17 17 17 17 1: 190**
**17 17 17 2 17 17 17 17 2 17 17 17 17 2: 193**
**17 17 17 3 17 17 17 17 3 17 17 17 17 3: 196**
**17 17 17 4 17 17 17 17 4 17 17 17 17 4: 199**
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: 205**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: 208**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: 211**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: 214**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: 217**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: 220**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
**17 17 17 13 17 17 17 17 13 17 17 17 17 13: 226**
**17 17 17 14 17 17 17 17 14 17 17 17 17 14: 229**
**17 17 17 15 17 17 17 17 15 17 17 17 17 15: 232**
**17 17 17 16 17 17 17 17 16 17 17 17 17 16: 235**

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   ||   || 436.__36__ || 109.__09__ || 0 ||=   ||
||   ||   ||   ||   ||   ||   ||   || 33\91 || 435.16484 || 105.494505 || 13.18681 ||   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] ||   || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   ||   ||   || 54\149 || 434.89933 || 104.69799 || 16.10738 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   ||   || 434.78261 || 104.34783 || 17.391 ||   ||
||   ||   ||   ||   ||   ||   ||   || 71\196 || 434.69388 || 104.08163 || 18.36735 ||   ||
||   ||   ||   ||   ||   ||   || 46\[[127edo|127]] ||   || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   ||   ||   ||   || 67\185 || 434.__594__ || 103.__783__ || 19.__459__ ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   ||   || 434.48276 || 103.448275 || 20.68966 ||   ||
||   ||   ||   ||   ||   ||   ||   || 80\221 || 434.38914 || 103.16742 || 21.71946 ||   ||
||   ||   ||   ||   ||   ||   || 59\[[163edo|163]] ||   || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   ||   ||   || 97\268 || 434.32836 || 102.985075 || 22.38806 ||   ||
||   ||   ||   ||   ||   || 38\[[105edo|105]] ||   ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   ||   || 93\257 || 434.241245 || 102.723735 || 23.3463 ||   ||
||   ||   ||   ||   ||   ||   || 55\[[152edo|152]] ||   || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   ||   ||   ||   ||   || 72\199 || 434.17085 || 102.51256 || 24.1206 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   ||   || 81\224 || 433.92857 || 101.78571 || 26.78571 ||   ||
||   ||   ||   ||   ||   ||   || 64\[[177edo|177]] ||   || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   ||   ||   || 111\307 || 433.87622 || 101.62866 || 27.36156 ||   ||
||   ||   ||   ||   ||   || 47\[[130edo|130]] ||   ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   ||   || 124\343 || 433.81924 || 101.45773 || 27.98834 ||   ||
||   ||   ||   ||   ||   ||   || 77\[[213edo|213]] ||   || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   ||   ||   ||   || 107\296 || 433.__783__ || 101.__351__ || 28.__378__ ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   ||   || 103\285 || 433.68421 || 101.05263 || 29.47368 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] ||   || 433.__6633__ || 100.__9900__ || 29.__7030__ || Since here are the optimal range Lufsur mode ||
||   ||   ||   ||   ||   ||   ||   || 116/321 || 433.64486 || 100.93458 || 29.96054 ||   ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   ||   || 99\274 || 433.57664 || 100.72993 || 30.65693 ||   ||
||   ||   ||   ||   ||   ||   || 56\[[155edo|155]] ||   || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   ||   ||   ||   ||   ||   || 69\191 || 433.50785 || 100.52356 || 31.41361 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   ||   || 433.__3__ || 100 || 33.__3__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 74\205 || 433.__17073__ || 99.__51219__ || 35.__12195__ ||   ||
||   ||   ||   ||   ||   ||   || 61\[[169edo|169]] ||   || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   ||   ||   || 109\302 || 433.11258 || 99.33775 || 35.76159 ||   ||
||   ||   ||   ||   ||   || 48\[[133edo|133]] ||   ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   ||   || 131\363 || 433.05785 || 99.17355 || 36.__36__ ||   ||
||   ||   ||   ||   ||   ||   || 83\[[230edo|230]] ||   || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   ||   ||   ||   || 118\327 || 433.02752 || 99.08257 || 36.69725 ||   ||
||   ||   ||   ||   || 35\[[97edo|97]] ||   ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   ||   || 127\352 || 432.95__45__ || 98.8__63__ || 37.5 ||   ||
||   ||   ||   ||   ||   ||   || 92\[[255edo|255]] ||   || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   ||   ||   || 149\413 || 432.92978 || 98.78935 || 37.7724 ||   ||
||   ||   ||   ||   ||   || 57\[[158edo|158]] ||   ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   ||   || 136\377 || 432.89125 || 98.67374 || 38.19629 ||   ||
||   ||   ||   ||   ||   ||   || 79\[[219edo|219]] ||   || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   ||   ||   ||   ||   || 101\280 || 432.85714 || 98.57143 || 38.57143 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   ||   || 97\269 || 432.713755 || 98.14126 || 40.1487 ||   ||
||   ||   ||   ||   ||   ||   || 75\[[208edo|208]] ||   || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   ||   ||   || 128\355 || 432.67606 || 98.02817 || 40.56338 ||   ||
||   ||   ||   ||   ||   || 53\[[147edo|147]] ||   ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   ||   || 137\380 || 432.63158 || 97.89474 || 41.05263 ||   ||
||   ||   ||   ||   ||   ||   || 84\[[233edo|233]] ||   || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   ||   ||   ||   || 115\319 || 432.60188 || 97.80564 || 41.37931 ||   ||
||   ||   ||   ||   || 31\[[86edo|86]] ||   ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   ||   || 102\283 || 432.50883 || 97.5265 || 42.40283 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[197edo|197]] ||   || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   ||   ||   || 111\308 || 432.46753 || 97.4026 || 42.85714 ||   ||
||   ||   ||   ||   ||   || 40\[[111edo|111]] ||   ||   || 432.__432__ || 97.__297__ || 43.__135__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 89\247 || 432.38866 || 97.16599 || 43.7247 ||   ||
||   ||   ||   ||   ||   ||   || 49\[[136edo|136]] ||   || 432.35294 || 97.05882 || 44.11765 ||   ||
||   ||   ||   ||   ||   ||   ||   || 58\[[161edo|161]] || 432.29814 || 96.89441 || 44.72051 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   ||   || 59\[[164edo|164]] || 431.__70731__ || 95.__12195__ || 51.__21951__ ||   ||
||   ||   ||   ||   ||   ||   || 50\[[139edo|139]] ||   || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   ||   ||   || 91\253 || 431.62055 || 94.86166 || 52.17391 ||   ||
||   ||   ||   ||   ||   || 41\[[114edo|114]] ||   ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   ||   || 114\317 || 431.54574 || 94.63722 || 52.996845 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[203edo|203]] ||   || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   ||   ||   ||   || 105\292 || 431.50685 || 94.52055 || 53.42466 ||   ||
||   ||   ||   ||   || 32\[[89edo|89]] ||   ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   ||   || 119\331 || 431.41664 || 94.29582 || 54.380665 ||   ||
||   ||   ||   ||   ||   ||   || 87\[[242edo|242]] ||   || 431.40496 || 94.21488 || 54.__54__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 142\395 || 431.392405 || 94.177215 || 54.68354 ||   ||
||   ||   ||   ||   ||   || 55\[[153edo|153]] ||   ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   ||   || 133\370 || 431.__351__ || 94.__054__ || 55.__135__ ||   ||
||   ||   ||   ||   ||   ||   || 78\[[217edo|217]] ||   || 431.33641 || 94.00922 || 55.29955 ||   ||
||   ||   ||   ||   ||   ||   ||   || 101\281 || 431.31673 || 93.95018 || 55.51601 ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   ||   || 106\295 || 431.18644 || 93.55932 || 56.94915 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[231edo|231]] ||   || 431.16833 || 93.50649 || 57.14286 ||   ||
||   ||   ||   ||   ||   ||   ||   || 143\398 || 431.15578 || 93.46734 || 57.28643 ||   ||
||   ||   ||   ||   ||   || 60\[[167edo|167]] ||   ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   ||   || 157\437 || 431.12128 || 93.36384 || 57.6659 ||   ||
||   ||   ||   ||   ||   ||   || 97\[[270edo|270]] ||   || 431.__1__ || 93.__3__ || 57.__8__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 134\373 || 431.0992 || 93.29759 || 57.90885 ||   ||
||   ||   ||   ||   || 37\[[103edo|103]] ||   ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   ||   || 125\348 || 431.03448 || 93.10345 || 58.60269 ||   ||
||   ||   ||   ||   ||   ||   || 88\[[245edo|245]] ||   || 431.02041 || 93.06122 || 58.77551 ||   ||
||   ||   ||   ||   ||   ||   ||   || 139\387 || 431.00775 || 93.02326 || 58.91473 ||   ||
||   ||   ||   ||   ||   || 51\[[142edo|142]] ||   ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   ||   || 116\323 || 430.95975 || 92.87926 || 59.44272 ||   ||
||   ||   ||   ||   ||   ||   || 65\[[181edo|181]] ||   || 430.93923 || 92.81768 || 59.66852 ||   ||
||   ||   ||   ||   ||   ||   ||   || 79\220 || 430.__90__ || 92.__72__ || 60 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   ||   || 75\209 || 430.62201 || 91.86603 || 63.157895 ||   ||
||   ||   ||   ||   ||   ||   || 61\[[170edo|170]] ||   || 430.58824 || 91.76471 || 62.54942 ||   ||
||   ||   ||   ||   ||   ||   ||   || 108\301 || 430.56478 || 91.69435 || 63.787375 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   ||   || 127\354 || 430.508475 || 91.52542 || 64.40678 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] ||   || 430.49327 || 91.47982 || 64.57505 || Until here are the optimal range Fuslur mode ||
||   ||   ||   ||   ||   ||   ||   || 113\315 || 430.47619 || 91.42857 || 64.761905 ||   ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   ||   || 118\329 || 430.39514 || 91.18541 || 65.653495 ||   ||
||   ||   ||   ||   ||   ||   || 85\[[237edo|237]] ||   || 430.37975 || 91.13924 || 65.82309 ||   ||
||   ||   ||   ||   ||   ||   ||   || 137\382 || 430.36649 || 91.09948 || 65.96859 ||   ||
||   ||   ||   ||   ||   || 52\[[145edo|145]] ||   ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   ||   || 123\343 || 430.3207 || 90.9621 || 66.4723 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[198edo|198]] ||   || 430.__30__ || 90.__90__ || 66.__67__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 90\251 || 430.27888 || 90.83665 || 66.93227 ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   ||   || 81\226 || 430.0885 || 90.26549 || 69.02655 ||   ||
||   ||   ||   ||   ||   ||   || 62\[[173edo|173]] ||   || 430.0578 || 90.17341 || 69.36416 ||   ||
||   ||   ||   ||   ||   ||   ||   || 105\293 || 430.03413 || 90.10239 || 69.62457 ||   ||
||   ||   ||   ||   ||   || 43\[[120edo|120]] ||   ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   ||   || 110\307 || 429.96743 || 89.90228 || 70.35831 ||   ||
||   ||   ||   ||   ||   ||   || 67\[[187edo|187]] ||   || 429.94652 || 89.83957 || 70.58824 ||   ||
||   ||   ||   ||   ||   ||   ||   || 91\254 || 429.92126 || 89.76378 || 70.86614 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   ||   || 77\215 || 429.76744 || 89.30233 || 72.55814 ||   ||
||   ||   ||   ||   ||   ||   || 53\[[148edo|148]] ||   || 429.__729__ || 89.__189__ || 72.__973__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 82\229 || 429.69432 || 89.08297 || 73.362445 ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   ||   || 429.__629__ || 88.__8__ || 74.__074__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 63\176 || 429.__54__ || 88.__63__ || 75 ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] ||   || 429.47368 || 88.42105 || 75.78947 ||   ||
||   ||   ||   ||   ||   ||   ||   || 39/109 || 429.3578 || 88.07339 || 77.06422 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   ||   || 428.57143 || 85.__714285__ || 85.__714285__ ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 1 5 5 5 5 1 5 5 5 5 1: 58&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 4 5 5 5 5 4 5 5 5 5 4: 67&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: 69&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: 81&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: 80&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: 95&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 1 8 8 8 8 1 8 8 8 8 1: 91&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 7 8 8 8 8 7 8 8 8 8 7: 109&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: 102&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: 105&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: 120&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: 123&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1: 113&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: 119&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: 131&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: 137&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 1 11 11 11 11 1 11 11 11 11 1: 124&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: 127&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: 130&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: 145&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :148&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: 151&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: 135&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: 147&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: 153&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: 165&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: 146&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: 149&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: 152&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: 155&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: 170&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: 173&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: 176&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: 179&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
14 14 14 1 14 14 14 14 1 14 14 14 14 1: 157&amp;lt;br /&amp;gt;
14 14 14 3 14 14 14 14 3 14 14 14 14 3: 163&amp;lt;br /&amp;gt;
14 14 14 5 14 14 14 14 5 14 14 14 14 5: 169&amp;lt;br /&amp;gt;
14 14 14 9 14 14 14 14 9 14 14 14 14 9: 181&amp;lt;br /&amp;gt;
14 14 14 11 14 14 14 14 11 14 14 14 14 11: 187&amp;lt;br /&amp;gt;
14 14 14 13 14 14 14 14 13 14 14 14 14 13: 193&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
15 15 15 1 15 15 15 15 1 15 15 15 15 1: 168&amp;lt;br /&amp;gt;
15 15 15 2 15 15 15 15 2 15 15 15 15 2: 171&amp;lt;br /&amp;gt;
15 15 15 4 15 15 15 15 4 15 15 15 15 4: 177&amp;lt;br /&amp;gt;
15 15 15 7 15 15 15 15 7 15 15 15 15 7: 186&amp;lt;br /&amp;gt;
15 15 15 8 15 15 15 15 8 15 15 15 15 8: 189&amp;lt;br /&amp;gt;
15 15 15 11 15 15 15 15 11 15 15 15 15 11: 198&amp;lt;br /&amp;gt;
15 15 15 13 15 15 15 15 13 15 15 15 15 13: 204&amp;lt;br /&amp;gt;
15 15 15 14 15 15 15 15 14 15 15 15 15 14: 207&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
16 16 16 1 16 16 16 16 1 16 16 16 16 1: 179&amp;lt;br /&amp;gt;
16 16 16 3 16 16 16 16 3 16 16 16 16 3: 185&amp;lt;br /&amp;gt;
16 16 16 5 16 16 16 16 5 16 16 16 16 5: 191&amp;lt;br /&amp;gt;
16 16 16 7 16 16 16 16 7 16 16 16 16 7: 197&amp;lt;br /&amp;gt;
16 16 16 9 16 16 16 16 9 16 16 16 16 9: 203&amp;lt;br /&amp;gt;
16 16 16 11 16 16 16 16 11 16 16 16 16 11: 209&amp;lt;br /&amp;gt;
16 16 16 13 16 16 16 16 13 16 16 16 16 13: 215&amp;lt;br /&amp;gt;
16 16 16 15 16 16 16 16 15 16 16 16 16 15: 221 &amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 1 17 17 17 17 1 17 17 17 17 1: 190&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 2 17 17 17 17 2 17 17 17 17 2: 193&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 3 17 17 17 17 3 17 17 17 17 3: 196&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 4 17 17 17 17 4 17 17 17 17 4: 199&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: 205&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: 208&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: 211&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: 214&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: 217&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: 220&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 13 17 17 17 17 13 17 17 17 17 13: 226&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 14 17 17 17 17 14 17 17 17 17 14: 229&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 15 17 17 17 17 15 17 17 17 17 15: 232&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 16 17 17 17 17 16 17 17 17 17 16: 235&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.&amp;lt;u&amp;gt;36&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.&amp;lt;u&amp;gt;09&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33\91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435.16484&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.494505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13.18681&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;54\149&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.89933&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.69799&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16.10738&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71\196&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.69388&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.08163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.36735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46\&amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.64567&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.93701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.89764&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;67\185&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.&amp;lt;u&amp;gt;594&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.&amp;lt;u&amp;gt;783&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19.&amp;lt;u&amp;gt;459&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448275&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
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      <sha1>c79dmg71fto1660kuu10x0kh64ymjlm</sha1>
    </revision>
    <revision>
      <id>4472</id>
      <timestamp>2012-06-21T22:20:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 347137464 - Original comment: **</comment>
      <origin>4472</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="96175" sha1="q9g881kc4g13jjf410e95nw2k0l9ht2" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-06-21 22:20:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;347137464&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

**5 5 5 1 5 5 5 5 1 5 5 5 5 1: 58**
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]**
**5 5 5 4 5 5 5 5 4 5 5 5 5 4: 67**

6 6 6 1 6 6 6 6 1 6 6 6 6 1: 69
6 6 6 5 6 6 6 6 5 6 6 6 6 5: 81

7 7 7 1 7 7 7 7 1 7 7 7 7 1: 80
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: 95

**8 8 8 1 8 8 8 8 1 8 8 8 8 1: 91**
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]]**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]**
**8 8 8 7 8 8 8 8 7 8 8 8 8 7: 109**

9 9 9 1 9 9 9 9 1 9 9 9 9 1: 102
9 9 9 2 9 9 9 9 2 9 9 9 9 2: 105
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: 120
9 9 9 8 9 9 9 9 8 9 9 9 9 8: 123

10 10 10 1 10 10 10 10 1 10 10 10 10 1: 113
10 10 10 3 10 10 10 10 3 10 10 10 10 3: 119
10 10 10 7 10 10 10 10 7 10 10 10 10 7: 131
10 10 10 9 10 10 10 10 9 10 10 10 10 9: 137

11 11 11 1 11 11 11 11 1 11 11 11 11 1: 124
11 11 11 2 11 11 11 11 2 11 11 11 11 2: 127
11 11 11 3 11 11 11 11 3 11 11 11 11 3: 130
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: 145
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :148
11 11 11 10 11 11 11 11 10 11 11 11 11 10: 151

12 12 12 1 12 12 12 12 1 12 12 12 12 1: 135
12 12 12 5 12 12 12 12 5 12 12 12 12 5: 147
12 12 12 7 12 12 12 12 7 12 12 12 12 7: 153
12 12 12 11 12 12 12 12 11 12 12 12 12 11: 165

13 13 13 1 13 13 13 13 1 13 13 13 13 1: 146
13 13 13 2 13 13 13 13 2 13 13 13 13 2: 149
13 13 13 3 13 13 13 13 3 13 13 13 13 3: 152
13 13 13 4 13 13 13 13 4 13 13 13 13 4: 155
13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]
13 13 13 9 13 13 13 13 9 13 13 13 13 9: 170
13 13 13 10 13 13 13 13 10 13 13 13 13 10: 173
13 13 13 11 13 13 13 13 11 13 13 13 13 11: 176
13 13 13 12 13 13 13 13 12 13 13 13 13 12: 179

**17 17 17 1 17 17 17 17 1 17 17 17 17 1: 190**
**17 17 17 2 17 17 17 17 2 17 17 17 17 2: 193**
**17 17 17 3 17 17 17 17 3 17 17 17 17 3: 196**
**17 17 17 4 17 17 17 17 4 17 17 17 17 4: 199**
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: 205**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: 208**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: 211**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: 214**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: 217**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: 220**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
**17 17 17 13 17 17 17 17 13 17 17 17 17 13: 226**
**17 17 17 14 17 17 17 17 14 17 17 17 17 14: 229**
**17 17 17 15 17 17 17 17 15 17 17 17 17 15: 232**
**17 17 17 16 17 17 17 17 16 17 17 17 17 16: 235**

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   ||   || 436.__36__ || 109.__09__ || 0 ||=   ||
||   ||   ||   ||   ||   ||   ||   || 33\91 || 435.16484 || 105.494505 || 13.18681 ||   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] ||   || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   ||   ||   || 54\149 || 434.89933 || 104.69799 || 16.10738 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   ||   || 434.78261 || 104.34783 || 17.391 ||   ||
||   ||   ||   ||   ||   ||   ||   || 71\196 || 434.69388 || 104.08163 || 18.36735 ||   ||
||   ||   ||   ||   ||   ||   || 46\[[127edo|127]] ||   || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   ||   ||   ||   || 67\185 || 434.__594__ || 103.__783__ || 19.__459__ ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   ||   || 434.48276 || 103.448275 || 20.68966 ||   ||
||   ||   ||   ||   ||   ||   ||   || 80\221 || 434.38914 || 103.16742 || 21.71946 ||   ||
||   ||   ||   ||   ||   ||   || 59\[[163edo|163]] ||   || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   ||   ||   || 97\268 || 434.32836 || 102.985075 || 22.38806 ||   ||
||   ||   ||   ||   ||   || 38\[[105edo|105]] ||   ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   ||   || 93\257 || 434.241245 || 102.723735 || 23.3463 ||   ||
||   ||   ||   ||   ||   ||   || 55\[[152edo|152]] ||   || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   ||   ||   ||   ||   || 72\199 || 434.17085 || 102.51256 || 24.1206 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   ||   || 81\224 || 433.92857 || 101.78571 || 26.78571 ||   ||
||   ||   ||   ||   ||   ||   || 64\[[177edo|177]] ||   || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   ||   ||   || 111\307 || 433.87622 || 101.62866 || 27.36156 ||   ||
||   ||   ||   ||   ||   || 47\[[130edo|130]] ||   ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   ||   || 124\343 || 433.81924 || 101.45773 || 27.98834 ||   ||
||   ||   ||   ||   ||   ||   || 77\[[213edo|213]] ||   || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   ||   ||   ||   || 107\296 || 433.__783__ || 101.__351__ || 28.__378__ ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   ||   || 103\285 || 433.68421 || 101.05263 || 29.47368 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] ||   || 433.__6633__ || 100.__9900__ || 29.__7030__ || Since here are the optimal range Lufsur mode ||
||   ||   ||   ||   ||   ||   ||   || 116/321 || 433.64486 || 100.93458 || 29.96054 ||   ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   ||   || 99\274 || 433.57664 || 100.72993 || 30.65693 ||   ||
||   ||   ||   ||   ||   ||   || 56\[[155edo|155]] ||   || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   ||   ||   ||   ||   ||   || 69\191 || 433.50785 || 100.52356 || 31.41361 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   ||   || 433.__3__ || 100 || 33.__3__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 74\205 || 433.__17073__ || 99.__51219__ || 35.__12195__ ||   ||
||   ||   ||   ||   ||   ||   || 61\[[169edo|169]] ||   || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   ||   ||   || 109\302 || 433.11258 || 99.33775 || 35.76159 ||   ||
||   ||   ||   ||   ||   || 48\[[133edo|133]] ||   ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   ||   || 131\363 || 433.05785 || 99.17355 || 36.__36__ ||   ||
||   ||   ||   ||   ||   ||   || 83\[[230edo|230]] ||   || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   ||   ||   ||   || 118\327 || 433.02752 || 99.08257 || 36.69725 ||   ||
||   ||   ||   ||   || 35\[[97edo|97]] ||   ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   ||   || 127\352 || 432.95__45__ || 98.8__63__ || 37.5 ||   ||
||   ||   ||   ||   ||   ||   || 92\[[255edo|255]] ||   || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   ||   ||   || 149\413 || 432.92978 || 98.78935 || 37.7724 ||   ||
||   ||   ||   ||   ||   || 57\[[158edo|158]] ||   ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   ||   || 136\377 || 432.89125 || 98.67374 || 38.19629 ||   ||
||   ||   ||   ||   ||   ||   || 79\[[219edo|219]] ||   || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   ||   ||   ||   ||   || 101\280 || 432.85714 || 98.57143 || 38.57143 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   ||   || 97\269 || 432.713755 || 98.14126 || 40.1487 ||   ||
||   ||   ||   ||   ||   ||   || 75\[[208edo|208]] ||   || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   ||   ||   || 128\355 || 432.67606 || 98.02817 || 40.56338 ||   ||
||   ||   ||   ||   ||   || 53\[[147edo|147]] ||   ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   ||   || 137\380 || 432.63158 || 97.89474 || 41.05263 ||   ||
||   ||   ||   ||   ||   ||   || 84\[[233edo|233]] ||   || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   ||   ||   ||   || 115\319 || 432.60188 || 97.80564 || 41.37931 ||   ||
||   ||   ||   ||   || 31\[[86edo|86]] ||   ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   ||   || 102\283 || 432.50883 || 97.5265 || 42.40283 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[197edo|197]] ||   || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   ||   ||   || 111\308 || 432.46753 || 97.4026 || 42.85714 ||   ||
||   ||   ||   ||   ||   || 40\[[111edo|111]] ||   ||   || 432.__432__ || 97.__297__ || 43.__135__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 89\247 || 432.38866 || 97.16599 || 43.7247 ||   ||
||   ||   ||   ||   ||   ||   || 49\[[136edo|136]] ||   || 432.35294 || 97.05882 || 44.11765 ||   ||
||   ||   ||   ||   ||   ||   ||   || 58\[[161edo|161]] || 432.29814 || 96.89441 || 44.72051 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   ||   || 59\[[164edo|164]] || 431.__70731__ || 95.__12195__ || 51.__21951__ ||   ||
||   ||   ||   ||   ||   ||   || 50\[[139edo|139]] ||   || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   ||   ||   || 91\253 || 431.62055 || 94.86166 || 52.17391 ||   ||
||   ||   ||   ||   ||   || 41\[[114edo|114]] ||   ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   ||   || 114\317 || 431.54574 || 94.63722 || 52.996845 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[203edo|203]] ||   || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   ||   ||   ||   || 105\292 || 431.50685 || 94.52055 || 53.42466 ||   ||
||   ||   ||   ||   || 32\[[89edo|89]] ||   ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   ||   || 119\331 || 431.41664 || 94.29582 || 54.380665 ||   ||
||   ||   ||   ||   ||   ||   || 87\[[242edo|242]] ||   || 431.40496 || 94.21488 || 54.__54__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 142\395 || 431.392405 || 94.177215 || 54.68354 ||   ||
||   ||   ||   ||   ||   || 55\[[153edo|153]] ||   ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   ||   || 133\370 || 431.__351__ || 94.__054__ || 55.__135__ ||   ||
||   ||   ||   ||   ||   ||   || 78\[[217edo|217]] ||   || 431.33641 || 94.00922 || 55.29955 ||   ||
||   ||   ||   ||   ||   ||   ||   || 101\281 || 431.31673 || 93.95018 || 55.51601 ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   ||   || 106\295 || 431.18644 || 93.55932 || 56.94915 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[231edo|231]] ||   || 431.16833 || 93.50649 || 57.14286 ||   ||
||   ||   ||   ||   ||   ||   ||   || 143\398 || 431.15578 || 93.46734 || 57.28643 ||   ||
||   ||   ||   ||   ||   || 60\[[167edo|167]] ||   ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   ||   || 157\437 || 431.12128 || 93.36384 || 57.6659 ||   ||
||   ||   ||   ||   ||   ||   || 97\[[270edo|270]] ||   || 431.__1__ || 93.__3__ || 57.__8__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 134\373 || 431.0992 || 93.29759 || 57.90885 ||   ||
||   ||   ||   ||   || 37\[[103edo|103]] ||   ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   ||   || 125\348 || 431.03448 || 93.10345 || 58.60269 ||   ||
||   ||   ||   ||   ||   ||   || 88\[[245edo|245]] ||   || 431.02041 || 93.06122 || 58.77551 ||   ||
||   ||   ||   ||   ||   ||   ||   || 139\387 || 431.00775 || 93.02326 || 58.91473 ||   ||
||   ||   ||   ||   ||   || 51\[[142edo|142]] ||   ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   ||   || 116\323 || 430.95975 || 92.87926 || 59.44272 ||   ||
||   ||   ||   ||   ||   ||   || 65\[[181edo|181]] ||   || 430.93923 || 92.81768 || 59.66852 ||   ||
||   ||   ||   ||   ||   ||   ||   || 79\220 || 430.__90__ || 92.__72__ || 60 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   ||   || 75\209 || 430.62201 || 91.86603 || 63.157895 ||   ||
||   ||   ||   ||   ||   ||   || 61\[[170edo|170]] ||   || 430.58824 || 91.76471 || 62.54942 ||   ||
||   ||   ||   ||   ||   ||   ||   || 108\301 || 430.56478 || 91.69435 || 63.787375 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   ||   || 127\354 || 430.508475 || 91.52542 || 64.40678 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] ||   || 430.49327 || 91.47982 || 64.57505 || Until here are the optimal range Fuslur mode ||
||   ||   ||   ||   ||   ||   ||   || 113\315 || 430.47619 || 91.42857 || 64.761905 ||   ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   || 85\[[237edo|237]] ||   || 430.37975 || 91.13924 || 65.82309 ||   ||
||   ||   ||   ||   ||   || 52\[[145edo|145]] ||   ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[198edo|198]] ||   || 430.__30__ || 90.__90__ || 66.__67__ ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   || 62\[[173edo|173]] ||   || 430.0578 || 90.17341 || 69.36416 ||   ||
||   ||   ||   ||   ||   || 43\[[120edo|120]] ||   ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   || 67\[[187edo|187]] ||   || 429.94652 || 89.83957 || 70.58824 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   || 53\[[148edo|148]] ||   || 429.__729__ || 89.__189__ || 72.__973__ ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   ||   || 429.__629__ || 88.__8__ || 74.__074__ ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] ||   || 429.47368 || 88.42105 || 75.78947 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   ||   || 428.57143 || 85.__714285__ || 85.__714285__ ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 1 5 5 5 5 1 5 5 5 5 1: 58&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 4 5 5 5 5 4 5 5 5 5 4: 67&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: 69&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: 81&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: 80&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: 95&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 1 8 8 8 8 1 8 8 8 8 1: 91&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 7 8 8 8 8 7 8 8 8 8 7: 109&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: 102&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: 105&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: 120&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: 123&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1: 113&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: 119&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: 131&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: 137&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 1 11 11 11 11 1 11 11 11 11 1: 124&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: 127&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: 130&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: 145&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :148&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: 151&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: 135&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: 147&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: 153&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: 165&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: 146&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: 149&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: 152&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: 155&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: 170&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: 173&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: 176&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: 179&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 1 17 17 17 17 1 17 17 17 17 1: 190&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 2 17 17 17 17 2 17 17 17 17 2: 193&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 3 17 17 17 17 3 17 17 17 17 3: 196&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 4 17 17 17 17 4 17 17 17 17 4: 199&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: 205&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: 208&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: 211&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: 214&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: 217&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: 220&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 13 17 17 17 17 13 17 17 17 17 13: 226&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 14 17 17 17 17 14 17 17 17 17 14: 229&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 15 17 17 17 17 15 17 17 17 17 15: 232&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 16 17 17 17 17 16 17 17 17 17 16: 235&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.&amp;lt;u&amp;gt;36&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.&amp;lt;u&amp;gt;09&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33\91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435.16484&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.494505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13.18681&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;54\149&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.89933&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.69799&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16.10738&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71\196&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.69388&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.08163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.36735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46\&amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.64567&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.93701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.89764&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;67\185&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.&amp;lt;u&amp;gt;594&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.&amp;lt;u&amp;gt;783&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19.&amp;lt;u&amp;gt;459&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448275&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20.68966&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;80\221&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.38914&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.16742&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21.71946&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;59\&amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.35583&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.06748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.0859&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97\268&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.32836&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.985075&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
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      <sha1>q9g881kc4g13jjf410e95nw2k0l9ht2</sha1>
    </revision>
    <revision>
      <id>4473</id>
      <timestamp>2012-06-20T20:55:43Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 346838218 - Original comment: **</comment>
      <origin>4473</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="83365" sha1="drxzif2o9grf5pzrts2eck2z4bm4enm" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-06-20 20:55:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;346838218&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

**5 5 5 1 5 5 5 5 1 5 5 5 5 1: 58** 
**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]**
**5 5 5 4 5 5 5 5 4 5 5 5 5 4: 67** 

6 6 6 1 6 6 6 6 1 6 6 6 6 1: 69
6 6 6 5 6 6 6 6 5 6 6 6 6 5: 81

7 7 7 1 7 7 7 7 1 7 7 7 7 1: 80
7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]
7 7 7 6 7 7 7 7 6 7 7 7 7 6: 95 

**8 8 8 1 8 8 8 8 1 8 8 8 8 1: 91** 
**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]]**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]**
**8 8 8 7 8 8 8 8 7 8 8 8 8 7: 109**

9 9 9 1 9 9 9 9 1 9 9 9 9 1: 102
9 9 9 2 9 9 9 9 2 9 9 9 9 2: 105 
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]
9 9 9 7 9 9 9 9 7 9 9 9 9 7: 120
9 9 9 8 9 9 9 9 8 9 9 9 9 8: 123 

10 10 10 1 10 10 10 10 1 10 10 10 10 1: 113
10 10 10 3 10 10 10 10 3 10 10 10 10 3: 119
10 10 10 7 10 10 10 10 7 10 10 10 10 7: 131
10 10 10 9 10 10 10 10 9 10 10 10 10 9: 137 

11 11 11 1 11 11 11 11 1 11 11 11 11 1: 124
11 11 11 2 11 11 11 11 2 11 11 11 11 2: 127
11 11 11 3 11 11 11 11 3 11 11 11 11 3: 130 
11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]
11 11 11 8 11 11 11 11 8 11 11 11 11 8: 145
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :148
11 11 11 10 11 11 11 11 10 11 11 11 11 10: 151

12 12 12 1 12 12 12 12 1 12 12 12 12 1: 135
12 12 12 5 12 12 12 12 5 12 12 12 12 5: 147
12 12 12 7 12 12 12 12 7 12 12 12 12 7: 153
12 12 12 11 12 12 12 12 11 12 12 12 12 11: 165 

13 13 13 1 13 13 13 13 1 13 13 13 13 1: 146
13 13 13 2 13 13 13 13 2 13 13 13 13 2: 149
13 13 13 3 13 13 13 13 3 13 13 13 13 3: 152
13 13 13 4 13 13 13 13 4 13 13 13 13 4: 155 
13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]
13 13 13 9 13 13 13 13 9 13 13 13 13 9: 170
13 13 13 10 13 13 13 13 10 13 13 13 13 10: 173
13 13 13 11 13 13 13 13 11 13 13 13 13 11: 176
13 13 13 12 13 13 13 13 12 13 13 13 13 12: 179 

**17 17 17 1 17 17 17 17 1 17 17 17 17 1: 190**
**17 17 17 2 17 17 17 17 2 17 17 17 17 2: 193**
**17 17 17 3 17 17 17 17 3 17 17 17 17 3: 196**
**17 17 17 4 17 17 17 17 4 17 17 17 17 4: 199** 
**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 6 17 17 17 17 6 17 17 17 17 6: 205**
**17 17 17 7 17 17 17 17 7 17 17 17 17 7: 208**
**17 17 17 8 17 17 17 17 8 17 17 17 17 8: 211**
**17 17 17 9 17 17 17 17 9 17 17 17 17 9: 214**
**17 17 17 10 17 17 17 17 10 17 17 17 17 10: 217**
**17 17 17 11 17 17 17 17 11 17 17 17 17 11: 220**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**
**17 17 17 13 17 17 17 17 13 17 17 17 17 13: 226**
**17 17 17 14 17 17 17 17 14 17 17 17 17 14: 229**
**17 17 17 15 17 17 17 17 15 17 17 17 17 15: 232**
**17 17 17 16 17 17 17 17 16 17 17 17 17 16: 235** 

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   ||   || 436.__36__ || 109.__09__ || 0 ||=   ||
||   ||   ||   ||   ||   ||   ||   || 33\91 || 435.16484 || 105.494505 || 13.18681 ||   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] ||   || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   ||   ||   || 54\149 || 434.89933 || 104.69799 || 16.10738 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   ||   || 434.78261 || 104.34783 || 17.391 ||   ||
||   ||   ||   ||   ||   ||   ||   || 71\196 || 434.69388 || 104.08163 || 18.36735 ||   ||
||   ||   ||   ||   ||   ||   || 46\[[127edo|127]] ||   || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   ||   ||   ||   || 67\185 || 434.__594__ || 103.__783__ || 19.__459__ ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   ||   || 434.48276 || 103.448275 || 20.68966 ||   ||
||   ||   ||   ||   ||   ||   ||   || 80\221 || 434.38914 || 103.16742 || 21.71946 ||   ||
||   ||   ||   ||   ||   ||   || 59\[[163edo|163]] ||   || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   ||   ||   || 97\268 || 434.32836 || 102.985075 || 22.38806 ||   ||
||   ||   ||   ||   ||   || 38\[[105edo|105]] ||   ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   ||   || 93\257 || 434.241245 || 102.723735 || 23.3463 ||   ||
||   ||   ||   ||   ||   ||   || 55\[[152edo|152]] ||   || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   ||   ||   ||   ||   || 72\199 || 434.17085 || 102.51256 || 24.1206 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   ||   || 81\224 || 433.92857 || 101.78571 || 26.78571 ||   ||
||   ||   ||   ||   ||   ||   || 64\[[177edo|177]] ||   || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   ||   ||   || 111\307 || 433.87622 || 101.62866 || 27.36156 ||   ||
||   ||   ||   ||   ||   || 47\[[130edo|130]] ||   ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   ||   || 124\343 || 433.81924 || 101.45773 || 27.98834 ||   ||
||   ||   ||   ||   ||   ||   || 77\[[213edo|213]] ||   || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   ||   ||   ||   || 107\296 || 433.__783__ || 101.__351__ || 28.__378__ ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   ||   || 103\285 || 433.68421 || 101.05263 || 29.47368 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] ||   || 433.__6633__ || 100.__9900__ || 29.__7030__ || Since here are the optimal range Lufsur mode ||
||   ||   ||   ||   ||   ||   ||   || 116/321 || 433.64486 || 100.93458 || 29.96054 ||   ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   ||   || 99\274 || 433.57664 || 100.72993 || 30.65693 ||   ||
||   ||   ||   ||   ||   ||   || 56\[[155edo|155]] ||   || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   ||   ||   ||   ||   ||   || 69\191 || 433.50785 || 100.52356 || 31.41361 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   ||   || 433.__3__ || 100 || 33.__3__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 74\205 || 433.__17073__ || 99.__51219__ || 35.__12195__ ||   ||
||   ||   ||   ||   ||   ||   || 61\[[169edo|169]] ||   || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   ||   ||   || 109\302 || 433.11258 || 99.33775 || 35.76159 ||   ||
||   ||   ||   ||   ||   || 48\[[133edo|133]] ||   ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   ||   || 131\363 || 433.05785 || 99.17355 || 36.__36__ ||   ||
||   ||   ||   ||   ||   ||   || 83\[[230edo|230]] ||   || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   ||   ||   ||   || 118\327 || 433.02752 || 99.08257 || 36.69725 ||   ||
||   ||   ||   ||   || 35\[[97edo|97]] ||   ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   ||   || 127\352 || 432.95__45__ || 98.8__63__ || 37.5 ||   ||
||   ||   ||   ||   ||   ||   || 92\[[255edo|255]] ||   || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   ||   ||   || 149\413 || 432.92978 || 98.78935 || 37.7724 ||   ||
||   ||   ||   ||   ||   || 57\[[158edo|158]] ||   ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   ||   || 136\377 || 432.89125 || 98.67374 || 38.19629 ||   ||
||   ||   ||   ||   ||   ||   || 79\[[219edo|219]] ||   || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   ||   ||   ||   ||   || 101\280 || 432.85714 || 98.57143 || 38.57143 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   ||   || 97\269 || 432.713755 || 98.14126 || 40.1487 ||   ||
||   ||   ||   ||   ||   ||   || 75\[[208edo|208]] ||   || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   ||   ||   || 128\355 || 432.67606 || 98.02817 || 40.56338 ||   ||
||   ||   ||   ||   ||   || 53\[[147edo|147]] ||   ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   ||   || 137\380 || 432.63158 || 97.89474 || 41.05263 ||   ||
||   ||   ||   ||   ||   ||   || 84\[[233edo|233]] ||   || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   ||   ||   ||   || 115\319 || 432.60188 || 97.80564 || 41.37931 ||   ||
||   ||   ||   ||   || 31\[[86edo|86]] ||   ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   ||   || 102\283 || 432.50883 || 97.5265 || 42.40283 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[197edo|197]] ||   || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   ||   ||   || 111\308 || 432.46753 || 97.4026 || 42.85714 ||   ||
||   ||   ||   ||   ||   || 40\[[111edo|111]] ||   ||   || 432.__432__ || 97.__297__ || 43.__135__ ||   ||
||   ||   ||   ||   ||   ||   ||   || 89\247 || 432.38866 || 97.16599 || 43.7247 ||   ||
||   ||   ||   ||   ||   ||   || 49\[[136edo|136]] ||   || 432.35294 || 97.05882 || 44.11765 ||   ||
||   ||   ||   ||   ||   ||   ||   || 58\[[161edo|161]] || 432.29814 || 96.89441 || 44.72051 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   ||   || 59\[[164edo|164]] || 431.__70731__ || 95.__12195__ || 51.__21951__ ||   ||
||   ||   ||   ||   ||   ||   || 50\[[139edo|139]] ||   || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   || 41\[[114edo|114]] ||   ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[203edo|203]] ||   || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   || 32\[[89edo|89]] ||   ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   || 87\[[242edo|242]] ||   || 431.40496 || 94.21488 || 54.54: ||   ||
||   ||   ||   ||   ||   || 55\[[153edo|153]] ||   ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   || 78\[[217edo|217]] ||   || 431.33641 || 94.00922 || 55.29955 ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[231edo|231]] ||   || 431.16833 || 93.50649 || 57.14286 ||   ||
||   ||   ||   ||   ||   || 60\[[167edo|167]] ||   ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   || 97\[[270edo|270]] ||   || 431.__1__ || 93.__3__ || 57.__8__ ||   ||
||   ||   ||   ||   || 37\[[103edo|103]] ||   ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   || 88\[[245edo|245]] ||   || 431.02041 || 93.06122 || 58.77551 ||   ||
||   ||   ||   ||   ||   || 51\[[142edo|142]] ||   ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   || 65\[[181edo|181]] ||   || 430.93923 || 92.81768 || 59.66852 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   || 61\[[170edo|170]] ||   || 430.58824 || 91.76471 || 62.54942 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] ||   || 430.49327 || 91.47982 || 64.57505 || Until here are the optimal range Fuslur mode ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   || 85\[[237edo|237]] ||   || 430.37975 || 91.13924 || 65.82309 ||   ||
||   ||   ||   ||   ||   || 52\[[145edo|145]] ||   ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[198edo|198]] ||   || 430.__30__ || 90.__90__ || 66.__67__ ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   || 62\[[173edo|173]] ||   || 430.0578 || 90.17341 || 69.36416 ||   ||
||   ||   ||   ||   ||   || 43\[[120edo|120]] ||   ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   || 67\[[187edo|187]] ||   || 429.94652 || 89.83957 || 70.58824 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   || 53\[[148edo|148]] ||   || 429.__729__ || 89.__189__ || 72.__973__ ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   ||   || 429.__629__ || 88.__8__ || 74.__074__ ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] ||   || 429.47368 || 88.42105 || 75.78947 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   ||   || 428.57143 || 85.__714285__ || 85.__714285__ ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 1 5 5 5 5 1 5 5 5 5 1: 58&amp;lt;/strong&amp;gt; &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 4 5 5 5 5 4 5 5 5 5 4: 67&amp;lt;/strong&amp;gt; &amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 1 6 6 6 6 1 6 6 6 6 1: 69&amp;lt;br /&amp;gt;
6 6 6 5 6 6 6 6 5 6 6 6 6 5: 81&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 1 7 7 7 7 1 7 7 7 7 1: 80&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 6 7 7 7 7 6 7 7 7 7 6: 95 &amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 1 8 8 8 8 1 8 8 8 8 1: 91&amp;lt;/strong&amp;gt; &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 7 8 8 8 8 7 8 8 8 8 7: 109&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 1 9 9 9 9 1 9 9 9 9 1: 102&amp;lt;br /&amp;gt;
9 9 9 2 9 9 9 9 2 9 9 9 9 2: 105 &amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 7 9 9 9 9 7 9 9 9 9 7: 120&amp;lt;br /&amp;gt;
9 9 9 8 9 9 9 9 8 9 9 9 9 8: 123 &amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
10 10 10 1 10 10 10 10 1 10 10 10 10 1: 113&amp;lt;br /&amp;gt;
10 10 10 3 10 10 10 10 3 10 10 10 10 3: 119&amp;lt;br /&amp;gt;
10 10 10 7 10 10 10 10 7 10 10 10 10 7: 131&amp;lt;br /&amp;gt;
10 10 10 9 10 10 10 10 9 10 10 10 10 9: 137 &amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 1 11 11 11 11 1 11 11 11 11 1: 124&amp;lt;br /&amp;gt;
11 11 11 2 11 11 11 11 2 11 11 11 11 2: 127&amp;lt;br /&amp;gt;
11 11 11 3 11 11 11 11 3 11 11 11 11 3: 130 &amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 8 11 11 11 11 8 11 11 11 11 8: 145&amp;lt;br /&amp;gt;
11 11 11 9 11 11 11 11 9 11 11 11 11 9 :148&amp;lt;br /&amp;gt;
11 11 11 10 11 11 11 11 10 11 11 11 11 10: 151&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12 12 12 1 12 12 12 12 1 12 12 12 12 1: 135&amp;lt;br /&amp;gt;
12 12 12 5 12 12 12 12 5 12 12 12 12 5: 147&amp;lt;br /&amp;gt;
12 12 12 7 12 12 12 12 7 12 12 12 12 7: 153&amp;lt;br /&amp;gt;
12 12 12 11 12 12 12 12 11 12 12 12 12 11: 165 &amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 1 13 13 13 13 1 13 13 13 13 1: 146&amp;lt;br /&amp;gt;
13 13 13 2 13 13 13 13 2 13 13 13 13 2: 149&amp;lt;br /&amp;gt;
13 13 13 3 13 13 13 13 3 13 13 13 13 3: 152&amp;lt;br /&amp;gt;
13 13 13 4 13 13 13 13 4 13 13 13 13 4: 155 &amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 9 13 13 13 13 9 13 13 13 13 9: 170&amp;lt;br /&amp;gt;
13 13 13 10 13 13 13 13 10 13 13 13 13 10: 173&amp;lt;br /&amp;gt;
13 13 13 11 13 13 13 13 11 13 13 13 13 11: 176&amp;lt;br /&amp;gt;
13 13 13 12 13 13 13 13 12 13 13 13 13 12: 179 &amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 1 17 17 17 17 1 17 17 17 17 1: 190&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 2 17 17 17 17 2 17 17 17 17 2: 193&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 3 17 17 17 17 3 17 17 17 17 3: 196&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 4 17 17 17 17 4 17 17 17 17 4: 199&amp;lt;/strong&amp;gt; &amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 6 17 17 17 17 6 17 17 17 17 6: 205&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 7 17 17 17 17 7 17 17 17 17 7: 208&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 8 17 17 17 17 8 17 17 17 17 8: 211&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 9 17 17 17 17 9 17 17 17 17 9: 214&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 10 17 17 17 17 10 17 17 17 17 10: 217&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 11 17 17 17 17 11 17 17 17 17 11: 220&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 13 17 17 17 17 13 17 17 17 17 13: 226&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 14 17 17 17 17 14 17 17 17 17 14: 229&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 15 17 17 17 17 15 17 17 17 17 15: 232&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 16 17 17 17 17 16 17 17 17 17 16: 235&amp;lt;/strong&amp;gt; &amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.&amp;lt;u&amp;gt;36&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.&amp;lt;u&amp;gt;09&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33\91&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435.16484&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105.494505&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13.18681&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;54\149&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.89933&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.69799&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16.10738&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.391&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71\196&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.69388&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.08163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.36735&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46\&amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.64567&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.93701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.89764&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;67\185&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.&amp;lt;u&amp;gt;594&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.&amp;lt;u&amp;gt;783&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19.&amp;lt;u&amp;gt;459&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448275&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20.68966&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;80\221&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.38914&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.16742&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21.71946&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;59\&amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.35583&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.06748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.0859&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97\268&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.32836&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.985075&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.38806&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;87\&amp;lt;a class="wiki_link" href="/242edo"&amp;gt;242&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;55\&amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;78\&amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;23\&amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;83\&amp;lt;a class="wiki_link" href="/231edo"&amp;gt;231&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.16833&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;93.50649&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;57.14286&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;60\&amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.13772&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;93.41317&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;57.48503&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;97\&amp;lt;a class="wiki_link" href="/270edo"&amp;gt;270&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;93.20388&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;58.25263&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;88\&amp;lt;a class="wiki_link" href="/245edo"&amp;gt;245&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;58.77551&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;51\&amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;61\&amp;lt;a class="wiki_link" href="/170edo"&amp;gt;170&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;47\&amp;lt;a class="wiki_link" href="/131edo"&amp;gt;131&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;430.53435&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;80\&amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;430.49327&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;91.47982&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;64.57505&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;Until here are the optimal range Fuslur mode&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;33\&amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;430.43478&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;91.30435&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;65.21739&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;85\&amp;lt;a class="wiki_link" href="/237edo"&amp;gt;237&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;430.37975&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;91.13924&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;65.82309&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;71\&amp;lt;a class="wiki_link" href="/198edo"&amp;gt;198&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;90.&amp;lt;u&amp;gt;90&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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      <sha1>drxzif2o9grf5pzrts2eck2z4bm4enm</sha1>
    </revision>
    <revision>
      <id>4474</id>
      <timestamp>2012-05-23T04:41:33Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 338654132 - Original comment: **</comment>
      <origin>4474</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="57137" sha1="cgkyxaiexbr22wm8618udjsv6sf75uy" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-05-23 04:41:33 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;338654132&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]**

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]]**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]**


9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]

**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   ||   || 436.__36__ || 109.__09__ || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] ||   || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   ||   || 434.78261 || 104.34783 || 17.39131 ||   ||
||   ||   ||   ||   ||   ||   || 46\[[127edo|127]] ||   || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   ||   || 434.48276 || 103.44828 || 22.03848 ||   ||
||   ||   ||   ||   ||   ||   || 59\[[163edo|163]] ||   || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   || 38\[[105edo|105]] ||   ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   || 55\[[152edo|152]] ||   || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   || 64\[[177edo|177]] ||   || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   || 47\[[130edo|130]] ||   ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   || 77\[[213edo|213]] ||   || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] ||   || 433.__6633__ || 100.__9900__ || 29.__7030__ || Since here are the optimal range Lufsur mode ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   || 56\[[155edo|155]] ||   || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   ||   || 433.__3__ || 100 || 33.__3__ ||   ||
||   ||   ||   ||   ||   ||   || 61\[[169edo|169]] ||   || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   || 48\[[133edo|133]] ||   ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[230edo|230]] ||   || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   || 35\[[97edo|97]] ||   ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   || 92\[[255edo|255]] ||   || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   || 57\[[158edo|158]] ||   ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   || 79\[[219edo|219]] ||   || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   || 75\[[208edo|208]] ||   || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   || 53\[[147edo|147]] ||   ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   || 84\[[233edo|233]] ||   || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   || 31\[[86edo|86]] ||   ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[197edo|197]] ||   || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   || 40\[[111edo|111]] ||   ||   || 432.__432__ || 97.__297__ || 43.__135__ ||   ||
||   ||   ||   ||   ||   ||   || 49\[[136edo|136]] ||   || 432.35294 || 97.05882 || 44.11765 ||   ||
||   ||   ||   ||   ||   ||   ||   || 58\[[161edo|161]] || 432.29814 || 96.89441 || 44.72051 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   ||   || 59\[[164edo|164]] || 431.__70731__ || 95.__12195__ || 51.__21951__ ||   ||
||   ||   ||   ||   ||   ||   || 50\[[139edo|139]] ||   || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   || 41\[[114edo|114]] ||   ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[203edo|203]] ||   || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   || 32\[[89edo|89]] ||   ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   || 87\[[242edo|242]] ||   || 431.40496 || 94.21488 || 54.54: ||   ||
||   ||   ||   ||   ||   || 55\[[153edo|153]] ||   ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   || 78\[[217edo|217]] ||   || 431.33641 || 94.00922 || 55.29955 ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[231edo|231]] ||   || 431.16833 || 93.50649 || 57.14286 ||   ||
||   ||   ||   ||   ||   || 60\[[167edo|167]] ||   ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   || 97\[[270edo|270]] ||   || 431.__1__ || 93.__3__ || 57.__8__ ||   ||
||   ||   ||   ||   || 37\[[103edo|103]] ||   ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   || 88\[[245edo|245]] ||   || 431.02041 || 93.06122 || 58.77551 ||   ||
||   ||   ||   ||   ||   || 51\[[142edo|142]] ||   ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   || 65\[[181edo|181]] ||   || 430.93923 || 92.81768 || 59.66852 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   || 61\[[170edo|170]] ||   || 430.58824 || 91.76471 || 62.54942 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] ||   || 430.49327 || 91.47982 || 64.57505 || Until here are the optimal range Fuslur mode ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   || 85\[[237edo|237]] ||   || 430.37975 || 91.13924 || 65.82309 ||   ||
||   ||   ||   ||   ||   || 52\[[145edo|145]] ||   ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[198edo|198]] ||   || 430.__30__ || 90.__90__ || 66.__67__ ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   || 62\[[173edo|173]] ||   || 430.0578 || 90.17341 || 69.36416 ||   ||
||   ||   ||   ||   ||   || 43\[[120edo|120]] ||   ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   || 67\[[187edo|187]] ||   || 429.94652 || 89.83957 || 70.58824 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   || 53\[[148edo|148]] ||   || 429.__729__ || 89.__189__ || 72.__973__ ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   ||   || 429.__629__ || 88.__8__ || 74.__074__ ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] ||   || 429.47368 || 88.42105 || 75.78947 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   ||   || 428.57143 || 85.__714285__ || 85.__714285__ ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.&amp;lt;u&amp;gt;36&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.&amp;lt;u&amp;gt;09&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.39131&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46\&amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.64567&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.93701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.89764&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.44828&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.03848&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;59\&amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.35583&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.06748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.0859&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;38\&amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.28571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;55\&amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;102.63158&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;23.68421&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;27.6923&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;77\&amp;lt;a class="wiki_link" href="/213edo"&amp;gt;213&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;28.16902&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;30\&amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;101.20842&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;29.&amp;lt;u&amp;gt;7030&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;43\&amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;100.84034&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;30.2521&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;57\&amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;79\&amp;lt;a class="wiki_link" href="/219edo"&amp;gt;219&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;22\&amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;39.34426&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;75\&amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.69231&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;98.07692&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;40.38439&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;53\&amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.65306&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;97.95918&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;40.81633&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;84\&amp;lt;a class="wiki_link" href="/233edo"&amp;gt;233&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.61803&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;97.85408&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;41.20172&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;31\&amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;71\&amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;40\&amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.&amp;lt;u&amp;gt;432&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;97.&amp;lt;u&amp;gt;297&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;43.&amp;lt;u&amp;gt;135&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;49\&amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;58\&amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;9\&amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;50\&amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;41\&amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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    <revision>
      <id>4475</id>
      <timestamp>2012-05-23T04:40:40Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 338653928 - Original comment: **</comment>
      <origin>4475</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="57109" sha1="sm8addw9wlxi44u4bnwmcx8dkzqps8f" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-05-23 04:40:40 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;338653928&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]**

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

**8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]]**
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]**
9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]

**17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)**
**17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)**

The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   ||   || 436.__36__ || 109.__09__ || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] ||   || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   ||   || 434.78261 || 104.34783 || 17.39131 ||   ||
||   ||   ||   ||   ||   ||   || 46\[[127edo|127]] ||   || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   ||   || 434.48276 || 103.44828 || 22.03848 ||   ||
||   ||   ||   ||   ||   ||   || 59\[[163edo|163]] ||   || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   || 38\[[105edo|105]] ||   ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   || 55\[[152edo|152]] ||   || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   || 64\[[177edo|177]] ||   || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   || 47\[[130edo|130]] ||   ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   || 77\[[213edo|213]] ||   || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] ||   || 433.__6633__ || 100.__9900__ || 29.__7030__ || Since here are the optimal range Lufsur mode ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   || 56\[[155edo|155]] ||   || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   ||   || 433.__3__ || 100 || 33.__3__ ||   ||
||   ||   ||   ||   ||   ||   || 61\[[169edo|169]] ||   || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   || 48\[[133edo|133]] ||   ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[230edo|230]] ||   || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   || 35\[[97edo|97]] ||   ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   || 92\[[255edo|255]] ||   || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   || 57\[[158edo|158]] ||   ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   || 79\[[219edo|219]] ||   || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   || 75\[[208edo|208]] ||   || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   || 53\[[147edo|147]] ||   ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   || 84\[[233edo|233]] ||   || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   || 31\[[86edo|86]] ||   ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[197edo|197]] ||   || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   || 40\[[111edo|111]] ||   ||   || 432.__432__ || 97.__297__ || 43.__135__ ||   ||
||   ||   ||   ||   ||   ||   || 49\[[136edo|136]] ||   || 432.35294 || 97.05882 || 44.11765 ||   ||
||   ||   ||   ||   ||   ||   ||   || 58\[[161edo|161]] || 432.29814 || 96.89441 || 44.72051 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   ||   || 59\[[164edo|164]] || 431.__70731__ || 95.__12195__ || 51.__21951__ ||   ||
||   ||   ||   ||   ||   ||   || 50\[[139edo|139]] ||   || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   || 41\[[114edo|114]] ||   ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[203edo|203]] ||   || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   || 32\[[89edo|89]] ||   ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   || 87\[[242edo|242]] ||   || 431.40496 || 94.21488 || 54.54: ||   ||
||   ||   ||   ||   ||   || 55\[[153edo|153]] ||   ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   || 78\[[217edo|217]] ||   || 431.33641 || 94.00922 || 55.29955 ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[231edo|231]] ||   || 431.16833 || 93.50649 || 57.14286 ||   ||
||   ||   ||   ||   ||   || 60\[[167edo|167]] ||   ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   || 97\[[270edo|270]] ||   || 431.__1__ || 93.__3__ || 57.__8__ ||   ||
||   ||   ||   ||   || 37\[[103edo|103]] ||   ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   || 88\[[245edo|245]] ||   || 431.02041 || 93.06122 || 58.77551 ||   ||
||   ||   ||   ||   ||   || 51\[[142edo|142]] ||   ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   || 65\[[181edo|181]] ||   || 430.93923 || 92.81768 || 59.66852 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   || 61\[[170edo|170]] ||   || 430.58824 || 91.76471 || 62.54942 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] ||   || 430.49327 || 91.47982 || 64.57505 || Until here are the optimal range Fuslur mode ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   || 85\[[237edo|237]] ||   || 430.37975 || 91.13924 || 65.82309 ||   ||
||   ||   ||   ||   ||   || 52\[[145edo|145]] ||   ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[198edo|198]] ||   || 430.__30__ || 90.__90__ || 66.__67__ ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   || 62\[[173edo|173]] ||   || 430.0578 || 90.17341 || 69.36416 ||   ||
||   ||   ||   ||   ||   || 43\[[120edo|120]] ||   ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   || 67\[[187edo|187]] ||   || 429.94652 || 89.83957 || 70.58824 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   || 53\[[148edo|148]] ||   || 429.__729__ || 89.__189__ || 72.__973__ ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   ||   || 429.__629__ || 88.__8__ || 74.__074__ ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] ||   || 429.47368 || 88.42105 || 75.78947 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   ||   || 428.57143 || 85.__714285__ || 85.__714285__ ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 5 17 17 17 17 5 17 17 17 17 5: &amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt; (Top limit for Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;17 17 17 12 17 17 17 17 12 17 17 17 17 12: &amp;lt;a class="wiki_link" href="/223edo"&amp;gt;223&amp;lt;/a&amp;gt; (Top limit for Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.&amp;lt;u&amp;gt;36&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.&amp;lt;u&amp;gt;09&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;59\&amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;22.0859&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;38\&amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;55\&amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;47\&amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;77\&amp;lt;a class="wiki_link" href="/213edo"&amp;gt;213&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;30.2521&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;58\&amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.29814&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;96.89441&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;44.72051&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;9\&amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;59\&amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.&amp;lt;u&amp;gt;70731&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;95.&amp;lt;u&amp;gt;12195&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;51.&amp;lt;u&amp;gt;21951&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;50\&amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.65468&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;94.96403&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;51.79856&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;41\&amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.57895&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;94.73684&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;52.63158&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.52709&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;94.58128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;53.20197&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;32\&amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.46067&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;94.38202&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;53.93258&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;87\&amp;lt;a class="wiki_link" href="/242edo"&amp;gt;242&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.40496&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;94.21488&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;54.54:&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;55\&amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.37255&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.11765&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;54.90196&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;78\&amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.33641&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;94.00922&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;55.29955&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;23\&amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;83\&amp;lt;a class="wiki_link" href="/231edo"&amp;gt;231&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.16833&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.50649&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;57.14286&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;60\&amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.13772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.41317&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;57.48503&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;97\&amp;lt;a class="wiki_link" href="/270edo"&amp;gt;270&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.&amp;lt;u&amp;gt;1&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;57.&amp;lt;u&amp;gt;8&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;37\&amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;24\&amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;429.85075&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;89.55224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71.64179&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;53\&amp;lt;a class="wiki_link" href="/148edo"&amp;gt;148&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.&amp;lt;u&amp;gt;729&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;89.&amp;lt;u&amp;gt;189&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;72.&amp;lt;u&amp;gt;973&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.&amp;lt;u&amp;gt;629&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.&amp;lt;u&amp;gt;8&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;74.&amp;lt;u&amp;gt;074&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34\&amp;lt;a class="wiki_link" href="/95edo"&amp;gt;95&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.47368&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.42105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;75.78947&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\&amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;428.57143&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.&amp;lt;u&amp;gt;714285&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.&amp;lt;u&amp;gt;714285&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>sm8addw9wlxi44u4bnwmcx8dkzqps8f</sha1>
    </revision>
    <revision>
      <id>4476</id>
      <timestamp>2012-05-06T17:48:38Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 330792672 - Original comment: **</comment>
      <origin>4476</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="55955" sha1="64z5xqgyvzrej65l3w4rzu352xr81ht" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-05-06 17:48:38 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;330792672&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
**3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)**
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

**4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]**
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
**4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]**

**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]**
**5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]**

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]

The next table below shows an extension of ED2s with respect to more generators:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   ||   || 436.__36__ || 109.__09__ || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] ||   || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   ||   || 434.78261 || 104.34783 || 17.39131 ||   ||
||   ||   ||   ||   ||   ||   || 46\[[127edo|127]] ||   || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   ||   || 434.48276 || 103.44828 || 22.03848 ||   ||
||   ||   ||   ||   ||   ||   || 59\[[163edo|163]] ||   || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   || 38\[[105edo|105]] ||   ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   || 55\[[152edo|152]] ||   || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   || 64\[[177edo|177]] ||   || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   || 47\[[130edo|130]] ||   ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   || 77\[[213edo|213]] ||   || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] ||   || 433.__6633__ || 100.__9900__ || 29.__7030__ ||   ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   || 56\[[155edo|155]] ||   || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   ||   || 433.__3__ || 100 || 33.__3__ ||   ||
||   ||   ||   ||   ||   ||   || 61\[[169edo|169]] ||   || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   || 48\[[133edo|133]] ||   ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[230edo|230]] ||   || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   || 35\[[97edo|97]] ||   ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   || 92\[[255edo|255]] ||   || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   || 57\[[158edo|158]] ||   ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   || 79\[[219edo|219]] ||   || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   || 75\[[208edo|208]] ||   || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   || 53\[[147edo|147]] ||   ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   || 84\[[233edo|233]] ||   || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   || 31\[[86edo|86]] ||   ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[197edo|197]] ||   || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   || 40\[[111edo|111]] ||   ||   || 432.__432__ || 97.__297__ || 43.__135__ ||   ||
||   ||   ||   ||   ||   ||   || 49\[[136edo|136]] ||   || 432.35294 || 97.05882 || 44.11765 ||   ||
||   ||   ||   ||   ||   ||   ||   || 58\[[161edo|161]] || 432.29814 || 96.89441 || 44.72051 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   ||   || 59\[[164edo|164]] || 431.__70731__ || 95.__12195__ || 51.__21951__ ||   ||
||   ||   ||   ||   ||   ||   || 50\[[139edo|139]] ||   || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   || 41\[[114edo|114]] ||   ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[203edo|203]] ||   || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   || 32\[[89edo|89]] ||   ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   || 87\[[242edo|242]] ||   || 431.40496 || 94.21488 || 54.54: ||   ||
||   ||   ||   ||   ||   || 55\[[153edo|153]] ||   ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   || 78\[[217edo|217]] ||   || 431.33641 || 94.00922 || 55.29955 ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[231edo|231]] ||   || 431.16833 || 93.50649 || 57.14286 ||   ||
||   ||   ||   ||   ||   || 60\[[167edo|167]] ||   ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   || 97\[[270edo|270]] ||   || 431.__1__ || 93.__3__ || 57.__8__ ||   ||
||   ||   ||   ||   || 37\[[103edo|103]] ||   ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   || 88\[[245edo|245]] ||   || 431.02041 || 93.06122 || 58.77551 ||   ||
||   ||   ||   ||   ||   || 51\[[142edo|142]] ||   ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   || 65\[[181edo|181]] ||   || 430.93923 || 92.81768 || 59.66852 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   || 61\[[170edo|170]] ||   || 430.58824 || 91.76471 || 62.54942 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] ||   || 430.49327 || 91.47982 || 64.57505 ||   ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   || 85\[[237edo|237]] ||   || 430.37975 || 91.13924 || 65.82309 ||   ||
||   ||   ||   ||   ||   || 52\[[145edo|145]] ||   ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[198edo|198]] ||   || 430.__30__ || 90.__90__ || 66.__67__ ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   || 62\[[173edo|173]] ||   || 430.0578 || 90.17341 || 69.36416 ||   ||
||   ||   ||   ||   ||   || 43\[[120edo|120]] ||   ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   || 67\[[187edo|187]] ||   || 429.94652 || 89.83957 || 70.58824 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   || 53\[[148edo|148]] ||   || 429.__729__ || 89.__189__ || 72.__973__ ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   ||   || 429.__629__ || 88.__8__ || 74.__074__ ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] ||   || 429.47368 || 88.42105 || 75.78947 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   ||   || 428.57143 || 85.__714285__ || 85.__714285__ ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s with respect to more generators:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.&amp;lt;u&amp;gt;36&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.&amp;lt;u&amp;gt;09&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.39131&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46\&amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.64567&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.93701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.89764&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.44828&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.03848&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;59\&amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.35583&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.06748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.0859&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;38\&amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.28571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;55\&amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.21053&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.63158&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23.68421&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17\&amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.04255&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.12766&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.53191&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64\&amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.89831&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.69492&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27.12965&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;47\&amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.84615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.53846&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27.6923&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;77\&amp;lt;a class="wiki_link" href="/213edo"&amp;gt;213&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.80282&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.40845&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.16902&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30\&amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.73494&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.20842&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.91566&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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        &amp;lt;td&amp;gt;31\&amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.55814&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;97.67442&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;41.86047&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;71\&amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.48731&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;42.63959&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;40\&amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.&amp;lt;u&amp;gt;432&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;97.&amp;lt;u&amp;gt;297&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;43.&amp;lt;u&amp;gt;135&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;49\&amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.35294&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;97.05882&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;44.11765&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;58\&amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.29814&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;96.89441&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;44.72051&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;9\&amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;59\&amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.&amp;lt;u&amp;gt;70731&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;95.&amp;lt;u&amp;gt;12195&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;51.&amp;lt;u&amp;gt;21951&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;50\&amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.65468&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;94.96403&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;51.79856&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;41\&amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;52.63158&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;53.93258&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;87\&amp;lt;a class="wiki_link" href="/242edo"&amp;gt;242&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.40496&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;94.21488&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;54.54:&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;55\&amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.37255&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.11765&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;54.90196&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;78\&amp;lt;a class="wiki_link" href="/217edo"&amp;gt;217&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.33641&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.00922&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;55.29955&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23\&amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;431.25&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>64z5xqgyvzrej65l3w4rzu352xr81ht</sha1>
    </revision>
    <revision>
      <id>4477</id>
      <timestamp>2012-05-04T14:53:21Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 330149350 - Original comment: **</comment>
      <origin>4477</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="55724" sha1="2j918j5f2tarf350wgkp89hexgzycm6" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-05-04 14:53:21 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;330149350&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

**4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]**
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
**4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]**

5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]
5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]

The next table below shows an extension of ED2s with respect to more generators:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   ||   || 436.__36__ || 109.__09__ || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] ||   || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   ||   || 434.78261 || 104.34783 || 17.39131 ||   ||
||   ||   ||   ||   ||   ||   || 46\[[127edo|127]] ||   || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   ||   || 434.48276 || 103.44828 || 22.03848 ||   ||
||   ||   ||   ||   ||   ||   || 59\[[163edo|163]] ||   || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   || 38\[[105edo|105]] ||   ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   || 55\[[152edo|152]] ||   || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   || 64\[[177edo|177]] ||   || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   || 47\[[130edo|130]] ||   ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   || 77\[[213edo|213]] ||   || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] ||   || 433.__6633__ || 100.__9900__ || 29.__7030__ ||   ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   || 56\[[155edo|155]] ||   || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   ||   || 433.__3__ || 100 || 33.__3__ ||   ||
||   ||   ||   ||   ||   ||   || 61\[[169edo|169]] ||   || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   || 48\[[133edo|133]] ||   ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[230edo|230]] ||   || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   || 35\[[97edo|97]] ||   ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   || 92\[[255edo|255]] ||   || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   || 57\[[158edo|158]] ||   ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   || 79\[[219edo|219]] ||   || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   || 75\[[208edo|208]] ||   || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   || 53\[[147edo|147]] ||   ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   || 84\[[233edo|233]] ||   || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   || 31\[[86edo|86]] ||   ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[197edo|197]] ||   || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   || 40\[[111edo|111]] ||   ||   || 432.__432__ || 97.__297__ || 43.__135__ ||   ||
||   ||   ||   ||   ||   ||   || 49\[[136edo|136]] ||   || 432.35294 || 97.05882 || 44.11765 ||   ||
||   ||   ||   ||   ||   ||   ||   || 58/161 || 432.29814 || 96.89441 || 44.72051 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   ||   || 59/164 || 431.__70731__ || 95.__12195__ || 51.__21951__ ||   ||
||   ||   ||   ||   ||   ||   || 50\[[139edo|139]] ||   || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   || 41\[[114edo|114]] ||   ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[203edo|203]] ||   || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   || 32\[[89edo|89]] ||   ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   || 87\[[242edo|242]] ||   || 431.40496 || 94.21488 || 54.54: ||   ||
||   ||   ||   ||   ||   || 55\[[153edo|153]] ||   ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   || 78\[[217edo|217]] ||   || 431.33641 || 94.00922 || 55.29955 ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[231edo|231]] ||   || 431.16833 || 93.50649 || 57.14286 ||   ||
||   ||   ||   ||   ||   || 60\[[167edo|167]] ||   ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   || 97\[[270edo|270]] ||   || 431.__1__ || 93.__3__ || 57.__8__ ||   ||
||   ||   ||   ||   || 37\[[103edo|103]] ||   ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   || 88\[[245edo|245]] ||   || 431.02041 || 93.06122 || 58.77551 ||   ||
||   ||   ||   ||   ||   || 51\[[142edo|142]] ||   ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   || 65\[[181edo|181]] ||   || 430.93923 || 92.81768 || 59.66852 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   || 61\[[170edo|170]] ||   || 430.58824 || 91.76471 || 62.54942 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] ||   || 430.49327 || 91.47982 || 64.57505 ||   ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   || 85\[[237edo|237]] ||   || 430.37975 || 91.13924 || 65.82309 ||   ||
||   ||   ||   ||   ||   || 52\[[145edo|145]] ||   ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[198edo|198]] ||   || 430.__30__ || 90.__90__ || 66.__67__ ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   || 62\[[173edo|173]] ||   || 430.0578 || 90.17341 || 69.36416 ||   ||
||   ||   ||   ||   ||   || 43\[[120edo|120]] ||   ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   || 67\[[187edo|187]] ||   || 429.94652 || 89.83957 || 70.58824 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   || 53\[[148edo|148]] ||   || 429.__729__ || 89.__189__ || 72.__973__ ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   ||   || 429.__629__ || 88.__8__ || 74.__074__ ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] ||   || 429.47368 || 88.42105 || 75.78947 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   ||   || 428.57143 || 85.__714285__ || 85.__714285__ ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s with respect to more generators:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.&amp;lt;u&amp;gt;36&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.&amp;lt;u&amp;gt;09&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.39131&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46\&amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.64567&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.93701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.89764&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.44828&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.03848&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;59\&amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.35583&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.06748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.0859&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;38\&amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.28571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;55\&amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.21053&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.63158&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23.68421&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17\&amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.04255&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.12766&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.53191&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;75\&amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;40.38439&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;53\&amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;84\&amp;lt;a class="wiki_link" href="/233edo"&amp;gt;233&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;31\&amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
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      <sha1>2j918j5f2tarf350wgkp89hexgzycm6</sha1>
    </revision>
    <revision>
      <id>4478</id>
      <timestamp>2012-05-03T19:08:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 329674008 - Original comment: **</comment>
      <origin>4478</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="50699" sha1="9a84yjvx7wej9fs8pm3yln0prso388l" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-05-03 19:08:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;329674008&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

**4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]**
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
**4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]**

5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]
5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]

The next table below shows an extension of ED2s with respect to more generators:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   || 436.36: || 109.09 || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   || 434.78261 || 104.34783 || 17.39131 ||   ||
||   ||   ||   ||   ||   ||   || 46\[[127edo|127]] || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   || 434.48276 || 103.44828 || 22.03848 ||   ||
||   ||   ||   ||   ||   ||   || 59\[[163edo|163]] || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   || 38\[[105edo|105]] ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   || 55\[[152edo|152]] || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   || 64\[[177edo|177]] || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   || 47\[[130edo|130]] ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   || 77\[[213edo|213]] || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] || 433.6633: || 100.9900: || 29.7030: ||   ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   || 56\[[155edo|155]] || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   || 433.3: || 100 || 33.3: ||   ||
||   ||   ||   ||   ||   ||   || 61\[[169edo|169]] || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   || 48\[[133edo|133]] ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[230edo|230]] || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   || 35\[[97edo|97]] ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   || 92\[[255edo|255]] || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   || 57\[[158edo|158]] ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   || 79\[[219edo|219]] || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   || 75\[[208edo|208]] || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   || 53\[[147edo|147]] ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   || 84\[[233edo|233]] || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   || 31\[[86edo|86]] ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[197edo|197]] || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   || 40\[[111edo|111]] ||   || 432.432: || 97.297: || 43.135: ||   ||
||   ||   ||   ||   ||   ||   || 49\[[136edo|136]] || 432.35294 || 97.05882 || 44.11765 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   || 50\[[139edo|139]] || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   || 41\[[114edo|114]] ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[203edo|203]] || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   || 32\[[89edo|89]] ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   || 87\[[242edo|242]] || 431.40496 || 94.21488 || 54.54: ||   ||
||   ||   ||   ||   ||   || 55\[[153edo|153]] ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   || 78\[[217edo|217]] || 431.33641 || 94.00922 || 55.29955 ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[231edo|231]] || 431.16833 || 93.50649 || 57.14286 ||   ||
||   ||   ||   ||   ||   || 60\[[167edo|167]] ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   || 97\[[270edo|270]] || 431.1: || 93.3: || 57.8: ||   ||
||   ||   ||   ||   || 37\[[103edo|103]] ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   || 88\[[245edo|245]] || 431.02041 || 93.06122 || 58.77551 ||   ||
||   ||   ||   ||   ||   || 51\[[142edo|142]] ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   || 65\[[181edo|181]] || 430.93923 || 92.81768 || 59.66852 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   || 61\[[170edo|170]] || 430.58824 || 91.76471 || 62.54942 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   || 80\[[223edo|223]] || 430.49327 || 91.47982 || 64.57505 ||   ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   || 85\[[237edo|237]] || 430.37975 || 91.13924 || 65.82309 ||   ||
||   ||   ||   ||   ||   || 52\[[145edo|145]] ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[198edo|198]] || 430.30: || 90.90: || 66.67: ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   || 62\[[173edo|173]] || 430.0578 || 90.17341 || 69.36416 ||   ||
||   ||   ||   ||   ||   || 43\[[120edo|120]] ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   || 67\[[187edo|187]] || 429.94652 || 89.83957 || 70.58824 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   || 53\[[148edo|148]] || 429.729: || 89.189: || 72.973: ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   || 429.629: || 88.8: || 74.074: ||   ||
||   ||   ||   ||   ||   ||   || 34\[[95edo|95]] || 429.47368 || 88.42105 || 75.78947 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   || 428.57143 || 85.714285: || 85.714285: ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s with respect to more generators:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.39131&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46\&amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.64567&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.93701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.89764&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.44828&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.03848&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;59\&amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.35583&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.06748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.0859&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;432.91139&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;79\&amp;lt;a class="wiki_link" href="/219edo"&amp;gt;219&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.87671&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;98.63014&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;38.35636&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;22\&amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.78689&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;98.36066&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;39.34426&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;75\&amp;lt;a class="wiki_link" href="/208edo"&amp;gt;208&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.69231&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;98.07692&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;40.38439&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;53\&amp;lt;a class="wiki_link" href="/147edo"&amp;gt;147&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.65306&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;97.95918&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;40.81633&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;84\&amp;lt;a class="wiki_link" href="/233edo"&amp;gt;233&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.61803&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;97.85408&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;31\&amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.55814&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;41.86047&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;71\&amp;lt;a class="wiki_link" href="/197edo"&amp;gt;197&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.48731&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;40\&amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.432:&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;97.297:&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;43.135:&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;49\&amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.35294&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;44.11765&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;9\&amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;50\&amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;51.79856&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;41\&amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;32\&amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;87\&amp;lt;a class="wiki_link" href="/242edo"&amp;gt;242&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;55\&amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;429.729:&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;429.47368&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9a84yjvx7wej9fs8pm3yln0prso388l</sha1>
    </revision>
    <revision>
      <id>4479</id>
      <timestamp>2012-05-03T16:28:47Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 329604054 - Original comment: **</comment>
      <origin>4479</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="49813" sha1="4qsa8fsbm5mhwyiet0rwc9m7nd167ph" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-05-03 16:28:47 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;329604054&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

**4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]**
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
**4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]**

5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]
5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]

The next table below shows an extension of ED2s with respect to more generators:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   || 436.36: || 109.09 || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   || 434.78261 || 104.34783 || 17.39131 ||   ||
||   ||   ||   ||   ||   ||   || 46\[[127edo|127]] || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   || 434.48276 || 103.44828 || 22.03848 ||   ||
||   ||   ||   ||   ||   ||   || 59\[[163edo|163]] || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   || 38\[[105edo|105]] ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   || 55\[[152edo|152]] || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   || 64\[[177edo|177]] || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   || 47\[[130edo|130]] ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   || 77\[[213edo|213]] || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] || 433.6633: || 100.9900: || 29.7030: ||   ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   || 56\[[155edo|155]] || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   || 433.3: || 100 || 33.3: ||   ||
||   ||   ||   ||   ||   ||   || 61\[[169edo|169]] || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   || 48\[[133edo|133]] ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[230edo|230]] || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   || 35\[[97edo|97]] ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   || 92\[[255edo|255]] || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   || 57\[[158edo|158]] ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   || 79\[[219edo|219]] || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   || 75\[[208edo|208]] || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   || 53\[[147edo|147]] ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   || 84\[[233edo|233]] || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   || 31\[[86edo|86]] ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[197edo|197]] || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   || 40\[[111edo|111]] ||   || 432.432: || 97.297: || 43.135: ||   ||
||   ||   ||   ||   ||   ||   || 49\[[136edo|136]] || 432.35294 || 97.05882 || 44.11765 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   || 50\[[139edo|139]] || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   || 41\[[114edo|114]] ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[203edo|203]] || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   || 32\[[89edo|89]] ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   || 87\242 || 431.40496 || 94.21488 || 54.54: ||   ||
||   ||   ||   ||   ||   || 55\[[153edo|153]] ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   || 78\217 || 431.33641 || 94.00922 || 55.29955 ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   || 83\231 || 431.16833 || 93.50649 || 57.14286 ||   ||
||   ||   ||   ||   ||   || 60\[[167edo|167]] ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   || 97\270 || 431.1: || 93.3: || 57.8: ||   ||
||   ||   ||   ||   || 37\[[103edo|103]] ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   || 88\245 || 431.02041 || 93.06122 || 58.77551 ||   ||
||   ||   ||   ||   ||   || 51\[[142edo|142]] ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   || 65\181 || 430.93923 || 92.81768 || 59.66852 ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   || 61\170 || 430.58824 || 91.76471 || 62.54942 ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   || 80\223 || 430.49327 || 91.47982 || 64.57505 ||   ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   || 85\237 || 430.37975 || 91.13924 || 65.82309 ||   ||
||   ||   ||   ||   ||   || 52\[[145edo|145]] ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   || 71\198 || 430.30: || 90.90: || 66.67: ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   || 62\173 || 430.0578 || 90.17341 || 69.36416 ||   ||
||   ||   ||   ||   ||   || 43\[[120edo|120]] ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   || 67\187 || 429.94652 || 89.83957 || 70.58824 ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   || 53\148 || 429.729: || 89.189: || 72.973: ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   || 429.629: || 88.8: || 74.074: ||   ||
||   ||   ||   ||   ||   ||   || 34\95 || 429.47368 || 88.42105 || 75.78947 ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   || 428.57143 || 85.714285: || 85.714285: ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s with respect to more generators:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.39131&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46\&amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.64567&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.93701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.89764&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.44828&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.03848&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;59\&amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.35583&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.06748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.0859&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;38\&amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.28571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;55\&amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.21053&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.63158&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23.68421&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17\&amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.04255&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.12766&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.53191&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64\&amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.89831&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.69492&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27.12965&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;24\&amp;lt;a class="wiki_link" href="/67edo"&amp;gt;67&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;429.85075&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;89.55224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71.64179&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;53\148&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.729:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;89.189:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;72.973:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/81edo"&amp;gt;81&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;429.629:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;88.8:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;74.074:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;34\95&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;429.47368&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;88.42105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;75.78947&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5\&amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;428.57143&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714285:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;85.714285:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>4qsa8fsbm5mhwyiet0rwc9m7nd167ph</sha1>
    </revision>
    <revision>
      <id>4480</id>
      <timestamp>2012-05-03T00:00:20Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 329102226 - Original comment: **</comment>
      <origin>4480</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="49051" sha1="e9mn8t3if6788aecu3sypzjj5159rat" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-05-03 00:00:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;329102226&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)**
3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)**

**4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]**
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
**4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]**

5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]
5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]

The next table below shows an extension of ED2s with respect to more generators:
|| 4\[[11edo|11]] ||   ||   ||   ||   ||   ||   || 436.36: || 109.09 || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29\[[80edo|80]] || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25\[[69edo|69]] ||   || 434.78261 || 104.34783 || 17.39131 ||   ||
||   ||   ||   ||   ||   ||   || 46\[[127edo|127]] || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   || 21\[[58edo|58]] ||   ||   || 434.48276 || 103.44828 || 22.03848 ||   ||
||   ||   ||   ||   ||   ||   || 59\[[163edo|163]] || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   || 38\[[105edo|105]] ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   || 55\[[152edo|152]] || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   || 17\[[47edo|47]] ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   || 64\[[177edo|177]] || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   || 47\[[130edo|130]] ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   || 77\[[213edo|213]] || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   || 30\[[83edo|83]] ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[202edo|202]] || 433.6633: || 100.9900: || 29.7030: ||   ||
||   ||   ||   ||   ||   || 43\[[119edo|119]] ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   || 56\[[155edo|155]] || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   || 13\[[36edo|36]] ||   ||   ||   ||   || 433.3: || 100 || 33.3: ||   ||
||   ||   ||   ||   ||   ||   || 61\[[169edo|169]] || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   || 48\[[133edo|133]] ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   || 83\[[230edo|230]] || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   || 35\[[97edo|97]] ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   || 92\[[255edo|255]] || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   || 57\[[158edo|158]] ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   || 79\[[219edo|219]] || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   || 22\[[61edo|61]] ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   || 75\[[208edo|208]] || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   || 53\[[147edo|147]] ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   || 84\[[233edo|233]] || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   || 31\[[86edo|86]] ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   || 71\[[197edo|197]] || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   || 40\[[111edo|111]] ||   || 432.432: || 97.297: || 43.135: ||   ||
||   ||   ||   ||   ||   ||   || 49\[[136edo|136]] || 432.35294 || 97.05882 || 44.11765 ||   ||
||   || 9\[[25edo|25]] ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   || 50\[[139edo|139]] || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   || 41\[[114edo|114]] ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   || 73\[[203edo|203]] || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   || 32\[[89edo|89]] ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 55\[[153edo|153]] ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   || 23\[[64edo|64]] ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 60\[[167edo|167]] ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   || 37\[[103edo|103]] ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 51\[[142edo|142]] ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   || 14\[[39edo|39]] ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 47\[[131edo|131]] ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   || 33\[[92edo|92]] ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 52\[[145edo|145]] ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   || 19\[[53edo|53]] ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 43\[[120edo|120]] ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   || 24\[[67edo|67]] ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 29\[[81edo|81]] ||   || 429.629: || 88.8: || 74.074: ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
|| 5\[[14edo|14]] ||   ||   ||   ||   ||   ||   || 428.57143 || 85.714285: || 85.714285: ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt; (Middle range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt; (Lufsur range)&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt; (Fuslur range)&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The next table below shows an extension of ED2s with respect to more generators:&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4\&amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29\&amp;lt;a class="wiki_link" href="/80edo"&amp;gt;80&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25\&amp;lt;a class="wiki_link" href="/69edo"&amp;gt;69&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.39131&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46\&amp;lt;a class="wiki_link" href="/127edo"&amp;gt;127&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.64567&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.93701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.89764&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21\&amp;lt;a class="wiki_link" href="/58edo"&amp;gt;58&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.44828&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.03848&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;59\&amp;lt;a class="wiki_link" href="/163edo"&amp;gt;163&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.35583&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.06748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.0859&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;38\&amp;lt;a class="wiki_link" href="/105edo"&amp;gt;105&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.28571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;55\&amp;lt;a class="wiki_link" href="/152edo"&amp;gt;152&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.21053&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.63158&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23.68421&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17\&amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.04255&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.12766&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.53191&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;64\&amp;lt;a class="wiki_link" href="/177edo"&amp;gt;177&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.89831&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.69492&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27.12965&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;47\&amp;lt;a class="wiki_link" href="/130edo"&amp;gt;130&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.84615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.53846&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27.6923&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;77\&amp;lt;a class="wiki_link" href="/213edo"&amp;gt;213&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.80282&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.40845&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.16902&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30\&amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.73494&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.20842&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.91566&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/202edo"&amp;gt;202&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.6633:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.9900:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29.7030:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43\&amp;lt;a class="wiki_link" href="/119edo"&amp;gt;119&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.61345&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.84034&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30.2521&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;56\&amp;lt;a class="wiki_link" href="/155edo"&amp;gt;155&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.54839&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.64516&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;9\&amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;50\&amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.65468&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;94.96403&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;51.79856&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;41\&amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.57895&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;94.73684&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;52.63158&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;73\&amp;lt;a class="wiki_link" href="/203edo"&amp;gt;203&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.52709&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;53.20197&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;32\&amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.46067&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;55\&amp;lt;a class="wiki_link" href="/153edo"&amp;gt;153&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.37255&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;94.11765&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;54.90196&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;23\&amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.25&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;60\&amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.13772&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;93.41317&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;57.48503&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;37\&amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;431.06796&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;51\&amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
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      <sha1>e9mn8t3if6788aecu3sypzjj5159rat</sha1>
    </revision>
    <revision>
      <id>4481</id>
      <timestamp>2012-05-02T21:47:54Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 329048920 - Original comment: **</comment>
      <origin>4481</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45606" sha1="iusbf2khsnqtv5hemifubmeeny5p3tf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-05-02 21:47:54 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;329048920&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]]**
3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]]
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]]**

**4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]**
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
**4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]**

5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]
5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]

|| 4/11 ||   ||   ||   ||   ||   ||   || 436.36: || 109.09 || 0 ||=   ||
||   ||   ||   ||   ||   ||   || 29/80 || 435 || 105 || 15 ||   ||
||   ||   ||   ||   ||   || 25/69 ||   || 434.78261 || 104.34783 || 17.39131 ||   ||
||   ||   ||   ||   ||   ||   || 46/127 || 434.64567 || 103.93701 || 18.89764 ||   ||
||   ||   ||   ||   || 21/58 ||   ||   || 434.48276 || 103.44828 || 22.03848 ||   ||
||   ||   ||   ||   ||   ||   || 59/163 || 434.35583 || 103.06748 || 22.0859 ||   ||
||   ||   ||   ||   ||   || 38/105 ||   || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   ||   ||   ||   || 55/152 || 434.21053 || 102.63158 || 23.68421 ||   ||
||   ||   ||   || 17/47 ||   ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   ||   || 64/177 || 433.89831 || 101.69492 || 27.12965 ||   ||
||   ||   ||   ||   ||   || 47/130 ||   || 433.84615 || 101.53846 || 27.6923 ||   ||
||   ||   ||   ||   ||   ||   || 77/213 || 433.80282 || 101.40845 || 28.16902 ||   ||
||   ||   ||   ||   || 30/83 ||   ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   ||   || 73/202 || 433.6633: || 100.9900: || 29.7030: ||   ||
||   ||   ||   ||   ||   || 43/119 ||   || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   ||   ||   ||   ||   || 56/155 || 433.54839 || 100.64516 || 30.96775 ||   ||
||   ||   || 13/36 ||   ||   ||   ||   || 433.3: || 100 || 33.3: ||   ||
||   ||   ||   ||   ||   ||   || 61/169 || 433.13609 || 99.40828 || 35.50295 ||   ||
||   ||   ||   ||   ||   || 48/133 ||   || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   ||   ||   || 83/230 || 433.04348 || 99.13043 || 36.52175 ||   ||
||   ||   ||   ||   || 35/97 ||   ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   ||   || 92/255 || 432.94118 || 98.82353 || 37.64702 ||   ||
||   ||   ||   ||   ||   || 57/158 ||   || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   ||   ||   ||   || 79/219 || 432.87671 || 98.63014 || 38.35636 ||   ||
||   ||   ||   || 22/61 ||   ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   ||   || 75/208 || 432.69231 || 98.07692 || 40.38439 ||   ||
||   ||   ||   ||   ||   || 53/147 ||   || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   ||   ||   || 84/233 || 432.61803 || 97.85408 || 41.20172 ||   ||
||   ||   ||   ||   || 31/86 ||   ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   ||   || 71/197 || 432.48731 || 97.46193 || 42.63959 ||   ||
||   ||   ||   ||   ||   || 40/111 ||   || 432.432: || 97.297: || 43.135: ||   ||
||   ||   ||   ||   ||   ||   || 49/136 || 432.35294 || 97.05882 || 44.11765 ||   ||
||   || 9/25 ||   ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   ||   || 50/139 || 431.65468 || 94.96403 || 51.79856 ||   ||
||   ||   ||   ||   ||   || 41/114 ||   || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   ||   ||   || 73/203 || 431.52709 || 94.58128 || 53.20197 ||   ||
||   ||   ||   ||   || 32/89 ||   ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 55/153 ||   || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   || 23/64 ||   ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 60/167 ||   || 431.13772 || 93.41317 || 57.48503 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   || 37/103 ||   ||   || 431.06796 || 93.20388 || 58.25263 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 51/142 ||   || 430.98592 || 92.95775 || 59.15493 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   || 14/39 ||   ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 47/131 ||   || 430.53435 || 91.60305 || 64.12214 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   || 33/92 ||   ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 52/145 ||   || 430.34483 || 91.03448 || 66.2069 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   || 19/53 ||   ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 43/120 ||   || 430 || 90 || 70 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   || 24/67 ||   ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 29/81 ||   || 429.629: || 88.8: || 74.074: ||   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
|| 5/14 ||   ||   ||   ||   ||   ||   || 428.57143 || 85.714285: || 85.714285: ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;435&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25/69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.39131&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46/127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.64567&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.93701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;18.89764&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21/58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.44828&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.03848&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;59/163&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.35583&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.06748&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.0859&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;38.35636&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;22/61&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;432.78689&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.36066&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;39.34426&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;75/208&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.69231&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.07692&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;40.38439&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;53/147&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.65306&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97.95918&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;40.81633&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;84/233&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.61803&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97.85408&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;41.20172&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;31/86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.55814&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97.67442&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;41.86047&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;71/197&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.48731&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97.46193&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;42.63959&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;40/111&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.432:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97.297:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43.135:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/136&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.35294&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97.05882&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;44.11765&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;50/139&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.65468&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.96403&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;51.79856&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;41/114&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.57895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.73684&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;52.63158&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;73/203&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.52709&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.58128&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;53.20197&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;32/89&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.46067&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.38202&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;53.93258&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;55/153&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.37255&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.11765&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;54.90196&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23/64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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      <sha1>iusbf2khsnqtv5hemifubmeeny5p3tf</sha1>
    </revision>
    <revision>
      <id>4482</id>
      <timestamp>2012-05-02T13:45:14Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 328787586 - Original comment: **</comment>
      <origin>4482</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="24774" sha1="qpxd4ny15kfxyy79n2m0pcg1jz51d8p" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2012-05-02 13:45:14 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;328787586&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]]**
3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]]
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]]**

**4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]**
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
**4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]**

5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]
5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]

|| 4/11 ||   ||   ||   ||   ||   || 436.36: || 109.09 || 0 ||=   ||
||   ||   ||   ||   ||   || 25/69 || 434.78261 || 104.34783 || 17.39131 ||   ||
||   ||   ||   ||   || 21/58 ||   || 434.48276 || 103.44828 || 22.03848 ||   ||
||   ||   ||   ||   ||   || 38/105 || 434.28571 || 102.85714 || 22.85714 ||   ||
||   ||   ||   || 17/47 ||   ||   || 434.04255 || 102.12766 || 25.53191 ||   ||
||   ||   ||   ||   ||   || 47/130 || 433.84615 || 101.53846 || 28.6938 ||   ||
||   ||   ||   ||   || 30/83 ||   || 433.73494 || 101.20842 || 28.91566 ||   ||
||   ||   ||   ||   ||   || 43/119 || 433.61345 || 100.84034 || 30.2521 ||   ||
||   ||   || 13/36 ||   ||   ||   || 433.3: || 100 || 33.3: ||   ||
||   ||   ||   ||   ||   || 48/133 || 433.08271 || 99.24812 || 36.09023 ||   ||
||   ||   ||   ||   || 35/97 ||   || 432.98969 || 98.96907 || 37.1134 ||   ||
||   ||   ||   ||   ||   || 57/158 || 432.91139 || 98.73418 || 37.97468 ||   ||
||   ||   ||   || 22/61 ||   ||   || 432.78689 || 98.36066 || 39.34426 ||   ||
||   ||   ||   ||   ||   || 53/147 || 432.65306 || 97.95918 || 40.81633 ||   ||
||   ||   ||   ||   || 31/86 ||   || 432.55814 || 97.67442 || 41.86047 ||   ||
||   ||   ||   ||   ||   || 40/111 || 432.432: || 97.297: || 43.135: ||   ||
||   || 9/25 ||   ||   ||   ||   || 432 || 96 || 48 ||= Boundary of propriety;
generators smaller than this are proper ||
||   ||   ||   ||   ||   || 41/114 || 431.57895 || 94.73684 || 52.63158 ||   ||
||   ||   ||   ||   || 32/89 ||   || 431.46067 || 94.38202 || 53.93258 ||   ||
||   ||   ||   ||   ||   || 55/153 || 431.37255 || 94.11765 || 54.90196 ||   ||
||   ||   ||   || 23/64 ||   ||   || 431.25 || 93.75 || 56.25 ||   ||
||   ||   ||   ||   ||   || 60/167 ||   ||   ||   ||   ||
||   ||   ||   ||   || 37/103 ||   || 431.06796 || 93.20388 || 59.25263 ||   ||
||   ||   ||   ||   ||   || 51/142 ||   ||   ||   ||   ||
||   ||   || 14/39 ||   ||   ||   || 430.76923 || 92.30769 || 61.53846 ||   ||
||   ||   ||   ||   ||   || 47/131 ||   ||   ||   ||   ||
||   ||   ||   ||   || 33/92 ||   || 430.43478 || 91.30435 || 65.21739 ||   ||
||   ||   ||   ||   ||   || 52/145 ||   ||   ||   ||   ||
||   ||   ||   || 19/53 ||   ||   || 430.18868 || 90.56604 || 67.92453 ||   ||
||   ||   ||   ||   ||   || 43/120 ||   ||   ||   ||   ||
||   ||   ||   ||   || 24/67 ||   || 429.85075 || 89.55224 || 71.64179 ||   ||
||   ||   ||   ||   ||   || 29/81 ||   ||   ||   ||   ||
|| 5/14 ||   ||   ||   ||   ||   || 428.57143 || 85.714285: || 85.714285: ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;436.36:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109.09&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25/69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.78261&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.34783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.39131&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;21/58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.48276&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.44828&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.03848&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;38/105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.28571&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22.85714&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;17/47&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;434.04255&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.12766&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25.53191&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;47/130&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.84615&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.53846&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.6938&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30/83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.73494&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;101.20842&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;28.91566&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43/119&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.61345&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100.84034&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;30.2521&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/36&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.3:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;33.3:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48/133&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;433.08271&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;99.24812&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;36.09023&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;35/97&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.98969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.96907&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;37.1134&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;57/158&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.91139&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.73418&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;37.97468&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22/61&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.78689&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;98.36066&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;39.34426&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;53/147&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.65306&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97.95918&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;40.81633&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;31/86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.55814&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97.67442&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;41.86047&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;40/111&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432.432:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97.297:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;43.135:&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;432&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Boundary of propriety;&amp;lt;br /&amp;gt;
generators smaller than this are proper&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;41/114&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.57895&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.73684&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;52.63158&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;32/89&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.46067&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.38202&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;53.93258&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;55/153&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.37255&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.11765&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;54.90196&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23/64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;431.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;56.25&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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      <sha1>qpxd4ny15kfxyy79n2m0pcg1jz51d8p</sha1>
    </revision>
    <revision>
      <id>4483</id>
      <timestamp>2012-04-30T03:40:32Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 327197824 - Original comment: **</comment>
      <origin>4483</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6026" sha1="qpape1q4y0wagbk61fswnpqnecjhhwz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-04-30 03:40:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;327197824&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]]**
3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]]
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]]**

**4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]**
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
**4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]**

5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]
5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this scheme: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>qpape1q4y0wagbk61fswnpqnecjhhwz</sha1>
    </revision>
    <revision>
      <id>4484</id>
      <timestamp>2012-04-30T03:32:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 327194864 - Original comment: **</comment>
      <origin>4484</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6022" sha1="65v5ogaqwkfurktnbwmqzvuftbpy1hd" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-04-30 03:32:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;327194864&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a type of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this form: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]]**
3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]]
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]]**

**4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]**
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
**4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]**

5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]
5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a type of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this form: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>65v5ogaqwkfurktnbwmqzvuftbpy1hd</sha1>
    </revision>
    <revision>
      <id>4485</id>
      <timestamp>2012-04-30T02:54:53Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 327182974 - Original comment: **</comment>
      <origin>4485</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6022" sha1="1wwr1uax6aytwzouuh87lfj3t15yg4h" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-04-30 02:54:53 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;327182974&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a kind of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this form: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]]**
3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]]
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]]**

**4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]**
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
**4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]**

5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]
5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]
13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a kind of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this form: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>1wwr1uax6aytwzouuh87lfj3t15yg4h</sha1>
    </revision>
    <revision>
      <id>4486</id>
      <timestamp>2012-04-30T02:53:02Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 327182380 - Original comment: **</comment>
      <origin>4486</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6220" sha1="cc16rd3z46knf0jeby9v72w7c8i40cu" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2012-04-30 02:53:02 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;327182380&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;= 

This is a kind of scale which denotes the use of a scale placed between [[11edo|11]] and [[14edo|14]] ED2's, employing a ratio generator between 41/32 ~ 9/7 (being [[25edo|25-ED2]] the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this form: LLLsLLLLsLLLLs.

__**ED2s that contains this scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]]**
3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]]
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]]**

**4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]**
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]
**4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]**

**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]]**
5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]
**7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]**
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

**9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]**
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]
**11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]**
11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

**13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]]**
**13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]**
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]
13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="The Ketradektriatoh Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;The Ketradektriatoh Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This is a kind of scale which denotes the use of a scale placed between &amp;lt;a class="wiki_link" href="/11edo"&amp;gt;11&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/14edo"&amp;gt;14&amp;lt;/a&amp;gt; ED2's, employing a ratio generator between 41/32 ~ 9/7 (being &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25-ED2&amp;lt;/a&amp;gt; the middle size of the Ketradektriatoh spectrum, in the 2;1 relation), resulting in a variant of tetradecatonic scale comforms by this form: LLLsLLLLsLLLLs.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;ED2s that contains this scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 7 11 11 11 11 7 11 11 11 11 7: &amp;lt;a class="wiki_link" href="/142edo"&amp;gt;142&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 8 13 13 13 13 8 13 13 13 13 8: &amp;lt;a class="wiki_link" href="/167edo"&amp;gt;167&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>cc16rd3z46knf0jeby9v72w7c8i40cu</sha1>
    </revision>
    <revision>
      <id>4487</id>
      <timestamp>2011-04-18T01:26:20Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 221061500 - Original comment: **</comment>
      <origin>4487</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6298" sha1="o6a1mjx9h6sabo7854ny6miz9hi1bd2" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2011-04-18 01:26:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;221061500&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;[WITNOTS] Scale&amp;lt;/span&amp;gt;= 

Can called too, the Tetradecimal Triatonic scale: L L L s L L L L s L L L L s.

__**EDOs that contains this 'special' scale:**__

**2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo]]**
3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo]]
**3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo]]**

**4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo]]**
4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo]]
**4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo]]**

**5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo]]**
5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo]]

6 6 6 2 6 6 6 6 2 6 6 6 6 2: [[72edo]]
**6 6 6 3 6 6 6 6 3 6 6 6 6 3: [[75edo]]**
6 6 6 4 6 6 6 6 4 6 6 6 6 4: [[78edo]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo]]
7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo]]
**7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo]]**
7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo]]

8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo]]
8 8 8 4 8 8 8 8 4 8 8 8 8 4: [[100edo]]
**8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo]]**

9 9 9 3 9 9 9 9 3 9 9 9 9 3: [[108edo]]
**9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo]]**
9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo]]
11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo]]
**11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo]]**

**13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo]]**
**13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo]]**
13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 3s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x[WITNOTS] Scale"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;&amp;lt;span style="color: #800061; font-size: 103%;"&amp;gt;[WITNOTS] Scale&amp;lt;/span&amp;gt;&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
Can called too, the Tetradecimal Triatonic scale: L L L s L L L L s L L L L s.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;&amp;lt;strong&amp;gt;EDOs that contains this 'special' scale:&amp;lt;/strong&amp;gt;&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;2 2 2 1 2 2 2 2 1 2 2 2 2 1: &amp;lt;a class="wiki_link" href="/25edo"&amp;gt;25edo&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
3 3 3 1 3 3 3 3 1 3 3 3 3 1: &amp;lt;a class="wiki_link" href="/36edo"&amp;gt;36edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;3 3 3 2 3 3 3 3 2 3 3 3 3 2: &amp;lt;a class="wiki_link" href="/39edo"&amp;gt;39edo&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 1 4 4 4 4 1 4 4 4 4 1: &amp;lt;a class="wiki_link" href="/47edo"&amp;gt;47edo&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
4 4 4 2 4 4 4 4 2 4 4 4 4 2: &amp;lt;a class="wiki_link" href="/50edo"&amp;gt;50edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;4 4 4 3 4 4 4 4 3 4 4 4 4 3: &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53edo&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;5 5 5 2 5 5 5 5 2 5 5 5 5 2: &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61edo&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
5 5 5 3 5 5 5 5 3 5 5 5 5 3: &amp;lt;a class="wiki_link" href="/64edo"&amp;gt;64edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
6 6 6 2 6 6 6 6 2 6 6 6 6 2: &amp;lt;a class="wiki_link" href="/72edo"&amp;gt;72edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;6 6 6 3 6 6 6 6 3 6 6 6 6 3: &amp;lt;a class="wiki_link" href="/75edo"&amp;gt;75edo&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
6 6 6 4 6 6 6 6 4 6 6 6 6 4: &amp;lt;a class="wiki_link" href="/78edo"&amp;gt;78edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
7 7 7 2 7 7 7 7 2 7 7 7 7 2: &amp;lt;a class="wiki_link" href="/83edo"&amp;gt;83edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 3 7 7 7 7 3 7 7 7 7 3: &amp;lt;a class="wiki_link" href="/86edo"&amp;gt;86edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;7 7 7 4 7 7 7 7 4 7 7 7 7 4: &amp;lt;a class="wiki_link" href="/89edo"&amp;gt;89edo&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
7 7 7 5 7 7 7 7 5 7 7 7 7 5: &amp;lt;a class="wiki_link" href="/92edo"&amp;gt;92edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
8 8 8 3 8 8 8 8 3 8 8 8 8 3: &amp;lt;a class="wiki_link" href="/97edo"&amp;gt;97edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
8 8 8 4 8 8 8 8 4 8 8 8 8 4: &amp;lt;a class="wiki_link" href="/100edo"&amp;gt;100edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;8 8 8 5 8 8 8 8 5 8 8 8 8 5: &amp;lt;a class="wiki_link" href="/103edo"&amp;gt;103edo&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
9 9 9 3 9 9 9 9 3 9 9 9 9 3: &amp;lt;a class="wiki_link" href="/108edo"&amp;gt;108edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;9 9 9 4 9 9 9 9 4 9 9 9 9 4: &amp;lt;a class="wiki_link" href="/111edo"&amp;gt;111edo&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
9 9 9 5 9 9 9 9 5 9 9 9 9 5: &amp;lt;a class="wiki_link" href="/114edo"&amp;gt;114edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
11 11 11 4 11 11 11 11 4 11 11 11 11 4: &amp;lt;a class="wiki_link" href="/133edo"&amp;gt;133edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
11 11 11 5 11 11 11 11 5 11 11 11 11 5: &amp;lt;a class="wiki_link" href="/136edo"&amp;gt;136edo&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;11 11 11 6 11 11 11 11 6 11 11 11 11 6: &amp;lt;a class="wiki_link" href="/139edo"&amp;gt;139edo&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 5 13 13 13 13 5 13 13 13 13 5: &amp;lt;a class="wiki_link" href="/158edo"&amp;gt;158edo&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;13 13 13 6 13 13 13 13 6 13 13 13 13 6: &amp;lt;a class="wiki_link" href="/161edo"&amp;gt;161edo&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
13 13 13 7 13 13 13 13 7 13 13 13 13 7: &amp;lt;a class="wiki_link" href="/164edo"&amp;gt;164edo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>o6a1mjx9h6sabo7854ny6miz9hi1bd2</sha1>
    </revision>
    <revision>
      <id>94397</id>
      <parentid>95</parentid>
      <timestamp>2022-08-16T03:57:58Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox, misc. edits</comment>
      <origin>94397</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8467" sha1="dy4ul02ujumpa1qmakv5vr1nd1iru1m" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Paucitonic = 4
| Pattern = LLLLsLLLLsLLLs
}}
The '''11L 3s''' [[MOS scale]] was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]].

This is a type of scale which denotes the use of a scale placed between [[11edo]] and [[14edo]], employing a ratio generator between 41/32 ~ 9/7 ([[25edo]] being the middle size of the Ketradektriatoh spectrum, in the 2:1 relation), resulting in a variant of tetradecatonic scale which conforms by this scheme: LLLLsLLLLsLLLs.

== Edos that contains this scale ==
'''2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)'''

'''3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)'''

'''3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)'''

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]

4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]

4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]

'''5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ'''

'''5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]''' '''φ'''

5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]

6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]

7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]

7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]

7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]

'''8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ'''

'''8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]''' '''φ'''

8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]

9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]

9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]

9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]

10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]

10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]

10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''' 11 11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''' 11 11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''': [[124edo|124]]

11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]

11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]

11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]

11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]

11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]

11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]

11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]

12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]

12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]

12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]

13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]

13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]

13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]

'''13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ'''

13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]

13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]

'''13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]''' '''φ'''

13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]

13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]

13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]

13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]

14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]

14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]

14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]

14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]

14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]

15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]

15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]

15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]

15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]

15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]

15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]

15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]

16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]

16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]

16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]

16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]

16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]

16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]

16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]

17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]

17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]

17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]

'''17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)'''

'''17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]'''

'''17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]'''

'''17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]'''

'''17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]'''

'''17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]'''

'''17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]'''

'''17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)'''

17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]

17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]

17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]

17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

== Scale tree ==
The next table below shows an extension of [[edo]]s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
{| class="wikitable"
|-
| 4\[[11edo|11]]
| 
| 
| 
| 
| 
| 
| 436.364
| 109.091
| 0
| style="text-align:center;" | 
|-
| 
| 
| 
| 
| 
| 
| 29\[[80edo|80]]
| 435
| 105
| 15
| 
|-
| 
| 
| 
| 
| 
| 25\[[69edo|69]]
| 
| 434.783
| 104.348
| 17.391
| 
|-
| 
| 
| 
| 
| 21\[[58edo|58]]
| 
| 
| 434.483
| 103.448
| 20.69
| 
|-
| 
| 
| 
| 17\[[47edo|47]]
| 
| 
| 
| 434.043
| 102.128
| 25.532
| 
|-
| 
| 
| 
| 
| 30\[[83edo|83]]
| 
| 
| 433.735
| 101.208
| 28.916
| 
|-
| 
| 
| 
| 
| 
| 
| 73\[[202edo|202]]
| 433.663
| 100.990
| 29.703
| Since here are the optimal range Lufsur mode (?)
|-
| 
| 
| 
| 
| 
| 43\[[119edo|119]]
| 
| 433.613
| 100.840
| 30.252
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.459
| 100.377
| 31.95
| 
|-
| 
| 
| 13\[[36edo|36]]
| 
| 
| 
| 
| 433.333
| 100
| 33.333
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.048
| 99.144
| 36.473
| 
|-
| 
| 
| 
| 
| 35\97
| 
| 
| 432.99
| 98.969
| 37.113
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 432.933
| 98.799
| 37.738
| 
|-
| 
| 
| 
| 22\[[61edo|61]]
| 
| 
| 
| 432.787
| 98.361
| 39.344
| 
|-
| 
| 9\[[25edo|25]]
| 
| 
| 
| 
| 
| 432
| 96
| 48
| style="text-align:center;" | Boundary of propriety;

generators smaller than this are proper
|-
| 
| 
| 
| 
| 
| 
| 
| 431.417
| 94.25
| 54.4155
| 
|-
| 
| 
| 
| 23\[[64edo|64]]
| 
| 
| 
| 431.25
| 93.75
| 56.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 431.1185
| 93.355
| 57.697
| 
|-
| 
| 
| 
| 
| 37\103
| 
| 
| 431.068
| 93.204
| 58.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 430.984
| 92.952
| 58.175
| 
|-
| 
| 
| 14\[[39edo|39]]
| 
| 
| 
| 
| 430.769
| 92.308
| 61.538
| 
|-
| 
| 
| 
| 
| 
| 47\[[131edo|131]]
| 
| 430.534
| 91.603
| 64.122
| 
|-
| 
| 
| 
| 
| 
| 
| 80\[[223edo|223]]
| 430.493
| 91.480
| 64.575
| Until here are the optimal range Fuslur mode (?)
|-
| 
| 
| 
| 
| 33\[[92edo|92]]
| 
| 
| 430.435
| 91.304
| 65.217
| 
|-
| 
| 
| 
| 19\[[53edo|53]]
| 
| 
| 
| 430.189
| 90.566
| 67.925
| 
|-
| 
| 
| 
| 
| 24\[[67edo|67]]
| 
| 
| 429.851
| 89.552
| 71.642
| 
|-
| 
| 
| 
| 
| 
| 29\[[81edo|81]]
| 
| 429.63
| 88.889
| 74.074
| 
|-
| 
| 
| 
| 
| 
| 
| 34\[[95edo|95]]
| 429.474
| 88.421
| 75.7895
| 
|-
| 5\[[14edo|14]]
| 
| 
| 
| 
| 
| 
| 428.571
| 85.714
| 85.714
| style="text-align:center;" | 
|}

{{todo|cleanup}}</text>
      <sha1>dy4ul02ujumpa1qmakv5vr1nd1iru1m</sha1>
    </revision>
    <revision>
      <id>94509</id>
      <parentid>94397</parentid>
      <timestamp>2022-08-17T09:20:54Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94509</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8466" sha1="slc525svyxxg24xdiegwrnq26b44ivn" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
The '''11L 3s''' [[MOS scale]] was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]].

This is a type of scale which denotes the use of a scale placed between [[11edo]] and [[14edo]], employing a ratio generator between 41/32 ~ 9/7 ([[25edo]] being the middle size of the Ketradektriatoh spectrum, in the 2:1 relation), resulting in a variant of tetradecatonic scale which conforms by this scheme: LLLLsLLLLsLLLs.

== Edos that contains this scale ==
'''2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)'''

'''3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)'''

'''3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)'''

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]

4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]

4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]

'''5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ'''

'''5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]''' '''φ'''

5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]

6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]

7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]

7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]

7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]

'''8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ'''

'''8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]''' '''φ'''

8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]

9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]

9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]

9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]

10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]

10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]

10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''' 11 11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''' 11 11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''': [[124edo|124]]

11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]

11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]

11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]

11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]

11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]

11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]

11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]

12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]

12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]

12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]

13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]

13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]

13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]

'''13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ'''

13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]

13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]

'''13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]''' '''φ'''

13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]

13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]

13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]

13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]

14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]

14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]

14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]

14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]

14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]

15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]

15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]

15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]

15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]

15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]

15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]

15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]

16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]

16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]

16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]

16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]

16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]

16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]

16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]

17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]

17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]

17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]

'''17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)'''

'''17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]'''

'''17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]'''

'''17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]'''

'''17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]'''

'''17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]'''

'''17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]'''

'''17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)'''

17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]

17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]

17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]

17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

== Scale tree ==
The next table below shows an extension of [[edo]]s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
{| class="wikitable"
|-
| 4\[[11edo|11]]
| 
| 
| 
| 
| 
| 
| 436.364
| 109.091
| 0
| style="text-align:center;" | 
|-
| 
| 
| 
| 
| 
| 
| 29\[[80edo|80]]
| 435
| 105
| 15
| 
|-
| 
| 
| 
| 
| 
| 25\[[69edo|69]]
| 
| 434.783
| 104.348
| 17.391
| 
|-
| 
| 
| 
| 
| 21\[[58edo|58]]
| 
| 
| 434.483
| 103.448
| 20.69
| 
|-
| 
| 
| 
| 17\[[47edo|47]]
| 
| 
| 
| 434.043
| 102.128
| 25.532
| 
|-
| 
| 
| 
| 
| 30\[[83edo|83]]
| 
| 
| 433.735
| 101.208
| 28.916
| 
|-
| 
| 
| 
| 
| 
| 
| 73\[[202edo|202]]
| 433.663
| 100.990
| 29.703
| Since here are the optimal range Lufsur mode (?)
|-
| 
| 
| 
| 
| 
| 43\[[119edo|119]]
| 
| 433.613
| 100.840
| 30.252
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.459
| 100.377
| 31.95
| 
|-
| 
| 
| 13\[[36edo|36]]
| 
| 
| 
| 
| 433.333
| 100
| 33.333
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.048
| 99.144
| 36.473
| 
|-
| 
| 
| 
| 
| 35\97
| 
| 
| 432.99
| 98.969
| 37.113
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 432.933
| 98.799
| 37.738
| 
|-
| 
| 
| 
| 22\[[61edo|61]]
| 
| 
| 
| 432.787
| 98.361
| 39.344
| 
|-
| 
| 9\[[25edo|25]]
| 
| 
| 
| 
| 
| 432
| 96
| 48
| style="text-align:center;" | Boundary of propriety;

generators smaller than this are proper
|-
| 
| 
| 
| 
| 
| 
| 
| 431.417
| 94.25
| 54.4155
| 
|-
| 
| 
| 
| 23\[[64edo|64]]
| 
| 
| 
| 431.25
| 93.75
| 56.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 431.1185
| 93.355
| 57.697
| 
|-
| 
| 
| 
| 
| 37\103
| 
| 
| 431.068
| 93.204
| 58.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 430.984
| 92.952
| 58.175
| 
|-
| 
| 
| 14\[[39edo|39]]
| 
| 
| 
| 
| 430.769
| 92.308
| 61.538
| 
|-
| 
| 
| 
| 
| 
| 47\[[131edo|131]]
| 
| 430.534
| 91.603
| 64.122
| 
|-
| 
| 
| 
| 
| 
| 
| 80\[[223edo|223]]
| 430.493
| 91.480
| 64.575
| Until here are the optimal range Fuslur mode (?)
|-
| 
| 
| 
| 
| 33\[[92edo|92]]
| 
| 
| 430.435
| 91.304
| 65.217
| 
|-
| 
| 
| 
| 19\[[53edo|53]]
| 
| 
| 
| 430.189
| 90.566
| 67.925
| 
|-
| 
| 
| 
| 
| 24\[[67edo|67]]
| 
| 
| 429.851
| 89.552
| 71.642
| 
|-
| 
| 
| 
| 
| 
| 29\[[81edo|81]]
| 
| 429.63
| 88.889
| 74.074
| 
|-
| 
| 
| 
| 
| 
| 
| 34\[[95edo|95]]
| 429.474
| 88.421
| 75.7895
| 
|-
| 5\[[14edo|14]]
| 
| 
| 
| 
| 
| 
| 428.571
| 85.714
| 85.714
| style="text-align:center;" | 
|}

{{todo|cleanup}}</text>
      <sha1>slc525svyxxg24xdiegwrnq26b44ivn</sha1>
    </revision>
    <revision>
      <id>101034</id>
      <parentid>94509</parentid>
      <timestamp>2023-01-06T16:15:27Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Copying from [[Category:Abstract MOS patterns]] to [[Category:14-tone scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>101034</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8495" sha1="cdosy5rd0ue78pwfw5wxf7sjk5dxqeq" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
The '''11L 3s''' [[MOS scale]] was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]].

This is a type of scale which denotes the use of a scale placed between [[11edo]] and [[14edo]], employing a ratio generator between 41/32 ~ 9/7 ([[25edo]] being the middle size of the Ketradektriatoh spectrum, in the 2:1 relation), resulting in a variant of tetradecatonic scale which conforms by this scheme: LLLLsLLLLsLLLs.

== Edos that contains this scale ==
'''2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)'''

'''3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)'''

'''3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)'''

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]

4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]

4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]

'''5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ'''

'''5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]''' '''φ'''

5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]

6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]

7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]

7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]

7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]

'''8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ'''

'''8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]''' '''φ'''

8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]

9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]

9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]

9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]

10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]

10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]

10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''' 11 11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''' 11 11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''': [[124edo|124]]

11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]

11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]

11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]

11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]

11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]

11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]

11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]

12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]

12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]

12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]

13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]

13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]

13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]

'''13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ'''

13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]

13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]

'''13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]''' '''φ'''

13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]

13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]

13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]

13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]

14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]

14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]

14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]

14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]

14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]

15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]

15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]

15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]

15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]

15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]

15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]

15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]

16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]

16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]

16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]

16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]

16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]

16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]

16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]

17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]

17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]

17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]

'''17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)'''

'''17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]'''

'''17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]'''

'''17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]'''

'''17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]'''

'''17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]'''

'''17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]'''

'''17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)'''

17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]

17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]

17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]

17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

== Scale tree ==
The next table below shows an extension of [[edo]]s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
{| class="wikitable"
|-
| 4\[[11edo|11]]
| 
| 
| 
| 
| 
| 
| 436.364
| 109.091
| 0
| style="text-align:center;" | 
|-
| 
| 
| 
| 
| 
| 
| 29\[[80edo|80]]
| 435
| 105
| 15
| 
|-
| 
| 
| 
| 
| 
| 25\[[69edo|69]]
| 
| 434.783
| 104.348
| 17.391
| 
|-
| 
| 
| 
| 
| 21\[[58edo|58]]
| 
| 
| 434.483
| 103.448
| 20.69
| 
|-
| 
| 
| 
| 17\[[47edo|47]]
| 
| 
| 
| 434.043
| 102.128
| 25.532
| 
|-
| 
| 
| 
| 
| 30\[[83edo|83]]
| 
| 
| 433.735
| 101.208
| 28.916
| 
|-
| 
| 
| 
| 
| 
| 
| 73\[[202edo|202]]
| 433.663
| 100.990
| 29.703
| Since here are the optimal range Lufsur mode (?)
|-
| 
| 
| 
| 
| 
| 43\[[119edo|119]]
| 
| 433.613
| 100.840
| 30.252
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.459
| 100.377
| 31.95
| 
|-
| 
| 
| 13\[[36edo|36]]
| 
| 
| 
| 
| 433.333
| 100
| 33.333
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.048
| 99.144
| 36.473
| 
|-
| 
| 
| 
| 
| 35\97
| 
| 
| 432.99
| 98.969
| 37.113
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 432.933
| 98.799
| 37.738
| 
|-
| 
| 
| 
| 22\[[61edo|61]]
| 
| 
| 
| 432.787
| 98.361
| 39.344
| 
|-
| 
| 9\[[25edo|25]]
| 
| 
| 
| 
| 
| 432
| 96
| 48
| style="text-align:center;" | Boundary of propriety;

generators smaller than this are proper
|-
| 
| 
| 
| 
| 
| 
| 
| 431.417
| 94.25
| 54.4155
| 
|-
| 
| 
| 
| 23\[[64edo|64]]
| 
| 
| 
| 431.25
| 93.75
| 56.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 431.1185
| 93.355
| 57.697
| 
|-
| 
| 
| 
| 
| 37\103
| 
| 
| 431.068
| 93.204
| 58.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 430.984
| 92.952
| 58.175
| 
|-
| 
| 
| 14\[[39edo|39]]
| 
| 
| 
| 
| 430.769
| 92.308
| 61.538
| 
|-
| 
| 
| 
| 
| 
| 47\[[131edo|131]]
| 
| 430.534
| 91.603
| 64.122
| 
|-
| 
| 
| 
| 
| 
| 
| 80\[[223edo|223]]
| 430.493
| 91.480
| 64.575
| Until here are the optimal range Fuslur mode (?)
|-
| 
| 
| 
| 
| 33\[[92edo|92]]
| 
| 
| 430.435
| 91.304
| 65.217
| 
|-
| 
| 
| 
| 19\[[53edo|53]]
| 
| 
| 
| 430.189
| 90.566
| 67.925
| 
|-
| 
| 
| 
| 
| 24\[[67edo|67]]
| 
| 
| 429.851
| 89.552
| 71.642
| 
|-
| 
| 
| 
| 
| 
| 29\[[81edo|81]]
| 
| 429.63
| 88.889
| 74.074
| 
|-
| 
| 
| 
| 
| 
| 
| 34\[[95edo|95]]
| 429.474
| 88.421
| 75.7895
| 
|-
| 5\[[14edo|14]]
| 
| 
| 
| 
| 
| 
| 428.571
| 85.714
| 85.714
| style="text-align:center;" | 
|}

{{todo|cleanup}}

[[Category:14-tone scales]]</text>
      <sha1>cdosy5rd0ue78pwfw5wxf7sjk5dxqeq</sha1>
    </revision>
    <revision>
      <id>109587</id>
      <parentid>101034</parentid>
      <timestamp>2023-04-26T07:50:59Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Reordered subheadings</comment>
      <origin>109587</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8494" sha1="01i5na8q2edj47ngxnl2x23dhw0orb4" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
The '''11L 3s''' [[MOS scale]] was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]]

This is a type of scale which denotes the use of a scale placed between [[11edo]] and [[14edo]], employing a ratio generator between 41/32 ~ 9/7 ([[25edo]] being the middle size of the Ketradektriatoh spectrum, in the 2:1 relation), resulting in a variant of tetradecatonic scale which conforms by this scheme: LLLLsLLLLsLLLs.

== Scale tree ==
The next table below shows an extension of [[edo]]s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
{| class="wikitable"
|-
| 4\[[11edo|11]]
| 
| 
| 
| 
| 
| 
| 436.364
| 109.091
| 0
| style="text-align:center;" | 
|-
| 
| 
| 
| 
| 
| 
| 29\[[80edo|80]]
| 435
| 105
| 15
| 
|-
| 
| 
| 
| 
| 
| 25\[[69edo|69]]
| 
| 434.783
| 104.348
| 17.391
| 
|-
| 
| 
| 
| 
| 21\[[58edo|58]]
| 
| 
| 434.483
| 103.448
| 20.69
| 
|-
| 
| 
| 
| 17\[[47edo|47]]
| 
| 
| 
| 434.043
| 102.128
| 25.532
| 
|-
| 
| 
| 
| 
| 30\[[83edo|83]]
| 
| 
| 433.735
| 101.208
| 28.916
| 
|-
| 
| 
| 
| 
| 
| 
| 73\[[202edo|202]]
| 433.663
| 100.990
| 29.703
| Since here are the optimal range Lufsur mode (?)
|-
| 
| 
| 
| 
| 
| 43\[[119edo|119]]
| 
| 433.613
| 100.840
| 30.252
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.459
| 100.377
| 31.95
| 
|-
| 
| 
| 13\[[36edo|36]]
| 
| 
| 
| 
| 433.333
| 100
| 33.333
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.048
| 99.144
| 36.473
| 
|-
| 
| 
| 
| 
| 35\97
| 
| 
| 432.99
| 98.969
| 37.113
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 432.933
| 98.799
| 37.738
| 
|-
| 
| 
| 
| 22\[[61edo|61]]
| 
| 
| 
| 432.787
| 98.361
| 39.344
| 
|-
| 
| 9\[[25edo|25]]
| 
| 
| 
| 
| 
| 432
| 96
| 48
| style="text-align:center;" | Boundary of propriety;

generators smaller than this are proper
|-
| 
| 
| 
| 
| 
| 
| 
| 431.417
| 94.25
| 54.4155
| 
|-
| 
| 
| 
| 23\[[64edo|64]]
| 
| 
| 
| 431.25
| 93.75
| 56.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 431.1185
| 93.355
| 57.697
| 
|-
| 
| 
| 
| 
| 37\103
| 
| 
| 431.068
| 93.204
| 58.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 430.984
| 92.952
| 58.175
| 
|-
| 
| 
| 14\[[39edo|39]]
| 
| 
| 
| 
| 430.769
| 92.308
| 61.538
| 
|-
| 
| 
| 
| 
| 
| 47\[[131edo|131]]
| 
| 430.534
| 91.603
| 64.122
| 
|-
| 
| 
| 
| 
| 
| 
| 80\[[223edo|223]]
| 430.493
| 91.480
| 64.575
| Until here are the optimal range Fuslur mode (?)
|-
| 
| 
| 
| 
| 33\[[92edo|92]]
| 
| 
| 430.435
| 91.304
| 65.217
| 
|-
| 
| 
| 
| 19\[[53edo|53]]
| 
| 
| 
| 430.189
| 90.566
| 67.925
| 
|-
| 
| 
| 
| 
| 24\[[67edo|67]]
| 
| 
| 429.851
| 89.552
| 71.642
| 
|-
| 
| 
| 
| 
| 
| 29\[[81edo|81]]
| 
| 429.63
| 88.889
| 74.074
| 
|-
| 
| 
| 
| 
| 
| 
| 34\[[95edo|95]]
| 429.474
| 88.421
| 75.7895
| 
|-
| 5\[[14edo|14]]
| 
| 
| 
| 
| 
| 
| 428.571
| 85.714
| 85.714
| style="text-align:center;" | 
|}

== Edos that contains this scale ==
'''2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)'''

'''3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)'''

'''3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)'''

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]

4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]

4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]

'''5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ'''

'''5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]''' '''φ'''

5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]

6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]

7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]

7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]

7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]

'''8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ'''

'''8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]''' '''φ'''

8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]

9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]

9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]

9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]

10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]

10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]

10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''' 11 11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''' 11 11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''': [[124edo|124]]

11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]

11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]

11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]

11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]

11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]

11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]

11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]

12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]

12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]

12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]

13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]

13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]

13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]

'''13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ'''

13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]

13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]

'''13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]''' '''φ'''

13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]

13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]

13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]

13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]

14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]

14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]

14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]

14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]

14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]

15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]

15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]

15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]

15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]

15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]

15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]

15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]

16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]

16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]

16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]

16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]

16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]

16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]

16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]

17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]

17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]

17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]

'''17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)'''

'''17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]'''

'''17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]'''

'''17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]'''

'''17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]'''

'''17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]'''

'''17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]'''

'''17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)'''

17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]

17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]

17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]

17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

{{todo|cleanup}}

[[Category:14-tone scales]]</text>
      <sha1>01i5na8q2edj47ngxnl2x23dhw0orb4</sha1>
    </revision>
    <revision>
      <id>109588</id>
      <parentid>109587</parentid>
      <timestamp>2023-04-26T07:51:15Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Scale tree */</comment>
      <origin>109588</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8489" sha1="55wptpohlc7my6zti8365uk3l7g1zuc" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
The '''11L 3s''' [[MOS scale]] was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]]

This is a type of scale which denotes the use of a scale placed between [[11edo]] and [[14edo]], employing a ratio generator between 41/32 ~ 9/7 ([[25edo]] being the middle size of the Ketradektriatoh spectrum, in the 2:1 relation), resulting in a variant of tetradecatonic scale which conforms by this scheme: LLLLsLLLLsLLLs.

== Scale tree ==
The table below shows an extension of [[edo]]s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
{| class="wikitable"
|-
| 4\[[11edo|11]]
| 
| 
| 
| 
| 
| 
| 436.364
| 109.091
| 0
| style="text-align:center;" | 
|-
| 
| 
| 
| 
| 
| 
| 29\[[80edo|80]]
| 435
| 105
| 15
| 
|-
| 
| 
| 
| 
| 
| 25\[[69edo|69]]
| 
| 434.783
| 104.348
| 17.391
| 
|-
| 
| 
| 
| 
| 21\[[58edo|58]]
| 
| 
| 434.483
| 103.448
| 20.69
| 
|-
| 
| 
| 
| 17\[[47edo|47]]
| 
| 
| 
| 434.043
| 102.128
| 25.532
| 
|-
| 
| 
| 
| 
| 30\[[83edo|83]]
| 
| 
| 433.735
| 101.208
| 28.916
| 
|-
| 
| 
| 
| 
| 
| 
| 73\[[202edo|202]]
| 433.663
| 100.990
| 29.703
| Since here are the optimal range Lufsur mode (?)
|-
| 
| 
| 
| 
| 
| 43\[[119edo|119]]
| 
| 433.613
| 100.840
| 30.252
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.459
| 100.377
| 31.95
| 
|-
| 
| 
| 13\[[36edo|36]]
| 
| 
| 
| 
| 433.333
| 100
| 33.333
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.048
| 99.144
| 36.473
| 
|-
| 
| 
| 
| 
| 35\97
| 
| 
| 432.99
| 98.969
| 37.113
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 432.933
| 98.799
| 37.738
| 
|-
| 
| 
| 
| 22\[[61edo|61]]
| 
| 
| 
| 432.787
| 98.361
| 39.344
| 
|-
| 
| 9\[[25edo|25]]
| 
| 
| 
| 
| 
| 432
| 96
| 48
| style="text-align:center;" | Boundary of propriety;

generators smaller than this are proper
|-
| 
| 
| 
| 
| 
| 
| 
| 431.417
| 94.25
| 54.4155
| 
|-
| 
| 
| 
| 23\[[64edo|64]]
| 
| 
| 
| 431.25
| 93.75
| 56.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 431.1185
| 93.355
| 57.697
| 
|-
| 
| 
| 
| 
| 37\103
| 
| 
| 431.068
| 93.204
| 58.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 430.984
| 92.952
| 58.175
| 
|-
| 
| 
| 14\[[39edo|39]]
| 
| 
| 
| 
| 430.769
| 92.308
| 61.538
| 
|-
| 
| 
| 
| 
| 
| 47\[[131edo|131]]
| 
| 430.534
| 91.603
| 64.122
| 
|-
| 
| 
| 
| 
| 
| 
| 80\[[223edo|223]]
| 430.493
| 91.480
| 64.575
| Until here are the optimal range Fuslur mode (?)
|-
| 
| 
| 
| 
| 33\[[92edo|92]]
| 
| 
| 430.435
| 91.304
| 65.217
| 
|-
| 
| 
| 
| 19\[[53edo|53]]
| 
| 
| 
| 430.189
| 90.566
| 67.925
| 
|-
| 
| 
| 
| 
| 24\[[67edo|67]]
| 
| 
| 429.851
| 89.552
| 71.642
| 
|-
| 
| 
| 
| 
| 
| 29\[[81edo|81]]
| 
| 429.63
| 88.889
| 74.074
| 
|-
| 
| 
| 
| 
| 
| 
| 34\[[95edo|95]]
| 429.474
| 88.421
| 75.7895
| 
|-
| 5\[[14edo|14]]
| 
| 
| 
| 
| 
| 
| 428.571
| 85.714
| 85.714
| style="text-align:center;" | 
|}

== Edos that contains this scale ==
'''2 2 2 1 2 2 2 2 1 2 2 2 2 1: [[25edo|25]] (Middle range)'''

'''3 3 3 1 3 3 3 3 1 3 3 3 3 1: [[36edo|36]] (Lufsur range)'''

'''3 3 3 2 3 3 3 3 2 3 3 3 3 2: [[39edo|39]] (Fuslur range)'''

4 4 4 1 4 4 4 4 1 4 4 4 4 1: [[47edo|47]]

4 4 4 2 4 4 4 4 2 4 4 4 4 2: [[50edo|50]]

4 4 4 3 4 4 4 4 3 4 4 4 4 3: [[53edo|53]]

5 5 5 1 5 5 5 5 1 5 5 5 5 1: [[58edo|58]]

'''5 5 5 2 5 5 5 5 2 5 5 5 5 2: [[61edo|61]] Split-φ'''

'''5 5 5 3 5 5 5 5 3 5 5 5 5 3: [[64edo|64]]''' '''φ'''

5 5 5 4 5 5 5 5 4 5 5 5 5 4: [[67edo|67]]

6 6 6 1 6 6 6 6 1 6 6 6 6 1: [[69edo|69]]

6 6 6 5 6 6 6 6 5 6 6 6 6 5: [[81edo|81]]

7 7 7 1 7 7 7 7 1 7 7 7 7 1: [[80edo|80]]

7 7 7 2 7 7 7 7 2 7 7 7 7 2: [[83edo|83]]

7 7 7 3 7 7 7 7 3 7 7 7 7 3: [[86edo|86]]

7 7 7 4 7 7 7 7 4 7 7 7 7 4: [[89edo|89]]

7 7 7 5 7 7 7 7 5 7 7 7 7 5: [[92edo|92]]

7 7 7 6 7 7 7 7 6 7 7 7 7 6: [[95edo|95]]

8 8 8 1 8 8 8 8 1 8 8 8 8 1: [[91edo|91]]

'''8 8 8 3 8 8 8 8 3 8 8 8 8 3: [[97edo|97]] Split-φ'''

'''8 8 8 5 8 8 8 8 5 8 8 8 8 5: [[103edo|103]]''' '''φ'''

8 8 8 7 8 8 8 8 7 8 8 8 8 7: [[109edo|109]]

9 9 9 1 9 9 9 9 1 9 9 9 9 1: [[102edo|102]]

9 9 9 2 9 9 9 9 2 9 9 9 9 2: [[105edo|105]]

9 9 9 4 9 9 9 9 4 9 9 9 9 4: [[111edo|111]]

9 9 9 5 9 9 9 9 5 9 9 9 9 5: [[114edo|114]]

9 9 9 7 9 9 9 9 7 9 9 9 9 7: [[120edo|120]]

9 9 9 8 9 9 9 9 8 9 9 9 9 8: [[123edo|123]]

10 10 10 1 10 10 10 10 1 10 10 10 10 1:[[113edo|113]]

10 10 10 3 10 10 10 10 3 10 10 10 10 3: [[119edo|119]]

10 10 10 7 10 10 10 10 7 10 10 10 10 7: [[131edo|131]]

10 10 10 9 10 10 10 10 9 10 10 10 10 9: [[137edo|137]]

11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''' 11 11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''' 11 11 11 11 '''&lt;span style="color: #006209;"&gt;1&lt;/span&gt;''': [[124edo|124]]

11 11 11 2 11 11 11 11 2 11 11 11 11 2: [[127edo|127]]

11 11 11 3 11 11 11 11 3 11 11 11 11 3: [[130edo|130]]

11 11 11 4 11 11 11 11 4 11 11 11 11 4: [[133edo|133]]

11 11 11 5 11 11 11 11 5 11 11 11 11 5: [[136edo|136]]

11 11 11 6 11 11 11 11 6 11 11 11 11 6: [[139edo|139]]

11 11 11 7 11 11 11 11 7 11 11 11 11 7: [[142edo|142]]

11 11 11 8 11 11 11 11 8 11 11 11 11 8: [[145edo|145]]

11 11 11 9 11 11 11 11 9 11 11 11 11 9 :[[148edo|148]]

11 11 11 10 11 11 11 11 10 11 11 11 11 10: [[151edo|151]]

12 12 12 1 12 12 12 12 1 12 12 12 12 1: [[135edo|135]]

12 12 12 5 12 12 12 12 5 12 12 12 12 5: [[147edo|147]]

12 12 12 7 12 12 12 12 7 12 12 12 12 7: [[153edo|153]]

12 12 12 11 12 12 12 12 11 12 12 12 12 11: [[165edo|165]]

13 13 13 1 13 13 13 13 1 13 13 13 13 1: [[146edo|146]]

13 13 13 2 13 13 13 13 2 13 13 13 13 2: [[149edo|149]]

13 13 13 3 13 13 13 13 3 13 13 13 13 3: [[152edo|152]]

13 13 13 4 13 13 13 13 4 13 13 13 13 4: [[155edo|155]]

'''13 13 13 5 13 13 13 13 5 13 13 13 13 5: [[158edo|158]] Split-φ'''

13 13 13 6 13 13 13 13 6 13 13 13 13 6: [[161edo|161]]

13 13 13 7 13 13 13 13 7 13 13 13 13 7: [[164edo|164]]

'''13 13 13 8 13 13 13 13 8 13 13 13 13 8: [[167edo|167]]''' '''φ'''

13 13 13 9 13 13 13 13 9 13 13 13 13 9: [[170edo|170]]

13 13 13 10 13 13 13 13 10 13 13 13 13 10: [[173edo|173]]

13 13 13 11 13 13 13 13 11 13 13 13 13 11: [[176edo|176]]

13 13 13 12 13 13 13 13 12 13 13 13 13 12: [[179edo|179]]

14 14 14 1 14 14 14 14 1 14 14 14 14 1: [[157edo|157]]

14 14 14 3 14 14 14 14 3 14 14 14 14 3: [[163edo|163]]

14 14 14 5 14 14 14 14 5 14 14 14 14 5: [[169edo|169]]

14 14 14 9 14 14 14 14 9 14 14 14 14 9: [[181edo|181]]

14 14 14 11 14 14 14 14 11 14 14 14 14 11: [[187edo|187]]

14 14 14 13 14 14 14 14 13 14 14 14 14 13: [[193edo|193]]

15 15 15 1 15 15 15 15 1 15 15 15 15 1: [[168edo|168]]

15 15 15 2 15 15 15 15 2 15 15 15 15 2: [[171edo|171]]

15 15 15 4 15 15 15 15 4 15 15 15 15 4: [[177edo|177]]

15 15 15 7 15 15 15 15 7 15 15 15 15 7: [[186edo|186]]

15 15 15 8 15 15 15 15 8 15 15 15 15 8: [[189edo|189]]

15 15 15 11 15 15 15 15 11 15 15 15 15 11: [[198edo|198]]

15 15 15 13 15 15 15 15 13 15 15 15 15 13: [[204edo|204]]

15 15 15 14 15 15 15 15 14 15 15 15 15 14: [[207edo|207]]

16 16 16 1 16 16 16 16 1 16 16 16 16 1: [[179edo|179]]

16 16 16 3 16 16 16 16 3 16 16 16 16 3: [[185edo|185]]

16 16 16 5 16 16 16 16 5 16 16 16 16 5: [[191edo|191]]

16 16 16 7 16 16 16 16 7 16 16 16 16 7: [[197edo|197]]

16 16 16 9 16 16 16 16 9 16 16 16 16 9: [[203edo|203]]

16 16 16 11 16 16 16 16 11 16 16 16 16 11: [[209edo|209]]

16 16 16 13 16 16 16 16 13 16 16 16 16 13: [[215edo|215]]

16 16 16 15 16 16 16 16 15 16 16 16 16 15: [[221edo|221]]

17 17 17 1 17 17 17 17 1 17 17 17 17 1: [[190edo|190]]

17 17 17 2 17 17 17 17 2 17 17 17 17 2: [[193edo|193]]

17 17 17 3 17 17 17 17 3 17 17 17 17 3: [[196edo|196]]

17 17 17 4 17 17 17 17 4 17 17 17 17 4: [[199edo|199]]

'''17 17 17 5 17 17 17 17 5 17 17 17 17 5: [[202edo|202]] (Top limit for Lufsur range)'''

'''17 17 17 6 17 17 17 17 6 17 17 17 17 6: [[205edo|205]]'''

'''17 17 17 7 17 17 17 17 7 17 17 17 17 7: [[208edo|208]]'''

'''17 17 17 8 17 17 17 17 8 17 17 17 17 8: [[211edo|211]]'''

'''17 17 17 9 17 17 17 17 9 17 17 17 17 9: [[214edo|214]]'''

'''17 17 17 10 17 17 17 17 10 17 17 17 17 10: [[217edo|217]]'''

'''17 17 17 11 17 17 17 17 11 17 17 17 17 11: [[220edo|220]]'''

'''17 17 17 12 17 17 17 17 12 17 17 17 17 12: [[223edo|223]] (Top limit for Fuslur range)'''

17 17 17 13 17 17 17 17 13 17 17 17 17 13: [[226edo|226]]

17 17 17 14 17 17 17 17 14 17 17 17 17 14: [[229edo|229]]

17 17 17 15 17 17 17 17 15 17 17 17 17 15: [[232edo|232]]

17 17 17 16 17 17 17 17 16 17 17 17 17 16: [[235edo|235]]

{{todo|cleanup}}

[[Category:14-tone scales]]</text>
      <sha1>55wptpohlc7my6zti8365uk3l7g1zuc</sha1>
    </revision>
    <revision>
      <id>109591</id>
      <parentid>109588</parentid>
      <timestamp>2023-04-26T08:16:56Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Cleaned up the page</comment>
      <origin>109591</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8744" sha1="sti7ijviu0vzx1dwgp2grsn9wh96qfb" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
The '''11L 3s''' [[MOS scale]] was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]]

This is a type of scale which denotes the use of a scale placed between [[11edo]] and [[14edo]], employing a ratio generator between 41/32 ~ 9/7 ([[25edo]] being the middle size of the Ketradektriatoh spectrum, in the 2:1 relation), resulting in a variant of tetradecatonic scale which conforms by this scheme: LLLLsLLLLsLLLs.







== Scale tree ==
The table below shows an extension of [[edo]]s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
{| class="wikitable"
|-
| 4\[[11edo|11]]
| 
| 
| 
| 
| 
| 
| 436.364
| 109.091
| 0
| style="text-align:center;" | 
|-
| 
| 
| 
| 
| 
| 
| 29\[[80edo|80]]
| 435
| 105
| 15
| 
|-
| 
| 
| 
| 
| 
| 25\[[69edo|69]]
| 
| 434.783
| 104.348
| 17.391
| 
|-
| 
| 
| 
| 
| 21\[[58edo|58]]
| 
| 
| 434.483
| 103.448
| 20.69
| 
|-
| 
| 
| 
| 17\[[47edo|47]]
| 
| 
| 
| 434.043
| 102.128
| 25.532
| 
|-
| 
| 
| 
| 
| 30\[[83edo|83]]
| 
| 
| 433.735
| 101.208
| 28.916
| 
|-
| 
| 
| 
| 
| 
| 
| 73\[[202edo|202]]
| 433.663
| 100.990
| 29.703
| Since here are the optimal range Lufsur mode (?)
|-
| 
| 
| 
| 
| 
| 43\[[119edo|119]]
| 
| 433.613
| 100.840
| 30.252
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.459
| 100.377
| 31.95
| 
|-
| 
| 
| 13\[[36edo|36]]
| 
| 
| 
| 
| 433.333
| 100
| 33.333
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.048
| 99.144
| 36.473
| 
|-
| 
| 
| 
| 
| 35\97
| 
| 
| 432.99
| 98.969
| 37.113
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 432.933
| 98.799
| 37.738
| 
|-
| 
| 
| 
| 22\[[61edo|61]]
| 
| 
| 
| 432.787
| 98.361
| 39.344
| 
|-
| 
| 9\[[25edo|25]]
| 
| 
| 
| 
| 
| 432
| 96
| 48
| style="text-align:center;" | Boundary of propriety;

generators smaller than this are proper
|-
| 
| 
| 
| 
| 
| 
| 
| 431.417
| 94.25
| 54.4155
| 
|-
| 
| 
| 
| 23\[[64edo|64]]
| 
| 
| 
| 431.25
| 93.75
| 56.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 431.1185
| 93.355
| 57.697
| 
|-
| 
| 
| 
| 
| 37\103
| 
| 
| 431.068
| 93.204
| 58.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 430.984
| 92.952
| 58.175
| 
|-
| 
| 
| 14\[[39edo|39]]
| 
| 
| 
| 
| 430.769
| 92.308
| 61.538
| 
|-
| 
| 
| 
| 
| 
| 47\[[131edo|131]]
| 
| 430.534
| 91.603
| 64.122
| 
|-
| 
| 
| 
| 
| 
| 
| 80\[[223edo|223]]
| 430.493
| 91.480
| 64.575
| Until here are the optimal range Fuslur mode (?)
|-
| 
| 
| 
| 
| 33\[[92edo|92]]
| 
| 
| 430.435
| 91.304
| 65.217
| 
|-
| 
| 
| 
| 19\[[53edo|53]]
| 
| 
| 
| 430.189
| 90.566
| 67.925
| 
|-
| 
| 
| 
| 
| 24\[[67edo|67]]
| 
| 
| 429.851
| 89.552
| 71.642
| 
|-
| 
| 
| 
| 
| 
| 29\[[81edo|81]]
| 
| 429.63
| 88.889
| 74.074
| 
|-
| 
| 
| 
| 
| 
| 
| 34\[[95edo|95]]
| 429.474
| 88.421
| 75.7895
| 
|-
| 5\[[14edo|14]]
| 
| 
| 
| 
| 
| 
| 428.571
| 85.714
| 85.714
| style="text-align:center;" | 
|}

== As an EDO subset ==
{| class="wikitable sortable"
|EDO
|Subset
|Special properties
|-
|[[25edo|25]]
|2 2 2 1 2 2 2 2 1 2 2 2 2 1
|Middle range
|-
|[[36edo|36]]
|3 3 3 1 3 3 3 3 1 3 3 3 3 1
|Lusfur range
|-
|[[39edo|39]]
|3 3 3 2 3 3 3 3 2 3 3 3 3 2
|Fuslur range
|-
|[[47edo|47]]
|4 4 4 1 4 4 4 4 1 4 4 4 4 1
|
|-
|[[50edo|50]]
|4 4 4 2 4 4 4 4 2 4 4 4 4 2
|
|-
|[[53edo|53]]
|4 4 4 3 4 4 4 4 3 4 4 4 4 3
|
|-
|[[58edo|58]]
|5 5 5 1 5 5 5 5 1 5 5 5 5 1
|
|-
|[[61edo|61]]
|5 5 5 2 5 5 5 5 2 5 5 5 5 2
|Split-φ
|-
|[[64edo|64]]
|5 5 5 3 5 5 5 5 3 5 5 5 5 3
|φ
|-
|[[67edo|67]]
|5 5 5 4 5 5 5 5 4 5 5 5 5 4
|
|-
|[[69edo|69]]
|6 6 6 1 6 6 6 6 1 6 6 6 6 1
|
|-
|[[81edo|81]]
|6 6 6 5 6 6 6 6 5 6 6 6 6 5
|
|-
|[[80edo|80]]
|7 7 7 1 7 7 7 7 1 7 7 7 7 1
|
|-
|[[83-limit|83]]
|7 7 7 2 7 7 7 7 2 7 7 7 7 2
|
|-
|[[86edo|86]]
|7 7 7 3 7 7 7 7 3 7 7 7 7 3
|
|-
|[[89edo|89]]
|7 7 7 4 7 7 7 7 4 7 7 7 7 4
|
|-
|[[92edo|92]]
|7 7 7 5 7 7 7 7 5 7 7 7 7 5
|
|-
|[[95edo|95]]
|7 7 7 6 7 7 7 7 6 7 7 7 7 6
|
|-
|[[91edo|91]]
|8 8 8 1 8 8 8 8 1 8 8 8 8 1
|
|-
|[[97edo|97]]
|8 8 8 3 8 8 8 8 3 8 8 8 8 3
|Split-φ
|-
|[[103edo|103]]
|8 8 8 5 8 8 8 8 5 8 8 8 8 5
|φ
|-
|[[109edo|109]]
|8 8 8 7 8 8 8 8 7 8 8 8 8 7
|
|-
|[[102edo|102]]
|9 9 9 1 9 9 9 9 1 9 9 9 9 1
|
|-
|[[105edo|105]]
|9 9 9 2 9 9 9 9 2 9 9 9 9 2
|
|-
|[[111edo|111]]
|9 9 9 4 9 9 9 9 4 9 9 9 9 4
|
|-
|[[114edo|114]]
|9 9 9 5 9 9 9 9 5 9 9 9 9 5
|
|-
|[[120edo|120]]
|9 9 9 7 9 9 9 9 7 9 9 9 9 7
|
|-
|[[123edo|123]]
|9 9 9 8 9 9 9 9 8 9 9 9 9 8
|
|-
|[[113edo|113]]
|10 10 10 1 10 10 10 10 1 10 10 10 10 1
|
|-
|[[119edo|119]]
|10 10 10 3 10 10 10 10 3 10 10 10 10 3
|
|-
|[[131edo|131]]
|10 10 10 7 10 10 10 10 7 10 10 10 10 7
|
|-
|[[137edo|137]]
|10 10 10 9 10 10 10 10 9 10 10 10 10 9
|
|-
|[[124edo|124]]
|11 11 11 1 11 11 11 11 1 11 11 11 11 1
|
|-
|[[127edo|127]]
|11 11 11 2 11 11 11 11 2 11 11 11 11 2
|
|-
|[[130edo|130]]
|11 11 11 3 11 11 11 11 3 11 11 11 11 3
|
|-
|[[133edo|133]]
|11 11 11 4 11 11 11 11 4 11 11 11 11 4
|
|-
|[[136edo|136]]
|11 11 11 5 11 11 11 11 5 11 11 11 11 5
|
|-
|[[139edo|139]]
|11 11 11 6 11 11 11 11 6 11 11 11 11 6
|
|-
|[[142edo|142]]
|11 11 11 7 11 11 11 11 7 11 11 11 11 7
|
|-
|[[145edo|145]]
|11 11 11 8 11 11 11 11 8 11 11 11 11 8
|
|-
|[[148edo|148]]
|11 11 11 9 11 11 11 11 9 11 11 11 11 9
|
|-
|[[151edo|151]]
|11 11 11 10 11 11 11 11 10 11 11 11 11 10
|
|-
|[[135edo|135]]
|12 12 12 1 12 12 12 12 1 12 12 12 12 1
|
|-
|[[147edo|147]]
|12 12 12 5 12 12 12 12 5 12 12 12 12 5
|
|-
|[[153edo|153]]
|12 12 12 7 12 12 12 12 7 12 12 12 12 7
|
|-
|[[165edo|165]]
|12 12 12 11 12 12 12 12 11 12 12 12 12 11
|
|-
|[[146edo|146]]
|13 13 13 1 13 13 13 13 1 13 13 13 13 1
|
|-
|[[149edo|149]]
|13 13 13 2 13 13 13 13 2 13 13 13 13 2
|
|-
|[[152edo|152]]
|13 13 13 3 13 13 13 13 3 13 13 13 13 3
|
|-
|[[155edo|155]]
|13 13 13 4 13 13 13 13 4 13 13 13 13 4
|
|-
|[[158edo|158]]
|13 13 13 5 13 13 13 13 5 13 13 13 13 5
|Split-φ
|-
|[[161edo|161]]
|13 13 13 6 13 13 13 13 6 13 13 13 13 6
|
|-
|[[164edo|164]]
|13 13 13 7 13 13 13 13 7 13 13 13 13 7
|
|-
|[[167edo|167]]
|13 13 13 8 13 13 13 13 8 13 13 13 13 8
|φ
|-
|[[170edo|170]]
|13 13 13 9 13 13 13 13 9 13 13 13 13 9
|
|-
|[[173edo|173]]
|13 13 13 10 13 13 13 13 10 13 13 13 13 10
|
|-
|[[176edo|176]]
|13 13 13 11 13 13 13 13 11 13 13 13 13 11
|
|-
|[[179edo|179]]
|13 13 13 12 13 13 13 13 12 13 13 13 13 12
|
|-
|[[157edo|157]]
|14 14 14 1 14 14 14 14 1 14 14 14 14 1
|
|-
|[[163edo|163]]
|14 14 14 3 14 14 14 14 3 14 14 14 14 3
|
|-
|[[169edo|169]]
|14 14 14 5 14 14 14 14 5 14 14 14 14 5
|
|-
|[[181edo|181]]
|14 14 14 9 14 14 14 14 9 14 14 14 14 9
|
|-
|[[187edo|187]]
|14 14 14 11 14 14 14 14 11 14 14 14 14 11
|
|-
|[[193edo|193]]
|14 14 14 13 14 14 14 14 13 14 14 14 14 13
|
|-
|[[168edo|168]]
|15 15 15 1 15 15 15 15 1 15 15 15 15 1
|
|-
|[[171edo|171]]
|15 15 15 2 15 15 15 15 2 15 15 15 15 2
|
|-
|[[177edo|177]]
|15 15 15 4 15 15 15 15 4 15 15 15 15 4
|
|-
|[[186edo|186]]
|15 15 15 7 15 15 15 15 7 15 15 15 15 7
|
|-
|[[189edo|189]]
|15 15 15 8 15 15 15 15 8 15 15 15 15 8
|
|-
|[[198edo|198]]
|15 15 15 11 15 15 15 15 11 15 15 15 15 11
|
|-
|[[204edo|204]]
|15 15 15 13 15 15 15 15 13 15 15 15 15 13
|
|-
|[[207edo|207]]
|15 15 15 14 15 15 15 15 14 15 15 15 15 14
|
|-
|[[179edo|179]]
|16 16 16 1 16 16 16 16 1 16 16 16 16 1
|
|-
|[[185edo|185]]
|16 16 16 3 16 16 16 16 3 16 16 16 16 3
|
|-
|[[191edo|191]]
|16 16 16 5 16 16 16 16 5 16 16 16 16 5
|
|-
|[[197edo|197]]
|16 16 16 7 16 16 16 16 7 16 16 16 16 7
|
|-
|[[203edo|203]]
|16 16 16 9 16 16 16 16 9 16 16 16 16 9
|
|-
|[[209edo|209]]
|16 16 16 11 16 16 16 16 11 16 16 16 16 11
|
|-
|[[215edo|215]]
|16 16 16 13 16 16 16 16 13 16 16 16 16 13
|
|-
|[[221edo|221]]
|16 16 16 15 16 16 16 16 15 16 16 16 16 15
|
|-
|[[190edo|190]]
|17 17 17 1 17 17 17 17 1 17 17 17 17 1
|
|-
|[[193edo|193]]
|17 17 17 2 17 17 17 17 2 17 17 17 17 2
|
|-
|[[196edo|196]]
|17 17 17 3 17 17 17 17 3 17 17 17 17 3
|
|-
|[[199edo|199]]
|17 17 17 4 17 17 17 17 4 17 17 17 17 4
|
|-
|[[202edo|202]]
|17 17 17 5 17 17 17 17 5 17 17 17 17 5
|Top limit for Lusfur range
|-
|[[205edo|205]]
|17 17 17 6 17 17 17 17 6 17 17 17 17 6
|
|-
|[[208edo|208]]
|17 17 17 7 17 17 17 17 7 17 17 17 17 7
|
|-
|[[211edo|211]]
|17 17 17 8 17 17 17 17 8 17 17 17 17 8
|
|-
|[[214edo|214]]
|17 17 17 9 17 17 17 17 9 17 17 17 17 9
|
|-
|[[217edo|217]]
|17 17 17 10 17 17 17 17 10 17 17 17 17 10
|
|-
|[[220edo|220]]
|17 17 17 11 17 17 17 17 11 17 17 17 17 11
|
|-
|[[223edo|223]]
|17 17 17 12 17 17 17 17 12 17 17 17 17 12
|Top limit for Fuslur range
|-
|[[226edo|226]]
|17 17 17 13 17 17 17 17 13 17 17 17 17 13
|
|-
|[[229edo|229]]
|17 17 17 14 17 17 17 17 14 17 17 17 17 14
|
|-
|[[232edo|232]]
|17 17 17 15 17 17 17 17 15 17 17 17 17 15
|
|-
|[[235edo|235]]
|17 17 17 16 17 17 17 17 16 17 17 17 17 16
|
|}
[[Category:14-tone scales]]</text>
      <sha1>sti7ijviu0vzx1dwgp2grsn9wh96qfb</sha1>
    </revision>
    <revision>
      <id>109592</id>
      <parentid>109591</parentid>
      <timestamp>2023-04-26T08:22:58Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>109592</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8753" sha1="ia8jvtwnstupyokmgdfj4ijswc92gdv" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
The '''11L 3s''' [[MOS scale]] was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]]

This is a type of scale which denotes the use of a scale placed between [[11edo]] and [[14edo]].

It employs a ratio generator between [[41/32]] and [[9/7]] ([[25edo]] being the middle size of the Ketradektriatoh spectrum, in the 2:1 relation).

This results in a variant of tetradecatonic scale which conforms by this scheme: LLLLsLLLLsLLLs.
== Scale tree ==
The table below shows an extension of [[edo]]s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
{| class="wikitable"
|-
| 4\[[11edo|11]]
| 
| 
| 
| 
| 
| 
| 436.364
| 109.091
| 0
| style="text-align:center;" | 
|-
| 
| 
| 
| 
| 
| 
| 29\[[80edo|80]]
| 435
| 105
| 15
| 
|-
| 
| 
| 
| 
| 
| 25\[[69edo|69]]
| 
| 434.783
| 104.348
| 17.391
| 
|-
| 
| 
| 
| 
| 21\[[58edo|58]]
| 
| 
| 434.483
| 103.448
| 20.69
| 
|-
| 
| 
| 
| 17\[[47edo|47]]
| 
| 
| 
| 434.043
| 102.128
| 25.532
| 
|-
| 
| 
| 
| 
| 30\[[83edo|83]]
| 
| 
| 433.735
| 101.208
| 28.916
| 
|-
| 
| 
| 
| 
| 
| 
| 73\[[202edo|202]]
| 433.663
| 100.990
| 29.703
| Since here are the optimal range Lufsur mode (?)
|-
| 
| 
| 
| 
| 
| 43\[[119edo|119]]
| 
| 433.613
| 100.840
| 30.252
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.459
| 100.377
| 31.95
| 
|-
| 
| 
| 13\[[36edo|36]]
| 
| 
| 
| 
| 433.333
| 100
| 33.333
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.048
| 99.144
| 36.473
| 
|-
| 
| 
| 
| 
| 35\97
| 
| 
| 432.99
| 98.969
| 37.113
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 432.933
| 98.799
| 37.738
| 
|-
| 
| 
| 
| 22\[[61edo|61]]
| 
| 
| 
| 432.787
| 98.361
| 39.344
| 
|-
| 
| 9\[[25edo|25]]
| 
| 
| 
| 
| 
| 432
| 96
| 48
| style="text-align:center;" | Boundary of propriety;

generators smaller than this are proper
|-
| 
| 
| 
| 
| 
| 
| 
| 431.417
| 94.25
| 54.4155
| 
|-
| 
| 
| 
| 23\[[64edo|64]]
| 
| 
| 
| 431.25
| 93.75
| 56.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 431.1185
| 93.355
| 57.697
| 
|-
| 
| 
| 
| 
| 37\103
| 
| 
| 431.068
| 93.204
| 58.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 430.984
| 92.952
| 58.175
| 
|-
| 
| 
| 14\[[39edo|39]]
| 
| 
| 
| 
| 430.769
| 92.308
| 61.538
| 
|-
| 
| 
| 
| 
| 
| 47\[[131edo|131]]
| 
| 430.534
| 91.603
| 64.122
| 
|-
| 
| 
| 
| 
| 
| 
| 80\[[223edo|223]]
| 430.493
| 91.480
| 64.575
| Until here are the optimal range Fuslur mode (?)
|-
| 
| 
| 
| 
| 33\[[92edo|92]]
| 
| 
| 430.435
| 91.304
| 65.217
| 
|-
| 
| 
| 
| 19\[[53edo|53]]
| 
| 
| 
| 430.189
| 90.566
| 67.925
| 
|-
| 
| 
| 
| 
| 24\[[67edo|67]]
| 
| 
| 429.851
| 89.552
| 71.642
| 
|-
| 
| 
| 
| 
| 
| 29\[[81edo|81]]
| 
| 429.63
| 88.889
| 74.074
| 
|-
| 
| 
| 
| 
| 
| 
| 34\[[95edo|95]]
| 429.474
| 88.421
| 75.7895
| 
|-
| 5\[[14edo|14]]
| 
| 
| 
| 
| 
| 
| 428.571
| 85.714
| 85.714
| style="text-align:center;" | 
|}

== As an EDO subset ==
{| class="wikitable sortable"
|EDO
|Subset
|Special properties
|-
|[[25edo|25]]
|2 2 2 1 2 2 2 2 1 2 2 2 2 1
|Middle range
|-
|[[36edo|36]]
|3 3 3 1 3 3 3 3 1 3 3 3 3 1
|Lusfur range
|-
|[[39edo|39]]
|3 3 3 2 3 3 3 3 2 3 3 3 3 2
|Fuslur range
|-
|[[47edo|47]]
|4 4 4 1 4 4 4 4 1 4 4 4 4 1
|
|-
|[[50edo|50]]
|4 4 4 2 4 4 4 4 2 4 4 4 4 2
|
|-
|[[53edo|53]]
|4 4 4 3 4 4 4 4 3 4 4 4 4 3
|
|-
|[[58edo|58]]
|5 5 5 1 5 5 5 5 1 5 5 5 5 1
|
|-
|[[61edo|61]]
|5 5 5 2 5 5 5 5 2 5 5 5 5 2
|Split-φ
|-
|[[64edo|64]]
|5 5 5 3 5 5 5 5 3 5 5 5 5 3
|φ
|-
|[[67edo|67]]
|5 5 5 4 5 5 5 5 4 5 5 5 5 4
|
|-
|[[69edo|69]]
|6 6 6 1 6 6 6 6 1 6 6 6 6 1
|
|-
|[[81edo|81]]
|6 6 6 5 6 6 6 6 5 6 6 6 6 5
|
|-
|[[80edo|80]]
|7 7 7 1 7 7 7 7 1 7 7 7 7 1
|
|-
|[[83-limit|83]]
|7 7 7 2 7 7 7 7 2 7 7 7 7 2
|
|-
|[[86edo|86]]
|7 7 7 3 7 7 7 7 3 7 7 7 7 3
|
|-
|[[89edo|89]]
|7 7 7 4 7 7 7 7 4 7 7 7 7 4
|
|-
|[[92edo|92]]
|7 7 7 5 7 7 7 7 5 7 7 7 7 5
|
|-
|[[95edo|95]]
|7 7 7 6 7 7 7 7 6 7 7 7 7 6
|
|-
|[[91edo|91]]
|8 8 8 1 8 8 8 8 1 8 8 8 8 1
|
|-
|[[97edo|97]]
|8 8 8 3 8 8 8 8 3 8 8 8 8 3
|Split-φ
|-
|[[103edo|103]]
|8 8 8 5 8 8 8 8 5 8 8 8 8 5
|φ
|-
|[[109edo|109]]
|8 8 8 7 8 8 8 8 7 8 8 8 8 7
|
|-
|[[102edo|102]]
|9 9 9 1 9 9 9 9 1 9 9 9 9 1
|
|-
|[[105edo|105]]
|9 9 9 2 9 9 9 9 2 9 9 9 9 2
|
|-
|[[111edo|111]]
|9 9 9 4 9 9 9 9 4 9 9 9 9 4
|
|-
|[[114edo|114]]
|9 9 9 5 9 9 9 9 5 9 9 9 9 5
|
|-
|[[120edo|120]]
|9 9 9 7 9 9 9 9 7 9 9 9 9 7
|
|-
|[[123edo|123]]
|9 9 9 8 9 9 9 9 8 9 9 9 9 8
|
|-
|[[113edo|113]]
|10 10 10 1 10 10 10 10 1 10 10 10 10 1
|
|-
|[[119edo|119]]
|10 10 10 3 10 10 10 10 3 10 10 10 10 3
|
|-
|[[131edo|131]]
|10 10 10 7 10 10 10 10 7 10 10 10 10 7
|
|-
|[[137edo|137]]
|10 10 10 9 10 10 10 10 9 10 10 10 10 9
|
|-
|[[124edo|124]]
|11 11 11 1 11 11 11 11 1 11 11 11 11 1
|
|-
|[[127edo|127]]
|11 11 11 2 11 11 11 11 2 11 11 11 11 2
|
|-
|[[130edo|130]]
|11 11 11 3 11 11 11 11 3 11 11 11 11 3
|
|-
|[[133edo|133]]
|11 11 11 4 11 11 11 11 4 11 11 11 11 4
|
|-
|[[136edo|136]]
|11 11 11 5 11 11 11 11 5 11 11 11 11 5
|
|-
|[[139edo|139]]
|11 11 11 6 11 11 11 11 6 11 11 11 11 6
|
|-
|[[142edo|142]]
|11 11 11 7 11 11 11 11 7 11 11 11 11 7
|
|-
|[[145edo|145]]
|11 11 11 8 11 11 11 11 8 11 11 11 11 8
|
|-
|[[148edo|148]]
|11 11 11 9 11 11 11 11 9 11 11 11 11 9
|
|-
|[[151edo|151]]
|11 11 11 10 11 11 11 11 10 11 11 11 11 10
|
|-
|[[135edo|135]]
|12 12 12 1 12 12 12 12 1 12 12 12 12 1
|
|-
|[[147edo|147]]
|12 12 12 5 12 12 12 12 5 12 12 12 12 5
|
|-
|[[153edo|153]]
|12 12 12 7 12 12 12 12 7 12 12 12 12 7
|
|-
|[[165edo|165]]
|12 12 12 11 12 12 12 12 11 12 12 12 12 11
|
|-
|[[146edo|146]]
|13 13 13 1 13 13 13 13 1 13 13 13 13 1
|
|-
|[[149edo|149]]
|13 13 13 2 13 13 13 13 2 13 13 13 13 2
|
|-
|[[152edo|152]]
|13 13 13 3 13 13 13 13 3 13 13 13 13 3
|
|-
|[[155edo|155]]
|13 13 13 4 13 13 13 13 4 13 13 13 13 4
|
|-
|[[158edo|158]]
|13 13 13 5 13 13 13 13 5 13 13 13 13 5
|Split-φ
|-
|[[161edo|161]]
|13 13 13 6 13 13 13 13 6 13 13 13 13 6
|
|-
|[[164edo|164]]
|13 13 13 7 13 13 13 13 7 13 13 13 13 7
|
|-
|[[167edo|167]]
|13 13 13 8 13 13 13 13 8 13 13 13 13 8
|φ
|-
|[[170edo|170]]
|13 13 13 9 13 13 13 13 9 13 13 13 13 9
|
|-
|[[173edo|173]]
|13 13 13 10 13 13 13 13 10 13 13 13 13 10
|
|-
|[[176edo|176]]
|13 13 13 11 13 13 13 13 11 13 13 13 13 11
|
|-
|[[179edo|179]]
|13 13 13 12 13 13 13 13 12 13 13 13 13 12
|
|-
|[[157edo|157]]
|14 14 14 1 14 14 14 14 1 14 14 14 14 1
|
|-
|[[163edo|163]]
|14 14 14 3 14 14 14 14 3 14 14 14 14 3
|
|-
|[[169edo|169]]
|14 14 14 5 14 14 14 14 5 14 14 14 14 5
|
|-
|[[181edo|181]]
|14 14 14 9 14 14 14 14 9 14 14 14 14 9
|
|-
|[[187edo|187]]
|14 14 14 11 14 14 14 14 11 14 14 14 14 11
|
|-
|[[193edo|193]]
|14 14 14 13 14 14 14 14 13 14 14 14 14 13
|
|-
|[[168edo|168]]
|15 15 15 1 15 15 15 15 1 15 15 15 15 1
|
|-
|[[171edo|171]]
|15 15 15 2 15 15 15 15 2 15 15 15 15 2
|
|-
|[[177edo|177]]
|15 15 15 4 15 15 15 15 4 15 15 15 15 4
|
|-
|[[186edo|186]]
|15 15 15 7 15 15 15 15 7 15 15 15 15 7
|
|-
|[[189edo|189]]
|15 15 15 8 15 15 15 15 8 15 15 15 15 8
|
|-
|[[198edo|198]]
|15 15 15 11 15 15 15 15 11 15 15 15 15 11
|
|-
|[[204edo|204]]
|15 15 15 13 15 15 15 15 13 15 15 15 15 13
|
|-
|[[207edo|207]]
|15 15 15 14 15 15 15 15 14 15 15 15 15 14
|
|-
|[[179edo|179]]
|16 16 16 1 16 16 16 16 1 16 16 16 16 1
|
|-
|[[185edo|185]]
|16 16 16 3 16 16 16 16 3 16 16 16 16 3
|
|-
|[[191edo|191]]
|16 16 16 5 16 16 16 16 5 16 16 16 16 5
|
|-
|[[197edo|197]]
|16 16 16 7 16 16 16 16 7 16 16 16 16 7
|
|-
|[[203edo|203]]
|16 16 16 9 16 16 16 16 9 16 16 16 16 9
|
|-
|[[209edo|209]]
|16 16 16 11 16 16 16 16 11 16 16 16 16 11
|
|-
|[[215edo|215]]
|16 16 16 13 16 16 16 16 13 16 16 16 16 13
|
|-
|[[221edo|221]]
|16 16 16 15 16 16 16 16 15 16 16 16 16 15
|
|-
|[[190edo|190]]
|17 17 17 1 17 17 17 17 1 17 17 17 17 1
|
|-
|[[193edo|193]]
|17 17 17 2 17 17 17 17 2 17 17 17 17 2
|
|-
|[[196edo|196]]
|17 17 17 3 17 17 17 17 3 17 17 17 17 3
|
|-
|[[199edo|199]]
|17 17 17 4 17 17 17 17 4 17 17 17 17 4
|
|-
|[[202edo|202]]
|17 17 17 5 17 17 17 17 5 17 17 17 17 5
|Top limit for Lusfur range
|-
|[[205edo|205]]
|17 17 17 6 17 17 17 17 6 17 17 17 17 6
|
|-
|[[208edo|208]]
|17 17 17 7 17 17 17 17 7 17 17 17 17 7
|
|-
|[[211edo|211]]
|17 17 17 8 17 17 17 17 8 17 17 17 17 8
|
|-
|[[214edo|214]]
|17 17 17 9 17 17 17 17 9 17 17 17 17 9
|
|-
|[[217edo|217]]
|17 17 17 10 17 17 17 17 10 17 17 17 17 10
|
|-
|[[220edo|220]]
|17 17 17 11 17 17 17 17 11 17 17 17 17 11
|
|-
|[[223edo|223]]
|17 17 17 12 17 17 17 17 12 17 17 17 17 12
|Top limit for Fuslur range
|-
|[[226edo|226]]
|17 17 17 13 17 17 17 17 13 17 17 17 17 13
|
|-
|[[229edo|229]]
|17 17 17 14 17 17 17 17 14 17 17 17 17 14
|
|-
|[[232edo|232]]
|17 17 17 15 17 17 17 17 15 17 17 17 17 15
|
|-
|[[235edo|235]]
|17 17 17 16 17 17 17 17 16 17 17 17 17 16
|
|}
[[Category:14-tone scales]]</text>
      <sha1>ia8jvtwnstupyokmgdfj4ijswc92gdv</sha1>
    </revision>
    <revision>
      <id>136571</id>
      <parentid>109592</parentid>
      <timestamp>2024-02-22T03:24:44Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Mos intro</comment>
      <origin>136571</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8806" sha1="4iwxs0k3ze3scfyh9z3mjw1hrlxqp9p" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
{{MOS intro|Other Names=the Ketradektriatoh scale}}

The '''11L 3s''' [[MOS scale]] was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]]

This is a type of scale which denotes the use of a scale placed between [[11edo]] and [[14edo]].

It employs a ratio generator between [[41/32]] and [[9/7]] ([[25edo]] being the middle size of the Ketradektriatoh spectrum, in the 2:1 relation).

This results in a variant of tetradecatonic scale which conforms by this scheme: LLLLsLLLLsLLLs.
== Scale tree ==
The table below shows an extension of [[edo]]s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
{| class="wikitable"
|-
| 4\[[11edo|11]]
| 
| 
| 
| 
| 
| 
| 436.364
| 109.091
| 0
| style="text-align:center;" | 
|-
| 
| 
| 
| 
| 
| 
| 29\[[80edo|80]]
| 435
| 105
| 15
| 
|-
| 
| 
| 
| 
| 
| 25\[[69edo|69]]
| 
| 434.783
| 104.348
| 17.391
| 
|-
| 
| 
| 
| 
| 21\[[58edo|58]]
| 
| 
| 434.483
| 103.448
| 20.69
| 
|-
| 
| 
| 
| 17\[[47edo|47]]
| 
| 
| 
| 434.043
| 102.128
| 25.532
| 
|-
| 
| 
| 
| 
| 30\[[83edo|83]]
| 
| 
| 433.735
| 101.208
| 28.916
| 
|-
| 
| 
| 
| 
| 
| 
| 73\[[202edo|202]]
| 433.663
| 100.990
| 29.703
| Since here are the optimal range Lufsur mode (?)
|-
| 
| 
| 
| 
| 
| 43\[[119edo|119]]
| 
| 433.613
| 100.840
| 30.252
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.459
| 100.377
| 31.95
| 
|-
| 
| 
| 13\[[36edo|36]]
| 
| 
| 
| 
| 433.333
| 100
| 33.333
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.048
| 99.144
| 36.473
| 
|-
| 
| 
| 
| 
| 35\97
| 
| 
| 432.99
| 98.969
| 37.113
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 432.933
| 98.799
| 37.738
| 
|-
| 
| 
| 
| 22\[[61edo|61]]
| 
| 
| 
| 432.787
| 98.361
| 39.344
| 
|-
| 
| 9\[[25edo|25]]
| 
| 
| 
| 
| 
| 432
| 96
| 48
| style="text-align:center;" | Boundary of propriety;

generators smaller than this are proper
|-
| 
| 
| 
| 
| 
| 
| 
| 431.417
| 94.25
| 54.4155
| 
|-
| 
| 
| 
| 23\[[64edo|64]]
| 
| 
| 
| 431.25
| 93.75
| 56.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 431.1185
| 93.355
| 57.697
| 
|-
| 
| 
| 
| 
| 37\103
| 
| 
| 431.068
| 93.204
| 58.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 430.984
| 92.952
| 58.175
| 
|-
| 
| 
| 14\[[39edo|39]]
| 
| 
| 
| 
| 430.769
| 92.308
| 61.538
| 
|-
| 
| 
| 
| 
| 
| 47\[[131edo|131]]
| 
| 430.534
| 91.603
| 64.122
| 
|-
| 
| 
| 
| 
| 
| 
| 80\[[223edo|223]]
| 430.493
| 91.480
| 64.575
| Until here are the optimal range Fuslur mode (?)
|-
| 
| 
| 
| 
| 33\[[92edo|92]]
| 
| 
| 430.435
| 91.304
| 65.217
| 
|-
| 
| 
| 
| 19\[[53edo|53]]
| 
| 
| 
| 430.189
| 90.566
| 67.925
| 
|-
| 
| 
| 
| 
| 24\[[67edo|67]]
| 
| 
| 429.851
| 89.552
| 71.642
| 
|-
| 
| 
| 
| 
| 
| 29\[[81edo|81]]
| 
| 429.63
| 88.889
| 74.074
| 
|-
| 
| 
| 
| 
| 
| 
| 34\[[95edo|95]]
| 429.474
| 88.421
| 75.7895
| 
|-
| 5\[[14edo|14]]
| 
| 
| 
| 
| 
| 
| 428.571
| 85.714
| 85.714
| style="text-align:center;" | 
|}

== As an EDO subset ==
{| class="wikitable sortable"
|EDO
|Subset
|Special properties
|-
|[[25edo|25]]
|2 2 2 1 2 2 2 2 1 2 2 2 2 1
|Middle range
|-
|[[36edo|36]]
|3 3 3 1 3 3 3 3 1 3 3 3 3 1
|Lusfur range
|-
|[[39edo|39]]
|3 3 3 2 3 3 3 3 2 3 3 3 3 2
|Fuslur range
|-
|[[47edo|47]]
|4 4 4 1 4 4 4 4 1 4 4 4 4 1
|
|-
|[[50edo|50]]
|4 4 4 2 4 4 4 4 2 4 4 4 4 2
|
|-
|[[53edo|53]]
|4 4 4 3 4 4 4 4 3 4 4 4 4 3
|
|-
|[[58edo|58]]
|5 5 5 1 5 5 5 5 1 5 5 5 5 1
|
|-
|[[61edo|61]]
|5 5 5 2 5 5 5 5 2 5 5 5 5 2
|Split-φ
|-
|[[64edo|64]]
|5 5 5 3 5 5 5 5 3 5 5 5 5 3
|φ
|-
|[[67edo|67]]
|5 5 5 4 5 5 5 5 4 5 5 5 5 4
|
|-
|[[69edo|69]]
|6 6 6 1 6 6 6 6 1 6 6 6 6 1
|
|-
|[[81edo|81]]
|6 6 6 5 6 6 6 6 5 6 6 6 6 5
|
|-
|[[80edo|80]]
|7 7 7 1 7 7 7 7 1 7 7 7 7 1
|
|-
|[[83-limit|83]]
|7 7 7 2 7 7 7 7 2 7 7 7 7 2
|
|-
|[[86edo|86]]
|7 7 7 3 7 7 7 7 3 7 7 7 7 3
|
|-
|[[89edo|89]]
|7 7 7 4 7 7 7 7 4 7 7 7 7 4
|
|-
|[[92edo|92]]
|7 7 7 5 7 7 7 7 5 7 7 7 7 5
|
|-
|[[95edo|95]]
|7 7 7 6 7 7 7 7 6 7 7 7 7 6
|
|-
|[[91edo|91]]
|8 8 8 1 8 8 8 8 1 8 8 8 8 1
|
|-
|[[97edo|97]]
|8 8 8 3 8 8 8 8 3 8 8 8 8 3
|Split-φ
|-
|[[103edo|103]]
|8 8 8 5 8 8 8 8 5 8 8 8 8 5
|φ
|-
|[[109edo|109]]
|8 8 8 7 8 8 8 8 7 8 8 8 8 7
|
|-
|[[102edo|102]]
|9 9 9 1 9 9 9 9 1 9 9 9 9 1
|
|-
|[[105edo|105]]
|9 9 9 2 9 9 9 9 2 9 9 9 9 2
|
|-
|[[111edo|111]]
|9 9 9 4 9 9 9 9 4 9 9 9 9 4
|
|-
|[[114edo|114]]
|9 9 9 5 9 9 9 9 5 9 9 9 9 5
|
|-
|[[120edo|120]]
|9 9 9 7 9 9 9 9 7 9 9 9 9 7
|
|-
|[[123edo|123]]
|9 9 9 8 9 9 9 9 8 9 9 9 9 8
|
|-
|[[113edo|113]]
|10 10 10 1 10 10 10 10 1 10 10 10 10 1
|
|-
|[[119edo|119]]
|10 10 10 3 10 10 10 10 3 10 10 10 10 3
|
|-
|[[131edo|131]]
|10 10 10 7 10 10 10 10 7 10 10 10 10 7
|
|-
|[[137edo|137]]
|10 10 10 9 10 10 10 10 9 10 10 10 10 9
|
|-
|[[124edo|124]]
|11 11 11 1 11 11 11 11 1 11 11 11 11 1
|
|-
|[[127edo|127]]
|11 11 11 2 11 11 11 11 2 11 11 11 11 2
|
|-
|[[130edo|130]]
|11 11 11 3 11 11 11 11 3 11 11 11 11 3
|
|-
|[[133edo|133]]
|11 11 11 4 11 11 11 11 4 11 11 11 11 4
|
|-
|[[136edo|136]]
|11 11 11 5 11 11 11 11 5 11 11 11 11 5
|
|-
|[[139edo|139]]
|11 11 11 6 11 11 11 11 6 11 11 11 11 6
|
|-
|[[142edo|142]]
|11 11 11 7 11 11 11 11 7 11 11 11 11 7
|
|-
|[[145edo|145]]
|11 11 11 8 11 11 11 11 8 11 11 11 11 8
|
|-
|[[148edo|148]]
|11 11 11 9 11 11 11 11 9 11 11 11 11 9
|
|-
|[[151edo|151]]
|11 11 11 10 11 11 11 11 10 11 11 11 11 10
|
|-
|[[135edo|135]]
|12 12 12 1 12 12 12 12 1 12 12 12 12 1
|
|-
|[[147edo|147]]
|12 12 12 5 12 12 12 12 5 12 12 12 12 5
|
|-
|[[153edo|153]]
|12 12 12 7 12 12 12 12 7 12 12 12 12 7
|
|-
|[[165edo|165]]
|12 12 12 11 12 12 12 12 11 12 12 12 12 11
|
|-
|[[146edo|146]]
|13 13 13 1 13 13 13 13 1 13 13 13 13 1
|
|-
|[[149edo|149]]
|13 13 13 2 13 13 13 13 2 13 13 13 13 2
|
|-
|[[152edo|152]]
|13 13 13 3 13 13 13 13 3 13 13 13 13 3
|
|-
|[[155edo|155]]
|13 13 13 4 13 13 13 13 4 13 13 13 13 4
|
|-
|[[158edo|158]]
|13 13 13 5 13 13 13 13 5 13 13 13 13 5
|Split-φ
|-
|[[161edo|161]]
|13 13 13 6 13 13 13 13 6 13 13 13 13 6
|
|-
|[[164edo|164]]
|13 13 13 7 13 13 13 13 7 13 13 13 13 7
|
|-
|[[167edo|167]]
|13 13 13 8 13 13 13 13 8 13 13 13 13 8
|φ
|-
|[[170edo|170]]
|13 13 13 9 13 13 13 13 9 13 13 13 13 9
|
|-
|[[173edo|173]]
|13 13 13 10 13 13 13 13 10 13 13 13 13 10
|
|-
|[[176edo|176]]
|13 13 13 11 13 13 13 13 11 13 13 13 13 11
|
|-
|[[179edo|179]]
|13 13 13 12 13 13 13 13 12 13 13 13 13 12
|
|-
|[[157edo|157]]
|14 14 14 1 14 14 14 14 1 14 14 14 14 1
|
|-
|[[163edo|163]]
|14 14 14 3 14 14 14 14 3 14 14 14 14 3
|
|-
|[[169edo|169]]
|14 14 14 5 14 14 14 14 5 14 14 14 14 5
|
|-
|[[181edo|181]]
|14 14 14 9 14 14 14 14 9 14 14 14 14 9
|
|-
|[[187edo|187]]
|14 14 14 11 14 14 14 14 11 14 14 14 14 11
|
|-
|[[193edo|193]]
|14 14 14 13 14 14 14 14 13 14 14 14 14 13
|
|-
|[[168edo|168]]
|15 15 15 1 15 15 15 15 1 15 15 15 15 1
|
|-
|[[171edo|171]]
|15 15 15 2 15 15 15 15 2 15 15 15 15 2
|
|-
|[[177edo|177]]
|15 15 15 4 15 15 15 15 4 15 15 15 15 4
|
|-
|[[186edo|186]]
|15 15 15 7 15 15 15 15 7 15 15 15 15 7
|
|-
|[[189edo|189]]
|15 15 15 8 15 15 15 15 8 15 15 15 15 8
|
|-
|[[198edo|198]]
|15 15 15 11 15 15 15 15 11 15 15 15 15 11
|
|-
|[[204edo|204]]
|15 15 15 13 15 15 15 15 13 15 15 15 15 13
|
|-
|[[207edo|207]]
|15 15 15 14 15 15 15 15 14 15 15 15 15 14
|
|-
|[[179edo|179]]
|16 16 16 1 16 16 16 16 1 16 16 16 16 1
|
|-
|[[185edo|185]]
|16 16 16 3 16 16 16 16 3 16 16 16 16 3
|
|-
|[[191edo|191]]
|16 16 16 5 16 16 16 16 5 16 16 16 16 5
|
|-
|[[197edo|197]]
|16 16 16 7 16 16 16 16 7 16 16 16 16 7
|
|-
|[[203edo|203]]
|16 16 16 9 16 16 16 16 9 16 16 16 16 9
|
|-
|[[209edo|209]]
|16 16 16 11 16 16 16 16 11 16 16 16 16 11
|
|-
|[[215edo|215]]
|16 16 16 13 16 16 16 16 13 16 16 16 16 13
|
|-
|[[221edo|221]]
|16 16 16 15 16 16 16 16 15 16 16 16 16 15
|
|-
|[[190edo|190]]
|17 17 17 1 17 17 17 17 1 17 17 17 17 1
|
|-
|[[193edo|193]]
|17 17 17 2 17 17 17 17 2 17 17 17 17 2
|
|-
|[[196edo|196]]
|17 17 17 3 17 17 17 17 3 17 17 17 17 3
|
|-
|[[199edo|199]]
|17 17 17 4 17 17 17 17 4 17 17 17 17 4
|
|-
|[[202edo|202]]
|17 17 17 5 17 17 17 17 5 17 17 17 17 5
|Top limit for Lusfur range
|-
|[[205edo|205]]
|17 17 17 6 17 17 17 17 6 17 17 17 17 6
|
|-
|[[208edo|208]]
|17 17 17 7 17 17 17 17 7 17 17 17 17 7
|
|-
|[[211edo|211]]
|17 17 17 8 17 17 17 17 8 17 17 17 17 8
|
|-
|[[214edo|214]]
|17 17 17 9 17 17 17 17 9 17 17 17 17 9
|
|-
|[[217edo|217]]
|17 17 17 10 17 17 17 17 10 17 17 17 17 10
|
|-
|[[220edo|220]]
|17 17 17 11 17 17 17 17 11 17 17 17 17 11
|
|-
|[[223edo|223]]
|17 17 17 12 17 17 17 17 12 17 17 17 17 12
|Top limit for Fuslur range
|-
|[[226edo|226]]
|17 17 17 13 17 17 17 17 13 17 17 17 17 13
|
|-
|[[229edo|229]]
|17 17 17 14 17 17 17 17 14 17 17 17 17 14
|
|-
|[[232edo|232]]
|17 17 17 15 17 17 17 17 15 17 17 17 17 15
|
|-
|[[235edo|235]]
|17 17 17 16 17 17 17 17 16 17 17 17 17 16
|
|}
[[Category:14-tone scales]]</text>
      <sha1>4iwxs0k3ze3scfyh9z3mjw1hrlxqp9p</sha1>
    </revision>
    <revision>
      <id>136572</id>
      <parentid>136571</parentid>
      <timestamp>2024-02-22T03:24:55Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <minor/>
      <comment>a space</comment>
      <origin>136572</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8805" sha1="e5j3080y52xqkn40fbcnnzpuqtso6ix" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
{{MOS intro|Other Names=the Ketradektriatoh scale}}
The '''11L 3s''' [[MOS scale]] was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]]

This is a type of scale which denotes the use of a scale placed between [[11edo]] and [[14edo]].

It employs a ratio generator between [[41/32]] and [[9/7]] ([[25edo]] being the middle size of the Ketradektriatoh spectrum, in the 2:1 relation).

This results in a variant of tetradecatonic scale which conforms by this scheme: LLLLsLLLLsLLLs.
== Scale tree ==
The table below shows an extension of [[edo]]s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
{| class="wikitable"
|-
| 4\[[11edo|11]]
| 
| 
| 
| 
| 
| 
| 436.364
| 109.091
| 0
| style="text-align:center;" | 
|-
| 
| 
| 
| 
| 
| 
| 29\[[80edo|80]]
| 435
| 105
| 15
| 
|-
| 
| 
| 
| 
| 
| 25\[[69edo|69]]
| 
| 434.783
| 104.348
| 17.391
| 
|-
| 
| 
| 
| 
| 21\[[58edo|58]]
| 
| 
| 434.483
| 103.448
| 20.69
| 
|-
| 
| 
| 
| 17\[[47edo|47]]
| 
| 
| 
| 434.043
| 102.128
| 25.532
| 
|-
| 
| 
| 
| 
| 30\[[83edo|83]]
| 
| 
| 433.735
| 101.208
| 28.916
| 
|-
| 
| 
| 
| 
| 
| 
| 73\[[202edo|202]]
| 433.663
| 100.990
| 29.703
| Since here are the optimal range Lufsur mode (?)
|-
| 
| 
| 
| 
| 
| 43\[[119edo|119]]
| 
| 433.613
| 100.840
| 30.252
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.459
| 100.377
| 31.95
| 
|-
| 
| 
| 13\[[36edo|36]]
| 
| 
| 
| 
| 433.333
| 100
| 33.333
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.048
| 99.144
| 36.473
| 
|-
| 
| 
| 
| 
| 35\97
| 
| 
| 432.99
| 98.969
| 37.113
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 432.933
| 98.799
| 37.738
| 
|-
| 
| 
| 
| 22\[[61edo|61]]
| 
| 
| 
| 432.787
| 98.361
| 39.344
| 
|-
| 
| 9\[[25edo|25]]
| 
| 
| 
| 
| 
| 432
| 96
| 48
| style="text-align:center;" | Boundary of propriety;

generators smaller than this are proper
|-
| 
| 
| 
| 
| 
| 
| 
| 431.417
| 94.25
| 54.4155
| 
|-
| 
| 
| 
| 23\[[64edo|64]]
| 
| 
| 
| 431.25
| 93.75
| 56.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 431.1185
| 93.355
| 57.697
| 
|-
| 
| 
| 
| 
| 37\103
| 
| 
| 431.068
| 93.204
| 58.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 430.984
| 92.952
| 58.175
| 
|-
| 
| 
| 14\[[39edo|39]]
| 
| 
| 
| 
| 430.769
| 92.308
| 61.538
| 
|-
| 
| 
| 
| 
| 
| 47\[[131edo|131]]
| 
| 430.534
| 91.603
| 64.122
| 
|-
| 
| 
| 
| 
| 
| 
| 80\[[223edo|223]]
| 430.493
| 91.480
| 64.575
| Until here are the optimal range Fuslur mode (?)
|-
| 
| 
| 
| 
| 33\[[92edo|92]]
| 
| 
| 430.435
| 91.304
| 65.217
| 
|-
| 
| 
| 
| 19\[[53edo|53]]
| 
| 
| 
| 430.189
| 90.566
| 67.925
| 
|-
| 
| 
| 
| 
| 24\[[67edo|67]]
| 
| 
| 429.851
| 89.552
| 71.642
| 
|-
| 
| 
| 
| 
| 
| 29\[[81edo|81]]
| 
| 429.63
| 88.889
| 74.074
| 
|-
| 
| 
| 
| 
| 
| 
| 34\[[95edo|95]]
| 429.474
| 88.421
| 75.7895
| 
|-
| 5\[[14edo|14]]
| 
| 
| 
| 
| 
| 
| 428.571
| 85.714
| 85.714
| style="text-align:center;" | 
|}

== As an EDO subset ==
{| class="wikitable sortable"
|EDO
|Subset
|Special properties
|-
|[[25edo|25]]
|2 2 2 1 2 2 2 2 1 2 2 2 2 1
|Middle range
|-
|[[36edo|36]]
|3 3 3 1 3 3 3 3 1 3 3 3 3 1
|Lusfur range
|-
|[[39edo|39]]
|3 3 3 2 3 3 3 3 2 3 3 3 3 2
|Fuslur range
|-
|[[47edo|47]]
|4 4 4 1 4 4 4 4 1 4 4 4 4 1
|
|-
|[[50edo|50]]
|4 4 4 2 4 4 4 4 2 4 4 4 4 2
|
|-
|[[53edo|53]]
|4 4 4 3 4 4 4 4 3 4 4 4 4 3
|
|-
|[[58edo|58]]
|5 5 5 1 5 5 5 5 1 5 5 5 5 1
|
|-
|[[61edo|61]]
|5 5 5 2 5 5 5 5 2 5 5 5 5 2
|Split-φ
|-
|[[64edo|64]]
|5 5 5 3 5 5 5 5 3 5 5 5 5 3
|φ
|-
|[[67edo|67]]
|5 5 5 4 5 5 5 5 4 5 5 5 5 4
|
|-
|[[69edo|69]]
|6 6 6 1 6 6 6 6 1 6 6 6 6 1
|
|-
|[[81edo|81]]
|6 6 6 5 6 6 6 6 5 6 6 6 6 5
|
|-
|[[80edo|80]]
|7 7 7 1 7 7 7 7 1 7 7 7 7 1
|
|-
|[[83-limit|83]]
|7 7 7 2 7 7 7 7 2 7 7 7 7 2
|
|-
|[[86edo|86]]
|7 7 7 3 7 7 7 7 3 7 7 7 7 3
|
|-
|[[89edo|89]]
|7 7 7 4 7 7 7 7 4 7 7 7 7 4
|
|-
|[[92edo|92]]
|7 7 7 5 7 7 7 7 5 7 7 7 7 5
|
|-
|[[95edo|95]]
|7 7 7 6 7 7 7 7 6 7 7 7 7 6
|
|-
|[[91edo|91]]
|8 8 8 1 8 8 8 8 1 8 8 8 8 1
|
|-
|[[97edo|97]]
|8 8 8 3 8 8 8 8 3 8 8 8 8 3
|Split-φ
|-
|[[103edo|103]]
|8 8 8 5 8 8 8 8 5 8 8 8 8 5
|φ
|-
|[[109edo|109]]
|8 8 8 7 8 8 8 8 7 8 8 8 8 7
|
|-
|[[102edo|102]]
|9 9 9 1 9 9 9 9 1 9 9 9 9 1
|
|-
|[[105edo|105]]
|9 9 9 2 9 9 9 9 2 9 9 9 9 2
|
|-
|[[111edo|111]]
|9 9 9 4 9 9 9 9 4 9 9 9 9 4
|
|-
|[[114edo|114]]
|9 9 9 5 9 9 9 9 5 9 9 9 9 5
|
|-
|[[120edo|120]]
|9 9 9 7 9 9 9 9 7 9 9 9 9 7
|
|-
|[[123edo|123]]
|9 9 9 8 9 9 9 9 8 9 9 9 9 8
|
|-
|[[113edo|113]]
|10 10 10 1 10 10 10 10 1 10 10 10 10 1
|
|-
|[[119edo|119]]
|10 10 10 3 10 10 10 10 3 10 10 10 10 3
|
|-
|[[131edo|131]]
|10 10 10 7 10 10 10 10 7 10 10 10 10 7
|
|-
|[[137edo|137]]
|10 10 10 9 10 10 10 10 9 10 10 10 10 9
|
|-
|[[124edo|124]]
|11 11 11 1 11 11 11 11 1 11 11 11 11 1
|
|-
|[[127edo|127]]
|11 11 11 2 11 11 11 11 2 11 11 11 11 2
|
|-
|[[130edo|130]]
|11 11 11 3 11 11 11 11 3 11 11 11 11 3
|
|-
|[[133edo|133]]
|11 11 11 4 11 11 11 11 4 11 11 11 11 4
|
|-
|[[136edo|136]]
|11 11 11 5 11 11 11 11 5 11 11 11 11 5
|
|-
|[[139edo|139]]
|11 11 11 6 11 11 11 11 6 11 11 11 11 6
|
|-
|[[142edo|142]]
|11 11 11 7 11 11 11 11 7 11 11 11 11 7
|
|-
|[[145edo|145]]
|11 11 11 8 11 11 11 11 8 11 11 11 11 8
|
|-
|[[148edo|148]]
|11 11 11 9 11 11 11 11 9 11 11 11 11 9
|
|-
|[[151edo|151]]
|11 11 11 10 11 11 11 11 10 11 11 11 11 10
|
|-
|[[135edo|135]]
|12 12 12 1 12 12 12 12 1 12 12 12 12 1
|
|-
|[[147edo|147]]
|12 12 12 5 12 12 12 12 5 12 12 12 12 5
|
|-
|[[153edo|153]]
|12 12 12 7 12 12 12 12 7 12 12 12 12 7
|
|-
|[[165edo|165]]
|12 12 12 11 12 12 12 12 11 12 12 12 12 11
|
|-
|[[146edo|146]]
|13 13 13 1 13 13 13 13 1 13 13 13 13 1
|
|-
|[[149edo|149]]
|13 13 13 2 13 13 13 13 2 13 13 13 13 2
|
|-
|[[152edo|152]]
|13 13 13 3 13 13 13 13 3 13 13 13 13 3
|
|-
|[[155edo|155]]
|13 13 13 4 13 13 13 13 4 13 13 13 13 4
|
|-
|[[158edo|158]]
|13 13 13 5 13 13 13 13 5 13 13 13 13 5
|Split-φ
|-
|[[161edo|161]]
|13 13 13 6 13 13 13 13 6 13 13 13 13 6
|
|-
|[[164edo|164]]
|13 13 13 7 13 13 13 13 7 13 13 13 13 7
|
|-
|[[167edo|167]]
|13 13 13 8 13 13 13 13 8 13 13 13 13 8
|φ
|-
|[[170edo|170]]
|13 13 13 9 13 13 13 13 9 13 13 13 13 9
|
|-
|[[173edo|173]]
|13 13 13 10 13 13 13 13 10 13 13 13 13 10
|
|-
|[[176edo|176]]
|13 13 13 11 13 13 13 13 11 13 13 13 13 11
|
|-
|[[179edo|179]]
|13 13 13 12 13 13 13 13 12 13 13 13 13 12
|
|-
|[[157edo|157]]
|14 14 14 1 14 14 14 14 1 14 14 14 14 1
|
|-
|[[163edo|163]]
|14 14 14 3 14 14 14 14 3 14 14 14 14 3
|
|-
|[[169edo|169]]
|14 14 14 5 14 14 14 14 5 14 14 14 14 5
|
|-
|[[181edo|181]]
|14 14 14 9 14 14 14 14 9 14 14 14 14 9
|
|-
|[[187edo|187]]
|14 14 14 11 14 14 14 14 11 14 14 14 14 11
|
|-
|[[193edo|193]]
|14 14 14 13 14 14 14 14 13 14 14 14 14 13
|
|-
|[[168edo|168]]
|15 15 15 1 15 15 15 15 1 15 15 15 15 1
|
|-
|[[171edo|171]]
|15 15 15 2 15 15 15 15 2 15 15 15 15 2
|
|-
|[[177edo|177]]
|15 15 15 4 15 15 15 15 4 15 15 15 15 4
|
|-
|[[186edo|186]]
|15 15 15 7 15 15 15 15 7 15 15 15 15 7
|
|-
|[[189edo|189]]
|15 15 15 8 15 15 15 15 8 15 15 15 15 8
|
|-
|[[198edo|198]]
|15 15 15 11 15 15 15 15 11 15 15 15 15 11
|
|-
|[[204edo|204]]
|15 15 15 13 15 15 15 15 13 15 15 15 15 13
|
|-
|[[207edo|207]]
|15 15 15 14 15 15 15 15 14 15 15 15 15 14
|
|-
|[[179edo|179]]
|16 16 16 1 16 16 16 16 1 16 16 16 16 1
|
|-
|[[185edo|185]]
|16 16 16 3 16 16 16 16 3 16 16 16 16 3
|
|-
|[[191edo|191]]
|16 16 16 5 16 16 16 16 5 16 16 16 16 5
|
|-
|[[197edo|197]]
|16 16 16 7 16 16 16 16 7 16 16 16 16 7
|
|-
|[[203edo|203]]
|16 16 16 9 16 16 16 16 9 16 16 16 16 9
|
|-
|[[209edo|209]]
|16 16 16 11 16 16 16 16 11 16 16 16 16 11
|
|-
|[[215edo|215]]
|16 16 16 13 16 16 16 16 13 16 16 16 16 13
|
|-
|[[221edo|221]]
|16 16 16 15 16 16 16 16 15 16 16 16 16 15
|
|-
|[[190edo|190]]
|17 17 17 1 17 17 17 17 1 17 17 17 17 1
|
|-
|[[193edo|193]]
|17 17 17 2 17 17 17 17 2 17 17 17 17 2
|
|-
|[[196edo|196]]
|17 17 17 3 17 17 17 17 3 17 17 17 17 3
|
|-
|[[199edo|199]]
|17 17 17 4 17 17 17 17 4 17 17 17 17 4
|
|-
|[[202edo|202]]
|17 17 17 5 17 17 17 17 5 17 17 17 17 5
|Top limit for Lusfur range
|-
|[[205edo|205]]
|17 17 17 6 17 17 17 17 6 17 17 17 17 6
|
|-
|[[208edo|208]]
|17 17 17 7 17 17 17 17 7 17 17 17 17 7
|
|-
|[[211edo|211]]
|17 17 17 8 17 17 17 17 8 17 17 17 17 8
|
|-
|[[214edo|214]]
|17 17 17 9 17 17 17 17 9 17 17 17 17 9
|
|-
|[[217edo|217]]
|17 17 17 10 17 17 17 17 10 17 17 17 17 10
|
|-
|[[220edo|220]]
|17 17 17 11 17 17 17 17 11 17 17 17 17 11
|
|-
|[[223edo|223]]
|17 17 17 12 17 17 17 17 12 17 17 17 17 12
|Top limit for Fuslur range
|-
|[[226edo|226]]
|17 17 17 13 17 17 17 17 13 17 17 17 17 13
|
|-
|[[229edo|229]]
|17 17 17 14 17 17 17 17 14 17 17 17 17 14
|
|-
|[[232edo|232]]
|17 17 17 15 17 17 17 17 15 17 17 17 17 15
|
|-
|[[235edo|235]]
|17 17 17 16 17 17 17 17 16 17 17 17 17 16
|
|}
[[Category:14-tone scales]]</text>
      <sha1>e5j3080y52xqkn40fbcnnzpuqtso6ix</sha1>
    </revision>
    <revision>
      <id>137597</id>
      <parentid>136572</parentid>
      <timestamp>2024-02-29T23:16:53Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Added todo-cleanup</comment>
      <origin>137597</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8878" sha1="hkpksbp6x66nwi7yj9ef4k3mgczplth" xml:space="preserve">{{todo|cleanup|inline=1|text=Replace scale tree, populate with entries}}
{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
{{MOS intro|Other Names=the Ketradektriatoh scale}}
The '''11L 3s''' [[MOS scale]] was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]]

This is a type of scale which denotes the use of a scale placed between [[11edo]] and [[14edo]].

It employs a ratio generator between [[41/32]] and [[9/7]] ([[25edo]] being the middle size of the Ketradektriatoh spectrum, in the 2:1 relation).

This results in a variant of tetradecatonic scale which conforms by this scheme: LLLLsLLLLsLLLs.
== Scale tree ==
The table below shows an extension of [[edo]]s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
{| class="wikitable"
|-
| 4\[[11edo|11]]
| 
| 
| 
| 
| 
| 
| 436.364
| 109.091
| 0
| style="text-align:center;" | 
|-
| 
| 
| 
| 
| 
| 
| 29\[[80edo|80]]
| 435
| 105
| 15
| 
|-
| 
| 
| 
| 
| 
| 25\[[69edo|69]]
| 
| 434.783
| 104.348
| 17.391
| 
|-
| 
| 
| 
| 
| 21\[[58edo|58]]
| 
| 
| 434.483
| 103.448
| 20.69
| 
|-
| 
| 
| 
| 17\[[47edo|47]]
| 
| 
| 
| 434.043
| 102.128
| 25.532
| 
|-
| 
| 
| 
| 
| 30\[[83edo|83]]
| 
| 
| 433.735
| 101.208
| 28.916
| 
|-
| 
| 
| 
| 
| 
| 
| 73\[[202edo|202]]
| 433.663
| 100.990
| 29.703
| Since here are the optimal range Lufsur mode (?)
|-
| 
| 
| 
| 
| 
| 43\[[119edo|119]]
| 
| 433.613
| 100.840
| 30.252
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.459
| 100.377
| 31.95
| 
|-
| 
| 
| 13\[[36edo|36]]
| 
| 
| 
| 
| 433.333
| 100
| 33.333
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.048
| 99.144
| 36.473
| 
|-
| 
| 
| 
| 
| 35\97
| 
| 
| 432.99
| 98.969
| 37.113
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 432.933
| 98.799
| 37.738
| 
|-
| 
| 
| 
| 22\[[61edo|61]]
| 
| 
| 
| 432.787
| 98.361
| 39.344
| 
|-
| 
| 9\[[25edo|25]]
| 
| 
| 
| 
| 
| 432
| 96
| 48
| style="text-align:center;" | Boundary of propriety;

generators smaller than this are proper
|-
| 
| 
| 
| 
| 
| 
| 
| 431.417
| 94.25
| 54.4155
| 
|-
| 
| 
| 
| 23\[[64edo|64]]
| 
| 
| 
| 431.25
| 93.75
| 56.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 431.1185
| 93.355
| 57.697
| 
|-
| 
| 
| 
| 
| 37\103
| 
| 
| 431.068
| 93.204
| 58.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 430.984
| 92.952
| 58.175
| 
|-
| 
| 
| 14\[[39edo|39]]
| 
| 
| 
| 
| 430.769
| 92.308
| 61.538
| 
|-
| 
| 
| 
| 
| 
| 47\[[131edo|131]]
| 
| 430.534
| 91.603
| 64.122
| 
|-
| 
| 
| 
| 
| 
| 
| 80\[[223edo|223]]
| 430.493
| 91.480
| 64.575
| Until here are the optimal range Fuslur mode (?)
|-
| 
| 
| 
| 
| 33\[[92edo|92]]
| 
| 
| 430.435
| 91.304
| 65.217
| 
|-
| 
| 
| 
| 19\[[53edo|53]]
| 
| 
| 
| 430.189
| 90.566
| 67.925
| 
|-
| 
| 
| 
| 
| 24\[[67edo|67]]
| 
| 
| 429.851
| 89.552
| 71.642
| 
|-
| 
| 
| 
| 
| 
| 29\[[81edo|81]]
| 
| 429.63
| 88.889
| 74.074
| 
|-
| 
| 
| 
| 
| 
| 
| 34\[[95edo|95]]
| 429.474
| 88.421
| 75.7895
| 
|-
| 5\[[14edo|14]]
| 
| 
| 
| 
| 
| 
| 428.571
| 85.714
| 85.714
| style="text-align:center;" | 
|}

== As an EDO subset ==
{| class="wikitable sortable"
|EDO
|Subset
|Special properties
|-
|[[25edo|25]]
|2 2 2 1 2 2 2 2 1 2 2 2 2 1
|Middle range
|-
|[[36edo|36]]
|3 3 3 1 3 3 3 3 1 3 3 3 3 1
|Lusfur range
|-
|[[39edo|39]]
|3 3 3 2 3 3 3 3 2 3 3 3 3 2
|Fuslur range
|-
|[[47edo|47]]
|4 4 4 1 4 4 4 4 1 4 4 4 4 1
|
|-
|[[50edo|50]]
|4 4 4 2 4 4 4 4 2 4 4 4 4 2
|
|-
|[[53edo|53]]
|4 4 4 3 4 4 4 4 3 4 4 4 4 3
|
|-
|[[58edo|58]]
|5 5 5 1 5 5 5 5 1 5 5 5 5 1
|
|-
|[[61edo|61]]
|5 5 5 2 5 5 5 5 2 5 5 5 5 2
|Split-φ
|-
|[[64edo|64]]
|5 5 5 3 5 5 5 5 3 5 5 5 5 3
|φ
|-
|[[67edo|67]]
|5 5 5 4 5 5 5 5 4 5 5 5 5 4
|
|-
|[[69edo|69]]
|6 6 6 1 6 6 6 6 1 6 6 6 6 1
|
|-
|[[81edo|81]]
|6 6 6 5 6 6 6 6 5 6 6 6 6 5
|
|-
|[[80edo|80]]
|7 7 7 1 7 7 7 7 1 7 7 7 7 1
|
|-
|[[83-limit|83]]
|7 7 7 2 7 7 7 7 2 7 7 7 7 2
|
|-
|[[86edo|86]]
|7 7 7 3 7 7 7 7 3 7 7 7 7 3
|
|-
|[[89edo|89]]
|7 7 7 4 7 7 7 7 4 7 7 7 7 4
|
|-
|[[92edo|92]]
|7 7 7 5 7 7 7 7 5 7 7 7 7 5
|
|-
|[[95edo|95]]
|7 7 7 6 7 7 7 7 6 7 7 7 7 6
|
|-
|[[91edo|91]]
|8 8 8 1 8 8 8 8 1 8 8 8 8 1
|
|-
|[[97edo|97]]
|8 8 8 3 8 8 8 8 3 8 8 8 8 3
|Split-φ
|-
|[[103edo|103]]
|8 8 8 5 8 8 8 8 5 8 8 8 8 5
|φ
|-
|[[109edo|109]]
|8 8 8 7 8 8 8 8 7 8 8 8 8 7
|
|-
|[[102edo|102]]
|9 9 9 1 9 9 9 9 1 9 9 9 9 1
|
|-
|[[105edo|105]]
|9 9 9 2 9 9 9 9 2 9 9 9 9 2
|
|-
|[[111edo|111]]
|9 9 9 4 9 9 9 9 4 9 9 9 9 4
|
|-
|[[114edo|114]]
|9 9 9 5 9 9 9 9 5 9 9 9 9 5
|
|-
|[[120edo|120]]
|9 9 9 7 9 9 9 9 7 9 9 9 9 7
|
|-
|[[123edo|123]]
|9 9 9 8 9 9 9 9 8 9 9 9 9 8
|
|-
|[[113edo|113]]
|10 10 10 1 10 10 10 10 1 10 10 10 10 1
|
|-
|[[119edo|119]]
|10 10 10 3 10 10 10 10 3 10 10 10 10 3
|
|-
|[[131edo|131]]
|10 10 10 7 10 10 10 10 7 10 10 10 10 7
|
|-
|[[137edo|137]]
|10 10 10 9 10 10 10 10 9 10 10 10 10 9
|
|-
|[[124edo|124]]
|11 11 11 1 11 11 11 11 1 11 11 11 11 1
|
|-
|[[127edo|127]]
|11 11 11 2 11 11 11 11 2 11 11 11 11 2
|
|-
|[[130edo|130]]
|11 11 11 3 11 11 11 11 3 11 11 11 11 3
|
|-
|[[133edo|133]]
|11 11 11 4 11 11 11 11 4 11 11 11 11 4
|
|-
|[[136edo|136]]
|11 11 11 5 11 11 11 11 5 11 11 11 11 5
|
|-
|[[139edo|139]]
|11 11 11 6 11 11 11 11 6 11 11 11 11 6
|
|-
|[[142edo|142]]
|11 11 11 7 11 11 11 11 7 11 11 11 11 7
|
|-
|[[145edo|145]]
|11 11 11 8 11 11 11 11 8 11 11 11 11 8
|
|-
|[[148edo|148]]
|11 11 11 9 11 11 11 11 9 11 11 11 11 9
|
|-
|[[151edo|151]]
|11 11 11 10 11 11 11 11 10 11 11 11 11 10
|
|-
|[[135edo|135]]
|12 12 12 1 12 12 12 12 1 12 12 12 12 1
|
|-
|[[147edo|147]]
|12 12 12 5 12 12 12 12 5 12 12 12 12 5
|
|-
|[[153edo|153]]
|12 12 12 7 12 12 12 12 7 12 12 12 12 7
|
|-
|[[165edo|165]]
|12 12 12 11 12 12 12 12 11 12 12 12 12 11
|
|-
|[[146edo|146]]
|13 13 13 1 13 13 13 13 1 13 13 13 13 1
|
|-
|[[149edo|149]]
|13 13 13 2 13 13 13 13 2 13 13 13 13 2
|
|-
|[[152edo|152]]
|13 13 13 3 13 13 13 13 3 13 13 13 13 3
|
|-
|[[155edo|155]]
|13 13 13 4 13 13 13 13 4 13 13 13 13 4
|
|-
|[[158edo|158]]
|13 13 13 5 13 13 13 13 5 13 13 13 13 5
|Split-φ
|-
|[[161edo|161]]
|13 13 13 6 13 13 13 13 6 13 13 13 13 6
|
|-
|[[164edo|164]]
|13 13 13 7 13 13 13 13 7 13 13 13 13 7
|
|-
|[[167edo|167]]
|13 13 13 8 13 13 13 13 8 13 13 13 13 8
|φ
|-
|[[170edo|170]]
|13 13 13 9 13 13 13 13 9 13 13 13 13 9
|
|-
|[[173edo|173]]
|13 13 13 10 13 13 13 13 10 13 13 13 13 10
|
|-
|[[176edo|176]]
|13 13 13 11 13 13 13 13 11 13 13 13 13 11
|
|-
|[[179edo|179]]
|13 13 13 12 13 13 13 13 12 13 13 13 13 12
|
|-
|[[157edo|157]]
|14 14 14 1 14 14 14 14 1 14 14 14 14 1
|
|-
|[[163edo|163]]
|14 14 14 3 14 14 14 14 3 14 14 14 14 3
|
|-
|[[169edo|169]]
|14 14 14 5 14 14 14 14 5 14 14 14 14 5
|
|-
|[[181edo|181]]
|14 14 14 9 14 14 14 14 9 14 14 14 14 9
|
|-
|[[187edo|187]]
|14 14 14 11 14 14 14 14 11 14 14 14 14 11
|
|-
|[[193edo|193]]
|14 14 14 13 14 14 14 14 13 14 14 14 14 13
|
|-
|[[168edo|168]]
|15 15 15 1 15 15 15 15 1 15 15 15 15 1
|
|-
|[[171edo|171]]
|15 15 15 2 15 15 15 15 2 15 15 15 15 2
|
|-
|[[177edo|177]]
|15 15 15 4 15 15 15 15 4 15 15 15 15 4
|
|-
|[[186edo|186]]
|15 15 15 7 15 15 15 15 7 15 15 15 15 7
|
|-
|[[189edo|189]]
|15 15 15 8 15 15 15 15 8 15 15 15 15 8
|
|-
|[[198edo|198]]
|15 15 15 11 15 15 15 15 11 15 15 15 15 11
|
|-
|[[204edo|204]]
|15 15 15 13 15 15 15 15 13 15 15 15 15 13
|
|-
|[[207edo|207]]
|15 15 15 14 15 15 15 15 14 15 15 15 15 14
|
|-
|[[179edo|179]]
|16 16 16 1 16 16 16 16 1 16 16 16 16 1
|
|-
|[[185edo|185]]
|16 16 16 3 16 16 16 16 3 16 16 16 16 3
|
|-
|[[191edo|191]]
|16 16 16 5 16 16 16 16 5 16 16 16 16 5
|
|-
|[[197edo|197]]
|16 16 16 7 16 16 16 16 7 16 16 16 16 7
|
|-
|[[203edo|203]]
|16 16 16 9 16 16 16 16 9 16 16 16 16 9
|
|-
|[[209edo|209]]
|16 16 16 11 16 16 16 16 11 16 16 16 16 11
|
|-
|[[215edo|215]]
|16 16 16 13 16 16 16 16 13 16 16 16 16 13
|
|-
|[[221edo|221]]
|16 16 16 15 16 16 16 16 15 16 16 16 16 15
|
|-
|[[190edo|190]]
|17 17 17 1 17 17 17 17 1 17 17 17 17 1
|
|-
|[[193edo|193]]
|17 17 17 2 17 17 17 17 2 17 17 17 17 2
|
|-
|[[196edo|196]]
|17 17 17 3 17 17 17 17 3 17 17 17 17 3
|
|-
|[[199edo|199]]
|17 17 17 4 17 17 17 17 4 17 17 17 17 4
|
|-
|[[202edo|202]]
|17 17 17 5 17 17 17 17 5 17 17 17 17 5
|Top limit for Lusfur range
|-
|[[205edo|205]]
|17 17 17 6 17 17 17 17 6 17 17 17 17 6
|
|-
|[[208edo|208]]
|17 17 17 7 17 17 17 17 7 17 17 17 17 7
|
|-
|[[211edo|211]]
|17 17 17 8 17 17 17 17 8 17 17 17 17 8
|
|-
|[[214edo|214]]
|17 17 17 9 17 17 17 17 9 17 17 17 17 9
|
|-
|[[217edo|217]]
|17 17 17 10 17 17 17 17 10 17 17 17 17 10
|
|-
|[[220edo|220]]
|17 17 17 11 17 17 17 17 11 17 17 17 17 11
|
|-
|[[223edo|223]]
|17 17 17 12 17 17 17 17 12 17 17 17 17 12
|Top limit for Fuslur range
|-
|[[226edo|226]]
|17 17 17 13 17 17 17 17 13 17 17 17 17 13
|
|-
|[[229edo|229]]
|17 17 17 14 17 17 17 17 14 17 17 17 17 14
|
|-
|[[232edo|232]]
|17 17 17 15 17 17 17 17 15 17 17 17 17 15
|
|-
|[[235edo|235]]
|17 17 17 16 17 17 17 17 16 17 17 17 17 16
|
|}
[[Category:14-tone scales]]</text>
      <sha1>hkpksbp6x66nwi7yj9ef4k3mgczplth</sha1>
    </revision>
    <revision>
      <id>137601</id>
      <parentid>137597</parentid>
      <timestamp>2024-03-01T00:19:02Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <origin>137601</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8895" sha1="1axbykv88u66skn5wg0wkc53r366cuk" xml:space="preserve">{{todo|cleanup|inline=1|text=Replace scale tree and populate with entries, rework intro}}
{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
{{MOS intro|Other Names=the Ketradektriatoh scale}}
The '''11L 3s''' [[MOS scale]] was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]]

This is a type of scale which denotes the use of a scale placed between [[11edo]] and [[14edo]].

It employs a ratio generator between [[41/32]] and [[9/7]] ([[25edo]] being the middle size of the Ketradektriatoh spectrum, in the 2:1 relation).

This results in a variant of tetradecatonic scale which conforms by this scheme: LLLLsLLLLsLLLs.
== Scale tree ==
The table below shows an extension of [[edo]]s which supports the Ketradektriatoh scale, with respect to the principal generator and their results for each L/s sizes:
{| class="wikitable"
|-
| 4\[[11edo|11]]
| 
| 
| 
| 
| 
| 
| 436.364
| 109.091
| 0
| style="text-align:center;" | 
|-
| 
| 
| 
| 
| 
| 
| 29\[[80edo|80]]
| 435
| 105
| 15
| 
|-
| 
| 
| 
| 
| 
| 25\[[69edo|69]]
| 
| 434.783
| 104.348
| 17.391
| 
|-
| 
| 
| 
| 
| 21\[[58edo|58]]
| 
| 
| 434.483
| 103.448
| 20.69
| 
|-
| 
| 
| 
| 17\[[47edo|47]]
| 
| 
| 
| 434.043
| 102.128
| 25.532
| 
|-
| 
| 
| 
| 
| 30\[[83edo|83]]
| 
| 
| 433.735
| 101.208
| 28.916
| 
|-
| 
| 
| 
| 
| 
| 
| 73\[[202edo|202]]
| 433.663
| 100.990
| 29.703
| Since here are the optimal range Lufsur mode (?)
|-
| 
| 
| 
| 
| 
| 43\[[119edo|119]]
| 
| 433.613
| 100.840
| 30.252
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.459
| 100.377
| 31.95
| 
|-
| 
| 
| 13\[[36edo|36]]
| 
| 
| 
| 
| 433.333
| 100
| 33.333
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 433.048
| 99.144
| 36.473
| 
|-
| 
| 
| 
| 
| 35\97
| 
| 
| 432.99
| 98.969
| 37.113
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 432.933
| 98.799
| 37.738
| 
|-
| 
| 
| 
| 22\[[61edo|61]]
| 
| 
| 
| 432.787
| 98.361
| 39.344
| 
|-
| 
| 9\[[25edo|25]]
| 
| 
| 
| 
| 
| 432
| 96
| 48
| style="text-align:center;" | Boundary of propriety;

generators smaller than this are proper
|-
| 
| 
| 
| 
| 
| 
| 
| 431.417
| 94.25
| 54.4155
| 
|-
| 
| 
| 
| 23\[[64edo|64]]
| 
| 
| 
| 431.25
| 93.75
| 56.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 431.1185
| 93.355
| 57.697
| 
|-
| 
| 
| 
| 
| 37\103
| 
| 
| 431.068
| 93.204
| 58.25
| 
|-
| 
| 
| 
| 
| 
| 
| 
| 430.984
| 92.952
| 58.175
| 
|-
| 
| 
| 14\[[39edo|39]]
| 
| 
| 
| 
| 430.769
| 92.308
| 61.538
| 
|-
| 
| 
| 
| 
| 
| 47\[[131edo|131]]
| 
| 430.534
| 91.603
| 64.122
| 
|-
| 
| 
| 
| 
| 
| 
| 80\[[223edo|223]]
| 430.493
| 91.480
| 64.575
| Until here are the optimal range Fuslur mode (?)
|-
| 
| 
| 
| 
| 33\[[92edo|92]]
| 
| 
| 430.435
| 91.304
| 65.217
| 
|-
| 
| 
| 
| 19\[[53edo|53]]
| 
| 
| 
| 430.189
| 90.566
| 67.925
| 
|-
| 
| 
| 
| 
| 24\[[67edo|67]]
| 
| 
| 429.851
| 89.552
| 71.642
| 
|-
| 
| 
| 
| 
| 
| 29\[[81edo|81]]
| 
| 429.63
| 88.889
| 74.074
| 
|-
| 
| 
| 
| 
| 
| 
| 34\[[95edo|95]]
| 429.474
| 88.421
| 75.7895
| 
|-
| 5\[[14edo|14]]
| 
| 
| 
| 
| 
| 
| 428.571
| 85.714
| 85.714
| style="text-align:center;" | 
|}

== As an EDO subset ==
{| class="wikitable sortable"
|EDO
|Subset
|Special properties
|-
|[[25edo|25]]
|2 2 2 1 2 2 2 2 1 2 2 2 2 1
|Middle range
|-
|[[36edo|36]]
|3 3 3 1 3 3 3 3 1 3 3 3 3 1
|Lusfur range
|-
|[[39edo|39]]
|3 3 3 2 3 3 3 3 2 3 3 3 3 2
|Fuslur range
|-
|[[47edo|47]]
|4 4 4 1 4 4 4 4 1 4 4 4 4 1
|
|-
|[[50edo|50]]
|4 4 4 2 4 4 4 4 2 4 4 4 4 2
|
|-
|[[53edo|53]]
|4 4 4 3 4 4 4 4 3 4 4 4 4 3
|
|-
|[[58edo|58]]
|5 5 5 1 5 5 5 5 1 5 5 5 5 1
|
|-
|[[61edo|61]]
|5 5 5 2 5 5 5 5 2 5 5 5 5 2
|Split-φ
|-
|[[64edo|64]]
|5 5 5 3 5 5 5 5 3 5 5 5 5 3
|φ
|-
|[[67edo|67]]
|5 5 5 4 5 5 5 5 4 5 5 5 5 4
|
|-
|[[69edo|69]]
|6 6 6 1 6 6 6 6 1 6 6 6 6 1
|
|-
|[[81edo|81]]
|6 6 6 5 6 6 6 6 5 6 6 6 6 5
|
|-
|[[80edo|80]]
|7 7 7 1 7 7 7 7 1 7 7 7 7 1
|
|-
|[[83-limit|83]]
|7 7 7 2 7 7 7 7 2 7 7 7 7 2
|
|-
|[[86edo|86]]
|7 7 7 3 7 7 7 7 3 7 7 7 7 3
|
|-
|[[89edo|89]]
|7 7 7 4 7 7 7 7 4 7 7 7 7 4
|
|-
|[[92edo|92]]
|7 7 7 5 7 7 7 7 5 7 7 7 7 5
|
|-
|[[95edo|95]]
|7 7 7 6 7 7 7 7 6 7 7 7 7 6
|
|-
|[[91edo|91]]
|8 8 8 1 8 8 8 8 1 8 8 8 8 1
|
|-
|[[97edo|97]]
|8 8 8 3 8 8 8 8 3 8 8 8 8 3
|Split-φ
|-
|[[103edo|103]]
|8 8 8 5 8 8 8 8 5 8 8 8 8 5
|φ
|-
|[[109edo|109]]
|8 8 8 7 8 8 8 8 7 8 8 8 8 7
|
|-
|[[102edo|102]]
|9 9 9 1 9 9 9 9 1 9 9 9 9 1
|
|-
|[[105edo|105]]
|9 9 9 2 9 9 9 9 2 9 9 9 9 2
|
|-
|[[111edo|111]]
|9 9 9 4 9 9 9 9 4 9 9 9 9 4
|
|-
|[[114edo|114]]
|9 9 9 5 9 9 9 9 5 9 9 9 9 5
|
|-
|[[120edo|120]]
|9 9 9 7 9 9 9 9 7 9 9 9 9 7
|
|-
|[[123edo|123]]
|9 9 9 8 9 9 9 9 8 9 9 9 9 8
|
|-
|[[113edo|113]]
|10 10 10 1 10 10 10 10 1 10 10 10 10 1
|
|-
|[[119edo|119]]
|10 10 10 3 10 10 10 10 3 10 10 10 10 3
|
|-
|[[131edo|131]]
|10 10 10 7 10 10 10 10 7 10 10 10 10 7
|
|-
|[[137edo|137]]
|10 10 10 9 10 10 10 10 9 10 10 10 10 9
|
|-
|[[124edo|124]]
|11 11 11 1 11 11 11 11 1 11 11 11 11 1
|
|-
|[[127edo|127]]
|11 11 11 2 11 11 11 11 2 11 11 11 11 2
|
|-
|[[130edo|130]]
|11 11 11 3 11 11 11 11 3 11 11 11 11 3
|
|-
|[[133edo|133]]
|11 11 11 4 11 11 11 11 4 11 11 11 11 4
|
|-
|[[136edo|136]]
|11 11 11 5 11 11 11 11 5 11 11 11 11 5
|
|-
|[[139edo|139]]
|11 11 11 6 11 11 11 11 6 11 11 11 11 6
|
|-
|[[142edo|142]]
|11 11 11 7 11 11 11 11 7 11 11 11 11 7
|
|-
|[[145edo|145]]
|11 11 11 8 11 11 11 11 8 11 11 11 11 8
|
|-
|[[148edo|148]]
|11 11 11 9 11 11 11 11 9 11 11 11 11 9
|
|-
|[[151edo|151]]
|11 11 11 10 11 11 11 11 10 11 11 11 11 10
|
|-
|[[135edo|135]]
|12 12 12 1 12 12 12 12 1 12 12 12 12 1
|
|-
|[[147edo|147]]
|12 12 12 5 12 12 12 12 5 12 12 12 12 5
|
|-
|[[153edo|153]]
|12 12 12 7 12 12 12 12 7 12 12 12 12 7
|
|-
|[[165edo|165]]
|12 12 12 11 12 12 12 12 11 12 12 12 12 11
|
|-
|[[146edo|146]]
|13 13 13 1 13 13 13 13 1 13 13 13 13 1
|
|-
|[[149edo|149]]
|13 13 13 2 13 13 13 13 2 13 13 13 13 2
|
|-
|[[152edo|152]]
|13 13 13 3 13 13 13 13 3 13 13 13 13 3
|
|-
|[[155edo|155]]
|13 13 13 4 13 13 13 13 4 13 13 13 13 4
|
|-
|[[158edo|158]]
|13 13 13 5 13 13 13 13 5 13 13 13 13 5
|Split-φ
|-
|[[161edo|161]]
|13 13 13 6 13 13 13 13 6 13 13 13 13 6
|
|-
|[[164edo|164]]
|13 13 13 7 13 13 13 13 7 13 13 13 13 7
|
|-
|[[167edo|167]]
|13 13 13 8 13 13 13 13 8 13 13 13 13 8
|φ
|-
|[[170edo|170]]
|13 13 13 9 13 13 13 13 9 13 13 13 13 9
|
|-
|[[173edo|173]]
|13 13 13 10 13 13 13 13 10 13 13 13 13 10
|
|-
|[[176edo|176]]
|13 13 13 11 13 13 13 13 11 13 13 13 13 11
|
|-
|[[179edo|179]]
|13 13 13 12 13 13 13 13 12 13 13 13 13 12
|
|-
|[[157edo|157]]
|14 14 14 1 14 14 14 14 1 14 14 14 14 1
|
|-
|[[163edo|163]]
|14 14 14 3 14 14 14 14 3 14 14 14 14 3
|
|-
|[[169edo|169]]
|14 14 14 5 14 14 14 14 5 14 14 14 14 5
|
|-
|[[181edo|181]]
|14 14 14 9 14 14 14 14 9 14 14 14 14 9
|
|-
|[[187edo|187]]
|14 14 14 11 14 14 14 14 11 14 14 14 14 11
|
|-
|[[193edo|193]]
|14 14 14 13 14 14 14 14 13 14 14 14 14 13
|
|-
|[[168edo|168]]
|15 15 15 1 15 15 15 15 1 15 15 15 15 1
|
|-
|[[171edo|171]]
|15 15 15 2 15 15 15 15 2 15 15 15 15 2
|
|-
|[[177edo|177]]
|15 15 15 4 15 15 15 15 4 15 15 15 15 4
|
|-
|[[186edo|186]]
|15 15 15 7 15 15 15 15 7 15 15 15 15 7
|
|-
|[[189edo|189]]
|15 15 15 8 15 15 15 15 8 15 15 15 15 8
|
|-
|[[198edo|198]]
|15 15 15 11 15 15 15 15 11 15 15 15 15 11
|
|-
|[[204edo|204]]
|15 15 15 13 15 15 15 15 13 15 15 15 15 13
|
|-
|[[207edo|207]]
|15 15 15 14 15 15 15 15 14 15 15 15 15 14
|
|-
|[[179edo|179]]
|16 16 16 1 16 16 16 16 1 16 16 16 16 1
|
|-
|[[185edo|185]]
|16 16 16 3 16 16 16 16 3 16 16 16 16 3
|
|-
|[[191edo|191]]
|16 16 16 5 16 16 16 16 5 16 16 16 16 5
|
|-
|[[197edo|197]]
|16 16 16 7 16 16 16 16 7 16 16 16 16 7
|
|-
|[[203edo|203]]
|16 16 16 9 16 16 16 16 9 16 16 16 16 9
|
|-
|[[209edo|209]]
|16 16 16 11 16 16 16 16 11 16 16 16 16 11
|
|-
|[[215edo|215]]
|16 16 16 13 16 16 16 16 13 16 16 16 16 13
|
|-
|[[221edo|221]]
|16 16 16 15 16 16 16 16 15 16 16 16 16 15
|
|-
|[[190edo|190]]
|17 17 17 1 17 17 17 17 1 17 17 17 17 1
|
|-
|[[193edo|193]]
|17 17 17 2 17 17 17 17 2 17 17 17 17 2
|
|-
|[[196edo|196]]
|17 17 17 3 17 17 17 17 3 17 17 17 17 3
|
|-
|[[199edo|199]]
|17 17 17 4 17 17 17 17 4 17 17 17 17 4
|
|-
|[[202edo|202]]
|17 17 17 5 17 17 17 17 5 17 17 17 17 5
|Top limit for Lusfur range
|-
|[[205edo|205]]
|17 17 17 6 17 17 17 17 6 17 17 17 17 6
|
|-
|[[208edo|208]]
|17 17 17 7 17 17 17 17 7 17 17 17 17 7
|
|-
|[[211edo|211]]
|17 17 17 8 17 17 17 17 8 17 17 17 17 8
|
|-
|[[214edo|214]]
|17 17 17 9 17 17 17 17 9 17 17 17 17 9
|
|-
|[[217edo|217]]
|17 17 17 10 17 17 17 17 10 17 17 17 17 10
|
|-
|[[220edo|220]]
|17 17 17 11 17 17 17 17 11 17 17 17 17 11
|
|-
|[[223edo|223]]
|17 17 17 12 17 17 17 17 12 17 17 17 17 12
|Top limit for Fuslur range
|-
|[[226edo|226]]
|17 17 17 13 17 17 17 17 13 17 17 17 17 13
|
|-
|[[229edo|229]]
|17 17 17 14 17 17 17 17 14 17 17 17 17 14
|
|-
|[[232edo|232]]
|17 17 17 15 17 17 17 17 15 17 17 17 17 15
|
|-
|[[235edo|235]]
|17 17 17 16 17 17 17 17 16 17 17 17 17 16
|
|}
[[Category:14-tone scales]]</text>
      <sha1>1axbykv88u66skn5wg0wkc53r366cuk</sha1>
    </revision>
    <revision>
      <id>137633</id>
      <parentid>137601</parentid>
      <timestamp>2024-03-01T02:09:21Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Replaced scale tree, removed edo table since scale tree displays edos already; removed "lufsur" and "fuslur" as it was never made clear what they meant, pending further investigation</comment>
      <origin>137633</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="344" sha1="pqsefvs4ekydytwownqxn2b5q5d9gm9" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
{{MOS intro|Other Names=the Ketradektriatoh scale}}
This scale was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]].
== Scale tree ==
{{Scale tree}}
[[Category:14-tone scales]]</text>
      <sha1>pqsefvs4ekydytwownqxn2b5q5d9gm9</sha1>
    </revision>
    <revision>
      <id>170575</id>
      <parentid>137633</parentid>
      <timestamp>2024-12-15T06:59:24Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Modes intervals</comment>
      <origin>170575</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="424" sha1="j6g9d8jvdlyqni66oxb1zppvxwnmr57" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
{{MOS intro|Other Names=the Ketradektriatoh scale}}
This scale was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]].

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree}}

{{todo|expand}}
[[Category:14-tone scales]]</text>
      <sha1>j6g9d8jvdlyqni66oxb1zppvxwnmr57</sha1>
    </revision>
    <revision>
      <id>183731</id>
      <parentid>170575</parentid>
      <timestamp>2025-02-28T16:41:20Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183731</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="434" sha1="1tfg1cg13g00xm10q0xydidp27h9s6p" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
{{MOS intro|Other Names=the Ketradektriatoh scale}}
This scale was named the "Ketradektriatoh scale" by [[Osmiorisbendi‎]].

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum}}

{{todo|expand}}

[[Category:14-tone scales]]</text>
      <sha1>1tfg1cg13g00xm10q0xydidp27h9s6p</sha1>
    </revision>
    <revision>
      <id>199902</id>
      <parentid>183731</parentid>
      <timestamp>2025-06-18T06:29:32Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>199902</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="532" sha1="perudky6rzcctayqjm49b3d8ukqa1er" xml:space="preserve">{{Infobox MOS
| Other names = Ketradektriatoh
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 3
| Equalized = 5
| Collapsed = 4
| Pattern = LLLLsLLLLsLLLs
}}
{{MOS intro|Other Names=}}

== Name ==
Vector Graphics proposes '''ketradekic''' as a name for this scale, based on the name "Ketradektriatoh scale" proposed by [[Osmiorisbendi‎]], adapted to fit scale naming conventions.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum}}

{{todo|expand}}

[[Category:14-tone scales]]</text>
      <sha1>perudky6rzcctayqjm49b3d8ukqa1er</sha1>
    </revision>
  </page>
  <page>
    <title>11L 4s</title>
    <ns>0</ns>
    <id>96</id>
    <revision>
      <id>96</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>96</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1182" sha1="icosmocs09nt2rugy64ppii3e65gpqx" xml:space="preserve">The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327.273). At its harmonic entropy minimum, 7 equals +3 generators.

{| class="wikitable"
|-
| | 4/15
| | 
| | 
| | 
| | 
| | 320
|-
| | 
| | 
| | 
| | 
| | 19/71
| | 321.127
|-
| | 
| | 
| | 
| | 15/56
| | 
| | 321.429
|-
| | 
| | 
| | 
| | 
| | 26/97
| | 321.6495
|-
| | 
| | 
| | 11/41
| | 
| | 
| | 321.951
|-
| | 
| | 
| | 
| | 
| | 
| | 322.146
|-
| | 
| | 
| | 
| | 
| | 29/108
| | 322.222
|-
| | 
| | 
| | 
| | 
| | 
| | 322.268
|-
| | 
| | 
| | 
| | 18/67
| | 
| | 322.388
|-
| | 
| | 
| | 
| | 
| | 
| | 322.5405
|-
| | 
| | 
| | 
| | 
| | 25/93
| | 322.581
|-
| | 
| | 7/26
| | 
| | 
| | 
| | 323.077
|-
| | 
| | 
| | 
| | 
| | 24/89
| | 323.5955
|-
| | 
| | 
| | 
| | 17/63
| | 
| | 323.8095
|-
| | 
| | 
| | 
| | 
| | 
| | 323.947
|-
| | 
| | 
| | 
| | 
| | 27/100
| | 324
|-
| | 
| | 
| | 
| | 
| | 
| | 324.055
|-
| | 
| | 
| | 10/37
| | 
| | 
| | 324.324
|-
| | 
| | 
| | 
| | 
| | 
| | 324.443
|-
| | 
| | 
| | 
| | 
| | 23/85
| | 324.706
|-
| | 
| | 
| | 
| | 13/48
| | 
| | 325
|-
| | 
| | 
| | 
| | 
| | 16/59
| | 325.424
|-
| | 3/11
| | 
| | 
| | 
| | 
| | 327.273
|}</text>
      <sha1>icosmocs09nt2rugy64ppii3e65gpqx</sha1>
    </revision>
    <revision>
      <id>4488</id>
      <timestamp>2015-11-14T15:44:22Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 566467359 - Original comment: **</comment>
      <origin>4488</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9449" sha1="6d6dk695xgn3b5208v70hyu7vs3mf3e" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-14 15:44:22 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566467359&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327.273). At its harmonic entropy minimum, 7 equals +3 generators.
|| 4/15 ||   ||   ||   ||   || 320 ||
||   ||   ||   ||   || 19/71 || 321.127 ||
||   ||   ||   || 15/56 ||   || 321.429 ||
||   ||   ||   ||   || 26/97 || 321.6495 ||
||   ||   || 11/41 ||   ||   || 321.951 ||
||   ||   ||   ||   ||   || 322.146 ||
||   ||   ||   ||   || 29/108 || 322.222 ||
||   ||   ||   ||   ||   || 322.268 ||
||   ||   ||   || 18/67 ||   || 322.388 ||
||   ||   ||   ||   ||   || 322.5405 ||
||   ||   ||   ||   || 25/93 || 322.581 ||
||   || 7/26 ||   ||   ||   || 323.077 ||
||   ||   ||   ||   || 24/89 || 323.5955 ||
||   ||   ||   || 17/63 ||   || 323.8095 ||
||   ||   ||   ||   ||   || 323.947 ||
||   ||   ||   ||   || 27/100 || 324 ||
||   ||   ||   ||   ||   || 324.055 ||
||   ||   || 10/37 ||   ||   || 324.324 ||
||   ||   ||   ||   ||   || 324.443 ||
||   ||   ||   ||   || 23/85 || 324.706 ||
||   ||   ||   || 13/48 ||   || 325 ||
||   ||   ||   ||   || 16/59 || 325.424 ||
|| 3/11 ||   ||   ||   ||   || 327.273 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 4s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327.273). At its harmonic entropy minimum, 7 equals +3 generators.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;320&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;321.127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;321.429&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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      <sha1>6d6dk695xgn3b5208v70hyu7vs3mf3e</sha1>
    </revision>
    <revision>
      <id>4489</id>
      <timestamp>2015-10-26T14:30:20Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 563957987 - Original comment: **</comment>
      <origin>4489</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9112" sha1="oxdu93xhoisv7oect2ixt2vfzlppts7" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-10-26 14:30:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;563957987&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327.273). At its harmonic entropy minimum, 7 equals +3 generators.
|| 4/15 ||   ||   ||   ||   || 320 ||
||   ||   ||   ||   || 19/71 || 321.127 ||
||   ||   ||   || 15/56 ||   || 321.429 ||
||   ||   ||   ||   || 26/97 || 321.6495 ||
||   ||   || 11/41 ||   ||   || 321.951 ||
||   ||   ||   ||   ||   || 322.146 ||
||   ||   ||   ||   || 29/108 || 322.222 ||
||   ||   ||   ||   ||   || 322.268 ||
||   ||   ||   || 18/67 ||   || 322.388 ||
||   ||   ||   ||   || 25/93 || 322.581 ||
||   || 7/26 ||   ||   ||   || 323.077 ||
||   ||   ||   ||   || 24/89 || 323.5955 ||
||   ||   ||   || 17/63 ||   || 323.8095 ||
||   ||   ||   ||   ||   || 323.947 ||
||   ||   ||   ||   || 27/100 || 324 ||
||   ||   ||   ||   ||   || 324.055 ||
||   ||   || 10/37 ||   ||   || 324.324 ||
||   ||   ||   ||   ||   || 324.443 ||
||   ||   ||   ||   || 23/85 || 324.706 ||
||   ||   ||   || 13/48 ||   || 325 ||
||   ||   ||   ||   || 16/59 || 325.424 ||
|| 3/11 ||   ||   ||   ||   || 327.273 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 4s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327.273). At its harmonic entropy minimum, 7 equals +3 generators.&amp;lt;br /&amp;gt;


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      <sha1>oxdu93xhoisv7oect2ixt2vfzlppts7</sha1>
    </revision>
    <revision>
      <id>4490</id>
      <timestamp>2015-10-26T14:30:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 563957907 - Original comment: **</comment>
      <origin>4490</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9114" sha1="rql4vapm06ucizvffstykq2t1a9fah9" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-10-26 14:30:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;563957907&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327 3/11). At its harmonic entropy minimum, 7 equals +3 generators.
|| 4/15 ||   ||   ||   ||   || 320 ||
||   ||   ||   ||   || 19/71 || 321.127 ||
||   ||   ||   || 15/56 ||   || 321.429 ||
||   ||   ||   ||   || 26/97 || 321.6495 ||
||   ||   || 11/41 ||   ||   || 321.951 ||
||   ||   ||   ||   ||   || 322.146 ||
||   ||   ||   ||   || 29/108 || 322.222 ||
||   ||   ||   ||   ||   || 322.268 ||
||   ||   ||   || 18/67 ||   || 322.388 ||
||   ||   ||   ||   || 25/93 || 322.581 ||
||   || 7/26 ||   ||   ||   || 323.077 ||
||   ||   ||   ||   || 24/89 || 323.5955 ||
||   ||   ||   || 17/63 ||   || 323.8095 ||
||   ||   ||   ||   ||   || 323.947 ||
||   ||   ||   ||   || 27/100 || 324 ||
||   ||   ||   ||   ||   || 324.055 ||
||   ||   || 10/37 ||   ||   || 324.324 ||
||   ||   ||   ||   ||   || 324.443 ||
||   ||   ||   ||   || 23/85 || 324.706 ||
||   ||   ||   || 13/48 ||   || 325 ||
||   ||   ||   ||   || 16/59 || 325.424 ||
|| 3/11 ||   ||   ||   ||   || 327.273 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 4s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327 3/11). At its harmonic entropy minimum, 7 equals +3 generators.&amp;lt;br /&amp;gt;


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      <sha1>rql4vapm06ucizvffstykq2t1a9fah9</sha1>
    </revision>
    <revision>
      <id>4491</id>
      <timestamp>2015-05-24T10:26:37Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 552005862 - Original comment: **</comment>
      <origin>4491</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8175" sha1="thy8lnqijhxj0hvhh8kpvyi3pmas7b2" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-24 10:26:37 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;552005862&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327 3/11). At its harmonic entropy minimum, 7 equals +3 generators.
|| 4/15 ||   ||   ||   ||   || 320 ||
||   ||   ||   ||   || 19/71 || 321.126761 ||
||   ||   ||   || 15/56 ||   || 321 3/7 ||
||   ||   ||   ||   || 26/97 || 321.6494845 ||
||   ||   || 11/41 ||   ||   || 321 39/41 ||
||   ||   ||   ||   || 29/108 || 322 2/9 ||
||   ||   ||   || 18/67 ||   || 322.38806 ||
||   ||   ||   ||   || 25/93 || 322.580645 ||
||   || 7/26 ||   ||   ||   || 323 1/13 ||
||   ||   ||   ||   || 24/89 || 323.595506 ||
||   ||   ||   || 17/63 ||   || 323.809524 ||
||   ||   ||   ||   || 27/100 || 324 ||
||   ||   ||   ||   ||   || 324.054811 ||
||   ||   || 10/37 ||   ||   || 324 12/37 ||
||   ||   ||   ||   ||   || 324.443424 ||
||   ||   ||   ||   || 23/85 || 324.705882 ||
||   ||   ||   || 13/48 ||   || 325 ||
||   ||   ||   ||   || 16/59 || 325.423729 ||
|| 3/11 ||   ||   ||   ||   || 327 3/11 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 4s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327 3/11). At its harmonic entropy minimum, 7 equals +3 generators.&amp;lt;br /&amp;gt;


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      <sha1>thy8lnqijhxj0hvhh8kpvyi3pmas7b2</sha1>
    </revision>
    <revision>
      <id>4492</id>
      <timestamp>2015-02-12T15:02:14Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540801068 - Original comment: **</comment>
      <origin>4492</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8173" sha1="5btt944bz5zq4jeveq8qimpsrjtgd3c" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-12 15:02:14 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540801068&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327 3/11). At its harmonic entropy minimum, 7 equals +3 generators.
|| 4/15 ||   ||   ||   ||   || 320 ||
||   ||   ||   ||   || 19/71 || 321.126761 ||
||   ||   ||   || 15/56 ||   || 321 3/7 ||
||   ||   ||   ||   || 26/97 || 321.6494845 ||
||   ||   || 11/41 ||   ||   || 321 39/41 ||
||   ||   ||   ||   || 29/108 || 322 2/9 ||
||   ||   ||   || 18/67 ||   || 322.38806 ||
||   ||   ||   ||   || 25/93 || 322.580645 ||
||   || 7/26 ||   ||   ||   || 323 1/13 ||
||   ||   ||   ||   || 24/89 || 323.595506 ||
||   ||   ||   || 17/63 ||   || 323.809524 ||
||   ||   ||   ||   || 27/100 || 324 ||
||   ||   ||   ||   ||   || 324.299933 ||
||   ||   || 10/37 ||   ||   || 324 12/37 ||
||   ||   ||   ||   ||   || 324.34854 ||
||   ||   ||   ||   || 23/85 || 324.705882 ||
||   ||   ||   || 13/48 ||   || 325 ||
||   ||   ||   ||   || 16/59 || 325.423729 ||
|| 3/11 ||   ||   ||   ||   || 327 3/11 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 4s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327 3/11). At its harmonic entropy minimum, 7 equals +3 generators.&amp;lt;br /&amp;gt;


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      <sha1>5btt944bz5zq4jeveq8qimpsrjtgd3c</sha1>
    </revision>
    <revision>
      <id>4493</id>
      <timestamp>2012-11-01T23:01:09Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 378529192 - Original comment: **</comment>
      <origin>4493</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7493" sha1="f2ayuqhlend9hdvgkzos7t4lsrvvkjw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-11-01 23:01:09 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;378529192&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327 3/11). At its harmonic entropy minimum, 7 equals +3 generators.
|| 4/15 ||   ||   ||   ||   || 320 ||
||   ||   ||   ||   || 19/71 || 321.126761 ||
||   ||   ||   || 15/56 ||   || 321 3/7 ||
||   ||   ||   ||   || 26/97 || 321.6494845 ||
||   ||   || 11/41 ||   ||   || 321 39/41 ||
||   ||   ||   ||   || 29/108 || 322 2/9 ||
||   ||   ||   || 18/67 ||   || 322.38806 ||
||   ||   ||   ||   || 25/93 || 322.580645 ||
||   || 7/26 ||   ||   ||   || 323 1/13 ||
||   ||   ||   ||   || 24/89 || 323.595506 ||
||   ||   ||   || 17/63 ||   || 323.809524 ||
||   ||   ||   ||   || 27/100 || 324 ||
||   ||   || 10/37 ||   ||   || 324 12/37 ||
||   ||   ||   ||   || 23/85 || 324.705882 ||
||   ||   ||   || 13/48 ||   || 325 ||
||   ||   ||   ||   || 16/59 || 325.423729 ||
|| 3/11 ||   ||   ||   ||   || 327 3/11 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;11L 4s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The interval which generates this MOS is a sharp 6/5 of height 4/15edo (320 cents) to 3/11edo (327 3/11). At its harmonic entropy minimum, 7 equals +3 generators.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;320&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;321.126761&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;321 3/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;26/97&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;321.6494845&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;321 39/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29/108&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;322 2/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;322.38806&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25/93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;322.580645&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;323 1/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;24/89&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;323.595506&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;323.809524&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27/100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;324&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;324 12/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23/85&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;324.705882&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/48&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;325&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;325.423729&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;327 3/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>f2ayuqhlend9hdvgkzos7t4lsrvvkjw</sha1>
    </revision>
    <revision>
      <id>94398</id>
      <parentid>96</parentid>
      <timestamp>2022-08-16T04:06:31Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox, name still missing, misc. edits</comment>
      <origin>94398</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1138" sha1="lno4vm377sk3gvln7tfa8bh3didayum" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 4
| Equalized = 4
| Paucitonic = 3
| Pattern = LLLLLsLLLLLs
}}
The '''11L 4s''' [[MOS scale]] is generated by a minor third slightly sharp of [[6/5]], between 4\15 (320{{cent}}) and 3\11 (327.273{{cent}}). At its harmonic entropy minimum, the octave-reduced 7th [[harmonic]] equals +3 generators.

== Scale tree ==
{| class="wikitable"
|-
| 4/15
| 
| 
| 
| 
| 320
|-
| 
| 
| 
| 
| 19/71
| 321.127
|-
| 
| 
| 
| 15/56
| 
| 321.429
|-
| 
| 
| 
| 
| 26/97
| 321.6495
|-
| 
| 
| 11/41
| 
| 
| 321.951
|-
| 
| 
| 
| 
| 
| 322.146
|-
| 
| 
| 
| 
| 29/108
| 322.222
|-
| 
| 
| 
| 
| 
| 322.268
|-
| 
| 
| 
| 18/67
| 
| 322.388
|-
| 
| 
| 
| 
| 
| 322.5405
|-
| 
| 
| 
| 
| 25/93
| 322.581
|-
| 
| 7/26
| 
| 
| 
| 323.077
|-
| 
| 
| 
| 
| 24/89
| 323.5955
|-
| 
| 
| 
| 17/63
| 
| 323.8095
|-
| 
| 
| 
| 
| 
| 323.947
|-
| 
| 
| 
| 
| 27/100
| 324
|-
| 
| 
| 
| 
| 
| 324.055
|-
| 
| 
| 10/37
| 
| 
| 324.324
|-
| 
| 
| 
| 
| 
| 324.443
|-
| 
| 
| 
| 
| 23/85
| 324.706
|-
| 
| 
| 
| 13/48
| 
| 325
|-
| 
| 
| 
| 
| 16/59
| 325.424
|-
| 3/11
| 
| 
| 
| 
| 327.273
|}</text>
      <sha1>lno4vm377sk3gvln7tfa8bh3didayum</sha1>
    </revision>
    <revision>
      <id>94510</id>
      <parentid>94398</parentid>
      <timestamp>2022-08-17T09:21:12Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94510</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1137" sha1="569wq9xgw48isff5w3tohvbxqnjd5lb" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 4
| Equalized = 4
| Collapsed = 3
| Pattern = LLLLLsLLLLLs
}}
The '''11L 4s''' [[MOS scale]] is generated by a minor third slightly sharp of [[6/5]], between 4\15 (320{{cent}}) and 3\11 (327.273{{cent}}). At its harmonic entropy minimum, the octave-reduced 7th [[harmonic]] equals +3 generators.

== Scale tree ==
{| class="wikitable"
|-
| 4/15
| 
| 
| 
| 
| 320
|-
| 
| 
| 
| 
| 19/71
| 321.127
|-
| 
| 
| 
| 15/56
| 
| 321.429
|-
| 
| 
| 
| 
| 26/97
| 321.6495
|-
| 
| 
| 11/41
| 
| 
| 321.951
|-
| 
| 
| 
| 
| 
| 322.146
|-
| 
| 
| 
| 
| 29/108
| 322.222
|-
| 
| 
| 
| 
| 
| 322.268
|-
| 
| 
| 
| 18/67
| 
| 322.388
|-
| 
| 
| 
| 
| 
| 322.5405
|-
| 
| 
| 
| 
| 25/93
| 322.581
|-
| 
| 7/26
| 
| 
| 
| 323.077
|-
| 
| 
| 
| 
| 24/89
| 323.5955
|-
| 
| 
| 
| 17/63
| 
| 323.8095
|-
| 
| 
| 
| 
| 
| 323.947
|-
| 
| 
| 
| 
| 27/100
| 324
|-
| 
| 
| 
| 
| 
| 324.055
|-
| 
| 
| 10/37
| 
| 
| 324.324
|-
| 
| 
| 
| 
| 
| 324.443
|-
| 
| 
| 
| 
| 23/85
| 324.706
|-
| 
| 
| 
| 13/48
| 
| 325
|-
| 
| 
| 
| 
| 16/59
| 325.424
|-
| 3/11
| 
| 
| 
| 
| 327.273
|}</text>
      <sha1>569wq9xgw48isff5w3tohvbxqnjd5lb</sha1>
    </revision>
    <revision>
      <id>101037</id>
      <parentid>94510</parentid>
      <timestamp>2023-01-06T16:16:17Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Copying from [[Category:Abstract MOS patterns]] to [[Category:15-tone scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>101037</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1166" sha1="99vdxgmqeobxz21hrmgk2kjgukfe2rq" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 4
| Equalized = 4
| Collapsed = 3
| Pattern = LLLLLsLLLLLs
}}
The '''11L 4s''' [[MOS scale]] is generated by a minor third slightly sharp of [[6/5]], between 4\15 (320{{cent}}) and 3\11 (327.273{{cent}}). At its harmonic entropy minimum, the octave-reduced 7th [[harmonic]] equals +3 generators.

== Scale tree ==
{| class="wikitable"
|-
| 4/15
| 
| 
| 
| 
| 320
|-
| 
| 
| 
| 
| 19/71
| 321.127
|-
| 
| 
| 
| 15/56
| 
| 321.429
|-
| 
| 
| 
| 
| 26/97
| 321.6495
|-
| 
| 
| 11/41
| 
| 
| 321.951
|-
| 
| 
| 
| 
| 
| 322.146
|-
| 
| 
| 
| 
| 29/108
| 322.222
|-
| 
| 
| 
| 
| 
| 322.268
|-
| 
| 
| 
| 18/67
| 
| 322.388
|-
| 
| 
| 
| 
| 
| 322.5405
|-
| 
| 
| 
| 
| 25/93
| 322.581
|-
| 
| 7/26
| 
| 
| 
| 323.077
|-
| 
| 
| 
| 
| 24/89
| 323.5955
|-
| 
| 
| 
| 17/63
| 
| 323.8095
|-
| 
| 
| 
| 
| 
| 323.947
|-
| 
| 
| 
| 
| 27/100
| 324
|-
| 
| 
| 
| 
| 
| 324.055
|-
| 
| 
| 10/37
| 
| 
| 324.324
|-
| 
| 
| 
| 
| 
| 324.443
|-
| 
| 
| 
| 
| 23/85
| 324.706
|-
| 
| 
| 
| 13/48
| 
| 325
|-
| 
| 
| 
| 
| 16/59
| 325.424
|-
| 3/11
| 
| 
| 
| 
| 327.273
|}

[[Category:15-tone scales]]</text>
      <sha1>99vdxgmqeobxz21hrmgk2kjgukfe2rq</sha1>
    </revision>
    <revision>
      <id>133182</id>
      <parentid>101037</parentid>
      <timestamp>2024-01-28T01:14:15Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added modes</comment>
      <origin>133182</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1190" sha1="8lohmcdyuho1flc5c13ykpi4o1ik90p" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 4
| Equalized = 4
| Collapsed = 3
| Pattern = LLLLLsLLLLLs
}}
The '''11L 4s''' [[MOS scale]] is generated by a minor third slightly sharp of [[6/5]], between 4\15 (320{{cent}}) and 3\11 (327.273{{cent}}). At its harmonic entropy minimum, the octave-reduced 7th [[harmonic]] equals +3 generators.
== Modes==
{{MOS modes}}
== Scale tree ==
{| class="wikitable"
|-
| 4/15
| 
| 
| 
| 
| 320
|-
| 
| 
| 
| 
| 19/71
| 321.127
|-
| 
| 
| 
| 15/56
| 
| 321.429
|-
| 
| 
| 
| 
| 26/97
| 321.6495
|-
| 
| 
| 11/41
| 
| 
| 321.951
|-
| 
| 
| 
| 
| 
| 322.146
|-
| 
| 
| 
| 
| 29/108
| 322.222
|-
| 
| 
| 
| 
| 
| 322.268
|-
| 
| 
| 
| 18/67
| 
| 322.388
|-
| 
| 
| 
| 
| 
| 322.5405
|-
| 
| 
| 
| 
| 25/93
| 322.581
|-
| 
| 7/26
| 
| 
| 
| 323.077
|-
| 
| 
| 
| 
| 24/89
| 323.5955
|-
| 
| 
| 
| 17/63
| 
| 323.8095
|-
| 
| 
| 
| 
| 
| 323.947
|-
| 
| 
| 
| 
| 27/100
| 324
|-
| 
| 
| 
| 
| 
| 324.055
|-
| 
| 
| 10/37
| 
| 
| 324.324
|-
| 
| 
| 
| 
| 
| 324.443
|-
| 
| 
| 
| 
| 23/85
| 324.706
|-
| 
| 
| 
| 13/48
| 
| 325
|-
| 
| 
| 
| 
| 16/59
| 325.424
|-
| 3/11
| 
| 
| 
| 
| 327.273
|}

[[Category:15-tone scales]]</text>
      <sha1>8lohmcdyuho1flc5c13ykpi4o1ik90p</sha1>
    </revision>
    <revision>
      <id>136560</id>
      <parentid>133182</parentid>
      <timestamp>2024-02-22T03:00:28Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Mos intro</comment>
      <origin>136560</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1165" sha1="kwhbwkamb8xfpxko6c8sv9qem6u46as" xml:space="preserve">{{MOS intro}}{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 4
| Equalized = 4
| Collapsed = 3
| Pattern = LLLLLsLLLLLs
}}
The bright generator represents a minor third slightly sharp of [[6/5]] (315.6¢). At its harmonic entropy minimum, the octave-reduced 7th [[harmonic]] (about 968¢) equals 3 of these generators.
== Modes==
{{MOS modes}}
== Scale tree ==
{| class="wikitable"
|-
| 4/15
| 
| 
| 
| 
| 320
|-
| 
| 
| 
| 
| 19/71
| 321.127
|-
| 
| 
| 
| 15/56
| 
| 321.429
|-
| 
| 
| 
| 
| 26/97
| 321.6495
|-
| 
| 
| 11/41
| 
| 
| 321.951
|-
| 
| 
| 
| 
| 
| 322.146
|-
| 
| 
| 
| 
| 29/108
| 322.222
|-
| 
| 
| 
| 
| 
| 322.268
|-
| 
| 
| 
| 18/67
| 
| 322.388
|-
| 
| 
| 
| 
| 
| 322.5405
|-
| 
| 
| 
| 
| 25/93
| 322.581
|-
| 
| 7/26
| 
| 
| 
| 323.077
|-
| 
| 
| 
| 
| 24/89
| 323.5955
|-
| 
| 
| 
| 17/63
| 
| 323.8095
|-
| 
| 
| 
| 
| 
| 323.947
|-
| 
| 
| 
| 
| 27/100
| 324
|-
| 
| 
| 
| 
| 
| 324.055
|-
| 
| 
| 10/37
| 
| 
| 324.324
|-
| 
| 
| 
| 
| 
| 324.443
|-
| 
| 
| 
| 
| 23/85
| 324.706
|-
| 
| 
| 
| 13/48
| 
| 325
|-
| 
| 
| 
| 
| 16/59
| 325.424
|-
| 3/11
| 
| 
| 
| 
| 327.273
|}

[[Category:15-tone scales]]</text>
      <sha1>kwhbwkamb8xfpxko6c8sv9qem6u46as</sha1>
    </revision>
    <revision>
      <id>136637</id>
      <parentid>136560</parentid>
      <timestamp>2024-02-22T18:16:51Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <minor/>
      <comment>Order of templates</comment>
      <origin>136637</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1166" sha1="7lhzuvnphf1zxbqj10axmei5igietg0" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 4
| Equalized = 4
| Collapsed = 3
| Pattern = LLLLLsLLLLLs
}}
{{MOS intro}}
The bright generator represents a minor third slightly sharp of [[6/5]] (315.6¢). At its harmonic entropy minimum, the octave-reduced 7th [[harmonic]] (about 968¢) equals 3 of these generators.
== Modes==
{{MOS modes}}
== Scale tree ==
{| class="wikitable"
|-
| 4/15
| 
| 
| 
| 
| 320
|-
| 
| 
| 
| 
| 19/71
| 321.127
|-
| 
| 
| 
| 15/56
| 
| 321.429
|-
| 
| 
| 
| 
| 26/97
| 321.6495
|-
| 
| 
| 11/41
| 
| 
| 321.951
|-
| 
| 
| 
| 
| 
| 322.146
|-
| 
| 
| 
| 
| 29/108
| 322.222
|-
| 
| 
| 
| 
| 
| 322.268
|-
| 
| 
| 
| 18/67
| 
| 322.388
|-
| 
| 
| 
| 
| 
| 322.5405
|-
| 
| 
| 
| 
| 25/93
| 322.581
|-
| 
| 7/26
| 
| 
| 
| 323.077
|-
| 
| 
| 
| 
| 24/89
| 323.5955
|-
| 
| 
| 
| 17/63
| 
| 323.8095
|-
| 
| 
| 
| 
| 
| 323.947
|-
| 
| 
| 
| 
| 27/100
| 324
|-
| 
| 
| 
| 
| 
| 324.055
|-
| 
| 
| 10/37
| 
| 
| 324.324
|-
| 
| 
| 
| 
| 
| 324.443
|-
| 
| 
| 
| 
| 23/85
| 324.706
|-
| 
| 
| 
| 13/48
| 
| 325
|-
| 
| 
| 
| 
| 16/59
| 325.424
|-
| 3/11
| 
| 
| 
| 
| 327.273
|}

[[Category:15-tone scales]]</text>
      <sha1>7lhzuvnphf1zxbqj10axmei5igietg0</sha1>
    </revision>
    <revision>
      <id>137508</id>
      <parentid>136637</parentid>
      <timestamp>2024-02-29T05:26:18Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Replaced scale tree</comment>
      <origin>137508</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="436" sha1="d5bo3mnexm2lxlypw071u2lx7dty98e" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 4
| Equalized = 4
| Collapsed = 3
| Pattern = LLLLLsLLLLLs
}}
{{MOS intro}}
The bright generator represents a minor third slightly sharp of [[6/5]] (315.6¢). At its harmonic entropy minimum, the octave-reduced 7th [[harmonic]] (about 968¢) equals 3 of these generators.
== Modes==
{{MOS modes}}
== Scale tree ==
{{Scale tree}}
[[Category:15-tone scales]]</text>
      <sha1>d5bo3mnexm2lxlypw071u2lx7dty98e</sha1>
    </revision>
    <revision>
      <id>161093</id>
      <parentid>137508</parentid>
      <timestamp>2024-10-19T06:10:37Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <comment>orgone[15] lol</comment>
      <origin>161093</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="558" sha1="026ym7mzy25hmnnx4ouou0s6xy244mt" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 4
| Equalized = 4
| Collapsed = 3
| Pattern = LLLLLsLLLLLs
}}
{{MOS intro}}
The bright generator represents a minor third slightly sharp of [[6/5]] (315.6¢), though this MOS is most reasonably interpreted in the 2.7.11 [[orgone]] temperament where it is identified as ~[[77/64]]. At its harmonic entropy minimum, the octave-reduced 7th [[harmonic]] (about 968¢) equals 3 of these generators.
== Modes==
{{MOS modes}}
== Scale tree ==
{{Scale tree}}
[[Category:15-tone scales]]</text>
      <sha1>026ym7mzy25hmnnx4ouou0s6xy244mt</sha1>
    </revision>
    <revision>
      <id>170576</id>
      <parentid>161093</parentid>
      <timestamp>2024-12-15T06:59:47Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Intervals</comment>
      <origin>170576</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="594" sha1="hc7fd3dnkq327seunzax6fivf4ivn10" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 4
| Equalized = 4
| Collapsed = 3
| Pattern = LLLLLsLLLLLs
}}
{{MOS intro}}
The bright generator represents a minor third slightly sharp of [[6/5]] (315.6¢), though this MOS is most reasonably interpreted in the 2.7.11 [[orgone]] temperament where it is identified as ~[[77/64]]. At its harmonic entropy minimum, the octave-reduced 7th [[harmonic]] (about 968¢) equals 3 of these generators.
== Modes==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree}}
[[Category:15-tone scales]]</text>
      <sha1>hc7fd3dnkq327seunzax6fivf4ivn10</sha1>
    </revision>
    <revision>
      <id>183732</id>
      <parentid>170576</parentid>
      <timestamp>2025-02-28T16:41:48Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183732</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="605" sha1="0vkg75r7lfdezzwo3m0pbgxtlnr5965" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 4
| Equalized = 4
| Collapsed = 3
| Pattern = LLLLLsLLLLLs
}}
{{MOS intro}}
The bright generator represents a minor third slightly sharp of [[6/5]] (315.6¢), though this MOS is most reasonably interpreted in the 2.7.11 [[orgone]] temperament where it is identified as ~[[77/64]]. At its harmonic entropy minimum, the octave-reduced 7th [[harmonic]] (about 968¢) equals 3 of these generators.

== Modes==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum}}

[[Category:15-tone scales]]</text>
      <sha1>0vkg75r7lfdezzwo3m0pbgxtlnr5965</sha1>
    </revision>
    <revision>
      <id>199887</id>
      <parentid>183732</parentid>
      <timestamp>2025-06-18T00:01:54Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>added temps</comment>
      <origin>199887</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="771" sha1="r1tx7iwrc8z1ucwug62bsfy2w47zaz4" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 1
| nLargeSteps = 11
| nSmallSteps = 4
| Equalized = 4
| Collapsed = 3
| Pattern = LLLLLsLLLLLs
}}
{{MOS intro}}
The bright generator represents a minor third slightly sharp of [[6/5]] (315.6¢), though this MOS is most reasonably interpreted in the 2.7.11 [[orgone]] temperament where it is identified as ~[[77/64]]. At its harmonic entropy minimum, the octave-reduced 7th [[harmonic]] (about 968¢) equals 3 of these generators.

== Modes==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 13/8 = Golden superklesimic
| 10/7 = [[Superkleismic]]
| 9/5 = [[Senator]]
| 9/4 = [[Orgone]]
| 5/2 = [[Hyperkleismic]]
| 5/1 = [[Oregon]]
| 6/1 = [[Ork]]/[[Orca]]}}

[[Category:15-tone scales]]</text>
      <sha1>r1tx7iwrc8z1ucwug62bsfy2w47zaz4</sha1>
    </revision>
  </page>
  <page>
    <title>12-22a</title>
    <ns>0</ns>
    <id>100</id>
    <revision>
      <id>100</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>100</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="668" sha1="rlmr23qoggna8w5sryao26pn3nrbixw" xml:space="preserve">! 12-22a.scl

!

12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000

12

!

54.54545

218.18182

272.72727

327.27273

490.90909

545.45455

709.09091

763.63636

818.18182

981.81818

1036.36364

2/1

=Music=
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 12of22studyPentUp4thsMstr]

[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 12of22gamelan1b]

[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)]

[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]

By [[Joel_Grant_Taylor|Joel Grant Taylor]]</text>
      <sha1>rlmr23qoggna8w5sryao26pn3nrbixw</sha1>
    </revision>
    <revision>
      <id>4513</id>
      <timestamp>2012-06-17T17:28:23Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 346006354 - Original comment: **</comment>
      <origin>4513</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3262" sha1="74qm6vjofyolejfk7iqm45mm3d3m82w" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-06-17 17:28:23 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;346006354&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12-22a.scl
!
12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000
 12
!
 54.54545
 218.18182
 272.72727
 327.27273
 490.90909
 545.45455
 709.09091
 763.63636
 818.18182
 981.81818
 1036.36364
 2/1

=Music=
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3|12of22studyPentUp4thsMstr]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3|12of22gamelan1b]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3|12of22study3 (children's story)]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3|12of22study7]]
By [[Joel Grant Taylor]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12-22a&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12-22a.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000&amp;lt;br /&amp;gt;
 12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
 54.54545&amp;lt;br /&amp;gt;
 218.18182&amp;lt;br /&amp;gt;
 272.72727&amp;lt;br /&amp;gt;
 327.27273&amp;lt;br /&amp;gt;
 490.90909&amp;lt;br /&amp;gt;
 545.45455&amp;lt;br /&amp;gt;
 709.09091&amp;lt;br /&amp;gt;
 763.63636&amp;lt;br /&amp;gt;
 818.18182&amp;lt;br /&amp;gt;
 981.81818&amp;lt;br /&amp;gt;
 1036.36364&amp;lt;br /&amp;gt;
 2/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3" rel="nofollow"&amp;gt;12of22studyPentUp4thsMstr&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3" rel="nofollow"&amp;gt;12of22gamelan1b&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3" rel="nofollow"&amp;gt;12of22study3 (children's story)&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3" rel="nofollow"&amp;gt;12of22study7&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
By &amp;lt;a class="wiki_link" href="/Joel%20Grant%20Taylor"&amp;gt;Joel Grant Taylor&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>74qm6vjofyolejfk7iqm45mm3d3m82w</sha1>
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    <revision>
      <id>4514</id>
      <timestamp>2012-06-17T16:16:59Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 345996502 - Original comment: **</comment>
      <origin>4514</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1826" sha1="igsk7gilr7u8ohm8s8c7cfrfemf2d0b" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-06-17 16:16:59 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;345996502&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12-22a.scl
!
12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000
 12
!
 54.54545
 218.18182
 272.72727
 327.27273
 490.90909
 545.45455
 709.09091
 763.63636
 818.18182
 981.81818
 1036.36364
 2/1
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12-22a&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12-22a.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000&amp;lt;br /&amp;gt;
 12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
 54.54545&amp;lt;br /&amp;gt;
 218.18182&amp;lt;br /&amp;gt;
 272.72727&amp;lt;br /&amp;gt;
 327.27273&amp;lt;br /&amp;gt;
 490.90909&amp;lt;br /&amp;gt;
 545.45455&amp;lt;br /&amp;gt;
 709.09091&amp;lt;br /&amp;gt;
 763.63636&amp;lt;br /&amp;gt;
 818.18182&amp;lt;br /&amp;gt;
 981.81818&amp;lt;br /&amp;gt;
 1036.36364&amp;lt;br /&amp;gt;
 2/1&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <parentid>100</parentid>
      <timestamp>2022-01-13T19:53:07Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup; +mode info; mark dead links</comment>
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      <text bytes="1047" sha1="8idf4sacmie50g126mes74szdu83rot" xml:space="preserve">&lt;pre&gt;
! 12-22a.scl
!
12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000
12
!
54.54545
218.18182
272.72727
327.27273
490.90909
545.45455
709.09091
763.63636
818.18182
981.81818
1036.36364
2/1
&lt;/pre&gt;

Mode: 2|9 (sLssLsLssLsL)

Suggested keyboard mappings: 
* Central B, providing notes from Ab through C#
* Central F#, providing notes from Eb through G#

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 12of22studyPentUp4thsMstr]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 12of22gamelan1b]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]{{dead link}}

By [[Joel Grant Taylor]]

[[Category:Pages with Scala files]]
[[Category:22edo]]
[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:MOS scales]]
[[Category:Superpyth]]
[[Category:P-chromatic]]</text>
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      <text bytes="1093" sha1="o34wcdjayf6o3kxqomxjumll4s99gwy" xml:space="preserve">The tuning of [[5L 7s]] provided by [[22edo]].&lt;pre&gt;
! 12-22a.scl
!
12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000
12
!
54.54545
218.18182
272.72727
327.27273
490.90909
545.45455
709.09091
763.63636
818.18182
981.81818
1036.36364
2/1
&lt;/pre&gt;

Mode: 2|9 (sLssLsLssLsL)

Suggested keyboard mappings: 
* Central B, providing notes from Ab through C#
* Central F#, providing notes from Eb through G#

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 12of22studyPentUp4thsMstr]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 12of22gamelan1b]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]{{dead link}}

By [[Joel Grant Taylor]]

[[Category:Pages with Scala files]]
[[Category:22edo]]
[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:MOS scales]]
[[Category:Superpyth]]
[[Category:P-chromatic]]</text>
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      <text bytes="1142" sha1="4nsiazsjymbthv565ji6u497ou8nsy2" xml:space="preserve">The tuning of [[5L 7s]] provided by [[22edo]].&lt;pre&gt;
! 12-22a.scl
!
12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000
12
!
54.54545
218.18182
272.72727
327.27273
490.90909
545.45455
709.09091
763.63636
818.18182
981.81818
1036.36364
2/1
&lt;/pre&gt;

Mode: 2|9 (sLssLsLssLsL)

Suggested keyboard mappings: 
* Central B, providing notes from Ab through C#
* Central F#, providing notes from Eb through G#

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 12of22studyPentUp4thsMstr]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 12of22gamelan1b]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]{{dead link}}

By [[Joel Grant Taylor]]

[[Category:Pages with Scala files]]
[[Category:22edo]]
[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:MOS scales]]
[[Category:Superpyth]]
[[Category:P-chromatic]]
[[Category:Pages with mostly numerical content]]</text>
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        <id>480</id>
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      <minor/>
      <comment>Text replacement - "[[Category:Pages with mostly numerical content]]" to ""</comment>
      <origin>201045</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1093" sha1="o34wcdjayf6o3kxqomxjumll4s99gwy" xml:space="preserve">The tuning of [[5L 7s]] provided by [[22edo]].&lt;pre&gt;
! 12-22a.scl
!
12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000
12
!
54.54545
218.18182
272.72727
327.27273
490.90909
545.45455
709.09091
763.63636
818.18182
981.81818
1036.36364
2/1
&lt;/pre&gt;

Mode: 2|9 (sLssLsLssLsL)

Suggested keyboard mappings: 
* Central B, providing notes from Ab through C#
* Central F#, providing notes from Eb through G#

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 12of22studyPentUp4thsMstr]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 12of22gamelan1b]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]{{dead link}}

By [[Joel Grant Taylor]]

[[Category:Pages with Scala files]]
[[Category:22edo]]
[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:MOS scales]]
[[Category:Superpyth]]
[[Category:P-chromatic]]</text>
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        <id>480</id>
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      <minor/>
      <comment>Text replacement - "[[Category:Tempered scales]]" to "[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>201267</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1142" sha1="rxje19ryflgmjuak2rsq29rwookcs9s" xml:space="preserve">The tuning of [[5L 7s]] provided by [[22edo]].&lt;pre&gt;
! 12-22a.scl
!
12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000
12
!
54.54545
218.18182
272.72727
327.27273
490.90909
545.45455
709.09091
763.63636
818.18182
981.81818
1036.36364
2/1
&lt;/pre&gt;

Mode: 2|9 (sLssLsLssLsL)

Suggested keyboard mappings: 
* Central B, providing notes from Ab through C#
* Central F#, providing notes from Eb through G#

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 12of22studyPentUp4thsMstr]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 12of22gamelan1b]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]{{dead link}}

By [[Joel Grant Taylor]]

[[Category:Pages with Scala files]]
[[Category:22edo]]
[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:MOS scales]]
[[Category:Superpyth]]
[[Category:P-chromatic]]</text>
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        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered scales]]↵[[Category:Pages with mostly numerical content]]" to "[[Category:Tempered_scales]]
[[Category:Pages with mostly numerical content]]"</comment>
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      <text bytes="1142" sha1="bcq8tppkgzpb3uxnsmppvzvsmorwadj" xml:space="preserve">The tuning of [[5L 7s]] provided by [[22edo]].&lt;pre&gt;
! 12-22a.scl
!
12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000
12
!
54.54545
218.18182
272.72727
327.27273
490.90909
545.45455
709.09091
763.63636
818.18182
981.81818
1036.36364
2/1
&lt;/pre&gt;

Mode: 2|9 (sLssLsLssLsL)

Suggested keyboard mappings: 
* Central B, providing notes from Ab through C#
* Central F#, providing notes from Eb through G#

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 12of22studyPentUp4thsMstr]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 12of22gamelan1b]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]{{dead link}}

By [[Joel Grant Taylor]]

[[Category:Pages with Scala files]]
[[Category:22edo]]
[[Category:12-tone scales]]
[[Category:Tempered_scales]]
[[Category:Pages with mostly numerical content]]
[[Category:MOS scales]]
[[Category:Superpyth]]
[[Category:P-chromatic]]</text>
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        <id>480</id>
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      <minor/>
      <comment>Text replacement - "[[Category:Tempered_scales]]↵[[Category:Pages with mostly numerical content]]" to "[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]"</comment>
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      <model>wikitext</model>
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      <text bytes="1142" sha1="rxje19ryflgmjuak2rsq29rwookcs9s" xml:space="preserve">The tuning of [[5L 7s]] provided by [[22edo]].&lt;pre&gt;
! 12-22a.scl
!
12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000
12
!
54.54545
218.18182
272.72727
327.27273
490.90909
545.45455
709.09091
763.63636
818.18182
981.81818
1036.36364
2/1
&lt;/pre&gt;

Mode: 2|9 (sLssLsLssLsL)

Suggested keyboard mappings: 
* Central B, providing notes from Ab through C#
* Central F#, providing notes from Eb through G#

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 12of22studyPentUp4thsMstr]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 12of22gamelan1b]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]{{dead link}}

By [[Joel Grant Taylor]]

[[Category:Pages with Scala files]]
[[Category:22edo]]
[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:MOS scales]]
[[Category:Superpyth]]
[[Category:P-chromatic]]</text>
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        <id>186</id>
      </contributor>
      <minor/>
      <comment>+ category</comment>
      <origin>220525</origin>
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      <format>text/x-wiki</format>
      <text bytes="1166" sha1="hpyvgx69xyio0riz00jgmii6v2szi3h" xml:space="preserve">The tuning of [[5L 7s]] provided by [[22edo]].&lt;pre&gt;
! 12-22a.scl
!
12 out of 22-tET, Pythagorean. Paul Erlich, TL 4-4-2000
12
!
54.54545
218.18182
272.72727
327.27273
490.90909
545.45455
709.09091
763.63636
818.18182
981.81818
1036.36364
2/1
&lt;/pre&gt;

Mode: 2|9 (sLssLsLssLsL)

Suggested keyboard mappings: 
* Central B, providing notes from Ab through C#
* Central F#, providing notes from Eb through G#

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22studyPentUp4thsMstr.mp3 12of22studyPentUp4thsMstr]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22gamelan1b.mp3 12of22gamelan1b]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study3.mp3 12of22study3 (children's story)]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of22study7.mp3 12of22study7]{{dead link}}

By [[Joel Grant Taylor]]

[[Category:Pages with Scala files]]
[[Category:22edo]]
[[Category:12-tone scales]]
[[Category:MOS scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Superpyth]]
[[Category:Quasisuper]]
[[Category:P-chromatic]]</text>
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  </page>
  <page>
    <title>22edo hexachordal scale (10L 2s)</title>
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      <text bytes="624" sha1="p6p8cg7nwlxgmbke6ncjrhnrqld8sqs" xml:space="preserve">! 12-22h.scl

!

Hexachordal 12-tone scale in 22-tET

12

!

109.09091

218.18182

327.27273

436.36364

490.90909

600.00000

709.09091

818.18182

927.27273

1036.36364

1145.45455

2/1

=Music=
[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge] and

[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina] both by [[Joel_Grant_Taylor|Joel Grant Taylor]].

[http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] by [http://chrisvaisvil.com/?p=2403 Chris Vaisvil].</text>
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      <timestamp>2012-06-17T17:18:39Z</timestamp>
      <contributor>
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        <id>0</id>
      </contributor>
      <comment>**Imported revision 346004692 - Original comment: **</comment>
      <origin>4515</origin>
      <model>wikitext</model>
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      <text bytes="3154" sha1="qvui2u9lx0xpk1htbqrwyyabudq6wbz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-06-17 17:18:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;346004692&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12-22h.scl
!
Hexachordal 12-tone scale in 22-tET
 12
!
 109.09091
 218.18182
 327.27273
 436.36364
 490.90909
 600.00000
 709.09091
 818.18182
 927.27273
 1036.36364
 1145.45455
 2/1

=Music=
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3|12-22hexachordal Dirge]] and
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3|12-22hexachordal Sonatina]] both by [[Joel Grant Taylor]].
[[http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3|Smoke Filled Bar]] by [[http://chrisvaisvil.com/?p=2403|Chris Vaisvil]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12-22h&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12-22h.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
Hexachordal 12-tone scale in 22-tET&amp;lt;br /&amp;gt;
 12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
 109.09091&amp;lt;br /&amp;gt;
 218.18182&amp;lt;br /&amp;gt;
 327.27273&amp;lt;br /&amp;gt;
 436.36364&amp;lt;br /&amp;gt;
 490.90909&amp;lt;br /&amp;gt;
 600.00000&amp;lt;br /&amp;gt;
 709.09091&amp;lt;br /&amp;gt;
 818.18182&amp;lt;br /&amp;gt;
 927.27273&amp;lt;br /&amp;gt;
 1036.36364&amp;lt;br /&amp;gt;
 1145.45455&amp;lt;br /&amp;gt;
 2/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3" rel="nofollow"&amp;gt;12-22hexachordal Dirge&amp;lt;/a&amp;gt; and&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3" rel="nofollow"&amp;gt;12-22hexachordal Sonatina&amp;lt;/a&amp;gt; both by &amp;lt;a class="wiki_link" href="/Joel%20Grant%20Taylor"&amp;gt;Joel Grant Taylor&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3" rel="nofollow"&amp;gt;Smoke Filled Bar&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/?p=2403" rel="nofollow"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>qvui2u9lx0xpk1htbqrwyyabudq6wbz</sha1>
    </revision>
    <revision>
      <id>93931</id>
      <parentid>101</parentid>
      <timestamp>2022-08-05T01:06:26Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Cleanup, misc. edits, categories</comment>
      <origin>93931</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="817" sha1="kymnkt4kxuh3c9pebvjmi4sum5stobp" xml:space="preserve">&lt;pre&gt;
! 12-22h.scl
!
Hexachordal 12-tone scale in 22-tET
12
!
109.09091
218.18182
327.27273
436.36364
490.90909
600.00000
709.09091
818.18182
927.27273
1036.36364
1145.45455
2/1
&lt;/pre&gt;

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge]{{dead link}} and
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina]{{dead link}}, both by [[Joel Grant Taylor]].
* [http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] ([http://chrisvaisvil.com/?p=2403 details]) by [[Chris Vaisvil]]

== See also ==
* [[10L 2s]]

[[Category:12-tone scales]]
[[Category:MOS scales]]
[[Category:Tempered scales]]
[[Category:22edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>kymnkt4kxuh3c9pebvjmi4sum5stobp</sha1>
    </revision>
    <revision>
      <id>186536</id>
      <parentid>93931</parentid>
      <timestamp>2025-03-17T04:01:04Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>186536</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="861" sha1="mycrbxln4va2uxlgvk3s7brca2jfwfo" xml:space="preserve">The tuning of [[10L 2s]] found in [[22edo]].&lt;pre&gt;
! 12-22h.scl
!
Hexachordal 12-tone scale in 22-tET
12
!
109.09091
218.18182
327.27273
436.36364
490.90909
600.00000
709.09091
818.18182
927.27273
1036.36364
1145.45455
2/1
&lt;/pre&gt;

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge]{{dead link}} and
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina]{{dead link}}, both by [[Joel Grant Taylor]].
* [http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] ([http://chrisvaisvil.com/?p=2403 details]) by [[Chris Vaisvil]]

== See also ==
* [[10L 2s]]

[[Category:12-tone scales]]
[[Category:MOS scales]]
[[Category:Tempered scales]]
[[Category:22edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>mycrbxln4va2uxlgvk3s7brca2jfwfo</sha1>
    </revision>
    <revision>
      <id>200470</id>
      <parentid>186536</parentid>
      <timestamp>2025-06-23T23:31:23Z</timestamp>
      <contributor>
        <username>Hkm</username>
        <id>551</id>
      </contributor>
      <origin>200470</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="910" sha1="r4x36k8x29n5ul5z13us48fj6hdnahj" xml:space="preserve">The tuning of [[10L 2s]] found in [[22edo]].&lt;pre&gt;
! 12-22h.scl
!
Hexachordal 12-tone scale in 22-tET
12
!
109.09091
218.18182
327.27273
436.36364
490.90909
600.00000
709.09091
818.18182
927.27273
1036.36364
1145.45455
2/1
&lt;/pre&gt;

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge]{{dead link}} and
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina]{{dead link}}, both by [[Joel Grant Taylor]].
* [http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] ([http://chrisvaisvil.com/?p=2403 details]) by [[Chris Vaisvil]]

== See also ==
* [[10L 2s]]

[[Category:12-tone scales]]
[[Category:MOS scales]]
[[Category:Tempered scales]]
[[Category:22edo]]
[[Category:Pages with Scala files]]
[[Category:Pages with mostly numerical content]]</text>
      <sha1>r4x36k8x29n5ul5z13us48fj6hdnahj</sha1>
    </revision>
    <revision>
      <id>200558</id>
      <parentid>200470</parentid>
      <timestamp>2025-06-24T00:22:44Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Pages with mostly numerical content]]" to ""</comment>
      <origin>200558</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="861" sha1="mycrbxln4va2uxlgvk3s7brca2jfwfo" xml:space="preserve">The tuning of [[10L 2s]] found in [[22edo]].&lt;pre&gt;
! 12-22h.scl
!
Hexachordal 12-tone scale in 22-tET
12
!
109.09091
218.18182
327.27273
436.36364
490.90909
600.00000
709.09091
818.18182
927.27273
1036.36364
1145.45455
2/1
&lt;/pre&gt;

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge]{{dead link}} and
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina]{{dead link}}, both by [[Joel Grant Taylor]].
* [http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] ([http://chrisvaisvil.com/?p=2403 details]) by [[Chris Vaisvil]]

== See also ==
* [[10L 2s]]

[[Category:12-tone scales]]
[[Category:MOS scales]]
[[Category:Tempered scales]]
[[Category:22edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>mycrbxln4va2uxlgvk3s7brca2jfwfo</sha1>
    </revision>
    <revision>
      <id>201140</id>
      <parentid>200558</parentid>
      <timestamp>2025-06-24T00:38:32Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered scales]]" to "[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>201140</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="910" sha1="37kv6t8we1f124buc74vnqtuztfshzp" xml:space="preserve">The tuning of [[10L 2s]] found in [[22edo]].&lt;pre&gt;
! 12-22h.scl
!
Hexachordal 12-tone scale in 22-tET
12
!
109.09091
218.18182
327.27273
436.36364
490.90909
600.00000
709.09091
818.18182
927.27273
1036.36364
1145.45455
2/1
&lt;/pre&gt;

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge]{{dead link}} and
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina]{{dead link}}, both by [[Joel Grant Taylor]].
* [http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] ([http://chrisvaisvil.com/?p=2403 details]) by [[Chris Vaisvil]]

== See also ==
* [[10L 2s]]

[[Category:12-tone scales]]
[[Category:MOS scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:22edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>37kv6t8we1f124buc74vnqtuztfshzp</sha1>
    </revision>
    <revision>
      <id>201405</id>
      <parentid>201140</parentid>
      <timestamp>2025-06-24T02:16:12Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered scales]]↵[[Category:Pages with mostly numerical content]]" to "[[Category:Tempered_scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>201405</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="910" sha1="94zjqn057jxph0fuxo8gcdm2oyh6m6i" xml:space="preserve">The tuning of [[10L 2s]] found in [[22edo]].&lt;pre&gt;
! 12-22h.scl
!
Hexachordal 12-tone scale in 22-tET
12
!
109.09091
218.18182
327.27273
436.36364
490.90909
600.00000
709.09091
818.18182
927.27273
1036.36364
1145.45455
2/1
&lt;/pre&gt;

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge]{{dead link}} and
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina]{{dead link}}, both by [[Joel Grant Taylor]].
* [http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] ([http://chrisvaisvil.com/?p=2403 details]) by [[Chris Vaisvil]]

== See also ==
* [[10L 2s]]

[[Category:12-tone scales]]
[[Category:MOS scales]]
[[Category:Tempered_scales]]
[[Category:Pages with mostly numerical content]]
[[Category:22edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>94zjqn057jxph0fuxo8gcdm2oyh6m6i</sha1>
    </revision>
    <revision>
      <id>202790</id>
      <parentid>201405</parentid>
      <timestamp>2025-06-24T02:53:55Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered_scales]]↵[[Category:Pages with mostly numerical content]]" to "[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>202790</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="910" sha1="37kv6t8we1f124buc74vnqtuztfshzp" xml:space="preserve">The tuning of [[10L 2s]] found in [[22edo]].&lt;pre&gt;
! 12-22h.scl
!
Hexachordal 12-tone scale in 22-tET
12
!
109.09091
218.18182
327.27273
436.36364
490.90909
600.00000
709.09091
818.18182
927.27273
1036.36364
1145.45455
2/1
&lt;/pre&gt;

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge]{{dead link}} and
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina]{{dead link}}, both by [[Joel Grant Taylor]].
* [http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] ([http://chrisvaisvil.com/?p=2403 details]) by [[Chris Vaisvil]]

== See also ==
* [[10L 2s]]

[[Category:12-tone scales]]
[[Category:MOS scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:22edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>37kv6t8we1f124buc74vnqtuztfshzp</sha1>
    </revision>
    <revision>
      <id>217475</id>
      <parentid>202790</parentid>
      <timestamp>2025-11-22T01:49:05Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <minor/>
      <comment>VectorGraphics moved page [[12-22h]] to [[22edo hexachordal scale (10L 2s)]]</comment>
      <origin>202790</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="910" sha1="37kv6t8we1f124buc74vnqtuztfshzp" xml:space="preserve">The tuning of [[10L 2s]] found in [[22edo]].&lt;pre&gt;
! 12-22h.scl
!
Hexachordal 12-tone scale in 22-tET
12
!
109.09091
218.18182
327.27273
436.36364
490.90909
600.00000
709.09091
818.18182
927.27273
1036.36364
1145.45455
2/1
&lt;/pre&gt;

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge]{{dead link}} and
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina]{{dead link}}, both by [[Joel Grant Taylor]].
* [http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] ([http://chrisvaisvil.com/?p=2403 details]) by [[Chris Vaisvil]]

== See also ==
* [[10L 2s]]

[[Category:12-tone scales]]
[[Category:MOS scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:22edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>37kv6t8we1f124buc74vnqtuztfshzp</sha1>
    </revision>
    <revision>
      <id>221645</id>
      <parentid>217475</parentid>
      <timestamp>2026-01-15T23:08:46Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>rewrite description</comment>
      <origin>221645</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="939" sha1="qaqlprgavr81ifvcsoxyztmog5xmmp0" xml:space="preserve">A modified [[10L 2s]], tuned to [[22edo]], containing two [[hexachord]]s.&lt;pre&gt;
! 12-22h.scl
!
Hexachordal 12-tone scale in 22-tET
12
!
109.09091
218.18182
327.27273
436.36364
490.90909
600.00000
709.09091
818.18182
927.27273
1036.36364
1145.45455
2/1
&lt;/pre&gt;

== Music ==
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 12-22hexachordal Dirge]{{dead link}} and
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 12-22hexachordal Sonatina]{{dead link}}, both by [[Joel Grant Taylor]].
* [http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 Smoke Filled Bar] ([http://chrisvaisvil.com/?p=2403 details]) by [[Chris Vaisvil]]

== See also ==
* [[10L 2s]]

[[Category:12-tone scales]]
[[Category:MOS scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:22edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>qaqlprgavr81ifvcsoxyztmog5xmmp0</sha1>
    </revision>
    <revision>
      <id>222650</id>
      <parentid>221645</parentid>
      <timestamp>2026-01-27T19:22:06Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Fix dead links</comment>
      <origin>222650</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1115" sha1="ctolnrb3k5hm8wjqlsh2qzrkune3zit" xml:space="preserve">A modified [[10L 2s]], tuned to [[22edo]], containing two [[hexachord]]s.&lt;pre&gt;
! 12-22h.scl
!
Hexachordal 12-tone scale in 22-tET
12
!
109.09091
218.18182
327.27273
436.36364
490.90909
600.00000
709.09091
818.18182
927.27273
1036.36364
1145.45455
2/1
&lt;/pre&gt;

== Music ==
; [[Joel Grant Taylor]]
* [https://web.archive.org/web/20201127012345/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Dirge.mp3 ''Dirge''] – in the hexachordal dodecatonic modmos, [[12-22h]]
* [https://web.archive.org/web/20201127012408/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12-22hexachordal%20Sonatina.mp3 ''Sonatina''] – ditto

; [[Chris Vaisvil]]
* ''Smoke Filled Bar'' (2012) – [https://www.chrisvaisvil.com/smoke-filled-bar/ blog] | [https://web.archive.org/web/20230530093324/http://micro.soonlabel.com/22-ET/20120616-12-22h.scl-smoke-filled-bar.mp3 play] – in 12-22h.

== See also ==
* [[10L 2s]]

[[Category:12-tone scales]]
[[Category:MOS scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:22edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>ctolnrb3k5hm8wjqlsh2qzrkune3zit</sha1>
    </revision>
  </page>
  <page>
    <title>12-27</title>
    <ns>0</ns>
    <id>102</id>
    <revision>
      <id>102</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>102</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="579" sha1="1v2ztvl39n9dvs1205wb5pljjdc0no8" xml:space="preserve">! 12-27.scl

!

12 out of 27, Herman Miller's Galticeran scale

12

!

133.33333

222.22222

311.11111

400.00000

533.33333

622.22222

711.11111

800.00000

933.33333

1022.22222

1111.11111

2/1

=Music=
[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 Sad Like Winter Leaves] by [http://soundcloud.com/cityoftheasleep/sad-like-winter-trees Igliashon Jones]

[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina] by [http://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Grant Taylor]</text>
      <sha1>1v2ztvl39n9dvs1205wb5pljjdc0no8</sha1>
    </revision>
    <revision>
      <id>4516</id>
      <timestamp>2012-06-17T17:23:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 346005590 - Original comment: **</comment>
      <origin>4516</origin>
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      <format>text/x-wiki</format>
      <text bytes="3026" sha1="m6u5rt8dthilcukx7c9twp52bqxk2wx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-06-17 17:23:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;346005590&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
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!
12 out of 27, Herman Miller's Galticeran scale
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=Music=
[[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3|Sad Like Winter Leaves]] by [[http://soundcloud.com/cityoftheasleep/sad-like-winter-trees|Igliashon Jones]]
[[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3|Galticeran Sonatina]] by [[http://soundcloud.com/joelgranttaylor/galticeran_sonatina|Joel Grant Taylor]]&lt;/pre&gt;&lt;/div&gt;
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12 out of 27, Herman Miller's Galticeran scale&amp;lt;br /&amp;gt;
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!&amp;lt;br /&amp;gt;
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&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Music"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Music&amp;lt;/h1&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3" rel="nofollow"&amp;gt;Sad Like Winter Leaves&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/cityoftheasleep/sad-like-winter-trees" rel="nofollow"&amp;gt;Igliashon Jones&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3" rel="nofollow"&amp;gt;Galticeran Sonatina&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://soundcloud.com/joelgranttaylor/galticeran_sonatina" rel="nofollow"&amp;gt;Joel Grant Taylor&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 345996578 - Original comment: **</comment>
      <origin>4517</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1809" sha1="lv4dctd0y6fe2pipaq455c51w2uh5m0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-06-17 16:17:38 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;345996578&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12-27.scl
!
12 out of 27, Herman Miller's Galticeran scale
 12
!
 133.33333
 222.22222
 311.11111
 400.00000
 533.33333
 622.22222
 711.11111
 800.00000
 933.33333
 1022.22222
 1111.11111
 2/1
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12-27&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12-27.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
12 out of 27, Herman Miller's Galticeran scale&amp;lt;br /&amp;gt;
 12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
 133.33333&amp;lt;br /&amp;gt;
 222.22222&amp;lt;br /&amp;gt;
 311.11111&amp;lt;br /&amp;gt;
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    <revision>
      <id>83929</id>
      <parentid>102</parentid>
      <timestamp>2022-01-14T14:31:04Z</timestamp>
      <contributor>
        <username>Dummy index</username>
        <id>318</id>
      </contributor>
      <comment>+category</comment>
      <origin>83929</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="597" sha1="n2q830vghhjwumh6gq26hoct3v42h5x" xml:space="preserve">&lt;pre&gt;
! 12-27.scl
!
12 out of 27, Herman Miller's Galticeran scale
12
!
133.33333
222.22222
311.11111
400.00000
533.33333
622.22222
711.11111
800.00000
933.33333
1022.22222
1111.11111
2/1
&lt;/pre&gt;

=Music=
[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 Sad Like Winter Leaves] by [http://soundcloud.com/cityoftheasleep/sad-like-winter-trees Igliashon Jones]

[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina] by [http://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Grant Taylor]

[[Category:Augene]]</text>
      <sha1>n2q830vghhjwumh6gq26hoct3v42h5x</sha1>
    </revision>
    <revision>
      <id>83930</id>
      <parentid>83929</parentid>
      <timestamp>2022-01-14T14:42:53Z</timestamp>
      <contributor>
        <username>Dummy index</username>
        <id>318</id>
      </contributor>
      <comment>+categories; mark dead links</comment>
      <origin>83930</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="687" sha1="ss6v9n5emc83p0i2zhn1tnqipw9h9qn" xml:space="preserve">&lt;pre&gt;
! 12-27.scl
!
12 out of 27, Herman Miller's Galticeran scale
12
!
133.33333
222.22222
311.11111
400.00000
533.33333
622.22222
711.11111
800.00000
933.33333
1022.22222
1111.11111
2/1
&lt;/pre&gt;

=Music=
[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 Sad Like Winter Leaves] by [http://soundcloud.com/cityoftheasleep/sad-like-winter-trees Igliashon Jones]{{dead link}}

[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina] by [http://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Grant Taylor]{{dead link}}

[[Category:Pages with Scala files]]
[[Category:12-tone scales]]
[[Category:Augene]]</text>
      <sha1>ss6v9n5emc83p0i2zhn1tnqipw9h9qn</sha1>
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    <revision>
      <id>186545</id>
      <parentid>83930</parentid>
      <timestamp>2025-03-17T04:06:29Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>186545</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="736" sha1="9gmvbxou9wazug76ve789cgbnzhfze1" xml:space="preserve">A soft tuning of [[3L 9s]] found in [[27edo]]. &lt;pre&gt;
! 12-27.scl
!
12 out of 27, Herman Miller's Galticeran scale
12
!
133.33333
222.22222
311.11111
400.00000
533.33333
622.22222
711.11111
800.00000
933.33333
1022.22222
1111.11111
2/1
&lt;/pre&gt;

=Music=
[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 Sad Like Winter Leaves] by [https://soundcloud.com/cityoftheasleep/sad-like-winter-trees Igliashon Jones]{{dead link}}

[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina] by [https://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Grant Taylor]{{dead link}}

[[Category:Pages with Scala files]]
[[Category:12-tone scales]]
[[Category:Augene]]</text>
      <sha1>9gmvbxou9wazug76ve789cgbnzhfze1</sha1>
    </revision>
    <revision>
      <id>222733</id>
      <parentid>186545</parentid>
      <timestamp>2026-01-28T08:32:51Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Recategorize</comment>
      <origin>222733</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="772" sha1="7b0bsl3etvxvr4bz0esdkgyryg45f4f" xml:space="preserve">A soft tuning of [[3L 9s]] found in [[27edo]]. &lt;pre&gt;
! 12-27.scl
!
12 out of 27, Herman Miller's Galticeran scale
12
!
133.33333
222.22222
311.11111
400.00000
533.33333
622.22222
711.11111
800.00000
933.33333
1022.22222
1111.11111
2/1
&lt;/pre&gt;

== Music ==
[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 Sad Like Winter Leaves] by [https://soundcloud.com/cityoftheasleep/sad-like-winter-trees Igliashon Jones]{{dead link}}

[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina] by [https://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Grant Taylor]{{dead link}}

[[Category:Pages with Scala files]]
[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Augmented]]</text>
      <sha1>7b0bsl3etvxvr4bz0esdkgyryg45f4f</sha1>
    </revision>
    <revision>
      <id>222740</id>
      <parentid>222733</parentid>
      <timestamp>2026-01-28T09:03:21Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Fix dead links</comment>
      <origin>222740</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="726" sha1="i8qvabamf35g8o5vbh5zsm29q0ow70l" xml:space="preserve">A soft tuning of [[3L 9s]] found in [[27edo]]. &lt;pre&gt;
! 12-27.scl
!
12 out of 27, Herman Miller's Galticeran scale
12
!
133.33333
222.22222
311.11111
400.00000
533.33333
622.22222
711.11111
800.00000
933.33333
1022.22222
1111.11111
2/1
&lt;/pre&gt;

== Music ==
; [[Igliashon Jones]]
* [https://web.archive.org/web/20201127012539/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 ''Sad Like Winter Leaves'']

; [[Joel Grant Taylor]]
* [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina'']

[[Category:Pages with Scala files]]
[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Augmented]]</text>
      <sha1>i8qvabamf35g8o5vbh5zsm29q0ow70l</sha1>
    </revision>
  </page>
  <page>
    <title>12-Adams</title>
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    <revision>
      <id>103</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>103</origin>
      <model>wikitext</model>
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      <text bytes="176" sha1="qexj28nkj5riroz41g4gkyn4sm2tp5z" xml:space="preserve">! 12-Adams.scl

!

12 out of 72-TET scale by Prent Rodgers

12

!

200.000

266.667

383.333

433.333

500.000

550.000

700.000

883.333

966.667

1050.000

1083.333

1200.000</text>
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    <revision>
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      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 464513604 - Original comment: **</comment>
      <origin>4518</origin>
      <model>wikitext</model>
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      <text bytes="1744" sha1="tnpj79zjsbjtuzhppsbkl6d0tcv07vv" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2013-10-30 12:28:35 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;464513604&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12-Adams.scl
!
12 out of 72-TET scale by Prent Rodgers
12
!
200.000
266.667
383.333
433.333
500.000
550.000
700.000
883.333
966.667
1050.000
1083.333
1200.000
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12-Adams&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12-Adams.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
12 out of 72-TET scale by Prent Rodgers&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
200.000&amp;lt;br /&amp;gt;
266.667&amp;lt;br /&amp;gt;
383.333&amp;lt;br /&amp;gt;
433.333&amp;lt;br /&amp;gt;
500.000&amp;lt;br /&amp;gt;
550.000&amp;lt;br /&amp;gt;
700.000&amp;lt;br /&amp;gt;
883.333&amp;lt;br /&amp;gt;
966.667&amp;lt;br /&amp;gt;
1050.000&amp;lt;br /&amp;gt;
1083.333&amp;lt;br /&amp;gt;
1200.000&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>tnpj79zjsbjtuzhppsbkl6d0tcv07vv</sha1>
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    <revision>
      <id>93932</id>
      <parentid>103</parentid>
      <timestamp>2022-08-05T02:02:21Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Cleanup, misc. edits, categories</comment>
      <origin>93932</origin>
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      <text bytes="322" sha1="fgaqzqzzia5dn0aiu5lxaceu9i90ddt" xml:space="preserve">&lt;pre&gt;
! 12-Adams.scl
!
12 out of 72-TET scale by Prent Rodgers
12
!
200.000
266.667
383.333
433.333
500.000
550.000
700.000
883.333
966.667
1050.000
1083.333
1200.000
&lt;/pre&gt;

== See also ==
* [[Prent Rodgers]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:72edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>fgaqzqzzia5dn0aiu5lxaceu9i90ddt</sha1>
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    <revision>
      <id>186542</id>
      <parentid>93932</parentid>
      <timestamp>2025-03-17T04:04:46Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
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      <origin>186542</origin>
      <model>wikitext</model>
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      <text bytes="424" sha1="nx0k4139hff48n84ftvk559hitg573n" xml:space="preserve">A 12-note scale in [[72edo]], with an unknown significance and a large number of different step sizes.&lt;pre&gt;
! 12-Adams.scl
!
12 out of 72-TET scale by Prent Rodgers
12
!
200.000
266.667
383.333
433.333
500.000
550.000
700.000
883.333
966.667
1050.000
1083.333
1200.000
&lt;/pre&gt;

== See also ==
* [[Prent Rodgers]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:72edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>nx0k4139hff48n84ftvk559hitg573n</sha1>
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    <revision>
      <id>200472</id>
      <parentid>186542</parentid>
      <timestamp>2025-06-23T23:31:53Z</timestamp>
      <contributor>
        <username>Hkm</username>
        <id>551</id>
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      <origin>200472</origin>
      <model>wikitext</model>
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      <text bytes="473" sha1="5gpvmdr3fci84yywcowywz2iza18krs" xml:space="preserve">A 12-note scale in [[72edo]], with an unknown significance and a large number of different step sizes.&lt;pre&gt;
! 12-Adams.scl
!
12 out of 72-TET scale by Prent Rodgers
12
!
200.000
266.667
383.333
433.333
500.000
550.000
700.000
883.333
966.667
1050.000
1083.333
1200.000
&lt;/pre&gt;

== See also ==
* [[Prent Rodgers]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:72edo]]
[[Category:Pages with Scala files]]
[[Category:Pages with mostly numerical content]]</text>
      <sha1>5gpvmdr3fci84yywcowywz2iza18krs</sha1>
    </revision>
    <revision>
      <id>200629</id>
      <parentid>200472</parentid>
      <timestamp>2025-06-24T00:24:18Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Pages with mostly numerical content]]" to ""</comment>
      <origin>200629</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="424" sha1="nx0k4139hff48n84ftvk559hitg573n" xml:space="preserve">A 12-note scale in [[72edo]], with an unknown significance and a large number of different step sizes.&lt;pre&gt;
! 12-Adams.scl
!
12 out of 72-TET scale by Prent Rodgers
12
!
200.000
266.667
383.333
433.333
500.000
550.000
700.000
883.333
966.667
1050.000
1083.333
1200.000
&lt;/pre&gt;

== See also ==
* [[Prent Rodgers]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:72edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>nx0k4139hff48n84ftvk559hitg573n</sha1>
    </revision>
    <revision>
      <id>200777</id>
      <parentid>200629</parentid>
      <timestamp>2025-06-24T00:27:53Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered scales]]" to "[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>200777</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="473" sha1="ffcnjx5t7xxpogymmw16stb3mgaps69" xml:space="preserve">A 12-note scale in [[72edo]], with an unknown significance and a large number of different step sizes.&lt;pre&gt;
! 12-Adams.scl
!
12 out of 72-TET scale by Prent Rodgers
12
!
200.000
266.667
383.333
433.333
500.000
550.000
700.000
883.333
966.667
1050.000
1083.333
1200.000
&lt;/pre&gt;

== See also ==
* [[Prent Rodgers]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:72edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>ffcnjx5t7xxpogymmw16stb3mgaps69</sha1>
    </revision>
    <revision>
      <id>201486</id>
      <parentid>200777</parentid>
      <timestamp>2025-06-24T02:17:37Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered scales]]↵[[Category:Pages with mostly numerical content]]" to "[[Category:Tempered_scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>201486</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="473" sha1="q2n14zegwunm7pyulvq1ev684l91o0l" xml:space="preserve">A 12-note scale in [[72edo]], with an unknown significance and a large number of different step sizes.&lt;pre&gt;
! 12-Adams.scl
!
12 out of 72-TET scale by Prent Rodgers
12
!
200.000
266.667
383.333
433.333
500.000
550.000
700.000
883.333
966.667
1050.000
1083.333
1200.000
&lt;/pre&gt;

== See also ==
* [[Prent Rodgers]]

[[Category:12-tone scales]]
[[Category:Tempered_scales]]
[[Category:Pages with mostly numerical content]]
[[Category:72edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>q2n14zegwunm7pyulvq1ev684l91o0l</sha1>
    </revision>
    <revision>
      <id>202785</id>
      <parentid>201486</parentid>
      <timestamp>2025-06-24T02:53:50Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered_scales]]↵[[Category:Pages with mostly numerical content]]" to "[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>202785</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="473" sha1="ffcnjx5t7xxpogymmw16stb3mgaps69" xml:space="preserve">A 12-note scale in [[72edo]], with an unknown significance and a large number of different step sizes.&lt;pre&gt;
! 12-Adams.scl
!
12 out of 72-TET scale by Prent Rodgers
12
!
200.000
266.667
383.333
433.333
500.000
550.000
700.000
883.333
966.667
1050.000
1083.333
1200.000
&lt;/pre&gt;

== See also ==
* [[Prent Rodgers]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:72edo]]
[[Category:Pages with Scala files]]</text>
      <sha1>ffcnjx5t7xxpogymmw16stb3mgaps69</sha1>
    </revision>
  </page>
  <page>
    <title>12-EDD</title>
    <ns>0</ns>
    <id>104</id>
    <redirect title="12edo" />
    <revision>
      <id>104</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>104</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="47" sha1="9vmkb8mtstodz8p33qi8sodwc4bydst" xml:space="preserve">#REDIRECT [http://purdal.wikispaces.com/12-EDD]</text>
      <sha1>9vmkb8mtstodz8p33qi8sodwc4bydst</sha1>
    </revision>
    <revision>
      <id>31074</id>
      <parentid>104</parentid>
      <timestamp>2018-08-19T19:55:13Z</timestamp>
      <contributor>
        <username>PreImportHandFixes&gt;Battaglia01</username>
        <id>0</id>
      </contributor>
      <comment>Redirected page to [[Purdal:12-EDD]]</comment>
      <origin>31074</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="28" sha1="sks2ho32zv8hlczea85hs9jmroivyyv" xml:space="preserve">#REDIRECT [[:purdal:12-EDD]]</text>
      <sha1>sks2ho32zv8hlczea85hs9jmroivyyv</sha1>
    </revision>
    <revision>
      <id>58185</id>
      <parentid>31074</parentid>
      <timestamp>2021-01-03T19:47:13Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>Changed redirect target from [[purdal:12-EDD]] to [[12edo]]</comment>
      <origin>58185</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19" sha1="r85u4zchybng6r81vws8zrx81xzvp75" xml:space="preserve">#REDIRECT [[12edo]]</text>
      <sha1>r85u4zchybng6r81vws8zrx81xzvp75</sha1>
    </revision>
    <revision>
      <id>73495</id>
      <parentid>58185</parentid>
      <timestamp>2021-06-30T23:31:20Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>skip double redirect</comment>
      <origin>73495</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20" sha1="0osge8pfmhsvtlubuvq2w176q7e2kju" xml:space="preserve">#REDIRECT [[12-EDO]]</text>
      <sha1>0osge8pfmhsvtlubuvq2w176q7e2kju</sha1>
    </revision>
    <revision>
      <id>76657</id>
      <parentid>73495</parentid>
      <timestamp>2021-08-21T16:02:35Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>Changed redirect target from [[12-EDO]] to [[12edo]]</comment>
      <origin>76657</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19" sha1="r85u4zchybng6r81vws8zrx81xzvp75" xml:space="preserve">#REDIRECT [[12edo]]</text>
      <sha1>r85u4zchybng6r81vws8zrx81xzvp75</sha1>
    </revision>
  </page>
  <page>
    <title>1200edo</title>
    <ns>0</ns>
    <id>105</id>
    <revision>
      <id>105</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>105</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1436" sha1="jdz09oomrors1farl07zarne7qars70" xml:space="preserve">The ''1200 division'' divides the octave in 1200 equal parts of exactly 1 [[cent|cent]] each. It is notable mostly because it is the equal division corresponding to cents.

1200edo is uniquely [[consistent|consistent]] through the [[11-limit|11-limit]], which means the intervals of the 11-limit[[Tonality_diamond| tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;lt;1200 1902 2786 3369 4141|. It is [[contorted|contorted]] in the [[5-limit|5-limit]], having the same mapping as 600edo. In the [[7-limit|7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo|171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo|494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit POTE tuning of [[Kleismic_family#Quadritikleismic|quadritikleismic temperament]]: &amp;lt;1200 1902 2785 3368|. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.
[[Category:cents]]
[[Category:edo]]
[[Category:notation]]
[[Category:theory]]</text>
      <sha1>jdz09oomrors1farl07zarne7qars70</sha1>
    </revision>
    <revision>
      <id>4519</id>
      <timestamp>2013-01-20T11:35:55Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 399910442 - Original comment: **</comment>
      <origin>4519</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4342" sha1="q105azb06xv1n14m3jgnqb232ew3c4d" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2013-01-20 11:35:55 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;399910442&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //1200 division// divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit[[tonality diamond| tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;lt;1200 1902 2786 3369 4141|. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit POTE tuning of [[Kleismic family#Quadritikleismic|quadritikleismic temperament]]: &amp;lt;1200 1902 2785 3368|. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1200edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;1200 division&amp;lt;/em&amp;gt; divides the octave in 1200 equal parts of exactly 1 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt; each. It is notable mostly because it is the equal division corresponding to cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
1200edo is uniquely &amp;lt;a class="wiki_link" href="/consistent"&amp;gt;consistent&amp;lt;/a&amp;gt; through the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, which means the intervals of the 11-limit&amp;lt;a class="wiki_link" href="/tonality%20diamond"&amp;gt; tonality diamond&amp;lt;/a&amp;gt;, and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;amp;lt;1200 1902 2786 3369 4141|. It is &amp;lt;a class="wiki_link" href="/contorted"&amp;gt;contorted&amp;lt;/a&amp;gt; in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, having the same mapping as 600edo. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171edo&amp;lt;/a&amp;gt;. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by &amp;lt;a class="wiki_link" href="/494edo"&amp;gt;494edo&amp;lt;/a&amp;gt;. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit POTE tuning of &amp;lt;a class="wiki_link" href="/Kleismic%20family#Quadritikleismic"&amp;gt;quadritikleismic temperament&amp;lt;/a&amp;gt;: &amp;amp;lt;1200 1902 2785 3368|. It also provides the optimal patent val for the 224&amp;amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>q105azb06xv1n14m3jgnqb232ew3c4d</sha1>
    </revision>
    <revision>
      <id>4520</id>
      <timestamp>2012-06-04T17:14:43Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 342579934 - Original comment: **</comment>
      <origin>4520</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4346" sha1="ggd18xszlgoivijw0x4pabk2cixizn4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-06-04 17:14:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;342579934&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //1200 division// divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit[[tonality diamond| tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;lt;1200 1902 2786 3369 4141|. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200cd, which is extremely closely close to the 7-limit POTE tuning of [[Kleismic family#Quadritikleismic|quadritikleismic temperament]]: &amp;lt;1200 1902 2785 3368|. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1200edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;1200 division&amp;lt;/em&amp;gt; divides the octave in 1200 equal parts of exactly 1 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt; each. It is notable mostly because it is the equal division corresponding to cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
1200edo is uniquely &amp;lt;a class="wiki_link" href="/consistent"&amp;gt;consistent&amp;lt;/a&amp;gt; through the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, which means the intervals of the 11-limit&amp;lt;a class="wiki_link" href="/tonality%20diamond"&amp;gt; tonality diamond&amp;lt;/a&amp;gt;, and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;amp;lt;1200 1902 2786 3369 4141|. It is &amp;lt;a class="wiki_link" href="/contorted"&amp;gt;contorted&amp;lt;/a&amp;gt; in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, having the same mapping as 600edo. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171edo&amp;lt;/a&amp;gt;. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by &amp;lt;a class="wiki_link" href="/494edo"&amp;gt;494edo&amp;lt;/a&amp;gt;. In the 7-limit, it provides a val, 1200cd, which is extremely closely close to the 7-limit POTE tuning of &amp;lt;a class="wiki_link" href="/Kleismic%20family#Quadritikleismic"&amp;gt;quadritikleismic temperament&amp;lt;/a&amp;gt;: &amp;amp;lt;1200 1902 2785 3368|. It also provides the optimal patent val for the 224&amp;amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ggd18xszlgoivijw0x4pabk2cixizn4</sha1>
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    <revision>
      <id>4521</id>
      <timestamp>2012-06-04T17:09:55Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 342578710 - Original comment: **</comment>
      <origin>4521</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4074" sha1="caqtfqhqykyg25f8f4uxcbsf060k8vc" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-06-04 17:09:55 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;342578710&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //1200 division// divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit[[tonality diamond| tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;lt;1200 1902 2786 3369 4141|. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200cd,  which is extremely closely close to the 7-limit POTE tuning of [[Kleismic family#Quadritikleismic|quadritikleismic temperament]]: &amp;lt;1200 1902 2785 3368|.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1200edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;1200 division&amp;lt;/em&amp;gt; divides the octave in 1200 equal parts of exactly 1 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt; each. It is notable mostly because it is the equal division corresponding to cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
1200edo is uniquely &amp;lt;a class="wiki_link" href="/consistent"&amp;gt;consistent&amp;lt;/a&amp;gt; through the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, which means the intervals of the 11-limit&amp;lt;a class="wiki_link" href="/tonality%20diamond"&amp;gt; tonality diamond&amp;lt;/a&amp;gt;, and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;amp;lt;1200 1902 2786 3369 4141|. It is &amp;lt;a class="wiki_link" href="/contorted"&amp;gt;contorted&amp;lt;/a&amp;gt; in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, having the same mapping as 600edo. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171edo&amp;lt;/a&amp;gt;. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by &amp;lt;a class="wiki_link" href="/494edo"&amp;gt;494edo&amp;lt;/a&amp;gt;. In the 7-limit, it provides a val, 1200cd,  which is extremely closely close to the 7-limit POTE tuning of &amp;lt;a class="wiki_link" href="/Kleismic%20family#Quadritikleismic"&amp;gt;quadritikleismic temperament&amp;lt;/a&amp;gt;: &amp;amp;lt;1200 1902 2785 3368|.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4522</id>
      <timestamp>2011-06-28T02:42:41Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 239085981 - Original comment: **</comment>
      <origin>4522</origin>
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      <format>text/x-wiki</format>
      <text bytes="3615" sha1="7wc9at7d1ldnltawnnqar44phy1geh3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2011-06-28 02:42:41 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239085981&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //1200 division// divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit[[tonality diamond| tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;lt;1200 1902 2786 3369 4141|. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1200edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;1200 division&amp;lt;/em&amp;gt; divides the octave in 1200 equal parts of exactly 1 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt; each. It is notable mostly because it is the equal division corresponding to cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
1200edo is uniquely &amp;lt;a class="wiki_link" href="/consistent"&amp;gt;consistent&amp;lt;/a&amp;gt; through the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, which means the intervals of the 11-limit&amp;lt;a class="wiki_link" href="/tonality%20diamond"&amp;gt; tonality diamond&amp;lt;/a&amp;gt;, and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;amp;lt;1200 1902 2786 3369 4141|. It is &amp;lt;a class="wiki_link" href="/contorted"&amp;gt;contorted&amp;lt;/a&amp;gt; in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, having the same mapping as 600edo. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171edo&amp;lt;/a&amp;gt;. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by &amp;lt;a class="wiki_link" href="/494edo"&amp;gt;494edo&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4523</id>
      <timestamp>2011-06-14T21:16:46Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 236716894 - Original comment: **</comment>
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      <text bytes="3482" sha1="tgh3qi8d5ginoof70k2ijv60egjpv17" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-14 21:16:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;236716894&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //1200 division// divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

1200edo is uniquely consistent through the [[11-limit]], which means the intervals of the 11-limit tonality diamond, and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;lt;1200 1902 2786 3369 4141|. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1200edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;1200 division&amp;lt;/em&amp;gt; divides the octave in 1200 equal parts of exactly 1 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt; each. It is notable mostly because it is the equal division corresponding to cents.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
1200edo is uniquely consistent through the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, which means the intervals of the 11-limit tonality diamond, and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;amp;lt;1200 1902 2786 3369 4141|. It is &amp;lt;a class="wiki_link" href="/contorted"&amp;gt;contorted&amp;lt;/a&amp;gt; in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, having the same mapping as 600edo. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by &amp;lt;a class="wiki_link" href="/171edo"&amp;gt;171edo&amp;lt;/a&amp;gt;. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by &amp;lt;a class="wiki_link" href="/494edo"&amp;gt;494edo&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>38676</id>
      <parentid>105</parentid>
      <timestamp>2018-11-27T22:48:45Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
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      <text bytes="1368" sha1="n6bzvw6rf089cu4fyiahjdggkresidk" xml:space="preserve">The '''1200 edo''' divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[Tonality_diamond|tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;lt;1200 1902 2786 3369 4141|. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit POTE tuning of [[Kleismic_family#Quadritikleismic|quadritikleismic temperament]]: &amp;lt;1200 1902 2785 3368|. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.
[[Category:cents]]
[[Category:edo]]
[[Category:notation]]
[[Category:theory]]</text>
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    <revision>
      <id>39515</id>
      <parentid>38676</parentid>
      <timestamp>2019-01-12T06:21:41Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <comment>fixed the edomapping of 11/1</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1368" sha1="7qgpchoi1p46s040zemb052nurc0ti5" xml:space="preserve">The '''1200 edo''' divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[Tonality_diamond|tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit patent val &amp;lt;1200 1902 2786 3369 4151|. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit POTE tuning of [[Kleismic_family#Quadritikleismic|quadritikleismic temperament]]: &amp;lt;1200 1902 2785 3368|. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.
[[Category:cents]]
[[Category:edo]]
[[Category:notation]]
[[Category:theory]]</text>
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    <revision>
      <id>45341</id>
      <parentid>39515</parentid>
      <timestamp>2020-05-30T18:51:57Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>removed cat notation, made link to temperament more specific via redirect, added some links, switched from inline val coding to [[template:val]]</comment>
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      <text bytes="1296" sha1="o0f4w9xnkluvqwdl4fwelks00xki5gi" xml:space="preserve">The '''1200 edo''' divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

[[Category:edo]]
[[Category:theory]]</text>
      <sha1>o0f4w9xnkluvqwdl4fwelks00xki5gi</sha1>
    </revision>
    <revision>
      <id>54969</id>
      <parentid>45341</parentid>
      <timestamp>2020-12-04T23:13:06Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54969</origin>
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      <text bytes="1322" sha1="bql0ol9rgh2ull2uwyfuphh2xij77rx" xml:space="preserve">The '''1200 edo''' divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

[[Category:Equal divisions of the octave]]
[[Category:theory]]</text>
      <sha1>bql0ol9rgh2ull2uwyfuphh2xij77rx</sha1>
    </revision>
    <revision>
      <id>78533</id>
      <parentid>54969</parentid>
      <timestamp>2021-09-25T09:04:41Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
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      <text bytes="1368" sha1="a9jb5skj8zmxxl8qq51k4zpii0e0e6i" xml:space="preserve">The '''1200 edo''' divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{primes in edo|1200|columns=15}}

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

[[Category:Equal divisions of the octave]]
[[Category:theory]]</text>
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    </revision>
    <revision>
      <id>78781</id>
      <parentid>78533</parentid>
      <timestamp>2021-09-29T19:47:50Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>add link for defactoring</comment>
      <origin>78781</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1388" sha1="6zxmzp9psbkt2fyw8ny3w75ah5tssuk" xml:space="preserve">The '''1200 edo''' divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{primes in edo|1200|columns=15}}

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] (or [[enfactored]]) in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

[[Category:Equal divisions of the octave]]
[[Category:theory]]</text>
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    </revision>
    <revision>
      <id>80705</id>
      <parentid>78781</parentid>
      <timestamp>2021-11-12T23:07:32Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
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      <comment>remove temporary link to defactoring</comment>
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      <text bytes="1368" sha1="a9jb5skj8zmxxl8qq51k4zpii0e0e6i" xml:space="preserve">The '''1200 edo''' divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{primes in edo|1200|columns=15}}

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it supports with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

[[Category:Equal divisions of the octave]]
[[Category:theory]]</text>
      <sha1>a9jb5skj8zmxxl8qq51k4zpii0e0e6i</sha1>
    </revision>
    <revision>
      <id>84781</id>
      <parentid>80705</parentid>
      <timestamp>2022-01-25T18:39:20Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>link to new page for Supports</comment>
      <origin>84781</origin>
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      <text bytes="1372" sha1="827m0hzbqyizjicmubhqb58as9loaah" xml:space="preserve">The '''1200 edo''' divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{primes in edo|1200|columns=15}}

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

[[Category:Equal divisions of the octave]]
[[Category:theory]]</text>
      <sha1>827m0hzbqyizjicmubhqb58as9loaah</sha1>
    </revision>
    <revision>
      <id>87440</id>
      <parentid>84781</parentid>
      <timestamp>2022-02-26T16:48:49Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87440</origin>
      <model>wikitext</model>
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      <text bytes="1352" sha1="47n5bjfra0jfszfqcqojte6t52ee5pm" xml:space="preserve">The '''1200 edo''' divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{primes in edo|1200|columns=15}}

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

[[Category:Equal divisions of the octave]]</text>
      <sha1>47n5bjfra0jfszfqcqojte6t52ee5pm</sha1>
    </revision>
    <revision>
      <id>89573</id>
      <parentid>87440</parentid>
      <timestamp>2022-03-15T20:05:40Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
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      <model>wikitext</model>
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      <text bytes="1671" sha1="lfmcfjdgrxwhokrq68iad40yzc6220d" xml:space="preserve">The '''1200 edo''' divides the octave in 1200 equal parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{primes in edo|1200|columns=15}}

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation which offers detuning by cents.

[[Category:Equal divisions of the octave]]</text>
      <sha1>lfmcfjdgrxwhokrq68iad40yzc6220d</sha1>
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    <revision>
      <id>90520</id>
      <parentid>89573</parentid>
      <timestamp>2022-03-29T10:21:20Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>substituting to change the text</comment>
      <origin>90520</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1882" sha1="818whmvdxshh875509zlv1maqciy4jl" xml:space="preserve">The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of about {{#expr:(1200/1200) round 3}} [[cent]]s each. . It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{primes in edo|1200|columns=15}}

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation which offers detuning by cents.

[[Category:Equal divisions of the octave]]</text>
      <sha1>818whmvdxshh875509zlv1maqciy4jl</sha1>
    </revision>
    <revision>
      <id>90521</id>
      <parentid>90520</parentid>
      <timestamp>2022-03-29T10:21:43Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>done</comment>
      <origin>90521</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1853" sha1="8enfny9icp1joem81tykikq252c1dxx" xml:space="preserve">The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{primes in edo|1200|columns=15}}

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation which offers detuning by cents.

[[Category:Equal divisions of the octave]]</text>
      <sha1>8enfny9icp1joem81tykikq252c1dxx</sha1>
    </revision>
    <revision>
      <id>90522</id>
      <parentid>90521</parentid>
      <timestamp>2022-03-29T10:38:15Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>90522</origin>
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      <text bytes="2058" sha1="rt45zt7y0l1e2a1edwuq0f3oqpgarpt" xml:space="preserve">The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{Harmonics in equal|1200}}
Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

[[Category:Equal divisions of the octave]]</text>
      <sha1>rt45zt7y0l1e2a1edwuq0f3oqpgarpt</sha1>
    </revision>
    <revision>
      <id>90525</id>
      <parentid>90522</parentid>
      <timestamp>2022-03-29T11:04:41Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>i'll add more when i can</comment>
      <origin>90525</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2537" sha1="h53fkflptp4f8o7986x54l8yq1mdvc2" xml:space="preserve">The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{Harmonics in equal|1200}}
Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |Subgroup
! rowspan="2" |[[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal
8ve stretch (¢)
! colspan="2" |Tuning error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3.7.17.31.41.47
|2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
|[{{val|1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
|0.0351
|3.51
|}
[[Category:Equal divisions of the octave]]</text>
      <sha1>h53fkflptp4f8o7986x54l8yq1mdvc2</sha1>
    </revision>
    <revision>
      <id>90539</id>
      <parentid>90525</parentid>
      <timestamp>2022-03-29T21:42:52Z</timestamp>
      <contributor>
        <username>Gud2228</username>
        <id>176</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>90539</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2618" sha1="ap9hvkd5gzekca7epcoqgi9i8lcgf6d" xml:space="preserve">The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{Harmonics in equal|1200}}
Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |Subgroup
! rowspan="2" |[[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal
8ve stretch (¢)
! colspan="2" |Tuning error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3.5.7.11
|9801/9800, 234375/234256, 825000/823543, 1711561/1769472
|
|
|
|
|-
|2.3.7.17.31.41.47
|2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
|[{{val|1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
|0.0351
|3.51
|}
[[Category:Equal divisions of the octave]]</text>
      <sha1>ap9hvkd5gzekca7epcoqgi9i8lcgf6d</sha1>
    </revision>
    <revision>
      <id>90540</id>
      <parentid>90539</parentid>
      <timestamp>2022-03-29T21:46:54Z</timestamp>
      <contributor>
        <username>Gud2228</username>
        <id>176</id>
      </contributor>
      <comment>/* Regular temperament properties */</comment>
      <origin>90540</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2653" sha1="m6axwj7hl5l6if3bd54qkb9vnqp3wp4" xml:space="preserve">The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{Harmonics in equal|1200}}
Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |Subgroup
! rowspan="2" |[[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal
8ve stretch (¢)
! colspan="2" |Tuning error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3.5.7.11
|9801/9800, 234375/234256, 825000/823543, 1711561/1769472
|[{{val|1200 1902 2786 3369 4151 }}]
|
|
|
|-
|2.3.7.17.31.41.47
|2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
|[{{val|1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
|0.0351
|3.51
|}
[[Category:Equal divisions of the octave]]</text>
      <sha1>m6axwj7hl5l6if3bd54qkb9vnqp3wp4</sha1>
    </revision>
    <revision>
      <id>90541</id>
      <parentid>90540</parentid>
      <timestamp>2022-03-29T21:48:56Z</timestamp>
      <contributor>
        <username>Gud2228</username>
        <id>176</id>
      </contributor>
      <comment>/* Regular temperament properties */</comment>
      <origin>90541</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2653" sha1="7zss4ph1fg9hoeb4irwkod8v1ur9n9w" xml:space="preserve">The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{Harmonics in equal|1200}}
Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |Subgroup
! rowspan="2" |[[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal
8ve stretch (¢)
! colspan="2" |Tuning error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3.5.7.11
|9801/9800, 234375/234256, 825000/823543, 1771561/1769472
|[{{val|1200 1902 2786 3369 4151 }}]
|
|
|
|-
|2.3.7.17.31.41.47
|2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
|[{{val|1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
|0.0351
|3.51
|}
[[Category:Equal divisions of the octave]]</text>
      <sha1>7zss4ph1fg9hoeb4irwkod8v1ur9n9w</sha1>
    </revision>
    <revision>
      <id>90602</id>
      <parentid>90541</parentid>
      <timestamp>2022-03-30T17:20:46Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ +7-limit</comment>
      <origin>90602</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2829" sha1="1kg0i20m1397lwbpbgrilixafxjk5ra" xml:space="preserve">The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{Harmonics in equal|1200}}
Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|-
| 2.3.7.17.31.41.47
| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
| 0.0351
| 3.51
|}

[[Category:Equal divisions of the octave]]</text>
      <sha1>1kg0i20m1397lwbpbgrilixafxjk5ra</sha1>
    </revision>
    <revision>
      <id>90650</id>
      <parentid>90602</parentid>
      <timestamp>2022-03-31T18:48:28Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>infobox</comment>
      <origin>90650</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3028" sha1="42h5ccomz66rerwmmicdo8dh75jzlt7" xml:space="preserve">{{Infobox ET
| Prime factorization = 2&lt;sup&gt;4&lt;/sup&gt; × 3 × 5&lt;sup&gt;2/sup&gt;
| Step size = 1¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 702\1200 = 702¢ (→[[200edo|117\200]])
| Major 2nd = 204\1200 = 204¢
}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{Harmonics in equal|1200}}
Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|-
| 2.3.7.17.31.41.47
| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
| 0.0351
| 3.51
|}

[[Category:Equal divisions of the octave]]</text>
      <sha1>42h5ccomz66rerwmmicdo8dh75jzlt7</sha1>
    </revision>
    <revision>
      <id>90651</id>
      <parentid>90650</parentid>
      <timestamp>2022-03-31T18:48:40Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <minor/>
      <origin>90651</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3029" sha1="lfxz7c88xi8z05625q302qhfj90ibnr" xml:space="preserve">{{Infobox ET
| Prime factorization = 2&lt;sup&gt;4&lt;/sup&gt; × 3 × 5&lt;sup&gt;2&lt;/sup&gt;
| Step size = 1¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 702\1200 = 702¢ (→[[200edo|117\200]])
| Major 2nd = 204\1200 = 204¢
}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

==Theory==
{{Harmonics in equal|1200}}
Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the EDOs whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|-
| 2.3.7.17.31.41.47
| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
| 0.0351
| 3.51
|}

[[Category:Equal divisions of the octave]]</text>
      <sha1>lfxz7c88xi8z05625q302qhfj90ibnr</sha1>
    </revision>
    <revision>
      <id>90653</id>
      <parentid>90651</parentid>
      <timestamp>2022-03-31T20:48:08Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Cleanup</comment>
      <origin>90653</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3063" sha1="oayd3otx4pix8386a5peh63v8y7bhzt" xml:space="preserve">{{Infobox ET
| Prime factorization = 2&lt;sup&gt;4&lt;/sup&gt; × 3 × 5&lt;sup&gt;2&lt;/sup&gt;
| Step size = 1¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 702\1200 (702.00¢) (→ [[200edo|117\200]])
| Semitones = 114:90 (114.00¢ : 90¢)
| Consistency = 11
}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
{{Harmonics in equal|1200}}

Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the edos whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|-
| 2.3.7.17.31.41.47
| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
| 0.0351
| 3.51
|}

[[Category:Equal divisions of the octave]]</text>
      <sha1>oayd3otx4pix8386a5peh63v8y7bhzt</sha1>
    </revision>
    <revision>
      <id>91777</id>
      <parentid>90653</parentid>
      <timestamp>2022-05-08T13:23:16Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>91777</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3098" sha1="fmcofhgq51080bwyil42usc8iugnzh5" xml:space="preserve">{{Infobox ET
| Prime factorization = 2&lt;sup&gt;4&lt;/sup&gt; × 3 × 5&lt;sup&gt;2&lt;/sup&gt;
| Step size = 1¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 702\1200 (702.00¢) (→ [[200edo|117\200]])
| Semitones = 114:90 (114.00¢ : 90¢)
| Consistency = 11
}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
{{Harmonics in equal|1200}}

Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the edos whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|-
| 2.3.7.17.31.41.47
| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
| 0.0351
| 3.51
|}

[[Category:Equal divisions of the octave]]</text>
      <sha1>fmcofhgq51080bwyil42usc8iugnzh5</sha1>
    </revision>
    <revision>
      <id>93193</id>
      <parentid>91777</parentid>
      <timestamp>2022-07-04T01:14:09Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>93193</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3127" sha1="blw3vmafpa61n9kx0l6e7rpqjnwpbv1" xml:space="preserve">{{Infobox ET
| Prime factorization = 2&lt;sup&gt;4&lt;/sup&gt; × 3 × 5&lt;sup&gt;2&lt;/sup&gt;
| Step size = 1¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 702\1200 (702.00¢) (→ [[200edo|117\200]])
| Semitones = 114:90 (114.00¢ : 90¢)
| Consistency = 11
}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
{{Harmonics in equal|1200}}

Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the edos whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|-
| 2.3.7.17.31.41.47
| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
| 0.0351
| 3.51
|}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>blw3vmafpa61n9kx0l6e7rpqjnwpbv1</sha1>
    </revision>
    <revision>
      <id>96086</id>
      <parentid>93193</parentid>
      <timestamp>2022-10-04T16:03:48Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET now computes most parameters automatically</comment>
      <origin>96086</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2911" sha1="1v70l4pxhv53nby383zruq4qpwwhjlk" xml:space="preserve">{{Infobox ET}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
{{Harmonics in equal|1200}}

Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the edos whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|-
| 2.3.7.17.31.41.47
| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
| 0.0351
| 3.51
|}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>1v70l4pxhv53nby383zruq4qpwwhjlk</sha1>
    </revision>
    <revision>
      <id>97306</id>
      <parentid>96086</parentid>
      <timestamp>2022-10-12T17:39:18Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>substituting to say 1200edo is 1 cent by definition, as how it previously used to be.</comment>
      <origin>97306</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2991" sha1="3rqpzthw0ansbk504tltrdnevx2tvpe" xml:space="preserve">{{#invoke:Infobox_ET|infobox_ET|tuning={{PAGENAME}}|Zeta=|Consistency=|Distinct consistency=}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
{{Harmonics in equal|1200}}

Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the edos whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|-
| 2.3.7.17.31.41.47
| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
| 0.0351
| 3.51
|}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>3rqpzthw0ansbk504tltrdnevx2tvpe</sha1>
    </revision>
    <revision>
      <id>97307</id>
      <parentid>97306</parentid>
      <timestamp>2022-10-12T17:39:59Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>wait what</comment>
      <origin>97307</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2986" sha1="da6426oulwpbovbxqam9h6y5mkttotv" xml:space="preserve">{{#invoke:Infobox_ET|infobox_ET|tuning=1200edo|Zeta=|Consistency=|Distinct consistency=}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
{{Harmonics in equal|1200}}

Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the edos whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|-
| 2.3.7.17.31.41.47
| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
| 0.0351
| 3.51
|}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>da6426oulwpbovbxqam9h6y5mkttotv</sha1>
    </revision>
    <revision>
      <id>97309</id>
      <parentid>97307</parentid>
      <timestamp>2022-10-12T17:40:58Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>nope, it doesn't work, I'll ask on discord</comment>
      <origin>97309</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2911" sha1="1v70l4pxhv53nby383zruq4qpwwhjlk" xml:space="preserve">{{Infobox ET}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
{{Harmonics in equal|1200}}

Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Uniquely, 47th harmonic is 6666 steps normally and 666 steps reduced. The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the edos whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]], which means the intervals of the 11-limit [[tonality diamond]], and hence their size in cents rounded to the nearest integer, can be found by applying the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|-
| 2.3.7.17.31.41.47
| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
| 0.0351
| 3.51
|}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>1v70l4pxhv53nby383zruq4qpwwhjlk</sha1>
    </revision>
    <revision>
      <id>100419</id>
      <parentid>97309</parentid>
      <timestamp>2022-12-24T15:45:55Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>style, reword, sectioning</comment>
      <origin>100419</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2942" sha1="ss0g3d0ixo2ck08zmoe0s0zbxeetix3" xml:space="preserve">{{Infobox ET}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
{{Harmonics in equal|1200}}

The divisors of 1200 are {{EDOs|1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, 600}}. These are all the edos whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

1200edo is uniquely [[consistent]] through the [[11-limit]]. As a corollary, this means that whole-cent approximations of the 11-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[contorted]] in the [[5-limit]], having the same mapping as 600edo. Upwards to the 47-limit, 1200edo offers good approximations (less than 17%, one standard deviation) for 2, 3, 7, 17, 31, 41, and 47 harmonics. Remarkably, 47th harmonic is 6666 steps normally and 666 steps reduced.

=== Regular temperament theory ===
In the [[7-limit]], it tempers out 2460375/2458624 and 95703125/95551488, leading to a temperament it [[support]]s with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200, also supported by [[171edo]]. 

In the 11-limit, it tempers out 9801/9800, 234375/234256 and 825000/823543, leading to a temperament with a half-octave period and an approximate 99/98 generator of 17\1200, also supported by [[494edo]]. In the 7-limit, it provides a val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|-
| 2.3.7.17.31.41.47
| 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416
| [{{val| 1200 1902 3369 4905 5945 6429 6666 }}]
| -0.0244
| 0.0351
| 3.51
|}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>ss0g3d0ixo2ck08zmoe0s0zbxeetix3</sha1>
    </revision>
    <revision>
      <id>108099</id>
      <parentid>100419</parentid>
      <timestamp>2023-04-18T04:50:37Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Rework and cleanup</comment>
      <origin>108099</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2676" sha1="6edrypdze4cgtv3e3ftwli84bkidie5" xml:space="preserve">{{Infobox ET}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
1200edo is distinctly [[consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactored]] in the [[5-limit]], having the same mapping as [[600edo]]. Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47. The 47th harmonic is 6666 steps and 666 steps reduced – a funny mathematical coincidence in our decimal system. 

1200et tempers out 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200]. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and which can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|}</text>
      <sha1>6edrypdze4cgtv3e3ftwli84bkidie5</sha1>
    </revision>
    <revision>
      <id>108174</id>
      <parentid>108099</parentid>
      <timestamp>2023-04-18T12:12:37Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
      <origin>108174</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2674" sha1="8y2r7pkrn4oc5xq0zvqm8rd7k82cq9o" xml:space="preserve">{{Infobox ET}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
1200edo is distinctly [[consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactored]] in the [[5-limit]], having the same mapping as [[600edo]]. Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47. The 47th harmonic is 6666 steps and 666 steps reduced – a funny mathematical coincidence in our decimal system. 

1200et tempers out 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|}</text>
      <sha1>8y2r7pkrn4oc5xq0zvqm8rd7k82cq9o</sha1>
    </revision>
    <revision>
      <id>108177</id>
      <parentid>108174</parentid>
      <timestamp>2023-04-18T12:16:45Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */ bring 47-limit commas back for reference still, even if they're not in the rtt table</comment>
      <origin>108177</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2803" sha1="imfu8s2ysfivd44gvkpsceazf45uvf5" xml:space="preserve">{{Infobox ET}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
1200edo is distinctly [[consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactored]] in the [[5-limit]], having the same mapping as [[600edo]]. Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 subgroup it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

1200et tempers out 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|}</text>
      <sha1>imfu8s2ysfivd44gvkpsceazf45uvf5</sha1>
    </revision>
    <revision>
      <id>108178</id>
      <parentid>108177</parentid>
      <timestamp>2023-04-18T12:17:33Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */ put this at the bottom so it first reads 5-limit, 7-limit, then higher limits</comment>
      <origin>108178</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2805" sha1="63qtba5aw64stfx041veueuurf5j2mp" xml:space="preserve">{{Infobox ET}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
1200edo is distinctly [[consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactored]] in the [[5-limit]], having the same mapping as [[600edo]]. 

1200et tempers out 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 subgroup it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|}</text>
      <sha1>63qtba5aw64stfx041veueuurf5j2mp</sha1>
    </revision>
    <revision>
      <id>110164</id>
      <parentid>108178</parentid>
      <timestamp>2023-04-28T15:41:19Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>Added music section</comment>
      <origin>110164</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2904" sha1="tr2qay7ufep0x1683o0xmw5ljarcd7u" xml:space="preserve">{{Infobox ET}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
1200edo is distinctly [[consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactored]] in the [[5-limit]], having the same mapping as [[600edo]]. 

1200et tempers out 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 subgroup it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|}

== Music ==
* [https://www.youtube.com/watch?v=lTT3QGTngIs Dream Up (Demo Version)] by [[Sevish]]</text>
      <sha1>tr2qay7ufep0x1683o0xmw5ljarcd7u</sha1>
    </revision>
    <revision>
      <id>119378</id>
      <parentid>110164</parentid>
      <timestamp>2023-07-09T05:18:43Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119378</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2915" sha1="s4zqsyp9bwz7sf2yza2tnj5hsqbwqsj" xml:space="preserve">{{novelty}}{{Infobox ET}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
1200edo is distinctly [[consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactored]] in the [[5-limit]], having the same mapping as [[600edo]]. 

1200et tempers out 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 subgroup it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|}

== Music ==
* [https://www.youtube.com/watch?v=lTT3QGTngIs Dream Up (Demo Version)] by [[Sevish]]</text>
      <sha1>s4zqsyp9bwz7sf2yza2tnj5hsqbwqsj</sha1>
    </revision>
    <revision>
      <id>119610</id>
      <parentid>119378</parentid>
      <timestamp>2023-07-09T06:51:20Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119610</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2904" sha1="tr2qay7ufep0x1683o0xmw5ljarcd7u" xml:space="preserve">{{Infobox ET}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
1200edo is distinctly [[consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactored]] in the [[5-limit]], having the same mapping as [[600edo]]. 

1200et tempers out 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate 225/224 of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224&amp;amp;752 temperament tempering out 2200/2197, 4096/4095, 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 subgroup it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| [{{val| 1200 1902 2786 3369 }}]
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| [{{val| 1200 1902 2786 3369 4151 }}]
| +0.0273
| 0.0743
| 7.43
|}

== Music ==
* [https://www.youtube.com/watch?v=lTT3QGTngIs Dream Up (Demo Version)] by [[Sevish]]</text>
      <sha1>tr2qay7ufep0x1683o0xmw5ljarcd7u</sha1>
    </revision>
    <revision>
      <id>126660</id>
      <parentid>119610</parentid>
      <timestamp>2023-10-31T10:08:18Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Misc. cleanup</comment>
      <origin>126660</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2990" sha1="7u3lotvm1jn2mvdd1b0ufrig4h0fygn" xml:space="preserve">{{Infobox ET}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
1200edo is [[consistency|distinctly consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactored]] in the [[5-limit]], having the same tuning as [[600edo]]. 

The equal temperament [[tempering out|tempers out]] 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate [[225/224]] of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224 &amp;amp; 976 temperament tempering out [[2200/2197]], [[4096/4095]], 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 [[subgroup]] it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The nontrivial divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| {{mapping| 1200 1902 2786 3369 }}
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| {{mapping| 1200 1902 2786 3369 4151 }}
| +0.0273
| 0.0743
| 7.43
|}

== Music ==
; [[Sevish]]
* [https://www.youtube.com/watch?v=lTT3QGTngIs ''Dream Up''] (2021, demo version)</text>
      <sha1>7u3lotvm1jn2mvdd1b0ufrig4h0fygn</sha1>
    </revision>
    <revision>
      <id>128220</id>
      <parentid>126660</parentid>
      <timestamp>2023-11-30T17:03:33Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>128220</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3002" sha1="38sb9irbqf8rbho4maz5govh7wlqzgs" xml:space="preserve">{{Infobox ET}}
The '''1200 equal divisions of the octave''' ('''1200edo'''), or the '''1200(-tone) equal temperament''' ('''1200tet''', '''1200et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 1200 [[equal]] parts of exactly 1 [[cent]] each, and a size close to [[1729/1728]]. It is notable mostly because it is the equal division corresponding to cents.

== Theory ==
1200edo is [[consistency|distinctly consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactoring|enfactored]] in the [[5-limit]], having the same tuning as [[600edo]]. 

The equal temperament [[tempering out|tempers out]] 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate [[225/224]] of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224 &amp;amp; 976 temperament tempering out [[2200/2197]], [[4096/4095]], 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 [[subgroup]] it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The nontrivial divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| {{mapping| 1200 1902 2786 3369 }}
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| {{mapping| 1200 1902 2786 3369 4151 }}
| +0.0273
| 0.0743
| 7.43
|}

== Music ==
; [[Sevish]]
* [https://www.youtube.com/watch?v=lTT3QGTngIs ''Dream Up''] (2021, demo version)</text>
      <sha1>38sb9irbqf8rbho4maz5govh7wlqzgs</sha1>
    </revision>
    <revision>
      <id>134214</id>
      <parentid>128220</parentid>
      <timestamp>2024-02-03T16:55:31Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>adopt intro</comment>
      <origin>134214</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2736" sha1="b2qzdv0gd6trluwibrpmwn1aabh1a85" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1200}}

1200edo is notable for being the equal division of the octave corresponding to '''[[cent]]s'''.
== Theory ==
1200edo is [[consistency|distinctly consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactoring|enfactored]] in the [[5-limit]], having the same tuning as [[600edo]]. 

The equal temperament [[tempering out|tempers out]] 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate [[225/224]] of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224 &amp;amp; 976 temperament tempering out [[2200/2197]], [[4096/4095]], 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 [[subgroup]] it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The nontrivial divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| {{mapping| 1200 1902 2786 3369 }}
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| {{mapping| 1200 1902 2786 3369 4151 }}
| +0.0273
| 0.0743
| 7.43
|}

== Music ==
; [[Sevish]]
* [https://www.youtube.com/watch?v=lTT3QGTngIs ''Dream Up''] (2021, demo version)</text>
      <sha1>b2qzdv0gd6trluwibrpmwn1aabh1a85</sha1>
    </revision>
    <revision>
      <id>135482</id>
      <parentid>134214</parentid>
      <timestamp>2024-02-13T18:25:04Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <minor/>
      <comment>+categories</comment>
      <origin>135482</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2757" sha1="rsa6i1ucmpgb57v0dur8rb0kehwfrzx" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1200}}

1200edo is notable for being the equal division of the octave corresponding to '''[[cent]]s'''.
== Theory ==
1200edo is [[consistency|distinctly consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactoring|enfactored]] in the [[5-limit]], having the same tuning as [[600edo]]. 

The equal temperament [[tempering out|tempers out]] 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate [[225/224]] of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224 &amp;amp; 976 temperament tempering out [[2200/2197]], [[4096/4095]], 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 [[subgroup]] it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The nontrivial divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| {{mapping| 1200 1902 2786 3369 }}
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| {{mapping| 1200 1902 2786 3369 4151 }}
| +0.0273
| 0.0743
| 7.43
|}

== Music ==
; [[Sevish]]
* [https://www.youtube.com/watch?v=lTT3QGTngIs ''Dream Up''] (2021, demo version)

[[Category:Listen]]</text>
      <sha1>rsa6i1ucmpgb57v0dur8rb0kehwfrzx</sha1>
    </revision>
    <revision>
      <id>145140</id>
      <parentid>135482</parentid>
      <timestamp>2024-05-28T12:56:19Z</timestamp>
      <contributor>
        <username>Francium</username>
        <id>381</id>
      </contributor>
      <comment>/* Music */ +music</comment>
      <origin>145140</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2856" sha1="nlnv9v4ldsoi4u9l12f4s3vgrx2260q" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1200}}

1200edo is notable for being the equal division of the octave corresponding to '''[[cent]]s'''.
== Theory ==
1200edo is [[consistency|distinctly consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactoring|enfactored]] in the [[5-limit]], having the same tuning as [[600edo]]. 

The equal temperament [[tempering out|tempers out]] 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate [[225/224]] of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224 &amp;amp; 976 temperament tempering out [[2200/2197]], [[4096/4095]], 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 [[subgroup]] it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The nontrivial divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| {{mapping| 1200 1902 2786 3369 }}
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| {{mapping| 1200 1902 2786 3369 4151 }}
| +0.0273
| 0.0743
| 7.43
|}

== Music ==
; [[Hideya]]
* [https://www.youtube.com/watch?v=FJhmgbuoRHA ''Like scattered blue light''] (2024)

; [[Sevish]]
* [https://www.youtube.com/watch?v=lTT3QGTngIs ''Dream Up''] (2021, demo version)

[[Category:Listen]]</text>
      <sha1>nlnv9v4ldsoi4u9l12f4s3vgrx2260q</sha1>
    </revision>
    <revision>
      <id>153938</id>
      <parentid>145140</parentid>
      <timestamp>2024-09-04T01:19:09Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Remove bold around "cent" (MOS:BOLD), rephrase</comment>
      <origin>153938</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2874" sha1="cd7r8vymdl99ocdnaa4xrsz8xpghb1a" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1200}}

1200edo is notable for being the equal division of the octave whose step size corresponds to exactly 1 [[cent]].

== Theory ==
1200edo is [[consistency|distinctly consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactoring|enfactored]] in the [[5-limit]], having the same tuning as [[600edo]]. 

The equal temperament [[tempering out|tempers out]] 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate [[225/224]] of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224 &amp;amp; 976 temperament tempering out [[2200/2197]], [[4096/4095]], 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 [[subgroup]] it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The nontrivial divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| {{mapping| 1200 1902 2786 3369 }}
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| {{mapping| 1200 1902 2786 3369 4151 }}
| +0.0273
| 0.0743
| 7.43
|}

== Music ==
; [[Hideya]]
* [https://www.youtube.com/watch?v=FJhmgbuoRHA ''Like scattered blue light''] (2024)

; [[Sevish]]
* [https://www.youtube.com/watch?v=lTT3QGTngIs ''Dream Up''] (2021, demo version)

[[Category:Listen]]</text>
      <sha1>cd7r8vymdl99ocdnaa4xrsz8xpghb1a</sha1>
    </revision>
    <revision>
      <id>181327</id>
      <parentid>153938</parentid>
      <timestamp>2025-02-18T16:55:25Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181327</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2868" sha1="jnnc65keecblw1ftqihaz5bkh7e1gs4" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

1200edo is notable for being the equal division of the octave whose step size corresponds to exactly 1 [[cent]].

== Theory ==
1200edo is [[consistency|distinctly consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactoring|enfactored]] in the [[5-limit]], having the same tuning as [[600edo]]. 

The equal temperament [[tempering out|tempers out]] 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate [[225/224]] of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224 &amp;amp; 976 temperament tempering out [[2200/2197]], [[4096/4095]], 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 [[subgroup]] it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The nontrivial divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| {{mapping| 1200 1902 2786 3369 }}
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| {{mapping| 1200 1902 2786 3369 4151 }}
| +0.0273
| 0.0743
| 7.43
|}

== Music ==
; [[Hideya]]
* [https://www.youtube.com/watch?v=FJhmgbuoRHA ''Like scattered blue light''] (2024)

; [[Sevish]]
* [https://www.youtube.com/watch?v=lTT3QGTngIs ''Dream Up''] (2021, demo version)

[[Category:Listen]]</text>
      <sha1>jnnc65keecblw1ftqihaz5bkh7e1gs4</sha1>
    </revision>
    <revision>
      <id>191060</id>
      <parentid>181327</parentid>
      <timestamp>2025-04-11T07:52:33Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ (''See [[regular temperament]] for more about what all this means and how to use it.'')</comment>
      <origin>191060</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2956" sha1="6u7pxv70torn2y97dqozdz8u5im5fef" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

1200edo is notable for being the equal division of the octave whose step size corresponds to exactly 1 [[cent]].

== Theory ==
1200edo is [[consistency|distinctly consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactoring|enfactored]] in the [[5-limit]], having the same tuning as [[600edo]]. 

The equal temperament [[tempering out|tempers out]] 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate [[225/224]] of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224 &amp;amp; 976 temperament tempering out [[2200/2197]], [[4096/4095]], 9801/9800 and 35750/35721. (''See [[regular temperament]] for more about what all this means and how to use it.'')

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 [[subgroup]] it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The nontrivial divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| {{mapping| 1200 1902 2786 3369 }}
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| {{mapping| 1200 1902 2786 3369 4151 }}
| +0.0273
| 0.0743
| 7.43
|}

== Music ==
; [[Hideya]]
* [https://www.youtube.com/watch?v=FJhmgbuoRHA ''Like scattered blue light''] (2024)

; [[Sevish]]
* [https://www.youtube.com/watch?v=lTT3QGTngIs ''Dream Up''] (2021, demo version)

[[Category:Listen]]</text>
      <sha1>6u7pxv70torn2y97dqozdz8u5im5fef</sha1>
    </revision>
    <revision>
      <id>191164</id>
      <parentid>191060</parentid>
      <timestamp>2025-04-11T11:20:22Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>No need to remind readers of what a regular temperament is everywhere</comment>
      <origin>191164</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2868" sha1="jnnc65keecblw1ftqihaz5bkh7e1gs4" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

1200edo is notable for being the equal division of the octave whose step size corresponds to exactly 1 [[cent]].

== Theory ==
1200edo is [[consistency|distinctly consistent]] through the [[11-odd-limit]]. This means that whole-cent approximations of the 11-odd-limit [[tonality diamond]] intervals are conveniently represented through the 11-limit [[patent val]] {{val| 1200 1902 2786 3369 4151 }}. It is [[enfactoring|enfactored]] in the [[5-limit]], having the same tuning as [[600edo]]. 

The equal temperament [[tempering out|tempers out]] 2460375/2458624 and 95703125/95551488 in the [[7-limit]], supporting the 171 &amp; 1029 temperament, with a period of 1/3 octave and a generator which is an approximate [[225/224]] of 7\1200. It tempers out [[9801/9800]], 234375/234256 and 825000/823543 in the 11-limit, supporting the 494 &amp; 706 temperament, with a half-octave period and an approximate 99/98 generator of 17\1200. 

It also provides a 7-limit val, 1200ccd, which is extremely closely close to the 7-limit [[POTE tuning]] of [[quadritikleismic temperament]]: {{val| 1200 1902 2785 3368 }}. It also provides the optimal patent val for the 224 &amp;amp; 976 temperament tempering out [[2200/2197]], [[4096/4095]], 9801/9800 and 35750/35721.

Upwards to the 47-limit, 1200edo offers relatively accurate approximations for 2, 3, 7, 17, 31, 41, and 47, and in the 2.3.7.17.31.41.47 [[subgroup]] it tempers out 2304/2303, 3808/3807, 6273/6272, 506447/506268, 632056/632043, 10218313/10214416. The 47th harmonic is 6666 steps and 666 steps reduced – a mathematical coincidence in our decimal system. 

=== Prime harmonics ===
{{Harmonics in equal|1200}}

=== Subsets and supersets ===
The nontrivial divisors of 1200 are {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400, and 600 }}. These are all the edos whose step size is an integer amount of cents, and thus can be played exactly on any digital audio workstation that offers detuning by cents.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2460375/2458624, 95703125/95551488, {{monzo| 36 -5 0 -10 }}
| {{mapping| 1200 1902 2786 3369 }}
| +0.0112
| 0.0748
| 7.48
|-
| 2.3.5.7.11
| 9801/9800, 234375/234256, 825000/823543, 1771561/1769472
| {{mapping| 1200 1902 2786 3369 4151 }}
| +0.0273
| 0.0743
| 7.43
|}

== Music ==
; [[Hideya]]
* [https://www.youtube.com/watch?v=FJhmgbuoRHA ''Like scattered blue light''] (2024)

; [[Sevish]]
* [https://www.youtube.com/watch?v=lTT3QGTngIs ''Dream Up''] (2021, demo version)

[[Category:Listen]]</text>
      <sha1>jnnc65keecblw1ftqihaz5bkh7e1gs4</sha1>
    </revision>
  </page>
  <page>
    <title>120edo</title>
    <ns>0</ns>
    <id>106</id>
    <revision>
      <id>106</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>106</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="589" sha1="tg08xv2qu7i5ucf0t6cd0xitis8rhy1" xml:space="preserve">120edo means division of the octave into equal parts of 10 cents each. Its patent val is contorted only through the 3-limit and does not temper out 81/80 in the 5-limit or 64/63 and 5120/5103 in the 7-limit. However, 5120/5103 is done about as badly as this interval can be done relative to an equal division, falling close to exactly in the middle of a step (1\120 is ~42.42 relative cents sharp of it). Being the simplest division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the fine relative cent of [[1edo|1edo]]</text>
      <sha1>tg08xv2qu7i5ucf0t6cd0xitis8rhy1</sha1>
    </revision>
    <revision>
      <id>4524</id>
      <timestamp>2016-10-29T15:45:11Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 597455206 - Original comment: **</comment>
      <origin>4524</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2500" sha1="sns6ulr60ahj8sejmy8xk4ij64mihqv" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-29 15:45:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;597455206&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;120edo means division of the octave into equal parts of 10 cents each. Its patent val is contorted only through the 3-limit and does not temper out 81/80 in the 5-limit or 64/63 and 5120/5103 in the 7-limit. However, 5120/5103 is done about as badly as this interval can be done relative to an equal division, falling close to exactly in the middle of a step (1\120 is ~42.42 relative cents sharp of it). Being the simplest division of the octave by the Germanic [[https://en.wikipedia.org/wiki/Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;120edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;120edo means division of the octave into equal parts of 10 cents each. Its patent val is contorted only through the 3-limit and does not temper out 81/80 in the 5-limit or 64/63 and 5120/5103 in the 7-limit. However, 5120/5103 is done about as badly as this interval can be done relative to an equal division, falling close to exactly in the middle of a step (1\120 is ~42.42 relative cents sharp of it). Being the simplest division of the octave by the Germanic &amp;lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Long_hundred" rel="nofollow"&amp;gt;long hundred&amp;lt;/a&amp;gt;, it has a unit step which is the fine relative cent of &amp;lt;a class="wiki_link" href="/1edo"&amp;gt;1edo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>sns6ulr60ahj8sejmy8xk4ij64mihqv</sha1>
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    <revision>
      <id>78766</id>
      <parentid>106</parentid>
      <timestamp>2021-09-29T19:45:08Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>add link for defactoring</comment>
      <origin>78766</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="617" sha1="5ulhjn9e4a2okiwzirimiidsxpbsupp" xml:space="preserve">120edo means division of the octave into equal parts of 10 cents each. Its [[patent val]] is [[contorted]] (or [[enfactored]]) only through the 3-limit and does not temper out 81/80 in the 5-limit or 64/63 and 5120/5103 in the 7-limit. However, 5120/5103 is done about as badly as this interval can be done relative to an equal division, falling close to exactly in the middle of a step (1\120 is ~42.42 relative cents sharp of it). Being the simplest division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the fine relative cent of [[1edo|1edo]]</text>
      <sha1>5ulhjn9e4a2okiwzirimiidsxpbsupp</sha1>
    </revision>
    <revision>
      <id>80720</id>
      <parentid>78766</parentid>
      <timestamp>2021-11-12T23:10:05Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>remove temporary link to defactoring</comment>
      <origin>80720</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="597" sha1="e8ik1ndypa671odd1k8jg9jc01hl3dm" xml:space="preserve">120edo means division of the octave into equal parts of 10 cents each. Its [[patent val]] is [[contorted]] only through the 3-limit and does not temper out 81/80 in the 5-limit or 64/63 and 5120/5103 in the 7-limit. However, 5120/5103 is done about as badly as this interval can be done relative to an equal division, falling close to exactly in the middle of a step (1\120 is ~42.42 relative cents sharp of it). Being the simplest division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the fine relative cent of [[1edo|1edo]]</text>
      <sha1>e8ik1ndypa671odd1k8jg9jc01hl3dm</sha1>
    </revision>
    <revision>
      <id>84316</id>
      <parentid>80720</parentid>
      <timestamp>2022-01-20T12:56:08Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>84316</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="826" sha1="mp3b3q64aqnkscab1boholcw9423num" xml:space="preserve">120edo means division of the octave into equal parts of 10 cents each. Its [[patent val]] is [[contorted]] only through the 3-limit and does not temper out 81/80 in the 5-limit or 64/63 and 5120/5103 in the 7-limit. However, 5120/5103 is done about as badly as this interval can be done relative to an equal division, falling close to exactly in the middle of a step (1\120 is ~42.42 relative cents sharp of it). Being the simplest division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the fine relative cent of [[1edo|1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

120edo is the 5th factorial EDO, and the 10th highly melodic EDO.

[[Category:Highly melodic]]</text>
      <sha1>mp3b3q64aqnkscab1boholcw9423num</sha1>
    </revision>
    <revision>
      <id>92930</id>
      <parentid>84316</parentid>
      <timestamp>2022-07-02T16:24:08Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Categories</comment>
      <origin>92930</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="897" sha1="5cywr9zsw6snuj2kfvswldkt0vlc9wf" xml:space="preserve">120edo means division of the octave into equal parts of 10 cents each. Its [[patent val]] is [[contorted]] only through the 3-limit and does not temper out 81/80 in the 5-limit or 64/63 and 5120/5103 in the 7-limit. However, 5120/5103 is done about as badly as this interval can be done relative to an equal division, falling close to exactly in the middle of a step (1\120 is ~42.42 relative cents sharp of it). Being the simplest division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the fine relative cent of [[1edo|1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

120edo is the 5th factorial EDO, and the 10th highly melodic EDO.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Highly melodic]]</text>
      <sha1>5cywr9zsw6snuj2kfvswldkt0vlc9wf</sha1>
    </revision>
    <revision>
      <id>96198</id>
      <parentid>92930</parentid>
      <timestamp>2022-10-04T18:53:04Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96198</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="912" sha1="dcfg88ajpartwjf6e6jjvvk29gkjtp4" xml:space="preserve">{{Infobox ET}}
120edo means division of the octave into equal parts of 10 cents each. Its [[patent val]] is [[contorted]] only through the 3-limit and does not temper out 81/80 in the 5-limit or 64/63 and 5120/5103 in the 7-limit. However, 5120/5103 is done about as badly as this interval can be done relative to an equal division, falling close to exactly in the middle of a step (1\120 is ~42.42 relative cents sharp of it). Being the simplest division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the fine relative cent of [[1edo|1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

120edo is the 5th factorial EDO, and the 10th highly melodic EDO.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Highly melodic]]</text>
      <sha1>dcfg88ajpartwjf6e6jjvvk29gkjtp4</sha1>
    </revision>
    <revision>
      <id>97103</id>
      <parentid>96198</parentid>
      <timestamp>2022-10-09T18:44:44Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Highly melodic]] to [[Category:Highly composite]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>97103</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="914" sha1="e7qc1mvhuukn1u6e4chtl1mm8zcnyq4" xml:space="preserve">{{Infobox ET}}
120edo means division of the octave into equal parts of 10 cents each. Its [[patent val]] is [[contorted]] only through the 3-limit and does not temper out 81/80 in the 5-limit or 64/63 and 5120/5103 in the 7-limit. However, 5120/5103 is done about as badly as this interval can be done relative to an equal division, falling close to exactly in the middle of a step (1\120 is ~42.42 relative cents sharp of it). Being the simplest division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the fine relative cent of [[1edo|1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

120edo is the 5th factorial EDO, and the 10th highly melodic EDO.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Highly composite]]</text>
      <sha1>e7qc1mvhuukn1u6e4chtl1mm8zcnyq4</sha1>
    </revision>
    <revision>
      <id>97304</id>
      <parentid>97103</parentid>
      <timestamp>2022-10-12T17:35:48Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>97304</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1089" sha1="p48rjdx5cirud3lmeqhj1u4d5dvfcwk" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo approximates with less than 25% error harmoincs: 2, 3, 7, 11, 13, 23, 29. Therefore, it's well suited for no-5s 13-limit.

Its [[patent val]] is [[contorted]] only through the 3-limit and does not temper out 81/80 in the 5-limit or 64/63 and 5120/5103 in the 7-limit. However, 5120/5103 is done about as badly as this interval can be done relative to an equal division, falling close to exactly in the middle of a step (1\120 is ~42.42 relative cents sharp of it). 

120edo is the 5th factorial EDO (120 = 1*2*3*4*5), and the 10th highly composite EDO.

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Highly composite]]</text>
      <sha1>p48rjdx5cirud3lmeqhj1u4d5dvfcwk</sha1>
    </revision>
    <revision>
      <id>100435</id>
      <parentid>97304</parentid>
      <timestamp>2022-12-25T23:12:39Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>100435</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1146" sha1="qhx8nss85vun5205kgpxgwo4ib8d9dy" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3*4*5 = 5!).

120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out [[243/242]].

120edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit [[superpyth]] temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which is [[marvel comma]] sharp of 3/2.

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Highly composite]]</text>
      <sha1>qhx8nss85vun5205kgpxgwo4ib8d9dy</sha1>
    </revision>
    <revision>
      <id>101730</id>
      <parentid>100435</parentid>
      <timestamp>2023-01-22T14:56:30Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>101730</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1224" sha1="ndrpablghpk8p6qxgclcu0tjh63y3b5" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3*4*5 = 5!).

120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out [[243/242]].

120edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit [[superpyth]] temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which is [[marvel comma]] sharp of 3/2. In the patent val it is also a tuning for the 7-limit [[decoid]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Highly composite]]</text>
      <sha1>ndrpablghpk8p6qxgclcu0tjh63y3b5</sha1>
    </revision>
    <revision>
      <id>114735</id>
      <parentid>101730</parentid>
      <timestamp>2023-05-26T14:24:57Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */ fix unclear pronoun</comment>
      <origin>114735</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1228" sha1="o1zemcmk9uttogad2b2mt86b0jf3qrj" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3*4*5 = 5!).

120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out [[243/242]].

120edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit [[superpyth]] temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which is [[marvel comma]] sharp of 3/2. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Highly composite]]</text>
      <sha1>o1zemcmk9uttogad2b2mt86b0jf3qrj</sha1>
    </revision>
    <revision>
      <id>116908</id>
      <parentid>114735</parentid>
      <timestamp>2023-06-16T08:26:28Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>116908</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8727" sha1="8hwewp8x0gbav6tfom7yy4vs0n59kpv" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3*4*5 = 5!).

120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out [[243/242]].

120edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit [[superpyth]] temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which is [[marvel comma]] sharp of 3/2. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|120}}

== Interval list ==
{|class="wikitable"
|-
!#
!Cents
![[Ups and downs notation]] (fifth 7\12)
![[Ups and downs notation]] (fifth 71\120)
|-
|0
|0
|{{UDnote|step=0}}
|{{UDnote|fifth=71|step=0}}
|-
|1
|10
|{{UDnote|step=1}}
|{{UDnote|fifth=71|step=1}}
|-
|2
|20
|{{UDnote|step=2}}
|{{UDnote|fifth=71|step=2}}
|-
|3
|30
|{{UDnote|step=3}}
|{{UDnote|fifth=71|step=3}}
|-
|4
|40
|{{UDnote|step=4}}
|{{UDnote|fifth=71|step=4}}
|-
|5
|50
|{{UDnote|step=5}}
|{{UDnote|fifth=71|step=5}}
|-
|6
|60
|{{UDnote|step=6}}
|{{UDnote|fifth=71|step=6}}
|-
|7
|70
|{{UDnote|step=7}}
|{{UDnote|fifth=71|step=7}}
|-
|8
|80
|{{UDnote|step=8}}
|{{UDnote|fifth=71|step=8}}
|-
|9
|90
|{{UDnote|step=9}}
|{{UDnote|fifth=71|step=9}}
|-
|10
|100
|{{UDnote|step=10}}
|{{UDnote|fifth=71|step=10}}
|-
|11
|110
|{{UDnote|step=11}}
|{{UDnote|fifth=71|step=11}}
|-
|12
|120
|{{UDnote|step=12}}
|{{UDnote|fifth=71|step=12}}
|-
|13
|130
|{{UDnote|step=13}}
|{{UDnote|fifth=71|step=13}}
|-
|14
|140
|{{UDnote|step=14}}
|{{UDnote|fifth=71|step=14}}
|-
|15
|150
|{{UDnote|step=15}}
|{{UDnote|fifth=71|step=15}}
|-
|16
|160
|{{UDnote|step=16}}
|{{UDnote|fifth=71|step=16}}
|-
|17
|170
|{{UDnote|step=17}}
|{{UDnote|fifth=71|step=17}}
|-
|18
|180
|{{UDnote|step=18}}
|{{UDnote|fifth=71|step=18}}
|-
|19
|190
|{{UDnote|step=19}}
|{{UDnote|fifth=71|step=19}}
|-
|20
|200
|{{UDnote|step=20}}
|{{UDnote|fifth=71|step=20}}
|-
|21
|210
|{{UDnote|step=21}}
|{{UDnote|fifth=71|step=21}}
|-
|22
|220
|{{UDnote|step=22}}
|{{UDnote|fifth=71|step=22}}
|-
|23
|230
|{{UDnote|step=23}}
|{{UDnote|fifth=71|step=23}}
|-
|24
|240
|{{UDnote|step=24}}
|{{UDnote|fifth=71|step=24}}
|-
|25
|250
|{{UDnote|step=25}}
|{{UDnote|fifth=71|step=25}}
|-
|26
|260
|{{UDnote|step=26}}
|{{UDnote|fifth=71|step=26}}
|-
|27
|270
|{{UDnote|step=27}}
|{{UDnote|fifth=71|step=27}}
|-
|28
|280
|{{UDnote|step=28}}
|{{UDnote|fifth=71|step=28}}
|-
|29
|290
|{{UDnote|step=29}}
|{{UDnote|fifth=71|step=29}}
|-
|30
|300
|{{UDnote|step=30}}
|{{UDnote|fifth=71|step=30}}
|-
|31
|310
|{{UDnote|step=31}}
|{{UDnote|fifth=71|step=31}}
|-
|32
|320
|{{UDnote|step=32}}
|{{UDnote|fifth=71|step=32}}
|-
|33
|330
|{{UDnote|step=33}}
|{{UDnote|fifth=71|step=33}}
|-
|34
|340
|{{UDnote|step=34}}
|{{UDnote|fifth=71|step=34}}
|-
|35
|350
|{{UDnote|step=35}}
|{{UDnote|fifth=71|step=35}}
|-
|36
|360
|{{UDnote|step=36}}
|{{UDnote|fifth=71|step=36}}
|-
|37
|370
|{{UDnote|step=37}}
|{{UDnote|fifth=71|step=37}}
|-
|38
|380
|{{UDnote|step=38}}
|{{UDnote|fifth=71|step=38}}
|-
|39
|390
|{{UDnote|step=39}}
|{{UDnote|fifth=71|step=39}}
|-
|40
|400
|{{UDnote|step=40}}
|{{UDnote|fifth=71|step=40}}
|-
|41
|410
|{{UDnote|step=41}}
|{{UDnote|fifth=71|step=41}}
|-
|42
|420
|{{UDnote|step=42}}
|{{UDnote|fifth=71|step=42}}
|-
|43
|430
|{{UDnote|step=43}}
|{{UDnote|fifth=71|step=43}}
|-
|44
|440
|{{UDnote|step=44}}
|{{UDnote|fifth=71|step=44}}
|-
|45
|450
|{{UDnote|step=45}}
|{{UDnote|fifth=71|step=45}}
|-
|46
|460
|{{UDnote|step=46}}
|{{UDnote|fifth=71|step=46}}
|-
|47
|470
|{{UDnote|step=47}}
|{{UDnote|fifth=71|step=47}}
|-
|48
|480
|{{UDnote|step=48}}
|{{UDnote|fifth=71|step=48}}
|-
|49
|490
|{{UDnote|step=49}}
|{{UDnote|fifth=71|step=49}}
|-
|50
|500
|{{UDnote|step=50}}
|{{UDnote|fifth=71|step=50}}
|-
|51
|510
|{{UDnote|step=51}}
|{{UDnote|fifth=71|step=51}}
|-
|52
|520
|{{UDnote|step=52}}
|{{UDnote|fifth=71|step=52}}
|-
|53
|530
|{{UDnote|step=53}}
|{{UDnote|fifth=71|step=53}}
|-
|54
|540
|{{UDnote|step=54}}
|{{UDnote|fifth=71|step=54}}
|-
|55
|550
|{{UDnote|step=55}}
|{{UDnote|fifth=71|step=55}}
|-
|56
|560
|{{UDnote|step=56}}
|{{UDnote|fifth=71|step=56}}
|-
|57
|570
|{{UDnote|step=57}}
|{{UDnote|fifth=71|step=57}}
|-
|58
|580
|{{UDnote|step=58}}
|{{UDnote|fifth=71|step=58}}
|-
|59
|590
|{{UDnote|step=59}}
|{{UDnote|fifth=71|step=59}}
|-
|60
|600
|{{UDnote|step=60}}
|{{UDnote|fifth=71|step=60}}
|-
|61
|610
|{{UDnote|step=61}}
|{{UDnote|fifth=71|step=61}}
|-
|62
|620
|{{UDnote|step=62}}
|{{UDnote|fifth=71|step=62}}
|-
|63
|630
|{{UDnote|step=63}}
|{{UDnote|fifth=71|step=63}}
|-
|64
|640
|{{UDnote|step=64}}
|{{UDnote|fifth=71|step=64}}
|-
|65
|650
|{{UDnote|step=65}}
|{{UDnote|fifth=71|step=65}}
|-
|66
|660
|{{UDnote|step=66}}
|{{UDnote|fifth=71|step=66}}
|-
|67
|670
|{{UDnote|step=67}}
|{{UDnote|fifth=71|step=67}}
|-
|68
|680
|{{UDnote|step=68}}
|{{UDnote|fifth=71|step=68}}
|-
|69
|690
|{{UDnote|step=69}}
|{{UDnote|fifth=71|step=69}}
|-
|70
|700
|{{UDnote|step=70}}
|{{UDnote|fifth=71|step=70}}
|-
|71
|710
|{{UDnote|step=71}}
|{{UDnote|fifth=71|step=71}}
|-
|72
|720
|{{UDnote|step=72}}
|{{UDnote|fifth=71|step=72}}
|-
|73
|730
|{{UDnote|step=73}}
|{{UDnote|fifth=71|step=73}}
|-
|74
|740
|{{UDnote|step=74}}
|{{UDnote|fifth=71|step=74}}
|-
|75
|750
|{{UDnote|step=75}}
|{{UDnote|fifth=71|step=75}}
|-
|76
|760
|{{UDnote|step=76}}
|{{UDnote|fifth=71|step=76}}
|-
|77
|770
|{{UDnote|step=77}}
|{{UDnote|fifth=71|step=77}}
|-
|78
|780
|{{UDnote|step=78}}
|{{UDnote|fifth=71|step=78}}
|-
|79
|790
|{{UDnote|step=79}}
|{{UDnote|fifth=71|step=79}}
|-
|80
|800
|{{UDnote|step=80}}
|{{UDnote|fifth=71|step=80}}
|-
|81
|810
|{{UDnote|step=81}}
|{{UDnote|fifth=71|step=81}}
|-
|82
|820
|{{UDnote|step=82}}
|{{UDnote|fifth=71|step=82}}
|-
|83
|830
|{{UDnote|step=83}}
|{{UDnote|fifth=71|step=83}}
|-
|84
|840
|{{UDnote|step=84}}
|{{UDnote|fifth=71|step=84}}
|-
|85
|850
|{{UDnote|step=85}}
|{{UDnote|fifth=71|step=85}}
|-
|86
|860
|{{UDnote|step=86}}
|{{UDnote|fifth=71|step=86}}
|-
|87
|870
|{{UDnote|step=87}}
|{{UDnote|fifth=71|step=87}}
|-
|88
|880
|{{UDnote|step=88}}
|{{UDnote|fifth=71|step=88}}
|-
|89
|890
|{{UDnote|step=89}}
|{{UDnote|fifth=71|step=89}}
|-
|90
|900
|{{UDnote|step=90}}
|{{UDnote|fifth=71|step=90}}
|-
|91
|910
|{{UDnote|step=91}}
|{{UDnote|fifth=71|step=91}}
|-
|92
|920
|{{UDnote|step=92}}
|{{UDnote|fifth=71|step=92}}
|-
|93
|930
|{{UDnote|step=93}}
|{{UDnote|fifth=71|step=93}}
|-
|94
|940
|{{UDnote|step=94}}
|{{UDnote|fifth=71|step=94}}
|-
|95
|950
|{{UDnote|step=95}}
|{{UDnote|fifth=71|step=95}}
|-
|96
|960
|{{UDnote|step=96}}
|{{UDnote|fifth=71|step=96}}
|-
|97
|970
|{{UDnote|step=97}}
|{{UDnote|fifth=71|step=97}}
|-
|98
|980
|{{UDnote|step=98}}
|{{UDnote|fifth=71|step=98}}
|-
|99
|990
|{{UDnote|step=99}}
|{{UDnote|fifth=71|step=99}}
|-
|100
|1000
|{{UDnote|step=100}}
|{{UDnote|fifth=71|step=100}}
|-
|101
|1010
|{{UDnote|step=101}}
|{{UDnote|fifth=71|step=101}}
|-
|102
|1020
|{{UDnote|step=102}}
|{{UDnote|fifth=71|step=102}}
|-
|103
|1030
|{{UDnote|step=103}}
|{{UDnote|fifth=71|step=103}}
|-
|104
|1040
|{{UDnote|step=104}}
|{{UDnote|fifth=71|step=104}}
|-
|105
|1050
|{{UDnote|step=105}}
|{{UDnote|fifth=71|step=105}}
|-
|106
|1060
|{{UDnote|step=106}}
|{{UDnote|fifth=71|step=106}}
|-
|107
|1070
|{{UDnote|step=107}}
|{{UDnote|fifth=71|step=107}}
|-
|108
|1080
|{{UDnote|step=108}}
|{{UDnote|fifth=71|step=108}}
|-
|109
|1090
|{{UDnote|step=109}}
|{{UDnote|fifth=71|step=109}}
|-
|110
|1100
|{{UDnote|step=110}}
|{{UDnote|fifth=71|step=110}}
|-
|111
|1110
|{{UDnote|step=111}}
|{{UDnote|fifth=71|step=111}}
|-
|112
|1120
|{{UDnote|step=112}}
|{{UDnote|fifth=71|step=112}}
|-
|113
|1130
|{{UDnote|step=113}}
|{{UDnote|fifth=71|step=113}}
|-
|114
|1140
|{{UDnote|step=114}}
|{{UDnote|fifth=71|step=114}}
|-
|115
|1150
|{{UDnote|step=115}}
|{{UDnote|fifth=71|step=115}}
|-
|116
|1160
|{{UDnote|step=116}}
|{{UDnote|fifth=71|step=116}}
|-
|117
|1170
|{{UDnote|step=117}}
|{{UDnote|fifth=71|step=117}}
|-
|118
|1180
|{{UDnote|step=118}}
|{{UDnote|fifth=71|step=118}}
|-
|119
|1190
|{{UDnote|step=119}}
|{{UDnote|fifth=71|step=119}}
|}
=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Highly composite]]</text>
      <sha1>8hwewp8x0gbav6tfom7yy4vs0n59kpv</sha1>
    </revision>
    <revision>
      <id>116909</id>
      <parentid>116908</parentid>
      <timestamp>2023-06-16T08:27:29Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>116909</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8793" sha1="crw9ufzg0w4z3v4r6d48fcfn37i3dkw" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3*4*5 = 5!).

120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out [[243/242]].

120edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit [[superpyth]] temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which is [[marvel comma]] sharp of 3/2. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

== Interval list ==
{|class="wikitable"
|-
!#
!Cents
![[Ups and downs notation]] (fifth 7\12)
![[Ups and downs notation]] (fifth 71\120)
|-
|0
|0
|{{UDnote|step=0}}
|{{UDnote|fifth=71|step=0}}
|-
|1
|10
|{{UDnote|step=1}}
|{{UDnote|fifth=71|step=1}}
|-
|2
|20
|{{UDnote|step=2}}
|{{UDnote|fifth=71|step=2}}
|-
|3
|30
|{{UDnote|step=3}}
|{{UDnote|fifth=71|step=3}}
|-
|4
|40
|{{UDnote|step=4}}
|{{UDnote|fifth=71|step=4}}
|-
|5
|50
|{{UDnote|step=5}}
|{{UDnote|fifth=71|step=5}}
|-
|6
|60
|{{UDnote|step=6}}
|{{UDnote|fifth=71|step=6}}
|-
|7
|70
|{{UDnote|step=7}}
|{{UDnote|fifth=71|step=7}}
|-
|8
|80
|{{UDnote|step=8}}
|{{UDnote|fifth=71|step=8}}
|-
|9
|90
|{{UDnote|step=9}}
|{{UDnote|fifth=71|step=9}}
|-
|10
|100
|{{UDnote|step=10}}
|{{UDnote|fifth=71|step=10}}
|-
|11
|110
|{{UDnote|step=11}}
|{{UDnote|fifth=71|step=11}}
|-
|12
|120
|{{UDnote|step=12}}
|{{UDnote|fifth=71|step=12}}
|-
|13
|130
|{{UDnote|step=13}}
|{{UDnote|fifth=71|step=13}}
|-
|14
|140
|{{UDnote|step=14}}
|{{UDnote|fifth=71|step=14}}
|-
|15
|150
|{{UDnote|step=15}}
|{{UDnote|fifth=71|step=15}}
|-
|16
|160
|{{UDnote|step=16}}
|{{UDnote|fifth=71|step=16}}
|-
|17
|170
|{{UDnote|step=17}}
|{{UDnote|fifth=71|step=17}}
|-
|18
|180
|{{UDnote|step=18}}
|{{UDnote|fifth=71|step=18}}
|-
|19
|190
|{{UDnote|step=19}}
|{{UDnote|fifth=71|step=19}}
|-
|20
|200
|{{UDnote|step=20}}
|{{UDnote|fifth=71|step=20}}
|-
|21
|210
|{{UDnote|step=21}}
|{{UDnote|fifth=71|step=21}}
|-
|22
|220
|{{UDnote|step=22}}
|{{UDnote|fifth=71|step=22}}
|-
|23
|230
|{{UDnote|step=23}}
|{{UDnote|fifth=71|step=23}}
|-
|24
|240
|{{UDnote|step=24}}
|{{UDnote|fifth=71|step=24}}
|-
|25
|250
|{{UDnote|step=25}}
|{{UDnote|fifth=71|step=25}}
|-
|26
|260
|{{UDnote|step=26}}
|{{UDnote|fifth=71|step=26}}
|-
|27
|270
|{{UDnote|step=27}}
|{{UDnote|fifth=71|step=27}}
|-
|28
|280
|{{UDnote|step=28}}
|{{UDnote|fifth=71|step=28}}
|-
|29
|290
|{{UDnote|step=29}}
|{{UDnote|fifth=71|step=29}}
|-
|30
|300
|{{UDnote|step=30}}
|{{UDnote|fifth=71|step=30}}
|-
|31
|310
|{{UDnote|step=31}}
|{{UDnote|fifth=71|step=31}}
|-
|32
|320
|{{UDnote|step=32}}
|{{UDnote|fifth=71|step=32}}
|-
|33
|330
|{{UDnote|step=33}}
|{{UDnote|fifth=71|step=33}}
|-
|34
|340
|{{UDnote|step=34}}
|{{UDnote|fifth=71|step=34}}
|-
|35
|350
|{{UDnote|step=35}}
|{{UDnote|fifth=71|step=35}}
|-
|36
|360
|{{UDnote|step=36}}
|{{UDnote|fifth=71|step=36}}
|-
|37
|370
|{{UDnote|step=37}}
|{{UDnote|fifth=71|step=37}}
|-
|38
|380
|{{UDnote|step=38}}
|{{UDnote|fifth=71|step=38}}
|-
|39
|390
|{{UDnote|step=39}}
|{{UDnote|fifth=71|step=39}}
|-
|40
|400
|{{UDnote|step=40}}
|{{UDnote|fifth=71|step=40}}
|-
|41
|410
|{{UDnote|step=41}}
|{{UDnote|fifth=71|step=41}}
|-
|42
|420
|{{UDnote|step=42}}
|{{UDnote|fifth=71|step=42}}
|-
|43
|430
|{{UDnote|step=43}}
|{{UDnote|fifth=71|step=43}}
|-
|44
|440
|{{UDnote|step=44}}
|{{UDnote|fifth=71|step=44}}
|-
|45
|450
|{{UDnote|step=45}}
|{{UDnote|fifth=71|step=45}}
|-
|46
|460
|{{UDnote|step=46}}
|{{UDnote|fifth=71|step=46}}
|-
|47
|470
|{{UDnote|step=47}}
|{{UDnote|fifth=71|step=47}}
|-
|48
|480
|{{UDnote|step=48}}
|{{UDnote|fifth=71|step=48}}
|-
|49
|490
|{{UDnote|step=49}}
|{{UDnote|fifth=71|step=49}}
|-
|50
|500
|{{UDnote|step=50}}
|{{UDnote|fifth=71|step=50}}
|-
|51
|510
|{{UDnote|step=51}}
|{{UDnote|fifth=71|step=51}}
|-
|52
|520
|{{UDnote|step=52}}
|{{UDnote|fifth=71|step=52}}
|-
|53
|530
|{{UDnote|step=53}}
|{{UDnote|fifth=71|step=53}}
|-
|54
|540
|{{UDnote|step=54}}
|{{UDnote|fifth=71|step=54}}
|-
|55
|550
|{{UDnote|step=55}}
|{{UDnote|fifth=71|step=55}}
|-
|56
|560
|{{UDnote|step=56}}
|{{UDnote|fifth=71|step=56}}
|-
|57
|570
|{{UDnote|step=57}}
|{{UDnote|fifth=71|step=57}}
|-
|58
|580
|{{UDnote|step=58}}
|{{UDnote|fifth=71|step=58}}
|-
|59
|590
|{{UDnote|step=59}}
|{{UDnote|fifth=71|step=59}}
|-
|60
|600
|{{UDnote|step=60}}
|{{UDnote|fifth=71|step=60}}
|-
|61
|610
|{{UDnote|step=61}}
|{{UDnote|fifth=71|step=61}}
|-
|62
|620
|{{UDnote|step=62}}
|{{UDnote|fifth=71|step=62}}
|-
|63
|630
|{{UDnote|step=63}}
|{{UDnote|fifth=71|step=63}}
|-
|64
|640
|{{UDnote|step=64}}
|{{UDnote|fifth=71|step=64}}
|-
|65
|650
|{{UDnote|step=65}}
|{{UDnote|fifth=71|step=65}}
|-
|66
|660
|{{UDnote|step=66}}
|{{UDnote|fifth=71|step=66}}
|-
|67
|670
|{{UDnote|step=67}}
|{{UDnote|fifth=71|step=67}}
|-
|68
|680
|{{UDnote|step=68}}
|{{UDnote|fifth=71|step=68}}
|-
|69
|690
|{{UDnote|step=69}}
|{{UDnote|fifth=71|step=69}}
|-
|70
|700
|{{UDnote|step=70}}
|{{UDnote|fifth=71|step=70}}
|-
|71
|710
|{{UDnote|step=71}}
|{{UDnote|fifth=71|step=71}}
|-
|72
|720
|{{UDnote|step=72}}
|{{UDnote|fifth=71|step=72}}
|-
|73
|730
|{{UDnote|step=73}}
|{{UDnote|fifth=71|step=73}}
|-
|74
|740
|{{UDnote|step=74}}
|{{UDnote|fifth=71|step=74}}
|-
|75
|750
|{{UDnote|step=75}}
|{{UDnote|fifth=71|step=75}}
|-
|76
|760
|{{UDnote|step=76}}
|{{UDnote|fifth=71|step=76}}
|-
|77
|770
|{{UDnote|step=77}}
|{{UDnote|fifth=71|step=77}}
|-
|78
|780
|{{UDnote|step=78}}
|{{UDnote|fifth=71|step=78}}
|-
|79
|790
|{{UDnote|step=79}}
|{{UDnote|fifth=71|step=79}}
|-
|80
|800
|{{UDnote|step=80}}
|{{UDnote|fifth=71|step=80}}
|-
|81
|810
|{{UDnote|step=81}}
|{{UDnote|fifth=71|step=81}}
|-
|82
|820
|{{UDnote|step=82}}
|{{UDnote|fifth=71|step=82}}
|-
|83
|830
|{{UDnote|step=83}}
|{{UDnote|fifth=71|step=83}}
|-
|84
|840
|{{UDnote|step=84}}
|{{UDnote|fifth=71|step=84}}
|-
|85
|850
|{{UDnote|step=85}}
|{{UDnote|fifth=71|step=85}}
|-
|86
|860
|{{UDnote|step=86}}
|{{UDnote|fifth=71|step=86}}
|-
|87
|870
|{{UDnote|step=87}}
|{{UDnote|fifth=71|step=87}}
|-
|88
|880
|{{UDnote|step=88}}
|{{UDnote|fifth=71|step=88}}
|-
|89
|890
|{{UDnote|step=89}}
|{{UDnote|fifth=71|step=89}}
|-
|90
|900
|{{UDnote|step=90}}
|{{UDnote|fifth=71|step=90}}
|-
|91
|910
|{{UDnote|step=91}}
|{{UDnote|fifth=71|step=91}}
|-
|92
|920
|{{UDnote|step=92}}
|{{UDnote|fifth=71|step=92}}
|-
|93
|930
|{{UDnote|step=93}}
|{{UDnote|fifth=71|step=93}}
|-
|94
|940
|{{UDnote|step=94}}
|{{UDnote|fifth=71|step=94}}
|-
|95
|950
|{{UDnote|step=95}}
|{{UDnote|fifth=71|step=95}}
|-
|96
|960
|{{UDnote|step=96}}
|{{UDnote|fifth=71|step=96}}
|-
|97
|970
|{{UDnote|step=97}}
|{{UDnote|fifth=71|step=97}}
|-
|98
|980
|{{UDnote|step=98}}
|{{UDnote|fifth=71|step=98}}
|-
|99
|990
|{{UDnote|step=99}}
|{{UDnote|fifth=71|step=99}}
|-
|100
|1000
|{{UDnote|step=100}}
|{{UDnote|fifth=71|step=100}}
|-
|101
|1010
|{{UDnote|step=101}}
|{{UDnote|fifth=71|step=101}}
|-
|102
|1020
|{{UDnote|step=102}}
|{{UDnote|fifth=71|step=102}}
|-
|103
|1030
|{{UDnote|step=103}}
|{{UDnote|fifth=71|step=103}}
|-
|104
|1040
|{{UDnote|step=104}}
|{{UDnote|fifth=71|step=104}}
|-
|105
|1050
|{{UDnote|step=105}}
|{{UDnote|fifth=71|step=105}}
|-
|106
|1060
|{{UDnote|step=106}}
|{{UDnote|fifth=71|step=106}}
|-
|107
|1070
|{{UDnote|step=107}}
|{{UDnote|fifth=71|step=107}}
|-
|108
|1080
|{{UDnote|step=108}}
|{{UDnote|fifth=71|step=108}}
|-
|109
|1090
|{{UDnote|step=109}}
|{{UDnote|fifth=71|step=109}}
|-
|110
|1100
|{{UDnote|step=110}}
|{{UDnote|fifth=71|step=110}}
|-
|111
|1110
|{{UDnote|step=111}}
|{{UDnote|fifth=71|step=111}}
|-
|112
|1120
|{{UDnote|step=112}}
|{{UDnote|fifth=71|step=112}}
|-
|113
|1130
|{{UDnote|step=113}}
|{{UDnote|fifth=71|step=113}}
|-
|114
|1140
|{{UDnote|step=114}}
|{{UDnote|fifth=71|step=114}}
|-
|115
|1150
|{{UDnote|step=115}}
|{{UDnote|fifth=71|step=115}}
|-
|116
|1160
|{{UDnote|step=116}}
|{{UDnote|fifth=71|step=116}}
|-
|117
|1170
|{{UDnote|step=117}}
|{{UDnote|fifth=71|step=117}}
|-
|118
|1180
|{{UDnote|step=118}}
|{{UDnote|fifth=71|step=118}}
|-
|119
|1190
|{{UDnote|step=119}}
|{{UDnote|fifth=71|step=119}}
|-
|120
|1200
|{{UDnote|step=120}}
|{{UDnote|fifth=71|step=120}}
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Highly composite]]</text>
      <sha1>crw9ufzg0w4z3v4r6d48fcfn37i3dkw</sha1>
    </revision>
    <revision>
      <id>118406</id>
      <parentid>116909</parentid>
      <timestamp>2023-07-05T03:04:39Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>118406</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1268" sha1="9477jcolj8bdnvabjr83bwvdmmlukpq" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3*4*5 = 5!).

120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out [[243/242]].

120edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit [[superpyth]] temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which is [[marvel comma]] sharp of 3/2. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

== Interval list ==
{{Interval table}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Highly composite]]</text>
      <sha1>9477jcolj8bdnvabjr83bwvdmmlukpq</sha1>
    </revision>
    <revision>
      <id>126572</id>
      <parentid>118406</parentid>
      <timestamp>2023-10-30T07:47:32Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>126572</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1361" sha1="dypytdcc7rfo6mjpmw8en26m9qqhrrg" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3*4*5 = 5!).

120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out [[243/242]].

120edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit [[superpyth]] temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which is [[marvel comma]] sharp of 3/2. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

The step size of this EDO is near the upper boundary of the [[just noticeable difference]].

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

== Interval list ==
{{Interval table}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Highly composite]]</text>
      <sha1>dypytdcc7rfo6mjpmw8en26m9qqhrrg</sha1>
    </revision>
    <revision>
      <id>134250</id>
      <parentid>126572</parentid>
      <timestamp>2024-02-03T23:08:42Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ Mentioned Newcome</comment>
      <origin>134250</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1461" sha1="56iepw31453y0tozlfm91vx0o5de6qj" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo is the 10th highly composite EDO and the 5th factorial EDO (120 = 1*2*3*4*5 = 5!).

120edo is an excellent tuning in the 2.3.7.11.13.23.29 subgroup. In the no-5s 11-limit, it tempers out [[243/242]].

120edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit [[superpyth]] temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which is [[marvel comma]] sharp of 3/2. It may be used as a ''de facto'' dual fifth in [[Substitute harmonic#Newcome|newcome]] temperament. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

The step size of this EDO is near the upper boundary of the [[just noticeable difference]].

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

== Interval list ==
{{Interval table}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Highly composite]]</text>
      <sha1>56iepw31453y0tozlfm91vx0o5de6qj</sha1>
    </revision>
    <revision>
      <id>145229</id>
      <parentid>134250</parentid>
      <timestamp>2024-05-29T08:49:57Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Anything about edostep size vs JND is useless due to how broad JND are interpreted (even the "upper bound" and "lower bound"); misc. cleanup</comment>
      <origin>145229</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1309" sha1="d2vqpu11ldxz7w3pltqiqt69e1r5brl" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo is an excellent tuning in the 2.3.7.11.13.23.29 [[subgroup]]. In the no-5's 11-limit, it tempers out [[243/242]].

120edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit [[superpyth]] temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which is [[marvel comma]] sharp of 3/2. It may be used as a ''de facto'' dual fifth in [[Substitute harmonic#Newcome|newcome]] temperament. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Subsets and supersets ===
120edo is the 10th highly composite edo and the 5th factorial edo (120 = 5! = 1 × 2 × 3 × 4 × 5).

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic [[wikipedia:Long_hundred|long hundred]], it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a concoctic generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

== Intervals ==
{{Interval table}}</text>
      <sha1>d2vqpu11ldxz7w3pltqiqt69e1r5brl</sha1>
    </revision>
    <revision>
      <id>145230</id>
      <parentid>145229</parentid>
      <timestamp>2024-05-29T08:52:46Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+subsets; misc. cleanup</comment>
      <origin>145230</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1375" sha1="7n9xu5kvo644z6itzulhy6hi93zbjhf" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo is an excellent tuning in the 2.3.7.11.13.23.29 [[subgroup]]. In the no-5's 11-limit, it tempers out [[243/242]].

120edo shares the perfect fifth with 12edo, tempering out the [[Pythagorean comma]]. The sharp fifth of 710 cents also has regular temperament interpretations. It is used in the 120b val for tuning the 5-limit [[superpyth]] temperament where it represents 3/2, and in the 120g val as a tuning for the 19-limit [[surmarvelpyth]] temperament where it represents 675/448, which is [[marvel comma]] sharp of 3/2. It may be used as a ''de facto'' dual fifth in [[Substitute harmonic#Newcome|newcome]] temperament. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Subsets and supersets ===
120edo is the 10th highly composite edo and the 5th factorial edo (120 = 5! = 1 × 2 × 3 × 4 × 5). It has many subsets: {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, and 60 }}. 

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic {{w|long hundred}}, it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a [[concoctic]] generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

== Intervals ==
{{Interval table}}</text>
      <sha1>7n9xu5kvo644z6itzulhy6hi93zbjhf</sha1>
    </revision>
    <revision>
      <id>145234</id>
      <parentid>145230</parentid>
      <timestamp>2024-05-29T09:18:25Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>-surmarvelpyth (pointless in 120edo as prime 5 is more than 120 steps away)</comment>
      <origin>145234</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1146" sha1="j2mc8tkt8w9a8bcvy09o4ejf4r98jlq" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo shares the [[perfect fifth]] with 12edo, [[tempering out]] the [[Pythagorean comma]]. 120edo is an excellent tuning in the 2.3.7.11.13.23.29 [[subgroup]]. In the no-5's 11-limit, it tempers out [[243/242]]. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

The 120bdd val is a tuning for [[superpyth]] where 3/2 is tuned to exactly 710 cents. It may be used as a ''de facto'' dual fifth in [[Substitute harmonic#Newcome|newcome]] temperament. 

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Subsets and supersets ===
120edo is the 10th highly composite edo and the 5th factorial edo (120 = 5! = 1 × 2 × 3 × 4 × 5). It has many subsets: {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, and 60 }}. 

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic {{w|long hundred}}, it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a [[concoctic]] generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

== Intervals ==
{{Interval table}}</text>
      <sha1>j2mc8tkt8w9a8bcvy09o4ejf4r98jlq</sha1>
    </revision>
    <revision>
      <id>149070</id>
      <parentid>145234</parentid>
      <timestamp>2024-07-29T23:45:03Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>149070</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1200" sha1="cwrp7rviv7bnb4xxhdg9jlomq6qbw3p" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo shares the [[perfect fifth]] with 12edo, [[tempering out]] the [[Pythagorean comma]]. 120edo is an excellent tuning in the 2.3.7.11.13.23.29 [[subgroup]]. In the no-5's 11-limit, it tempers out [[243/242]]. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

The 120bdd val is a tuning for [[superpyth]] where 3/2 is tuned to exactly 710 cents. It may be used as a ''de facto'' dual fifth in [[Substitute harmonic#Newcome|newcome]] temperament. 

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Subsets and supersets ===
120edo is the 10th highly composite edo and the 5th factorial edo (120 = 5! = 1 × 2 × 3 × 4 × 5). It has many subsets: {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, and 60 }}. 

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic {{w|long hundred}}, it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a [[concoctic]] generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

== JI approximation ==
{{Q-odd-limit intervals|120}}

== Intervals ==
{{Interval table}}</text>
      <sha1>cwrp7rviv7bnb4xxhdg9jlomq6qbw3p</sha1>
    </revision>
    <revision>
      <id>179624</id>
      <parentid>149070</parentid>
      <timestamp>2025-02-06T19:19:30Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>+ Ups and downs notation</comment>
      <origin>179624</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1471" sha1="tpgwv45fgk0xjc9vuzmnykdmm72hewv" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo shares the [[perfect fifth]] with 12edo, [[tempering out]] the [[Pythagorean comma]]. 120edo is an excellent tuning in the 2.3.7.11.13.23.29 [[subgroup]]. In the no-5's 11-limit, it tempers out [[243/242]]. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

The 120bdd val is a tuning for [[superpyth]] where 3/2 is tuned to exactly 710{{c}}. It may be used as a ''de facto'' dual fifth in [[Substitute harmonic#Newcome|newcome]] temperament. 

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Subsets and supersets ===
120edo is the 10th highly composite edo and the 5th factorial edo (since {{nowrap|120 {{=}} 5!}} {{nowrap|{{=}} 1 × 2 × 3 × 4 × 5}}). It has many subsets: {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, and 60 }}. 

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic {{w|long hundred}}, it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a [[concoctic]] generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

== JI approximation ==
{{Q-odd-limit intervals|120}}

== Intervals ==
{{Interval table}}

== Notation ==
=== Ups and downs notation ===
120edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals and Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals:

{{Sharpness-sharp10|120}}</text>
      <sha1>tpgwv45fgk0xjc9vuzmnykdmm72hewv</sha1>
    </revision>
    <revision>
      <id>179625</id>
      <parentid>179624</parentid>
      <timestamp>2025-02-06T19:19:47Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Notation */</comment>
      <origin>179625</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1475" sha1="3uwlsh1ealfqho8z82x9do1nx4395na" xml:space="preserve">{{Infobox ET}}
{{EDO intro|120}}

== Theory ==
120edo shares the [[perfect fifth]] with 12edo, [[tempering out]] the [[Pythagorean comma]]. 120edo is an excellent tuning in the 2.3.7.11.13.23.29 [[subgroup]]. In the no-5's 11-limit, it tempers out [[243/242]]. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

The 120bdd val is a tuning for [[superpyth]] where 3/2 is tuned to exactly 710{{c}}. It may be used as a ''de facto'' dual fifth in [[Substitute harmonic#Newcome|newcome]] temperament. 

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Subsets and supersets ===
120edo is the 10th highly composite edo and the 5th factorial edo (since {{nowrap|120 {{=}} 5!}} {{nowrap|{{=}} 1 × 2 × 3 × 4 × 5}}). It has many subsets: {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, and 60 }}. 

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic {{w|long hundred}}, it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a [[concoctic]] generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

== JI approximation ==
{{Q-odd-limit intervals|120}}

== Intervals ==
{{Interval table}}

== Notation ==
=== Ups and downs notation ===
120edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals and Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals:

{{Sharpness-sharp10-qt1|120}}</text>
      <sha1>3uwlsh1ealfqho8z82x9do1nx4395na</sha1>
    </revision>
    <revision>
      <id>181328</id>
      <parentid>179625</parentid>
      <timestamp>2025-02-18T16:55:46Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181328</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1470" sha1="estsevvochi31z0mem3vf7up8vhuj4y" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
120edo shares the [[perfect fifth]] with 12edo, [[tempering out]] the [[Pythagorean comma]]. 120edo is an excellent tuning in the 2.3.7.11.13.23.29 [[subgroup]]. In the no-5's 11-limit, it tempers out [[243/242]]. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

The 120bdd val is a tuning for [[superpyth]] where 3/2 is tuned to exactly 710{{c}}. It may be used as a ''de facto'' dual fifth in [[Substitute harmonic#Newcome|newcome]] temperament. 

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Subsets and supersets ===
120edo is the 10th highly composite edo and the 5th factorial edo (since {{nowrap|120 {{=}} 5!}} {{nowrap|{{=}} 1 × 2 × 3 × 4 × 5}}). It has many subsets: {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, and 60 }}. 

=== Miscellaneous properties ===
Being the simplest division of the octave by the Germanic {{w|long hundred}}, it has a unit step which is the fine relative cent of [[1edo]].

120edo also has a [[concoctic]] generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

== JI approximation ==
{{Q-odd-limit intervals|120}}

== Intervals ==
{{Interval table}}

== Notation ==
=== Ups and downs notation ===
120edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals and Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals:

{{Sharpness-sharp10-qt1|120}}</text>
      <sha1>estsevvochi31z0mem3vf7up8vhuj4y</sha1>
    </revision>
    <revision>
      <id>203835</id>
      <parentid>181328</parentid>
      <timestamp>2025-06-29T06:28:38Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>wtf is the "long hundred" thing doing here, pretty sure that's MMTM's doing</comment>
      <origin>203835</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1327" sha1="7r8o7kqn45fza9mg75sh0lnqds55z5i" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
120edo shares the [[perfect fifth]] with 12edo, [[tempering out]] the [[Pythagorean comma]]. 120edo is an excellent tuning in the 2.3.7.11.13.23.29 [[subgroup]]. In the no-5's 11-limit, it tempers out [[243/242]]. In the patent val 120edo is also a tuning for the 7-limit [[decoid]] temperament.

The 120bdd val is a tuning for [[superpyth]] where 3/2 is tuned to exactly 710{{c}}. It may be used as a ''de facto'' dual fifth in [[Substitute harmonic#Newcome|newcome]] temperament. 

=== Prime harmonics ===
{{Harmonics in equal|120}}

=== Subsets and supersets ===
120edo is the 10th highly composite edo and the 5th factorial edo (since {{nowrap|120 {{=}} 5!}} {{nowrap|{{=}} 1 × 2 × 3 × 4 × 5}}). It has many subsets: {{EDOs| 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 30, 40, and 60 }}. 

=== Miscellaneous properties ===
120edo also has a [[concoctic]] generator that resembles the leap day excess of earth, 29\120 corresponding to 5 hours and 48 minutes.

== JI approximation ==
{{Q-odd-limit intervals|120}}

== Intervals ==
{{Interval table}}

== Notation ==
=== Ups and downs notation ===
120edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals and Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals:

{{Sharpness-sharp10-qt1|120}}</text>
      <sha1>7r8o7kqn45fza9mg75sh0lnqds55z5i</sha1>
    </revision>
  </page>
  <page>
    <title>121edo</title>
    <ns>0</ns>
    <id>107</id>
    <revision>
      <id>107</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>107</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2357" sha1="dp05ggr06lqoc1hxspu8c9pncje05r0" xml:space="preserve">The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Mirkwai_clan|grendel temperament]]. It is [[consistent|consistent]] through to the 19 odd limit and uniquely consistent to the 15 odd limit.

Because it tempers out 540/539 it allows [[swetismic_chords|swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic_chords|huntmic chords]], because it tempers out 352/351 it allows [[minthmic_chords|minthmic chords]], because it tempers out 364/363 it allows [[gentle_chords|gentle chords]], because it tempers out 676/675 it allows [[island_tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic_tetrad|nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=13-limit detempering of 121et=
[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]
[[Category:gentle]]
[[Category:grendel]]
[[Category:huntma]]
[[Category:island]]
[[Category:marveltwin]]
[[Category:minthma]]
[[Category:swetisma]]</text>
      <sha1>dp05ggr06lqoc1hxspu8c9pncje05r0</sha1>
    </revision>
    <revision>
      <id>4525</id>
      <timestamp>2014-01-26T13:37:06Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 485361746 - Original comment: **</comment>
      <origin>4525</origin>
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      <text bytes="6211" sha1="9sd4klovkrmghqi62kym4vqfw1u3lzl" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2014-01-26 13:37:06 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;485361746&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai clan|grendel temperament]]. It is [[consistent]] through to the 19 odd limit and uniquely consistent to the 15 odd limit.

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=13-limit detempering of 121et=
[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;121edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Mirkwai%20clan"&amp;gt;grendel temperament&amp;lt;/a&amp;gt;. It is &amp;lt;a class="wiki_link" href="/consistent"&amp;gt;consistent&amp;lt;/a&amp;gt; through to the 19 odd limit and uniquely consistent to the 15 odd limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Because it tempers out 540/539 it allows &amp;lt;a class="wiki_link" href="/swetismic%20chords"&amp;gt;swetismic chords&amp;lt;/a&amp;gt;, because it tempers out 325/324 it allows &amp;lt;a class="wiki_link" href="/marveltwin%20triad"&amp;gt;marveltwin chords&amp;lt;/a&amp;gt;, because it tempers out 640/637 it allows &amp;lt;a class="wiki_link" href="/huntmic%20chords"&amp;gt;huntmic chords&amp;lt;/a&amp;gt;, because it tempers out 352/351 it allows &amp;lt;a class="wiki_link" href="/minthmic%20chords"&amp;gt;minthmic chords&amp;lt;/a&amp;gt;, because it tempers out 364/363 it allows &amp;lt;a class="wiki_link" href="/gentle%20chords"&amp;gt;gentle chords&amp;lt;/a&amp;gt;, because it tempers out 676/675 it allows &amp;lt;a class="wiki_link" href="/island%20tetrad"&amp;gt;island chords&amp;lt;/a&amp;gt; and because it tempers out 1575/1573 it allows the &amp;lt;a class="wiki_link" href="/nicolic%20tetrad"&amp;gt;nicolic tetrad&amp;lt;/a&amp;gt;. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x13-limit detempering of 121et"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;13-limit detempering of 121et&amp;lt;/h1&amp;gt;
[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9sd4klovkrmghqi62kym4vqfw1u3lzl</sha1>
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    <revision>
      <id>4526</id>
      <timestamp>2011-08-06T03:35:49Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 244584971 - Original comment: **</comment>
      <origin>4526</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4308" sha1="ng5inavg76f0jpn80z23g6o2kb4k6kl" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-06 03:35:49 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;244584971&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai clan|grendel temperament]]. It is [[consistent]] through to the 19 odd limit and uniquely consistent to the 15 odd limit.

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;121edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Mirkwai%20clan"&amp;gt;grendel temperament&amp;lt;/a&amp;gt;. It is &amp;lt;a class="wiki_link" href="/consistent"&amp;gt;consistent&amp;lt;/a&amp;gt; through to the 19 odd limit and uniquely consistent to the 15 odd limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Because it tempers out 540/539 it allows &amp;lt;a class="wiki_link" href="/swetismic%20chords"&amp;gt;swetismic chords&amp;lt;/a&amp;gt;, because it tempers out 325/324 it allows &amp;lt;a class="wiki_link" href="/marveltwin%20triad"&amp;gt;marveltwin chords&amp;lt;/a&amp;gt;, because it tempers out 640/637 it allows &amp;lt;a class="wiki_link" href="/huntmic%20chords"&amp;gt;huntmic chords&amp;lt;/a&amp;gt;, because it tempers out 352/351 it allows &amp;lt;a class="wiki_link" href="/minthmic%20chords"&amp;gt;minthmic chords&amp;lt;/a&amp;gt;, because it tempers out 364/363 it allows &amp;lt;a class="wiki_link" href="/gentle%20chords"&amp;gt;gentle chords&amp;lt;/a&amp;gt;, because it tempers out 676/675 it allows &amp;lt;a class="wiki_link" href="/island%20tetrad"&amp;gt;island chords&amp;lt;/a&amp;gt; and because it tempers out 1575/1573 it allows the &amp;lt;a class="wiki_link" href="/nicolic%20tetrad"&amp;gt;nicolic tetrad&amp;lt;/a&amp;gt;. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <timestamp>2011-08-05T14:00:08Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
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      <comment>**Imported revision 244515527 - Original comment: **</comment>
      <origin>4527</origin>
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      <text bytes="4068" sha1="c8m0nj7jwj09p2fxym8e4dhizcfuv77" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-05 14:00:08 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;244515527&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai clan|grendel temperament]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;121edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Mirkwai%20clan"&amp;gt;grendel temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Because it tempers out 540/539 it allows &amp;lt;a class="wiki_link" href="/swetismic%20chords"&amp;gt;swetismic chords&amp;lt;/a&amp;gt;, because it tempers out 325/324 it allows &amp;lt;a class="wiki_link" href="/marveltwin%20triad"&amp;gt;marveltwin chords&amp;lt;/a&amp;gt;, because it tempers out 640/637 it allows &amp;lt;a class="wiki_link" href="/huntmic%20chords"&amp;gt;huntmic chords&amp;lt;/a&amp;gt;, because it tempers out 352/351 it allows &amp;lt;a class="wiki_link" href="/minthmic%20chords"&amp;gt;minthmic chords&amp;lt;/a&amp;gt;, because it tempers out 364/363 it allows &amp;lt;a class="wiki_link" href="/gentle%20chords"&amp;gt;gentle chords&amp;lt;/a&amp;gt;, because it tempers out 676/675 it allows &amp;lt;a class="wiki_link" href="/island%20tetrad"&amp;gt;island chords&amp;lt;/a&amp;gt; and because it tempers out 1575/1573 it allows the &amp;lt;a class="wiki_link" href="/nicolic%20tetrad"&amp;gt;nicolic tetrad&amp;lt;/a&amp;gt;. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <timestamp>2011-08-02T23:49:04Z</timestamp>
      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 244050165 - Original comment: **</comment>
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      <text bytes="3878" sha1="ozzxn3q3xp91ascbd1d56gvlfcn4ya0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-02 23:49:04 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;244050165&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai clan|grendel temperament]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], and because it tempers out 676/675 it allows [[island tetrad|island chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;121edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Mirkwai%20clan"&amp;gt;grendel temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Because it tempers out 540/539 it allows &amp;lt;a class="wiki_link" href="/swetismic%20chords"&amp;gt;swetismic chords&amp;lt;/a&amp;gt;, because it tempers out 325/324 it allows &amp;lt;a class="wiki_link" href="/marveltwin%20triad"&amp;gt;marveltwin chords&amp;lt;/a&amp;gt;, because it tempers out 640/637 it allows &amp;lt;a class="wiki_link" href="/huntmic%20chords"&amp;gt;huntmic chords&amp;lt;/a&amp;gt;, because it tempers out 352/351 it allows &amp;lt;a class="wiki_link" href="/minthmic%20chords"&amp;gt;minthmic chords&amp;lt;/a&amp;gt;, because it tempers out 364/363 it allows &amp;lt;a class="wiki_link" href="/gentle%20chords"&amp;gt;gentle chords&amp;lt;/a&amp;gt;, and because it tempers out 676/675 it allows &amp;lt;a class="wiki_link" href="/island%20tetrad"&amp;gt;island chords&amp;lt;/a&amp;gt;. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <timestamp>2011-08-02T23:32:06Z</timestamp>
      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 244048181 - Original comment: **</comment>
      <origin>4529</origin>
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      <text bytes="3313" sha1="40gaww5gdnd5cxu2dx6lgf9auuiwy1l" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-02 23:32:06 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;244048181&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai clan|grendel temperament]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin triad|marveltwin chords]], because it tempers out 352/351 it allows [[minthmic chords]] and because it tempers out 364/363 it allows [[gentle chords]]. That makes for a very flexible system.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;121edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Mirkwai%20clan"&amp;gt;grendel temperament&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Because it tempers out 540/539 it allows &amp;lt;a class="wiki_link" href="/swetismic%20chords"&amp;gt;swetismic chords&amp;lt;/a&amp;gt;, because it tempers out 325/324 it allows &amp;lt;a class="wiki_link" href="/marveltwin%20triad"&amp;gt;marveltwin chords&amp;lt;/a&amp;gt;, because it tempers out 352/351 it allows &amp;lt;a class="wiki_link" href="/minthmic%20chords"&amp;gt;minthmic chords&amp;lt;/a&amp;gt; and because it tempers out 364/363 it allows &amp;lt;a class="wiki_link" href="/gentle%20chords"&amp;gt;gentle chords&amp;lt;/a&amp;gt;. That makes for a very flexible system.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 213482580 - Original comment: **</comment>
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      <text bytes="2452" sha1="kzzi01ebeq37milh6gjj9slw1ax32ye" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-24 02:48:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;213482580&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai clan|grendel temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;121edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 13-limit &amp;lt;a class="wiki_link" href="/Mirkwai%20clan"&amp;gt;grendel temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 213482302 - Original comment: **</comment>
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      <text bytes="2452" sha1="fvofgxm5oaruwzqgem0qxb033nm05lh" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-24 02:46:21 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;213482302&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the [[optimal patent val]] for 11-limit [[Mirkwai clan|grendel temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;121edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 11-limit &amp;lt;a class="wiki_link" href="/Mirkwai%20clan"&amp;gt;grendel temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>fvofgxm5oaruwzqgem0qxb033nm05lh</sha1>
    </revision>
    <revision>
      <id>40210</id>
      <parentid>107</parentid>
      <timestamp>2019-02-13T01:51:06Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>40210</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2562" sha1="3u0haf1rt9k8cfn9pralk655rrflrjs" xml:space="preserve">The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each and being the square closest to division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the quadratic (fine) relative cent of [[1edo]]. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Mirkwai_clan|grendel temperament]]. It is [[consistent|consistent]] through to the 19 odd limit and uniquely consistent to the 15 odd limit.

Because it tempers out 540/539 it allows [[swetismic_chords|swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic_chords|huntmic chords]], because it tempers out 352/351 it allows [[minthmic_chords|minthmic chords]], because it tempers out 364/363 it allows [[gentle_chords|gentle chords]], because it tempers out 676/675 it allows [[island_tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic_tetrad|nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=13-limit detempering of 121et=
[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]
[[Category:gentle]]
[[Category:grendel]]
[[Category:huntma]]
[[Category:island]]
[[Category:marveltwin]]
[[Category:minthma]]
[[Category:swetisma]]</text>
      <sha1>3u0haf1rt9k8cfn9pralk655rrflrjs</sha1>
    </revision>
    <revision>
      <id>43515</id>
      <parentid>40210</parentid>
      <timestamp>2020-02-10T01:20:55Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>/* 13-limit detempering of 121et */</comment>
      <origin>43515</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2596" sha1="ms2zziz5xrpy6psx1g72n6ywwyypt5t" xml:space="preserve">The 121 equal temperament divides the octave into 121 equal steps of 9.917 cents each and being the square closest to division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the quadratic (fine) relative cent of [[1edo]]. It has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out 15625/15552 in the 5-limit; 4000/3969, 6144/6125, 10976/10935 in the 7-limit; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the 13-limit; 256/255, 375/374  and 442/441 in the 17-limit; 190/189 and 361/360 in the 19-limit. It also serves as the [[Optimal_patent_val|optimal patent val]] for 13-limit [[Mirkwai_clan|grendel temperament]]. It is [[consistent|consistent]] through to the 19 odd limit and uniquely consistent to the 15 odd limit.

Because it tempers out 540/539 it allows [[swetismic_chords|swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic_chords|huntmic chords]], because it tempers out 352/351 it allows [[minthmic_chords|minthmic chords]], because it tempers out 364/363 it allows [[gentle_chords|gentle chords]], because it tempers out 676/675 it allows [[island_tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic_tetrad|nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=13-limit detempering of 121et=
See [[Detempering|detempering]].

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]
[[Category:gentle]]
[[Category:grendel]]
[[Category:huntma]]
[[Category:island]]
[[Category:marveltwin]]
[[Category:minthma]]
[[Category:swetisma]]</text>
      <sha1>ms2zziz5xrpy6psx1g72n6ywwyypt5t</sha1>
    </revision>
    <revision>
      <id>49728</id>
      <parentid>43515</parentid>
      <timestamp>2020-09-17T03:58:57Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Cleanup and rename some categories</comment>
      <origin>49728</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2548" sha1="47m6f71ex04arewrc8et5cy9eavn4t7" xml:space="preserve">The '''121 equal temperament''' divides the octave into 121 equal steps of 9.917 cents each and being the square closest to division of the octave by the Germanic [[Wikipedia:Long_hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo]]. 

121edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island_tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

== 13-limit detempering of 121et ==
: ''See also: [[Detempering]].''

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Edo]]
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Swetismic]]</text>
      <sha1>47m6f71ex04arewrc8et5cy9eavn4t7</sha1>
    </revision>
    <revision>
      <id>54971</id>
      <parentid>49728</parentid>
      <timestamp>2020-12-04T23:13:07Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54971</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2574" sha1="jeibbr7gek34m3r3satknqtvq34aukb" xml:space="preserve">The '''121 equal temperament''' divides the octave into 121 equal steps of 9.917 cents each and being the square closest to division of the octave by the Germanic [[Wikipedia:Long_hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo]]. 

121edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island_tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

== 13-limit detempering of 121et ==
: ''See also: [[Detempering]].''

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Equal divisions of the octave]]
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Swetismic]]</text>
      <sha1>jeibbr7gek34m3r3satknqtvq34aukb</sha1>
    </revision>
    <revision>
      <id>70991</id>
      <parentid>54971</parentid>
      <timestamp>2021-05-29T12:27:04Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>added prime errors</comment>
      <origin>70991</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2615" sha1="qwwvqtntk7owt59fsi96str8iq59gud" xml:space="preserve">The '''121 equal temperament''' divides the octave into 121 equal steps of 9.917 cents each and being the square closest to division of the octave by the Germanic [[Wikipedia:Long_hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo]]. 

121edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island_tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

{{Primes in edo|121|columns=10|prec=2}}

== 13-limit detempering of 121et ==
: ''See also: [[Detempering]].''

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Equal divisions of the octave]]
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Swetismic]]</text>
      <sha1>qwwvqtntk7owt59fsi96str8iq59gud</sha1>
    </revision>
    <revision>
      <id>74122</id>
      <parentid>70991</parentid>
      <timestamp>2021-07-11T11:57:00Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>the same prec is now estimated by EDO magnitude</comment>
      <origin>74122</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2608" sha1="miwfhni8tzmchjfo8aeaj5wplpizcxm" xml:space="preserve">The '''121 equal temperament''' divides the octave into 121 equal steps of 9.917 cents each and being the square closest to division of the octave by the Germanic [[Wikipedia:Long_hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo]]. 

121edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island_tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

{{Primes in edo|121|columns=10}}

== 13-limit detempering of 121et ==
: ''See also: [[Detempering]].''

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Equal divisions of the octave]]
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Swetismic]]</text>
      <sha1>miwfhni8tzmchjfo8aeaj5wplpizcxm</sha1>
    </revision>
    <revision>
      <id>77502</id>
      <parentid>74122</parentid>
      <timestamp>2021-09-06T20:04:20Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Improve intro and sectioning</comment>
      <origin>77502</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2803" sha1="gmwa05bidq75rq255n6w24jwy6ldq49" xml:space="preserve">The '''121 equal divisions of the octave''' ('''121edo'''), or the '''121(-tone) equal temperament''' ('''121tet''', '''121et''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.917 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo]]. 

== Theory ==
121edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Primes in edo|121|columns=10}}

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Equal divisions of the octave]]
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Swetismic]]</text>
      <sha1>gmwa05bidq75rq255n6w24jwy6ldq49</sha1>
    </revision>
    <revision>
      <id>78457</id>
      <parentid>77502</parentid>
      <timestamp>2021-09-23T20:58:28Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+RTT tables</comment>
      <origin>78457</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4855" sha1="n5vb28hkqrrsjymyayg1unbpifb02aa" xml:space="preserve">The '''121 equal divisions of the octave''' ('''121edo'''), or the '''121(-tone) equal temperament''' ('''121tet''', '''121et''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.917 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo]]. 

== Theory ==
121edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Primes in edo|121|columns=10}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 442/441, 540/539
| [{{val| 121 192 281 340 419 448  }}]
| -0.787
| 0.480
| 4.85
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Hanson]] / [[metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 98304/78125
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46/121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47/121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48/121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Equal divisions of the octave]]
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Swetismic]]</text>
      <sha1>n5vb28hkqrrsjymyayg1unbpifb02aa</sha1>
    </revision>
    <revision>
      <id>78486</id>
      <parentid>78457</parentid>
      <timestamp>2021-09-24T13:13:39Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */</comment>
      <origin>78486</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4849" sha1="c955fn5klw03jvsg4e7w0fr93x8vyh3" xml:space="preserve">The '''121 equal divisions of the octave''' ('''121edo'''), or the '''121(-tone) equal temperament''' ('''121tet''', '''121et''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.917 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo]]. 

== Theory ==
121edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Primes in edo|121|columns=10}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 442/441, 540/539
| [{{val| 121 192 281 340 419 448  }}]
| -0.787
| 0.480
| 4.85
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Hanson]] / [[metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46/121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47/121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48/121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Equal divisions of the octave]]
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Swetismic]]</text>
      <sha1>c955fn5klw03jvsg4e7w0fr93x8vyh3</sha1>
    </revision>
    <revision>
      <id>78509</id>
      <parentid>78486</parentid>
      <timestamp>2021-09-24T19:58:29Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ +tertiathirds</comment>
      <origin>78509</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4901" sha1="cn78armhrtd0uctdqjuykh2cise39yv" xml:space="preserve">The '''121 equal divisions of the octave''' ('''121edo'''), or the '''121(-tone) equal temperament''' ('''121tet''', '''121et''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.917 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo]]. 

== Theory ==
121edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Primes in edo|121|columns=10}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 442/441, 540/539
| [{{val| 121 192 281 340 419 448  }}]
| -0.787
| 0.480
| 4.85
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Hanson]] / [[metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46/121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47/121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48/121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Equal divisions of the octave]]
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Swetismic]]</text>
      <sha1>cn78armhrtd0uctdqjuykh2cise39yv</sha1>
    </revision>
    <revision>
      <id>79303</id>
      <parentid>78509</parentid>
      <timestamp>2021-10-10T10:07:51Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>79303</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5118" sha1="q83sg8dhbamw8paebg1koxjrfjj2qpg" xml:space="preserve">The '''121 equal divisions of the octave''' ('''121EDO'''), or the '''121(-tone) equal temperament''' ('''121TET''', '''121ET''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.9174 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo|1EDO]]. 

== Theory ==
121EDO has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121ET, it is a system only associated with 121.

=== Prime harmonics ===
{{Primes in edo|121|columns=10}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 442/441, 540/539
| [{{val| 121 192 281 340 419 448 495 }}]
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| [{{val| 121 192 281 340 419 448 495 514 }}]
| -0.689
| 0.519
| 5.23
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Hanson]] / [[metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121ET ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Equal divisions of the octave]]
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Swetismic]]</text>
      <sha1>q83sg8dhbamw8paebg1koxjrfjj2qpg</sha1>
    </revision>
    <revision>
      <id>79307</id>
      <parentid>79303</parentid>
      <timestamp>2021-10-10T10:48:43Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <minor/>
      <origin>79307</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5142" sha1="h8i0w6qtdfbjbglpk0gerc78e02buy0" xml:space="preserve">The '''121 equal divisions of the octave''' ('''121EDO'''), or the '''121(-tone) equal temperament''' ('''121TET''', '''121ET''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.9174 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo|1EDO]]. 

== Theory ==
121EDO has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121ET, it is a system only associated with 121.

=== Prime harmonics ===
{{Primes in edo|121|columns=10}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| [{{val| 121 192 281 340 419 448 495 }}]
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| [{{val| 121 192 281 340 419 448 495 514 }}]
| -0.689
| 0.519
| 5.23
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Hanson]] / [[metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121ET ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Equal divisions of the octave]]
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Quintupole]]
[[Category:Swetismic]]</text>
      <sha1>h8i0w6qtdfbjbglpk0gerc78e02buy0</sha1>
    </revision>
    <revision>
      <id>91489</id>
      <parentid>79307</parentid>
      <timestamp>2022-04-22T03:56:30Z</timestamp>
      <contributor>
        <username>AthiTrydhen</username>
        <id>314</id>
      </contributor>
      <comment>/* 13-limit detempering of 121ET */</comment>
      <origin>91489</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5327" sha1="2nzcg4f8y9o5qpfx4hbgc099radl2yg" xml:space="preserve">The '''121 equal divisions of the octave''' ('''121EDO'''), or the '''121(-tone) equal temperament''' ('''121TET''', '''121ET''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.9174 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo|1EDO]]. 

== Theory ==
121EDO has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121ET, it is a system only associated with 121.

=== Prime harmonics ===
{{Primes in edo|121|columns=10}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| [{{val| 121 192 281 340 419 448 495 }}]
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| [{{val| 121 192 281 340 419 448 495 514 }}]
| -0.689
| 0.519
| 5.23
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Hanson]] / [[metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121ET ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

==Miscellany==

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates peppermint temperament. This makes it suitable for neo-Gothic tunings.

[[Category:Equal divisions of the octave]]
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Quintupole]]
[[Category:Swetismic]]</text>
      <sha1>2nzcg4f8y9o5qpfx4hbgc099radl2yg</sha1>
    </revision>
    <revision>
      <id>92931</id>
      <parentid>91489</parentid>
      <timestamp>2022-07-02T16:25:51Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92931</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5355" sha1="6bb4u3rmr82v5jzlmbct2o4znwvaq5e" xml:space="preserve">The '''121 equal divisions of the octave''' ('''121EDO'''), or the '''121(-tone) equal temperament''' ('''121TET''', '''121ET''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.9174 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo|1EDO]]. 

== Theory ==
121EDO has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121ET, it is a system only associated with 121.

=== Prime harmonics ===
{{Primes in edo|121|columns=10}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| [{{val| 121 192 281 340 419 448 495 }}]
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| [{{val| 121 192 281 340 419 448 495 514 }}]
| -0.689
| 0.519
| 5.23
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Hanson]] / [[metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121ET ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

==Miscellany==

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates peppermint temperament. This makes it suitable for neo-Gothic tunings.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Quintupole]]
[[Category:Swetismic]]</text>
      <sha1>6bb4u3rmr82v5jzlmbct2o4znwvaq5e</sha1>
    </revision>
    <revision>
      <id>96199</id>
      <parentid>92931</parentid>
      <timestamp>2022-10-04T18:53:17Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96199</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5370" sha1="i02fn5sp2cgy0ufgb5g0eajuikjz7cr" xml:space="preserve">{{Infobox ET}}
The '''121 equal divisions of the octave''' ('''121EDO'''), or the '''121(-tone) equal temperament''' ('''121TET''', '''121ET''') when viewed from a [[regular temperament]] perspective, divides the octave into 121 [[equal]] steps of 9.9174 [[cent]]s each, and being the square closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the quadratic (fine) relative cent of [[1edo|1EDO]]. 

== Theory ==
121EDO has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121ET, it is a system only associated with 121.

=== Prime harmonics ===
{{Primes in edo|121|columns=10}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| [{{val| 121 192 281 340 419 448 495 }}]
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| [{{val| 121 192 281 340 419 448 495 514 }}]
| -0.689
| 0.519
| 5.23
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Hanson]] / [[metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121ET ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

==Miscellany==

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates peppermint temperament. This makes it suitable for neo-Gothic tunings.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Quintupole]]
[[Category:Swetismic]]</text>
      <sha1>i02fn5sp2cgy0ufgb5g0eajuikjz7cr</sha1>
    </revision>
    <revision>
      <id>107138</id>
      <parentid>96199</parentid>
      <timestamp>2023-04-07T12:47:48Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>edo intro, remove mmtmism as there's little to no recorded usage of number 121 as a subdivision unit in human history</comment>
      <origin>107138</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4937" sha1="8i0uxtrszyk9spi6hb6sllewfk1mtrj" xml:space="preserve">{{Infobox ET}}
{{EDO intro|121}}

== Theory ==
121EDO has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121ET, it is a system only associated with 121.

=== Prime harmonics ===
{{Primes in edo|121|columns=10}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| [{{val| 121 192 281 340 419 448 495 }}]
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| [{{val| 121 192 281 340 419 448 495 514 }}]
| -0.689
| 0.519
| 5.23
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Hanson]] / [[metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121ET ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

==Miscellany==

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates peppermint temperament. This makes it suitable for neo-Gothic tunings.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Quintupole]]
[[Category:Swetismic]]</text>
      <sha1>8i0uxtrszyk9spi6hb6sllewfk1mtrj</sha1>
    </revision>
    <revision>
      <id>107139</id>
      <parentid>107138</parentid>
      <timestamp>2023-04-07T12:48:07Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Rank-2 temperaments */</comment>
      <origin>107139</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4934" sha1="i2e8jtqaqrsl7gruuqnb6rues2nute1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|121}}

== Theory ==
121EDO has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, 6144/6125, 10976/10935 in the [[7-limit]]; 540/539, 896/891 and 1375/1372 in the 11-limit; 325/324, 352/351, 364/363 and 625/624 in the [[13-limit]]; 256/255, 375/374 and 442/441 in the [[17-limit]]; 190/189 and 361/360 in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[Mirkwai_clan #Grendel|grendel temperament]]. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin_triad|marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island tetrad|island chords]] and because it tempers out 1575/1573 it allows the [[nicolic tetrad]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121ET, it is a system only associated with 121.

=== Prime harmonics ===
{{Primes in edo|121|columns=10}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| [{{val| 121 192 281 340 419 448 495 }}]
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| [{{val| 121 192 281 340 419 448 495 514 }}]
| -0.689
| 0.519
| 5.23
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Hanson]] / [[metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121ET ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

==Miscellany==

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates peppermint temperament. This makes it suitable for neo-Gothic tunings.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Quintupole]]
[[Category:Swetismic]]</text>
      <sha1>i2e8jtqaqrsl7gruuqnb6rues2nute1</sha1>
    </revision>
    <revision>
      <id>123000</id>
      <parentid>107139</parentid>
      <timestamp>2023-08-29T06:15:36Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup; linking; style</comment>
      <origin>123000</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4856" sha1="ejpu5kh0qnnxaw68r704qgkydid6ida" xml:space="preserve">{{Infobox ET}}
{{EDO intro|121}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and 1375/1372 in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Harmonics in equal|121}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| [{{val| 121 192 281 340 419 448 495 }}]
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| [{{val| 121 192 281 340 419 448 495 514 }}]
| -0.689
| 0.519
| 5.23
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

== Miscellany ==

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates peppermint temperament. This makes it suitable for neo-Gothic tunings.

[[Category:Gentle]]
[[Category:Grendel]]
[[Category:Huntmic]]
[[Category:Island]]
[[Category:Marveltwin]]
[[Category:Minthmic]]
[[Category:Quintupole]]
[[Category:Swetismic]]</text>
      <sha1>ejpu5kh0qnnxaw68r704qgkydid6ida</sha1>
    </revision>
    <revision>
      <id>131907</id>
      <parentid>123000</parentid>
      <timestamp>2024-01-15T15:00:22Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Address a name change; review categories</comment>
      <origin>131907</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4740" sha1="ovi6ifhl4rfl3umoe4mryeh5h0ugqq1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|121}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and 1375/1372 in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Harmonics in equal|121}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| [{{val| 121 192 281 340 419 448 495 }}]
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| [{{val| 121 192 281 340 419 448 495 514 }}]
| -0.689
| 0.519
| 5.23
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

== Miscellany ==

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates peppermint temperament. This makes it suitable for neo-Gothic tunings.

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>ovi6ifhl4rfl3umoe4mryeh5h0ugqq1</sha1>
    </revision>
    <revision>
      <id>132307</id>
      <parentid>131907</parentid>
      <timestamp>2024-01-21T07:27:32Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>/* Miscellany */ Added links to peppermint and neo-Gothic</comment>
      <origin>132307</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4748" sha1="l3224ap0h832crj64c9a5vagar7tig7" xml:space="preserve">{{Infobox ET}}
{{EDO intro|121}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd primes from 3 to 19 all have sharp tunings. It tempers out [[15625/15552]] in the [[5-limit]]; 4000/3969, [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and 1375/1372 in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Harmonics in equal|121}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| [{{val| 121 192 }}]
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| [{{val| 121 192 281 }}]
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| [{{val| 121 192 281 340 }}]
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| [{{val| 121 192 281 340 419 }}]
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| [{{val| 121 192 281 340 419 448 }}]
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| [{{val| 121 192 281 340 419 448 495 }}]
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| [{{val| 121 192 281 340 419 448 495 514 }}]
| -0.689
| 0.519
| 5.23
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

== Miscellany ==

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>l3224ap0h832crj64c9a5vagar7tig7</sha1>
    </revision>
    <revision>
      <id>145225</id>
      <parentid>132307</parentid>
      <timestamp>2024-05-29T08:38:09Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>145225</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4971" sha1="6hcx0ffy22m1lfqd4flntmmaz5x1j4l" xml:space="preserve">{{Infobox ET}}
{{EDO intro|121}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. The equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Harmonics in equal|121}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| {{mapping| 121 192 }}
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| -0.689
| 0.519
| 5.23
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

== Miscellany ==

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>6hcx0ffy22m1lfqd4flntmmaz5x1j4l</sha1>
    </revision>
    <revision>
      <id>145226</id>
      <parentid>145225</parentid>
      <timestamp>2024-05-29T08:41:48Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+subsets and supersets</comment>
      <origin>145226</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5106" sha1="suztfsuazgg9abj7uk5xuuwuzxyh8ki" xml:space="preserve">{{Infobox ET}}
{{EDO intro|121}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. The equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Harmonics in equal|121}}

=== Subsets and supersets ===
Since 121 factors into {{factorization|121}}, 121edo contains [[11edo]] as its only nontrivial subset. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| {{mapping| 121 192 }}
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| -0.689
| 0.519
| 5.23
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

== Miscellany ==

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>suztfsuazgg9abj7uk5xuuwuzxyh8ki</sha1>
    </revision>
    <revision>
      <id>145701</id>
      <parentid>145226</parentid>
      <timestamp>2024-06-04T13:28:53Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ 13- to 23-limit notability</comment>
      <origin>145701</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5377" sha1="d0t1tpgyaiwkqxnzxiljszbspjnu2z0" xml:space="preserve">{{Infobox ET}}
{{EDO intro|121}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. The equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Harmonics in equal|121}}

=== Subsets and supersets ===
Since 121 factors into {{factorization|121}}, 121edo contains [[11edo]] as its only nontrivial subset. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| {{mapping| 121 192 }}
| -0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| -0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| -0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| -0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| -0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| -0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| -0.689
| 0.519
| 5.23
|}
* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]]. 

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

== Miscellany ==

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>d0t1tpgyaiwkqxnzxiljszbspjnu2z0</sha1>
    </revision>
    <revision>
      <id>165985</id>
      <parentid>145701</parentid>
      <timestamp>2024-11-16T06:08:21Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>165985</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4889" sha1="bbbiijwp32msfa2gkfph67mvjmuzsqw" xml:space="preserve">{{Infobox ET}}
{{EDO intro|121}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. The equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Harmonics in equal|121}}

=== Subsets and supersets ===
Since 121 factors into {{factorization|121}}, 121edo contains [[11edo]] as its only nontrivial subset. 

== Regular temperament properties ==
{{comma basis begin}}
|-
| 2.3
| {{monzo| 192 -121 }}
| {{mapping| 121 192 }}
| &amp;minus;0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| &amp;minus;0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| &amp;minus;0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| &amp;minus;0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| &amp;minus;0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| &amp;minus;0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| &amp;minus;0.689
| 0.519
| 5.23
{{comma basis end}}
* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]]. 

=== Rank-2 temperaments ===
{{rank-2 begin}}
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br /&gt;(5\121)
| 495.87&lt;br /&gt;(49.59)
| 4/3&lt;br /&gt;(36/35)
| [[Hendecatonic]]
{{rank-2 end}}
{{orf}}

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

== Miscellany ==
Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>bbbiijwp32msfa2gkfph67mvjmuzsqw</sha1>
    </revision>
    <revision>
      <id>166095</id>
      <parentid>165985</parentid>
      <timestamp>2024-11-16T11:42:33Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>166095</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5445" sha1="phgt0b99uazoxqd19lh28u11fm9qqyw" xml:space="preserve">{{Infobox ET}}
{{EDO intro|121}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. The equal temperament [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Harmonics in equal|121}}

=== Subsets and supersets ===
Since 121 factors into {{factorization|121}}, 121edo contains [[11edo]] as its only nontrivial subset. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| {{mapping| 121 192 }}
| &amp;minus;0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| &amp;minus;0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| &amp;minus;0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| &amp;minus;0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| &amp;minus;0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| &amp;minus;0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| &amp;minus;0.689
| 0.519
| 5.23
|}
* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]]. 

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br /&gt;(5\121)
| 495.87&lt;br /&gt;(49.59)
| 4/3&lt;br /&gt;(36/35)
| [[Hendecatonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

== Miscellany ==
Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>phgt0b99uazoxqd19lh28u11fm9qqyw</sha1>
    </revision>
    <revision>
      <id>176493</id>
      <parentid>166095</parentid>
      <timestamp>2025-01-16T15:03:51Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176493</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5378" sha1="gazfga1aaze26lp5ipcb5c47v0idur4" xml:space="preserve">{{Infobox ET}}
{{EDO intro|121}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Harmonics in equal|121}}

=== Subsets and supersets ===
Since 121 factors into {{factorization|121}}, 121edo contains [[11edo]] as its only nontrivial subset. 

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| {{mapping| 121 192 }}
| −0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| −0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| −0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| −0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| −0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| −0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| −0.689
| 0.519
| 5.23
|}
* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]]. 

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br /&gt;(5\121)
| 495.87&lt;br /&gt;(49.59)
| 4/3&lt;br /&gt;(36/35)
| [[Hendecatonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

== Miscellany ==
Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>gazfga1aaze26lp5ipcb5c47v0idur4</sha1>
    </revision>
    <revision>
      <id>176752</id>
      <parentid>176493</parentid>
      <timestamp>2025-01-17T15:19:20Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ cleanup</comment>
      <origin>176752</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5384" sha1="96iymb3ijq7lzq0v9smechthxe65drw" xml:space="preserve">{{Infobox ET}}
{{EDO intro|121}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Harmonics in equal|121}}

=== Subsets and supersets ===
Since 121 factors into 11&lt;sup&gt;2&lt;/sup&gt;, 121edo contains [[11edo]] as its only nontrivial subset.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| {{mapping| 121 192 }}
| −0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| −0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| −0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| −0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| −0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| −0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| −0.689
| 0.519
| 5.23
|}
* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]]. 

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br /&gt;(5\121)
| 495.87&lt;br /&gt;(49.59)
| 4/3&lt;br /&gt;(36/35)
| [[Hendecatonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

== Miscellany ==
Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>96iymb3ijq7lzq0v9smechthxe65drw</sha1>
    </revision>
    <revision>
      <id>181330</id>
      <parentid>176752</parentid>
      <timestamp>2025-02-18T16:57:52Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181330</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5379" sha1="93l9lmyalcrbbyjgkqsitaak7oa0ult" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Prime harmonics ===
{{Harmonics in equal|121}}

=== Subsets and supersets ===
Since 121 factors into 11&lt;sup&gt;2&lt;/sup&gt;, 121edo contains [[11edo]] as its only nontrivial subset.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| {{mapping| 121 192 }}
| −0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| −0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| −0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| −0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| −0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| −0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| −0.689
| 0.519
| 5.23
|}
* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]]. 

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br /&gt;(5\121)
| 495.87&lt;br /&gt;(49.59)
| 4/3&lt;br /&gt;(36/35)
| [[Hendecatonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

== Miscellany ==
Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>93l9lmyalcrbbyjgkqsitaak7oa0ult</sha1>
    </revision>
    <revision>
      <id>203477</id>
      <parentid>181330</parentid>
      <timestamp>2025-06-24T15:38:42Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>121 isn't consistent in the 21-odd-limit, hence the table shows odd harmonics as per the template doc</comment>
      <origin>203477</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5377" sha1="0elpgnhzhwavkyd68d5qx2azuxfmbei" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

=== Odd harmonics ===
{{Harmonics in equal|121}}

=== Subsets and supersets ===
Since 121 factors into 11&lt;sup&gt;2&lt;/sup&gt;, 121edo contains [[11edo]] as its only nontrivial subset.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| {{mapping| 121 192 }}
| −0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| −0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| −0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| −0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| −0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| −0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| −0.689
| 0.519
| 5.23
|}
* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]]. 

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br /&gt;(5\121)
| 495.87&lt;br /&gt;(49.59)
| 4/3&lt;br /&gt;(36/35)
| [[Hendecatonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

== Miscellany ==
Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-Gothic]] tunings.

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>0elpgnhzhwavkyd68d5qx2azuxfmbei</sha1>
    </revision>
    <revision>
      <id>219965</id>
      <parentid>203477</parentid>
      <timestamp>2025-12-27T07:36:23Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <minor/>
      <comment>merge section with theory</comment>
      <origin>219965</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5360" sha1="inu4ha850zi36xe7xkfree6xy07joq0" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-gothic]] tunings.

=== Odd harmonics ===
{{Harmonics in equal|121}}

=== Subsets and supersets ===
Since 121 factors into 11&lt;sup&gt;2&lt;/sup&gt;, 121edo contains [[11edo]] as its only nontrivial subset.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| {{mapping| 121 192 }}
| −0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| −0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| −0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| −0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| −0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| −0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| −0.689
| 0.519
| 5.23
|}
* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]]. 

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br /&gt;(5\121)
| 495.87&lt;br /&gt;(49.59)
| 4/3&lt;br /&gt;(36/35)
| [[Hendecatonic]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>inu4ha850zi36xe7xkfree6xy07joq0</sha1>
    </revision>
    <revision>
      <id>225947</id>
      <parentid>219965</parentid>
      <timestamp>2026-03-13T13:33:16Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct" to "[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct"</comment>
      <origin>225947</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5415" sha1="8cu6egpkmuvls2lixcsy0ss7sjt6l4f" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-gothic]] tunings.

=== Odd harmonics ===
{{Harmonics in equal|121}}

=== Subsets and supersets ===
Since 121 factors into 11&lt;sup&gt;2&lt;/sup&gt;, 121edo contains [[11edo]] as its only nontrivial subset.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 192 -121 }}
| {{mapping| 121 192 }}
| −0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{mapping| 121 192 281 }}
| −0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{mapping| 121 192 281 340 }}
| −0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{mapping| 121 192 281 340 419 }}
| −0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{mapping| 121 192 281 340 419 448 }}
| −0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{mapping| 121 192 281 340 419 448 495 }}
| −0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{mapping| 121 192 281 340 419 448 495 514 }}
| −0.689
| 0.519
| 5.23
|}
* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]]. 

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperament
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Leapday]] / [[polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br /&gt;(5\121)
| 495.87&lt;br /&gt;(49.59)
| 4/3&lt;br /&gt;(36/35)
| [[Hendecatonic]]
|}
&lt;nowiki /&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Grendel]]
[[Category:Quintupole]]</text>
      <sha1>8cu6egpkmuvls2lixcsy0ss7sjt6l4f</sha1>
    </revision>
    <revision>
      <id>227294</id>
      <parentid>225947</parentid>
      <timestamp>2026-04-05T10:41:36Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Update links</comment>
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      <text bytes="5415" sha1="8nqb6sewr82bbp94wcmuairumn96oip" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
121edo has a distinctly sharp tendency, in that the odd [[harmonic]]s from 3 to 19 all have sharp tunings. It [[tempering out|tempers out]] 15625/15552 ([[15625/15552|kleisma]]) in the [[5-limit]]; [[4000/3969]], [[6144/6125]], [[10976/10935]] in the [[7-limit]]; [[540/539]], [[896/891]] and [[1375/1372]] in the 11-limit; [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the [[13-limit]]; [[256/255]], [[375/374]] and [[442/441]] in the [[17-limit]]; [[190/189]] and [[361/360]] in the [[19-limit]]. It also serves as the [[optimal patent val]] for 13-limit [[grendel]] temperament. It is [[consistent]] through to the [[19-odd-limit]] and uniquely consistent to the [[15-odd-limit]].

Because it tempers out 540/539 it allows [[swetismic chords]], because it tempers out 325/324 it allows [[marveltwin chords]], because it tempers out 640/637 it allows [[huntmic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic chords]], because it tempers out 676/675 it allows [[island chords]] and because it tempers out 1575/1573 it allows [[nicolic chords]]. That makes for a very flexible system, and since this suite of commas defines 13-limit 121et, it is a system only associated with 121.

Since 121 is part of the Fibonacci sequence beginning with 5 and 12, 121edo closely approximates [[peppermint]] temperament. This makes it suitable for [[neo-gothic]] tunings.

=== Odd harmonics ===
{{Harmonics in equal|121}}

=== Subsets and supersets ===
Since 121 factors into 11&lt;sup&gt;2&lt;/sup&gt;, 121edo contains [[11edo]] as its only nontrivial subset.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 192 -121 }}
| {{Mapping| 121 192 }}
| −0.687
| 0.687
| 6.93
|-
| 2.3.5
| 15625/15552, {{monzo| 31 -21 1 }}
| {{Mapping| 121 192 281 }}
| −0.524
| 0.606
| 6.11
|-
| 2.3.5.7
| 4000/3969, 6144/6125, 10976/10935
| {{Mapping| 121 192 281 340 }}
| −0.667
| 0.580
| 5.85
|-
| 2.3.5.7.11
| 540/539, 896/891, 1375/1372, 4375/4356
| {{Mapping| 121 192 281 340 419 }}
| −0.768
| 0.556
| 5.61
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 540/539, 625/624
| {{Mapping| 121 192 281 340 419 448 }}
| −0.750
| 0.510
| 5.14
|-
| 2.3.5.7.11.13.17
| 256/255, 325/324, 352/351, 364/363, 375/374, 442/441
| {{Mapping| 121 192 281 340 419 448 495 }}
| −0.787
| 0.480
| 4.85
|-
| 2.3.5.7.11.13.17.19
| 190/189, 256/255, 325/324, 352/351, 361/360, 364/363, 375/374
| {{Mapping| 121 192 281 340 419 448 495 514 }}
| −0.689
| 0.519
| 5.23
|}
* 121et (121i val) has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19-, and 23-limit, beating [[111edo|111]] before being superseded by [[130edo|130]] in all those limits except for the 17-limit, where it is superseded by [[140edo|140]]. 

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperament
|-
| 1
| 9\121
| 89.26
| 21/20
| [[Slithy]]
|-
| 1
| 10\121
| 99.17
| 18/17
| [[Quintupole]]
|-
| 1
| 12\121
| 119.01
| 15/14
| [[Subsedia]]
|-
| 1
| 13\121
| 128.93
| 14/13
| [[Tertiathirds]]
|-
| 1
| 16\121
| 158.68
| 35/32
| [[Hemikleismic]]
|-
| 1
| 27\121
| 267.77
| 7/6
| [[Hemimaquila]]
|-
| 1
| 32\121
| 317.36
| 6/5
| [[Metakleismic]]
|-
| 1
| 39\121
| 386.78
| 5/4
| [[Grendel]]
|-
| 1
| 40\121
| 396.69
| 44/35
| [[Squarschmidt]]
|-
| 1
| 42\121
| 416.53
| 14/11
| [[Sqrtphi]]
|-
| 1
| 46\121
| 456.20
| 125/96
| [[Qak]]
|-
| 1
| 47\121
| 466.12
| 55/42
| [[Hemiseptisix]]
|-
| 1
| 48\121
| 476.03
| 21/16
| [[Subfourth]]
|-
| 1
| 50\121
| 495.87
| 4/3
| [[Polypyth]]
|-
| 1
| 51\121
| 505.79
| 75/56
| [[Marfifths]] / marf / diatessic
|-
| 1
| 54\121
| 535.54
| 512/375
| [[Maquila]]
|-
| 1
| 59\121
| 585.12
| 7/5
| [[Pluto]]
|-
| 11
| 50\121&lt;br&gt;(5\121)
| 495.87&lt;br&gt;(49.59)
| 4/3&lt;br&gt;(36/35)
| [[Hendecatonic (temperament)|Hendecatonic]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== 13-limit detempering of 121et ==
{{See also| Detempering }}

[100/99, 64/63, 50/49, 40/39, 36/35, 28/27, 25/24, 22/21, 21/20, 35/33, 16/15, 15/14, 14/13, 13/12, 12/11, 35/32, 11/10, 10/9, 39/35, 28/25, 9/8, 25/22, 8/7, 55/48, 15/13, 64/55, 7/6, 75/64, 13/11, 25/21, 105/88, 6/5, 63/52, 40/33, 11/9, 16/13, 26/21, 56/45, 5/4, 44/35, 63/50, 14/11, 32/25, 9/7, 35/27, 13/10, 55/42, 21/16, 33/25, 4/3, 75/56, 35/26, 27/20, 15/11, 48/35, 11/8, 18/13, 39/28, 7/5, 45/32, 64/45, 10/7, 56/39, 13/9, 16/11, 35/24, 22/15, 40/27, 49/33, 112/75, 3/2, 50/33, 32/21, 55/36, 20/13, 54/35, 14/9, 25/16, 11/7, 63/40, 35/22, 8/5, 45/28, 21/13, 13/8, 18/11, 33/20, 104/63, 5/3, 117/70, 42/25, 22/13, 75/44, 12/7, 55/32, 26/15, 96/55, 7/4, 44/25, 16/9, 25/14, 70/39, 9/5, 20/11, 64/35, 11/6, 24/13, 13/7, 28/15, 15/8, 49/26, 40/21, 21/11, 25/13, 27/14, 35/18, 39/20, 49/25, 63/32, 99/50, 2]

[[Category:Grendel]]
[[Category:Quintupole]]</text>
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  <page>
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      <text bytes="708" sha1="5yrhht7y1f07kujr8wp523xhrzqsh3y" xml:space="preserve">The 12276 division divides the octave into equal steps of size 0.097752 cents. This creates a unit known as the '''prima,''' useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three '''tuning units'''. 

12276 is a strong 11-limit system, with a lower 11-limit relative error than any division aside from [[6691edo|6691]]. It factors as 12276 = 2^2 * 3^2 * 11 * 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. 12276 tempers out the atom, so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.</text>
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      <comment>**Imported revision 614809973 - Original comment: **</comment>
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      <text bytes="2948" sha1="lgyxk6ko0o9yw7uozaqu850xd7zwhq4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hearneg|hearneg]] and made on &lt;tt&gt;2017-06-17 05:15:55 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;614809973&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 12276 division divides the octave into equal steps of size 0.097752 cents. This creates a unit known as the **prima,** useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three **tuning units**. 

12276 is a strong 11-limit system, with a lower 11-limit relative error than any division aside from [[6691edo|6691]]. It factors as 12276 = 2^2 * 3^2 * 11 * 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. 12276 tempers out the atom, so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively. &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12276edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 12276 division divides the octave into equal steps of size 0.097752 cents. This creates a unit known as the &amp;lt;strong&amp;gt;prima,&amp;lt;/strong&amp;gt; useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three &amp;lt;strong&amp;gt;tuning units&amp;lt;/strong&amp;gt;. &amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12276 is a strong 11-limit system, with a lower 11-limit relative error than any division aside from &amp;lt;a class="wiki_link" href="/6691edo"&amp;gt;6691&amp;lt;/a&amp;gt;. It factors as 12276 = 2^2 * 3^2 * 11 * 31, and among its divisors are &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/99edo"&amp;gt;99&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/198edo"&amp;gt;198&amp;lt;/a&amp;gt;. 12276 tempers out the atom, so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <text bytes="2155" sha1="c3kngdrw2iwwc31vaw7kowqy5s6pcm8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-08-22 19:48:39 UTC&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 12276 division divides the octave into equal steps of size 0.097752 cents. It is a strong 11-limit system, with a lower 11-limit relative error than any division aside from [[6691edo|6691]]. It factors as 12276 = 2^2 * 3^2 * 11 * 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]].&lt;/pre&gt;&lt;/div&gt;
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      <text bytes="2112" sha1="iyunruyzyycbluzj3etbehf8qf8x0ly" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-08-22 19:48:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;557183271&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 12276 division divides the octave into equal steps of size 0.097752 cents. It is a strong 11-limit system, with a lower 11-limit relative error than any division aside from [[6691edo|6691]]. It factors as 12276 = 2^2 * 3^2 * 11 * 31, and among its divisors are [[12edo|12]], [[22edo|22], [[31edo|31]], [[99edo|99]] and [[198edo|198]].&lt;/pre&gt;&lt;/div&gt;
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      <text bytes="775" sha1="jwh2sck4voflavr1882k5pbyuqa4l9b" xml:space="preserve">'''12276EDO''' is the [[EDO|equal division of the octave]] into 12276 parts of exact 0.09775171 cents each. This creates a unit known as the '''prima,''' useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three '''tuning units'''. 

12276 is a strong 11-limit system, with a lower 11-limit relative error than any division aside from [[6691edo|6691]]. It factors as 12276 = 2^2 * 3^2 * 11 * 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. 12276 tempers out the atom, so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.

[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>jwh2sck4voflavr1882k5pbyuqa4l9b</sha1>
    </revision>
    <revision>
      <id>41858</id>
      <parentid>41257</parentid>
      <timestamp>2019-06-01T10:45:39Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <comment>minor edit</comment>
      <origin>41858</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="791" sha1="6te5uutesq8da6tf6edi7vprvjrc6so" xml:space="preserve">'''12276EDO''' is the [[EDO|equal division of the octave]] into 12276 parts of exact 0.09775171 cents each. This creates a unit known as the '''prima''', useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three '''[[Tuning unit|tuning units]]'''. 

12276 is a strong 11-limit system, with a lower 11-limit relative error than any division aside from [[6691edo|6691]]. It factors as 12276 = 2^2 * 3^2 * 11 * 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. 12276 tempers out the atom, so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.

[[Category:Edo]]
[[Category:Theory]]</text>
      <sha1>6te5uutesq8da6tf6edi7vprvjrc6so</sha1>
    </revision>
    <revision>
      <id>54970</id>
      <parentid>41858</parentid>
      <timestamp>2020-12-04T23:13:07Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54970</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="817" sha1="ouy0fdwd0fc05l06tkzj51vcac104ek" xml:space="preserve">'''12276EDO''' is the [[EDO|equal division of the octave]] into 12276 parts of exact 0.09775171 cents each. This creates a unit known as the '''prima''', useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three '''[[Tuning unit|tuning units]]'''. 

12276 is a strong 11-limit system, with a lower 11-limit relative error than any division aside from [[6691edo|6691]]. It factors as 12276 = 2^2 * 3^2 * 11 * 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. 12276 tempers out the atom, so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.

[[Category:Equal divisions of the octave]]
[[Category:Theory]]</text>
      <sha1>ouy0fdwd0fc05l06tkzj51vcac104ek</sha1>
    </revision>
    <revision>
      <id>70299</id>
      <parentid>54970</parentid>
      <timestamp>2021-05-21T19:20:27Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>70299</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="823" sha1="m048ozc5y61xy0o6mzi85hzn35h8j85" xml:space="preserve">'''12276EDO''' is the [[EDO|equal division of the octave]] into 12276 parts of exact 0.09775171 cents each. This creates a unit known as the '''prima''', useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three '''[[Tuning unit|tuning units]]'''. 

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. It factors as 12276 = 2^2 * 3^2 * 11 * 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. 12276 tempers out the atom, so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.

[[Category:Equal divisions of the octave]]
[[Category:Theory]]</text>
      <sha1>m048ozc5y61xy0o6mzi85hzn35h8j85</sha1>
    </revision>
    <revision>
      <id>70300</id>
      <parentid>70299</parentid>
      <timestamp>2021-05-21T19:21:08Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>70300</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="825" sha1="chk7pzk1wshkhy45lq1ehsecuvetc6p" xml:space="preserve">'''12276EDO''' is the [[EDO|equal division of the octave]] into 12276 parts of exactly 0.09775171 cents each. This creates a unit known as the '''prima''', useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three '''[[Tuning unit|tuning units]]'''. 

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. It factors as 12276 = 2^2 * 3^2 * 11 * 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. 12276 tempers out the atom, so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.

[[Category:Equal divisions of the octave]]
[[Category:Theory]]</text>
      <sha1>chk7pzk1wshkhy45lq1ehsecuvetc6p</sha1>
    </revision>
    <revision>
      <id>70320</id>
      <parentid>70300</parentid>
      <timestamp>2021-05-22T03:51:06Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Cleanup</comment>
      <origin>70320</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="848" sha1="4doccstqy95ch048n5p3rx1s3zktms8" xml:space="preserve">'''12276EDO''' is the [[EDO|equal division of the octave]] into 12276 parts of exactly 0.09775171 cents each. This creates a unit known as the '''prima''', useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three '''[[Tuning unit|tuning units]]'''. 

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. It factors as 12276 = 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. 12276 tempers out the atom, so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.

[[Category:Equal divisions of the octave]]
[[Category:Theory]]</text>
      <sha1>4doccstqy95ch048n5p3rx1s3zktms8</sha1>
    </revision>
    <revision>
      <id>70407</id>
      <parentid>70320</parentid>
      <timestamp>2021-05-23T20:53:01Z</timestamp>
      <contributor>
        <username>Keenan Pepper</username>
        <id>25</id>
      </contributor>
      <minor/>
      <comment>link atom</comment>
      <origin>70407</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="870" sha1="0w6o7i4t68id6ed593284ypl2bmrxtt" xml:space="preserve">'''12276EDO''' is the [[EDO|equal division of the octave]] into 12276 parts of exactly 0.09775171 cents each. This creates a unit known as the '''prima''', useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three '''[[Tuning unit|tuning units]]'''. 

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. It factors as 12276 = 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. 12276 tempers out the [[Kirnberger's atom|atom]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.

[[Category:Equal divisions of the octave]]
[[Category:Theory]]</text>
      <sha1>0w6o7i4t68id6ed593284ypl2bmrxtt</sha1>
    </revision>
    <revision>
      <id>87441</id>
      <parentid>70407</parentid>
      <timestamp>2022-02-26T16:48:49Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87441</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="850" sha1="7xvog67hhrmnhp665blkz816061xcve" xml:space="preserve">'''12276EDO''' is the [[EDO|equal division of the octave]] into 12276 parts of exactly 0.09775171 cents each. This creates a unit known as the '''prima''', useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three '''[[Tuning unit|tuning units]]'''. 

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. It factors as 12276 = 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. 12276 tempers out the [[Kirnberger's atom|atom]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.

[[Category:Equal divisions of the octave]]</text>
      <sha1>7xvog67hhrmnhp665blkz816061xcve</sha1>
    </revision>
    <revision>
      <id>93253</id>
      <parentid>87441</parentid>
      <timestamp>2022-07-04T01:41:26Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>93253</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="880" sha1="c5tf8c396hywvq6nbr4bpr3ar3gqev3" xml:space="preserve">'''12276EDO''' is the [[EDO|equal division of the octave]] into 12276 parts of exactly 0.09775171 cents each. This creates a unit known as the '''prima''', useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three '''[[Tuning unit|tuning units]]'''. 

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. It factors as 12276 = 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. 12276 tempers out the [[Kirnberger's atom|atom]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.

[[Category:Equal divisions of the octave|#####]] &lt;!-- 5-digit number --&gt;</text>
      <sha1>c5tf8c396hywvq6nbr4bpr3ar3gqev3</sha1>
    </revision>
    <revision>
      <id>96529</id>
      <parentid>93253</parentid>
      <timestamp>2022-10-04T22:21:50Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
      <origin>96529</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="895" sha1="apojgtae02xsns2tlrhhhn9yks3tn3r" xml:space="preserve">{{Infobox ET}}
'''12276EDO''' is the [[EDO|equal division of the octave]] into 12276 parts of exactly 0.09775171 cents each. This creates a unit known as the '''prima''', useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three '''[[Tuning unit|tuning units]]'''. 

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. It factors as 12276 = 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. 12276 tempers out the [[Kirnberger's atom|atom]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.

[[Category:Equal divisions of the octave|#####]] &lt;!-- 5-digit number --&gt;</text>
      <sha1>apojgtae02xsns2tlrhhhn9yks3tn3r</sha1>
    </revision>
    <revision>
      <id>100244</id>
      <parentid>96529</parentid>
      <timestamp>2022-12-21T15:36:56Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>cleanup</comment>
      <origin>100244</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="970" sha1="lr2kkqy6sxxj0xrge9badv2uuck40z3" xml:space="preserve">{{Infobox ET}}
{{EDO intro|12276}}
==Theory==
12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.
===Interval size measure===
12276edo factors as 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the '''prima''', useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three [[Tuning unit|tuning units]]. 

In addition, 12276edo tempers out the [[septimal ruthenia]], meaning that [[64/63]] is exactly 1/44th of the octave, or 279 primas.



[[Category:Equal divisions of the octave|#####]] &lt;!-- 5-digit number --&gt;</text>
      <sha1>lr2kkqy6sxxj0xrge9badv2uuck40z3</sha1>
    </revision>
    <revision>
      <id>100442</id>
      <parentid>100244</parentid>
      <timestamp>2022-12-26T06:40:26Z</timestamp>
      <contributor>
        <username>Lhearne</username>
        <id>270</id>
      </contributor>
      <minor/>
      <origin>100442</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="974" sha1="k04z82v428cppy3a4gikcmuunhgizip" xml:space="preserve">{{Infobox ET}}
{{EDO intro|12276}}
==Theory==
12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively.
===Interval size measure===
12276edo factors as 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the '''[[prima]]''', useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three [[Tuning unit|tuning units]]. 

In addition, 12276edo tempers out the [[septimal ruthenia]], meaning that [[64/63]] is exactly 1/44th of the octave, or 279 primas.



[[Category:Equal divisions of the octave|#####]] &lt;!-- 5-digit number --&gt;</text>
      <sha1>k04z82v428cppy3a4gikcmuunhgizip</sha1>
    </revision>
    <revision>
      <id>100446</id>
      <parentid>100442</parentid>
      <timestamp>2022-12-26T11:06:04Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>-section title for now since there's no other sections; +prime error table; +categories</comment>
      <origin>100446</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1059" sha1="q8mkqkyahhx4sgcyok3zuejzab515mt" xml:space="preserve">{{Infobox ET}}
{{EDO intro|12276}}

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas respectively. In addition, 12276edo tempers out the [[septimal ruthenia]], meaning that [[64/63]] is exactly 1/44 of the octave, or 279 primas.

=== Prime harmonics ===
{{Harmonics in equal|12276|columns=11}}

=== Interval size measure ===
12276edo factors as 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. The Pythagorean comma is represented by 240 prima, and the syntonic comma by 220. A prima is almost exactly three [[tuning unit]]s. 

[[Category:Equal divisions of the octave|#####]] &lt;!-- 5-digit number --&gt;
[[Category:12276edo| ]] &lt;!-- main article --&gt;</text>
      <sha1>q8mkqkyahhx4sgcyok3zuejzab515mt</sha1>
    </revision>
    <revision>
      <id>116237</id>
      <parentid>100446</parentid>
      <timestamp>2023-06-10T12:56:51Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>not sure why reduplicate how much are commas in steps in different sections, write that more concisely</comment>
      <origin>116237</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="964" sha1="4e0gu8k2ctqbhsezsbyqlobh5mqmeec" xml:space="preserve">{{Infobox ET}}
{{EDO intro|12276}}

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]] and the [[septimal ruthenia]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps.

=== Prime harmonics ===
{{Harmonics in equal|12276|columns=11}}

=== Interval size measure ===
12276edo factors as 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. A prima is almost exactly three [[tuning unit]]s. 

[[Category:Equal divisions of the octave|#####]] &lt;!-- 5-digit number --&gt;
[[Category:12276edo| ]] &lt;!-- main article --&gt;</text>
      <sha1>4e0gu8k2ctqbhsezsbyqlobh5mqmeec</sha1>
    </revision>
    <revision>
      <id>119267</id>
      <parentid>116237</parentid>
      <timestamp>2023-07-09T04:16:13Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119267</origin>
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      <text bytes="983" sha1="jjrj1vwdivlsdng4smgnt0tc2iu1ph0" xml:space="preserve">{{novelty}}{{stub}}{{Infobox ET}}
{{EDO intro|12276}}

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]] and the [[septimal ruthenia]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps.

=== Prime harmonics ===
{{Harmonics in equal|12276|columns=11}}

=== Interval size measure ===
12276edo factors as 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. A prima is almost exactly three [[tuning unit]]s. 

[[Category:Equal divisions of the octave|#####]] &lt;!-- 5-digit number --&gt;
[[Category:12276edo| ]] &lt;!-- main article --&gt;</text>
      <sha1>jjrj1vwdivlsdng4smgnt0tc2iu1ph0</sha1>
    </revision>
    <revision>
      <id>119719</id>
      <parentid>119267</parentid>
      <timestamp>2023-07-09T07:11:53Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119719</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="964" sha1="4e0gu8k2ctqbhsezsbyqlobh5mqmeec" xml:space="preserve">{{Infobox ET}}
{{EDO intro|12276}}

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]] and the [[septimal ruthenia]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps.

=== Prime harmonics ===
{{Harmonics in equal|12276|columns=11}}

=== Interval size measure ===
12276edo factors as 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. A prima is almost exactly three [[tuning unit]]s. 

[[Category:Equal divisions of the octave|#####]] &lt;!-- 5-digit number --&gt;
[[Category:12276edo| ]] &lt;!-- main article --&gt;</text>
      <sha1>4e0gu8k2ctqbhsezsbyqlobh5mqmeec</sha1>
    </revision>
    <revision>
      <id>181331</id>
      <parentid>119719</parentid>
      <timestamp>2025-02-18T16:58:19Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181331</origin>
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      <text bytes="957" sha1="g4wo88krlx543zyo7zy32wwrxfum2gz" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]] and the [[septimal ruthenia]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps.

=== Prime harmonics ===
{{Harmonics in equal|12276|columns=11}}

=== Interval size measure ===
12276edo factors as 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. A prima is almost exactly three [[tuning unit]]s. 

[[Category:Equal divisions of the octave|#####]] &lt;!-- 5-digit number --&gt;
[[Category:12276edo| ]] &lt;!-- main article --&gt;</text>
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    <revision>
      <id>205272</id>
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      <timestamp>2025-07-28T16:34:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
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      <minor/>
      <comment>Cleanup &amp; + category</comment>
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      <text bytes="926" sha1="3wafvccfe9tq9d8mu17gczwvwrlutce" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]] and the [[septimal ruthenia]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps.

=== Prime harmonics ===
{{Harmonics in equal|12276|columns=11}}

=== Subsets and supersets ===
12276edo factors into primes as {{nowrap| 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31 }}, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. A prima is almost exactly three [[tuning unit]]s. 

[[Category:3-limit record edos|#####]] &lt;!-- 5-digit number --&gt;</text>
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    <revision>
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      <timestamp>2026-02-16T08:57:23Z</timestamp>
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        <id>347</id>
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      <text bytes="1022" sha1="2jyfqf30381gt14wbz9eitp3y53tgvt" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

12276 is a strong 11-limit system, with a lower 11-limit relative error than any lower division aside from [[6691edo|6691]]. 12276 tempers out the [[Kirnberger's atom|atom]] and the [[septimal ruthenia]], so that the Pythagorean and syntonic commas an be approximated by 12 and 11 schismas, 240 and 220 steps respectively, and septimal comma is represented by 1/44 of the octave, 279 steps. It is the smallest [[atomic]] EDO inside its [[5-odd-limit]] [[diamond tradeoff]] tuning range.

=== Prime harmonics ===
{{Harmonics in equal|12276|columns=11}}

=== Subsets and supersets ===
12276edo factors into primes as {{nowrap| 2&lt;sup&gt;2&lt;/sup&gt; × 3&lt;sup&gt;2&lt;/sup&gt; × 11 × 31 }}, and among its divisors are [[12edo|12]], [[22edo|22]], [[31edo|31]], [[99edo|99]] and [[198edo|198]]. This creates a unit known as the ''[[prima]]'', useful for measurement of 11-limit intervals and commas. A prima is almost exactly three [[tuning unit]]s. 

[[Category:3-limit record edos|#####]] &lt;!-- 5-digit number --&gt;</text>
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  <page>
    <title>122edo</title>
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    <revision>
      <id>109</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
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        <id>0</id>
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      <text bytes="708" sha1="sygc357o9jlbkew2sbz5kike2ymh8ys" xml:space="preserve">'''122edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 122 parts of 9.836 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for 7-limit [[Marvel_temperaments|tritonic temperament]] and 11-limit [[Marvel_temperaments|tritoni temperament]], and the planar [[Squalentine_temperament|squalentine temperament]]. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit|5-limit]], 225/224 in the [[7-limit|7-limit]], 385/384 and 4000/3993 in the [[11-limit|11-limit]], and 351/350 and 364/363 in the [[13-limit|13-limit]].

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * [[61edo|61]].
[[Category:edo]]
[[Category:theory]]</text>
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    <revision>
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      <timestamp>2011-08-21T17:38:09Z</timestamp>
      <contributor>
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      <comment>**Imported revision 247458473 - Original comment: **</comment>
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      <text bytes="3017" sha1="ccjbx9ncu1nda7pq9iqxhc2ricdcv95" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-21 17:38:09 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;247458473&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**122edo** is the [[equal division of the octave]] into 122 parts of 9.836 [[cent]]s each. It is the [[optimal patent val]] for 7-limit [[Marvel temperaments|tritonic temperament]] and 11-limit [[Marvel temperaments|tritoni temperament]], and the planar [[squalentine temperament]]. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]], 225/224 in the [[7-limit]], 385/384 and 4000/3993 in the [[11-limit]], and 351/350 and 364/363 in the [[13-limit]].

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * [[61edo|61]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;br /&amp;gt;
122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 240697103 - Original comment: **</comment>
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      <text bytes="2864" sha1="8n2r5b730dp1j6hueqyfajsrrdjhxg8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-07-10 10:00:05 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;240697103&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**122edo** is the [[equal division of the octave]] into 122 parts of 9.836 [[cent]]s each. It is the [[optimal patent val]] for 7-limit [[Marvel temperaments|tritonic temperament]] and 11-limit [[Marvel temperaments|tritoni temperament]], and [[tempering out|tempers out]] 78732/78125 in the [[5-limit]], 225/224 in the [[7-limit]], 385/384 and 4000/3993 in the [[11-limit]], and 351/350 and 364/363 in the [[13-limit]].

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * [[61edo|61]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;122edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;122edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 122 parts of 9.836 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 7-limit &amp;lt;a class="wiki_link" href="/Marvel%20temperaments"&amp;gt;tritonic temperament&amp;lt;/a&amp;gt; and 11-limit &amp;lt;a class="wiki_link" href="/Marvel%20temperaments"&amp;gt;tritoni temperament&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, 225/224 in the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, 385/384 and 4000/3993 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, and 351/350 and 364/363 in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 239306591 - Original comment: **</comment>
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      <text bytes="2711" sha1="o4b244f88cuzdr2edy8ref3ltefks6l" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 08:57:13 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239306591&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**122edo** is the [[equal division of the octave]] into 122 parts of 9.836 [[cent]]s each. It is the [[optimal patent val]] for 7- and 11-limit [[Marvel temperaments|tritonic temperament]], and [[tempering out|tempers out]] 78732/78125 in the [[5-limit]], 225/224 in the [[7-limit]], 385/384 and 4000/3993 in the [[11-limit]], and 351/350 and 364/363 in the [[13-limit]].

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * [[61edo|61]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;122edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;122edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 122 parts of 9.836 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 7- and 11-limit &amp;lt;a class="wiki_link" href="/Marvel%20temperaments"&amp;gt;tritonic temperament&amp;lt;/a&amp;gt;, and &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 78732/78125 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, 225/224 in the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, 385/384 and 4000/3993 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, and 351/350 and 364/363 in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <comment>**Imported revision 239306385 - Original comment: **</comment>
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      <text bytes="2192" sha1="do6cin23zt242fkrqmegmzm0je43yqg" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 08:55:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239306385&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //122 equal division// divides the octave into 122 equal parts of 9.836 cents each. It is the optimal patent val for 7- and 11-limit [[Marvel temperaments|tritonic temperament]], and tempers out 78732/78125 in the 5-limit, 225/224 in the 7-limit, 385/384 and 4000/3993 in the 11-limit, and 351/350 and 364/363 in the 13-limit.

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * [[61edo|61]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;122edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;122 equal division&amp;lt;/em&amp;gt; divides the octave into 122 equal parts of 9.836 cents each. It is the optimal patent val for 7- and 11-limit &amp;lt;a class="wiki_link" href="/Marvel%20temperaments"&amp;gt;tritonic temperament&amp;lt;/a&amp;gt;, and tempers out 78732/78125 in the 5-limit, 225/224 in the 7-limit, 385/384 and 4000/3993 in the 11-limit, and 351/350 and 364/363 in the 13-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * &amp;lt;a class="wiki_link" href="/61edo"&amp;gt;61&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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        <id>0</id>
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      <comment>**Imported revision 214069270 - Original comment: **</comment>
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      <text bytes="2143" sha1="bogu9vlooa0ip97i7long23sq4ixhxh" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-25 15:49:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214069270&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //122 equal division// divides the octave into 122 equal parts of 9.836 cents each. It is the optimal patent val for 7- and 11-limit [[Marvel temperaments|tritonic temperament]], and tempers out 78732/78125 in the 5-limit, 225/224 in the 7-limit, 385/384 and 4000/3993 in the 11-limit, and 351/350 and 364/363 in the 13-limit.

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * 61.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;122edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;122 equal division&amp;lt;/em&amp;gt; divides the octave into 122 equal parts of 9.836 cents each. It is the optimal patent val for 7- and 11-limit &amp;lt;a class="wiki_link" href="/Marvel%20temperaments"&amp;gt;tritonic temperament&amp;lt;/a&amp;gt;, and tempers out 78732/78125 in the 5-limit, 225/224 in the 7-limit, 385/384 and 4000/3993 in the 11-limit, and 351/350 and 364/363 in the 13-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * 61.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>bogu9vlooa0ip97i7long23sq4ixhxh</sha1>
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    <revision>
      <id>4540</id>
      <timestamp>2011-03-25T15:26:59Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 214063972 - Original comment: **</comment>
      <origin>4540</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1949" sha1="apkziuebh3p1372dqvm13gi1ora3gfs" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-25 15:26:59 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214063972&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //122 equal division// divides the octave into 122 equal parts of 9.836 cents each. It is the optimal patent val for 7- and 11-limit [[Marvel temperaments|tritonic temperament]], and tempers out 78732/78125 in the 5-limit, 225/224 in the 7-limit, 385/384 and 4000/3993 in the 11-limit, and 351/350 and 364/363 in the 13-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;122edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;122 equal division&amp;lt;/em&amp;gt; divides the octave into 122 equal parts of 9.836 cents each. It is the optimal patent val for 7- and 11-limit &amp;lt;a class="wiki_link" href="/Marvel%20temperaments"&amp;gt;tritonic temperament&amp;lt;/a&amp;gt;, and tempers out 78732/78125 in the 5-limit, 225/224 in the 7-limit, 385/384 and 4000/3993 in the 11-limit, and 351/350 and 364/363 in the 13-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>apkziuebh3p1372dqvm13gi1ora3gfs</sha1>
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    <revision>
      <id>54972</id>
      <parentid>109</parentid>
      <timestamp>2020-12-04T23:13:07Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54972</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="734" sha1="psi0qfxepoa7nkrsy713w8l2zeyzl55" xml:space="preserve">'''122edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 122 parts of 9.836 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for 7-limit [[Marvel_temperaments|tritonic temperament]] and 11-limit [[Marvel_temperaments|tritoni temperament]], and the planar [[Squalentine_temperament|squalentine temperament]]. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit|5-limit]], 225/224 in the [[7-limit|7-limit]], 385/384 and 4000/3993 in the [[11-limit|11-limit]], and 351/350 and 364/363 in the [[13-limit|13-limit]].

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * [[61edo|61]].
[[Category:Equal divisions of the octave]]
[[Category:theory]]</text>
      <sha1>psi0qfxepoa7nkrsy713w8l2zeyzl55</sha1>
    </revision>
    <revision>
      <id>72940</id>
      <parentid>54972</parentid>
      <timestamp>2021-06-22T22:21:28Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <comment>Prime column, Relation to other tunings.</comment>
      <origin>72940</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="980" sha1="gbjve3tl2bf0ftor5ou37brz9jyco15" xml:space="preserve">'''122edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 122 parts of 9.836 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for 7-limit [[Marvel_temperaments|tritonic temperament]] and 11-limit [[Marvel_temperaments|tritoni temperament]], and the planar [[Squalentine_temperament|squalentine temperament]]. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit|5-limit]], 225/224 in the [[7-limit|7-limit]], 385/384 and 4000/3993 in the [[11-limit|11-limit]], and 351/350 and 364/363 in the [[13-limit|13-limit]].

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * [[61edo|61]]. 122=[[55edo|55]]+[[67edo|67]], and so using the c val it is the convergent towards [[1/6_syntonic_comma_meantone|1/6 comma meantone]], with a fifth just a hundredth of a cent flatter.
{{primes in edo|122|columns=10|prec=3}}
[[Category:Equal divisions of the octave]]
[[Category:theory]]
[[Category:Meantone]]</text>
      <sha1>gbjve3tl2bf0ftor5ou37brz9jyco15</sha1>
    </revision>
    <revision>
      <id>87439</id>
      <parentid>72940</parentid>
      <timestamp>2022-02-26T16:48:49Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87439</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="960" sha1="cklv82kc7cv9ru8rgnzhni6i3lcxg53" xml:space="preserve">'''122edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 122 parts of 9.836 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for 7-limit [[Marvel_temperaments|tritonic temperament]] and 11-limit [[Marvel_temperaments|tritoni temperament]], and the planar [[Squalentine_temperament|squalentine temperament]]. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit|5-limit]], 225/224 in the [[7-limit|7-limit]], 385/384 and 4000/3993 in the [[11-limit|11-limit]], and 351/350 and 364/363 in the [[13-limit|13-limit]].

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * [[61edo|61]]. 122=[[55edo|55]]+[[67edo|67]], and so using the c val it is the convergent towards [[1/6_syntonic_comma_meantone|1/6 comma meantone]], with a fifth just a hundredth of a cent flatter.
{{primes in edo|122|columns=10|prec=3}}
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]</text>
      <sha1>cklv82kc7cv9ru8rgnzhni6i3lcxg53</sha1>
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    <revision>
      <id>92932</id>
      <parentid>87439</parentid>
      <timestamp>2022-07-02T16:26:00Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92932</origin>
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      <format>text/x-wiki</format>
      <text bytes="989" sha1="jev2a9gqeogn7u2kdasrymzwplda6fu" xml:space="preserve">'''122edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 122 parts of 9.836 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for 7-limit [[Marvel_temperaments|tritonic temperament]] and 11-limit [[Marvel_temperaments|tritoni temperament]], and the planar [[Squalentine_temperament|squalentine temperament]]. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit|5-limit]], 225/224 in the [[7-limit|7-limit]], 385/384 and 4000/3993 in the [[11-limit|11-limit]], and 351/350 and 364/363 in the [[13-limit|13-limit]].

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * [[61edo|61]]. 122=[[55edo|55]]+[[67edo|67]], and so using the c val it is the convergent towards [[1/6_syntonic_comma_meantone|1/6 comma meantone]], with a fifth just a hundredth of a cent flatter.
{{primes in edo|122|columns=10|prec=3}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Meantone]]</text>
      <sha1>jev2a9gqeogn7u2kdasrymzwplda6fu</sha1>
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    <revision>
      <id>96200</id>
      <parentid>92932</parentid>
      <timestamp>2022-10-04T18:53:28Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
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      <origin>96200</origin>
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      <text bytes="1004" sha1="mi728u86y17sfijqak6vj2i3v15i5ud" xml:space="preserve">{{Infobox ET}}
'''122edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 122 parts of 9.836 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for 7-limit [[Marvel_temperaments|tritonic temperament]] and 11-limit [[Marvel_temperaments|tritoni temperament]], and the planar [[Squalentine_temperament|squalentine temperament]]. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit|5-limit]], 225/224 in the [[7-limit|7-limit]], 385/384 and 4000/3993 in the [[11-limit|11-limit]], and 351/350 and 364/363 in the [[13-limit|13-limit]].

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * [[61edo|61]]. 122=[[55edo|55]]+[[67edo|67]], and so using the c val it is the convergent towards [[1/6_syntonic_comma_meantone|1/6 comma meantone]], with a fifth just a hundredth of a cent flatter.
{{primes in edo|122|columns=10|prec=3}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Meantone]]</text>
      <sha1>mi728u86y17sfijqak6vj2i3v15i5ud</sha1>
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    <revision>
      <id>124922</id>
      <parentid>96200</parentid>
      <timestamp>2023-10-01T09:24:29Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <minor/>
      <comment>Bypass redirect.</comment>
      <origin>124922</origin>
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      <text bytes="976" sha1="kpdcpxmxghr2bktef2lo9gqk9fuhgdn" xml:space="preserve">{{Infobox ET}}
'''122edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 122 parts of 9.836 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for 7-limit [[Marvel_temperaments|tritonic temperament]] and 11-limit [[Marvel_temperaments|tritoni temperament]], and the planar [[Squalentine_temperament|squalentine temperament]]. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit|5-limit]], 225/224 in the [[7-limit|7-limit]], 385/384 and 4000/3993 in the [[11-limit|11-limit]], and 351/350 and 364/363 in the [[13-limit|13-limit]].

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * [[61edo|61]]. 122=[[55edo|55]]+[[67edo|67]], and so using the c val it is the convergent towards [[1/6-comma meantone]], with a fifth just a hundredth of a cent flatter.
{{primes in edo|122|columns=10|prec=3}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Meantone]]</text>
      <sha1>kpdcpxmxghr2bktef2lo9gqk9fuhgdn</sha1>
    </revision>
    <revision>
      <id>132863</id>
      <parentid>124922</parentid>
      <timestamp>2024-01-25T07:46:05Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Replace deprecated template</comment>
      <origin>132863</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="963" sha1="toanchdjdowpii82unbfdc77njdwkrb" xml:space="preserve">{{Infobox ET}}
'''122edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 122 parts of 9.836 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for 7-limit [[Marvel_temperaments|tritonic temperament]] and 11-limit [[Marvel_temperaments|tritoni temperament]], and the planar [[Squalentine_temperament|squalentine temperament]]. It [[tempering_out|tempers out]] 78732/78125 in the [[5-limit|5-limit]], 225/224 in the [[7-limit|7-limit]], 385/384 and 4000/3993 in the [[11-limit|11-limit]], and 351/350 and 364/363 in the [[13-limit|13-limit]].

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 * [[61edo|61]]. 122=[[55edo|55]]+[[67edo|67]], and so using the c val it is the convergent towards [[1/6-comma meantone]], with a fifth just a hundredth of a cent flatter.
{{harmonics in equal|122}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Meantone]]</text>
      <sha1>toanchdjdowpii82unbfdc77njdwkrb</sha1>
    </revision>
    <revision>
      <id>145222</id>
      <parentid>132863</parentid>
      <timestamp>2024-05-29T08:29:15Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Adopt template: EDO intro; -redundant categories; misc. cleanup</comment>
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      <text bytes="747" sha1="ohpxsvn1ujzqh7wgauzme4ucupkh83m" xml:space="preserve">{{Infobox ET}}
{{EDO intro}}

It is the [[optimal patent val]] for the 7-limit [[tritonic]] temperament and the 11-limit [[Marvel temperaments #Tritoni|tritoni]] temperament, and the planar [[squalentine]] temperament. It [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224 in the [[7-limit]]; [[385/384]] and [[4000/3993]] in the [[11-limit]]; and [[351/350]] and [[364/363]] in the [[13-limit]].

122 is flat in tendency, with the odd primes from 3 to 13 tuned flat. 122 = 2 × [[61edo|61]]. 122 = [[55edo|55]] + [[67edo|67]], and so using the 122c val it is the convergent towards [[1/6-comma meantone]], with a fifth just a hundredth of a cent flatter.

=== Odd harmonics ===
{{Harmonics in equal|122}}

[[Category:Meantone]]</text>
      <sha1>ohpxsvn1ujzqh7wgauzme4ucupkh83m</sha1>
    </revision>
    <revision>
      <id>145224</id>
      <parentid>145222</parentid>
      <timestamp>2024-05-29T08:34:27Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Rework</comment>
      <origin>145224</origin>
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      <text bytes="935" sha1="9hm0aehyvrbgh0nor24d1li6lwehp3z" xml:space="preserve">{{Infobox ET}}
{{EDO intro}}

122 is flat in tendency, with the [[prime harmonic]]s from 3 to 13 tuned flat. The equal temperament [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224 in the [[7-limit]]; [[385/384]] and [[4000/3993]] in the [[11-limit]]; and [[351/350]] and [[364/363]] in the [[13-limit]]. It provides the [[optimal patent val]] for the 7-limit [[tritonic]] temperament and the 11-limit [[Marvel temperaments #Tritoni|tritoni]] temperament, and the [[rank-3 temperament|planar]] [[squalentine]] temperament. 

122 = [[55edo|55]] + [[67edo|67]], and so using the 122c val it is the convergent towards [[1/6-comma meantone]], with a fifth just a hundredth of a cent flatter.

=== Odd harmonics ===
{{Harmonics in equal|122}}

=== Subsets and supersets ===
Since 122 factors into {{factorization|122}}, 122edo contains [[2edo]] and [[61edo]] as its subsets. 

[[Category:Tritonic]]
[[Category:Meantone]]</text>
      <sha1>9hm0aehyvrbgh0nor24d1li6lwehp3z</sha1>
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    <revision>
      <id>159517</id>
      <parentid>145224</parentid>
      <timestamp>2024-10-11T20:03:40Z</timestamp>
      <contributor>
        <username>MisterShafXen</username>
        <id>509</id>
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      <minor/>
      <origin>159517</origin>
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      <text bytes="1016" sha1="tno7rnvzyektayvqfye8ufu4dsd4yu3" xml:space="preserve">{{Infobox ET}}
{{EDO intro}}

122 is flat in tendency, with the [[prime harmonic]]s from 3 to 13 tuned flat. The equal temperament [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224 in the [[7-limit]]; [[385/384]] and [[4000/3993]] in the [[11-limit]]; and [[351/350]] and [[364/363]] in the [[13-limit]]. It provides the [[optimal patent val]] for the 7-limit [[tritonic]] temperament and the 11-limit [[Marvel temperaments #Tritoni|tritoni]] temperament, and the [[rank-3 temperament|planar]] [[squalentine]] temperament. 

122 = [[55edo|55]] + [[67edo|67]], and so using the 122c val it is the convergent towards [[1/6-comma meantone]], with a fifth just a hundredth of a cent flatter.

=== Odd harmonics ===
{{Harmonics in equal|122|23=}}

Harmonic 25 (two 5/4 major thirds) is about halfway between 122edo's steps.

=== Subsets and supersets ===
Since 122 factors into {{factorization|122}}, 122edo contains [[2edo]] and [[61edo]] as its subsets. 

[[Category:Tritonic]]
[[Category:Meantone]]</text>
      <sha1>tno7rnvzyektayvqfye8ufu4dsd4yu3</sha1>
    </revision>
    <revision>
      <id>159518</id>
      <parentid>159517</parentid>
      <timestamp>2024-10-11T20:05:41Z</timestamp>
      <contributor>
        <username>MisterShafXen</username>
        <id>509</id>
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      <minor/>
      <origin>159518</origin>
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      <text bytes="1082" sha1="dkyf91dpkgpbka0ujpdflicwtykf5az" xml:space="preserve">{{Infobox ET}}
{{EDO intro}}

122 is flat in tendency, with the [[prime harmonic]]s from 3 to 13 tuned flat. The equal temperament [[tempering out|tempers out]] 78732/78125 in the [[5-limit]]; 225/224 in the [[7-limit]]; [[385/384]] and [[4000/3993]] in the [[11-limit]]; and [[351/350]] and [[364/363]] in the [[13-limit]]. It provides the [[optimal patent val]] for the 7-limit [[tritonic]] temperament and the 11-limit [[Marvel temperaments #Tritoni|tritoni]] temperament, and the [[rank-3 temperament|planar]] [[squalentine]] temperament. 

122 = [[55edo|55]] + [[67edo|67]], and so using the 122c val it is the convergent towards [[1/6-comma meantone]], with a fifth just a hundredth of a cent flatter.

=== Odd harmonics ===
{{Harmonics in equal|122|23=}}

Harmonic 25 (two 5/4 major thirds) is about halfway between 122edo's steps.

=== Subsets and supersets ===
Since 122 factors into {{factorization|122}}, 122edo contains [[2edo]] and [[61edo]] as its subsets. 244edo (double 122edo) provides a good correction to harmonic 25. 

[[Category:Tritonic]]
[[Category:Meantone]]</text>
      <sha1>dkyf91dpkgpbka0ujpdflicwtykf5az</sha1>
    </revision>
    <revision>
      <id>174952</id>
      <parentid>159518</parentid>
      <timestamp>2025-01-07T05:25:32Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Add links</comment>
      <origin>174952</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1069" sha1="ghwr2d4zb5nz3tkf3c8prb3hehek0mw" xml:space="preserve">{{Infobox ET}}
{{EDO intro}}

122 is flat in tendency, with the [[prime harmonic]]s from 3 to 13 tuned flat. This equal temperament [[tempers out]] 78732/78125 in the [[5-limit]]; 225/224 in the [[7-limit]]; [[385/384]] and [[4000/3993]] in the [[11-limit]]; and [[351/350]] and [[364/363]] in the [[13-limit]]. It provides the [[optimal patent val]] for the 7-limit [[tritonic]] temperament and the 11-limit [[Marvel temperaments #Tritoni|tritoni]] temperament, and the [[rank-3|planar]] temperament [[squalentine]]. 

122 = [[55edo|55]] + [[67edo|67]], and so using the 122c [[val]] it is the [[convergent]] towards [[1/6-comma meantone]], with a fifth just a hundredth of a cent flatter.

=== Odd harmonics ===
{{Harmonics in equal|122|23=}}

Harmonic 25 (two 5/4 major thirds) is about halfway between 122edo's steps.

=== Subsets and supersets ===
Since 122 factors into {{factorization|122}}, 122edo contains [[2edo]] and [[61edo]] as its subsets. [[244edo]] (double 122edo) provides a good correction to harmonic 25. 

[[Category:Tritonic]]
[[Category:Meantone]]</text>
      <sha1>ghwr2d4zb5nz3tkf3c8prb3hehek0mw</sha1>
    </revision>
    <revision>
      <id>176753</id>
      <parentid>174952</parentid>
      <timestamp>2025-01-17T15:21:07Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Cleanup</comment>
      <origin>176753</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1110" sha1="0pgbildi0qyd76380tfok49go0xinqa" xml:space="preserve">{{Infobox ET}}
{{EDO intro}}

122 is flat in tendency, with the [[prime harmonic]]s from 3 to 13 tuned flat. As an equal temperament, it [[tempering out|tempers out]] 78732/78125 ([[sensipent comma]]) in the [[5-limit]]; [[225/224]] in the [[7-limit]]; [[385/384]] and [[4000/3993]] in the [[11-limit]]; and [[351/350]] and [[364/363]] in the [[13-limit]]. It provides the [[optimal patent val]] for the 7-limit [[tritonic]] temperament and the 11-limit [[Marvel temperaments #Tritoni|tritoni]] temperament, and the [[rank-3|planar]] temperament [[squalentine]]. 

122 = [[55edo|55]] + [[67edo|67]], and so using the 122c [[val]] it is the [[convergent]] towards [[1/6-comma meantone]], with a fifth just a hundredth of a cent flatter.

=== Odd harmonics ===
{{Harmonics in equal|122}}

Harmonic 25 (two 5/4 major thirds) is about halfway between 122edo's steps.

=== Subsets and supersets ===
Since 122 factors into {{factorization|122}}, 122edo contains [[2edo]] and [[61edo]] as its subsets. [[244edo]] (double 122edo) provides a good correction to harmonic 25. 

[[Category:Tritonic]]
[[Category:Meantone]]</text>
      <sha1>0pgbildi0qyd76380tfok49go0xinqa</sha1>
    </revision>
    <revision>
      <id>181332</id>
      <parentid>176753</parentid>
      <timestamp>2025-02-18T16:58:38Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181332</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1109" sha1="j254ut91cv6fcv9tzj4l8hv52b63vfe" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

122 is flat in tendency, with the [[prime harmonic]]s from 3 to 13 tuned flat. As an equal temperament, it [[tempering out|tempers out]] 78732/78125 ([[sensipent comma]]) in the [[5-limit]]; [[225/224]] in the [[7-limit]]; [[385/384]] and [[4000/3993]] in the [[11-limit]]; and [[351/350]] and [[364/363]] in the [[13-limit]]. It provides the [[optimal patent val]] for the 7-limit [[tritonic]] temperament and the 11-limit [[Marvel temperaments #Tritoni|tritoni]] temperament, and the [[rank-3|planar]] temperament [[squalentine]]. 

122 = [[55edo|55]] + [[67edo|67]], and so using the 122c [[val]] it is the [[convergent]] towards [[1/6-comma meantone]], with a fifth just a hundredth of a cent flatter.

=== Odd harmonics ===
{{Harmonics in equal|122}}

Harmonic 25 (two 5/4 major thirds) is about halfway between 122edo's steps.

=== Subsets and supersets ===
Since 122 factors into {{factorization|122}}, 122edo contains [[2edo]] and [[61edo]] as its subsets. [[244edo]] (double 122edo) provides a good correction to harmonic 25. 

[[Category:Tritonic]]
[[Category:Meantone]]</text>
      <sha1>j254ut91cv6fcv9tzj4l8hv52b63vfe</sha1>
    </revision>
    <revision>
      <id>226313</id>
      <parentid>181332</parentid>
      <timestamp>2026-03-18T08:21:52Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Subsets and supersets */ 244 also corrects harmonic 7</comment>
      <origin>226313</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1116" sha1="9uya3c83yd9scinvcrae2cybpsyppl2" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

122 is flat in tendency, with the [[prime harmonic]]s from 3 to 13 tuned flat. As an equal temperament, it [[tempering out|tempers out]] 78732/78125 ([[sensipent comma]]) in the [[5-limit]]; [[225/224]] in the [[7-limit]]; [[385/384]] and [[4000/3993]] in the [[11-limit]]; and [[351/350]] and [[364/363]] in the [[13-limit]]. It provides the [[optimal patent val]] for the 7-limit [[tritonic]] temperament and the 11-limit [[Marvel temperaments #Tritoni|tritoni]] temperament, and the [[rank-3|planar]] temperament [[squalentine]]. 

122 = [[55edo|55]] + [[67edo|67]], and so using the 122c [[val]] it is the [[convergent]] towards [[1/6-comma meantone]], with a fifth just a hundredth of a cent flatter.

=== Odd harmonics ===
{{Harmonics in equal|122}}

Harmonic 25 (two 5/4 major thirds) is about halfway between 122edo's steps.

=== Subsets and supersets ===
Since 122 factors into {{factorization|122}}, 122edo contains [[2edo]] and [[61edo]] as its subsets. [[244edo]] (double 122edo) provides a good correction to harmonics 7 and 25. 

[[Category:Tritonic]]
[[Category:Meantone]]</text>
      <sha1>9uya3c83yd9scinvcrae2cybpsyppl2</sha1>
    </revision>
  </page>
  <page>
    <title>1236edo</title>
    <ns>0</ns>
    <id>110</id>
    <revision>
      <id>110</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>110</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="431" sha1="0bndubkvqcycn2f8mr1yhf2q9v90dee" xml:space="preserve">The 1236 division of the octave divides it into 1236 equal parts of 0.9709 cents each. It is a  [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely consistent through the 17-limit, with a 17-limit comma basis of 2601/2600, 5832/5831, 9801/9800, 10648/10647, 14875/14872 and 105644/105625. It is divisible by 12 (12 * 103 = 1236).</text>
      <sha1>0bndubkvqcycn2f8mr1yhf2q9v90dee</sha1>
    </revision>
    <revision>
      <id>4541</id>
      <timestamp>2015-08-15T12:11:29Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 556730349 - Original comment: **</comment>
      <origin>4541</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2143" sha1="6ybuh0use8ejdj2sati941c5mtfqsv4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2015-08-15 12:11:29 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;556730349&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 1236 division of the octave divides it into 1236 equal parts of 0.9709 cents each. It is a  [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely consistent through the 17-limit, with a 17-limit comma basis of 2601/2600, 5832/5831, 9801/9800, 10648/10647, 14875/14872 and 105644/105625. It is divisible by 12 (12 * 103 = 1236).&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;1236edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 1236 division of the octave divides it into 1236 equal parts of 0.9709 cents each. It is a  &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta peak edo&amp;lt;/a&amp;gt;, though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely consistent through the 17-limit, with a 17-limit comma basis of 2601/2600, 5832/5831, 9801/9800, 10648/10647, 14875/14872 and 105644/105625. It is divisible by 12 (12 * 103 = 1236).&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>6ybuh0use8ejdj2sati941c5mtfqsv4</sha1>
    </revision>
    <revision>
      <id>41173</id>
      <parentid>110</parentid>
      <timestamp>2019-03-26T02:01:11Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>41173</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="484" sha1="cr6kfsy6yfvla44aoxuuqbimi0oqwfv" xml:space="preserve">The 1236 division of the octave divides it into 1236 equal parts of 0.9709 cents each. It is a  [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely consistent through the 17-limit, with a 17-limit comma basis of 2601/2600, 5832/5831, 9801/9800, 10648/10647, 14875/14872 and 105644/105625. It is divisible by 12, which is also the sum of its digits (1 + 2 + 3 + 6 = 12 * 103 = 1236).</text>
      <sha1>cr6kfsy6yfvla44aoxuuqbimi0oqwfv</sha1>
    </revision>
    <revision>
      <id>80486</id>
      <parentid>41173</parentid>
      <timestamp>2021-11-08T22:41:55Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>hyphenate and link comma-basis</comment>
      <origin>80486</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="488" sha1="iqtrkcqnbmslivrtpmh0qexkvbf12nf" xml:space="preserve">The 1236 division of the octave divides it into 1236 equal parts of 0.9709 cents each. It is a  [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely consistent through the 17-limit, with a 17-limit [[comma-basis]] of 2601/2600, 5832/5831, 9801/9800, 10648/10647, 14875/14872 and 105644/105625. It is divisible by 12, which is also the sum of its digits (1 + 2 + 3 + 6 = 12 * 103 = 1236).</text>
      <sha1>iqtrkcqnbmslivrtpmh0qexkvbf12nf</sha1>
    </revision>
    <revision>
      <id>80631</id>
      <parentid>80486</parentid>
      <timestamp>2021-11-12T20:17:29Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>unhyphenate "comma basis"</comment>
      <origin>80631</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="488" sha1="kgat25rndxz09q2kiyegn7e0rh19xdk" xml:space="preserve">The 1236 division of the octave divides it into 1236 equal parts of 0.9709 cents each. It is a  [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely consistent through the 17-limit, with a 17-limit [[comma basis]] of 2601/2600, 5832/5831, 9801/9800, 10648/10647, 14875/14872 and 105644/105625. It is divisible by 12, which is also the sum of its digits (1 + 2 + 3 + 6 = 12 * 103 = 1236).</text>
      <sha1>kgat25rndxz09q2kiyegn7e0rh19xdk</sha1>
    </revision>
    <revision>
      <id>91417</id>
      <parentid>80631</parentid>
      <timestamp>2022-04-18T15:50:13Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+prime error table, +links, +category, and misc</comment>
      <origin>91417</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="610" sha1="n03svrx1me62ucftvv2xzcup6edbsie" xml:space="preserve">The '''1236 divisions of the octave''' divides the [[octave]] into 1236 [[equal]] parts of 0.9709 [[cent]]s each. It is a  [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely [[consistent]] through the 17-limit, with a 17-limit [[comma basis]] of [[2601/2600]], [[5832/5831]], [[9801/9800]], [[10648/10647]], 14875/14872 and 105644/105625. It is divisible by 12, which is also the sum of its digits (1 + 2 + 3 + 6 = 12 × 103 = 1236).

{{Harmonics in equal|1236}}

[[Category:Equal divisions of the octave]]</text>
      <sha1>n03svrx1me62ucftvv2xzcup6edbsie</sha1>
    </revision>
    <revision>
      <id>91466</id>
      <parentid>91417</parentid>
      <timestamp>2022-04-19T16:32:07Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Clarify, simplify comma basis, note it being an atomic system, and replace irrelevant number theory with factorization and subset edos</comment>
      <origin>91466</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="661" sha1="11xqunuswr9khwk6ag7l106xko0pm8y" xml:space="preserve">The '''1236 divisions of the octave''' divides the [[octave]] into 1236 [[equal]] parts of 0.9709 [[cent]]s each. It is a  [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely [[consistent]] through the 17-odd-limit, with a 17-limit [[comma basis]] of {[[2601/2600]], [[4096/4095]], [[6656/6655]], [[5832/5831]], [[9801/9800]], 105644/105625}. 

1236 = 2&lt;sup&gt;2&lt;/sup&gt; × 3 × 103, with subset edos 2, 3, 6, 12, 103, 206, 309, and 618. It is divisible by 12, and is an [[atomic]] system. 

{{Harmonics in equal|1236}}

[[Category:Equal divisions of the octave]]</text>
      <sha1>11xqunuswr9khwk6ag7l106xko0pm8y</sha1>
    </revision>
    <revision>
      <id>91467</id>
      <parentid>91466</parentid>
      <timestamp>2022-04-19T16:33:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Hmmm</comment>
      <origin>91467</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="661" sha1="p7g3h4bkv0t2bm1v9nmp23e9pasrzct" xml:space="preserve">The '''1236 divisions of the octave''' divides the [[octave]] into 1236 [[equal]] parts of 0.9709 [[cent]]s each. It is a  [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely [[consistent]] through the 17-odd-limit, with a 17-limit [[comma basis]] of {[[2601/2600]], [[4096/4095]], [[5832/5831]], [[6656/6655]], [[9801/9800]], 105644/105625}. 

1236 = 2&lt;sup&gt;2&lt;/sup&gt; × 3 × 103, with subset edos 2, 3, 6, 12, 103, 206, 309, and 618. It is divisible by 12, and is an [[atomic]] system. 

{{Harmonics in equal|1236}}

[[Category:Equal divisions of the octave]]</text>
      <sha1>p7g3h4bkv0t2bm1v9nmp23e9pasrzct</sha1>
    </revision>
    <revision>
      <id>93195</id>
      <parentid>91467</parentid>
      <timestamp>2022-07-04T01:14:29Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>93195</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="690" sha1="l53m9vzpffjrex4q2h7pvw2ohbguii3" xml:space="preserve">The '''1236 divisions of the octave''' divides the [[octave]] into 1236 [[equal]] parts of 0.9709 [[cent]]s each. It is a  [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely [[consistent]] through the 17-odd-limit, with a 17-limit [[comma basis]] of {[[2601/2600]], [[4096/4095]], [[5832/5831]], [[6656/6655]], [[9801/9800]], 105644/105625}. 

1236 = 2&lt;sup&gt;2&lt;/sup&gt; × 3 × 103, with subset edos 2, 3, 6, 12, 103, 206, 309, and 618. It is divisible by 12, and is an [[atomic]] system. 

{{Harmonics in equal|1236}}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>l53m9vzpffjrex4q2h7pvw2ohbguii3</sha1>
    </revision>
    <revision>
      <id>96480</id>
      <parentid>93195</parentid>
      <timestamp>2022-10-04T22:09:01Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
      <origin>96480</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="705" sha1="knzzg1db08164zmtr0i47zmzcd77ecb" xml:space="preserve">{{Infobox ET}}
The '''1236 divisions of the octave''' divides the [[octave]] into 1236 [[equal]] parts of 0.9709 [[cent]]s each. It is a  [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely [[consistent]] through the 17-odd-limit, with a 17-limit [[comma basis]] of {[[2601/2600]], [[4096/4095]], [[5832/5831]], [[6656/6655]], [[9801/9800]], 105644/105625}. 

1236 = 2&lt;sup&gt;2&lt;/sup&gt; × 3 × 103, with subset edos 2, 3, 6, 12, 103, 206, 309, and 618. It is divisible by 12, and is an [[atomic]] system. 

{{Harmonics in equal|1236}}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>knzzg1db08164zmtr0i47zmzcd77ecb</sha1>
    </revision>
    <revision>
      <id>98671</id>
      <parentid>96480</parentid>
      <timestamp>2022-10-29T09:19:32Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>sectioning and one more column for the prime error table</comment>
      <origin>98671</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="739" sha1="74pnsl54ttkdun34znvopyaf6lhppi7" xml:space="preserve">{{Infobox ET}}
The '''1236 divisions of the octave''' divides the [[octave]] into 1236 [[equal]] parts of 0.9709 [[cent]]s each. It is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely [[consistent]] through the 17-odd-limit, with a 17-limit [[comma basis]] of {[[2601/2600]], [[4096/4095]], [[5832/5831]], [[6656/6655]], [[9801/9800]], 105644/105625}. 

1236 = 2&lt;sup&gt;2&lt;/sup&gt; × 3 × 103, with subset edos 2, 3, 6, 12, 103, 206, 309, and 618. It is divisible by 12, and is an [[atomic]] system. 

=== Prime harmonics ===
{{Harmonics in equal|1236|columns=11}}

[[Category:Equal divisions of the octave|####]] &lt;!-- 4-digit number --&gt;</text>
      <sha1>74pnsl54ttkdun34znvopyaf6lhppi7</sha1>
    </revision>
    <revision>
      <id>101093</id>
      <parentid>98671</parentid>
      <timestamp>2023-01-08T04:35:02Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Improve intro and separate a section for divisors</comment>
      <origin>101093</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="597" sha1="58wj8dht2h8u3whodl7xbhjhxub37o6" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1236}}

1236edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely [[consistent]] through the [[17-odd-limit]], with a 17-limit [[comma basis]] of {[[2601/2600]], [[4096/4095]], [[5832/5831]], [[6656/6655]], [[9801/9800]], 105644/105625}. 

=== Prime harmonics ===
{{Harmonics in equal|1236|columns=11}}

=== Divisors ===
1236 = 2&lt;sup&gt;2&lt;/sup&gt; × 3 × 103, with subset edos 2, 3, 6, 12, 103, 206, 309, and 618. It is divisible by 12, and is an [[atomic]] system.</text>
      <sha1>58wj8dht2h8u3whodl7xbhjhxub37o6</sha1>
    </revision>
    <revision>
      <id>119374</id>
      <parentid>101093</parentid>
      <timestamp>2023-07-09T05:16:50Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119374</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="616" sha1="8t8ice3jo5tueqnkm3s22kiijy3yp4h" xml:space="preserve">{{novelty}}{{stub}}{{Infobox ET}}
{{EDO intro|1236}}

1236edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely [[consistent]] through the [[17-odd-limit]], with a 17-limit [[comma basis]] of {[[2601/2600]], [[4096/4095]], [[5832/5831]], [[6656/6655]], [[9801/9800]], 105644/105625}. 

=== Prime harmonics ===
{{Harmonics in equal|1236|columns=11}}

=== Divisors ===
1236 = 2&lt;sup&gt;2&lt;/sup&gt; × 3 × 103, with subset edos 2, 3, 6, 12, 103, 206, 309, and 618. It is divisible by 12, and is an [[atomic]] system.</text>
      <sha1>8t8ice3jo5tueqnkm3s22kiijy3yp4h</sha1>
    </revision>
    <revision>
      <id>119614</id>
      <parentid>119374</parentid>
      <timestamp>2023-07-09T06:52:00Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>119614</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="597" sha1="58wj8dht2h8u3whodl7xbhjhxub37o6" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1236}}

1236edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]], though not zeta integral nor zeta gap. It is a strong 17-limit system and uniquely [[consistent]] through the [[17-odd-limit]], with a 17-limit [[comma basis]] of {[[2601/2600]], [[4096/4095]], [[5832/5831]], [[6656/6655]], [[9801/9800]], 105644/105625}. 

=== Prime harmonics ===
{{Harmonics in equal|1236|columns=11}}

=== Divisors ===
1236 = 2&lt;sup&gt;2&lt;/sup&gt; × 3 × 103, with subset edos 2, 3, 6, 12, 103, 206, 309, and 618. It is divisible by 12, and is an [[atomic]] system.</text>
      <sha1>58wj8dht2h8u3whodl7xbhjhxub37o6</sha1>
    </revision>
    <revision>
      <id>126657</id>
      <parentid>119614</parentid>
      <timestamp>2023-10-31T09:54:28Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Adopt template: Factorization; misc. cleanup</comment>
      <origin>126657</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="642" sha1="my5d01awd3yaz7mthpdgm7w8jlslu0p" xml:space="preserve">{{Infobox ET}}
{{EDO intro|1236}}

1236edo is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. It is a strong 17-limit system and [[consistency|distinctly consistent]] through the [[17-odd-limit]], with a 17-limit [[comma basis]] of {[[2601/2600]], [[4096/4095]], [[5832/5831]], [[6656/6655]], [[9801/9800]], 105644/105625}. 

=== Prime harmonics ===
{{Harmonics in equal|1236|columns=11}}

=== Subsets and supersets ===
Since 1236 factors into {{factorization|1236}}, 1236edo has subset edos {{EDOs| 2, 3, 6, 12, 103, 206, 309, and 618 }}. It is divisible by 12, and is an [[atomic]] system.</text>
      <sha1>my5d01awd3yaz7mthpdgm7w8jlslu0p</sha1>
    </revision>
    <revision>
      <id>181333</id>
      <parentid>126657</parentid>
      <timestamp>2025-02-18T16:58:53Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181333</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="636" sha1="orv643iltfwa2qzo8y78ouebqvevr58" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

1236edo is a [[zeta peak edo]], though not [[zeta integral edo|zeta integral]] nor [[zeta gap edo|zeta gap]]. It is a strong 17-limit system and [[consistency|distinctly consistent]] through the [[17-odd-limit]], with a 17-limit [[comma basis]] of {[[2601/2600]], [[4096/4095]], [[5832/5831]], [[6656/6655]], [[9801/9800]], 105644/105625}. 

=== Prime harmonics ===
{{Harmonics in equal|1236|columns=11}}

=== Subsets and supersets ===
Since 1236 factors into {{factorization|1236}}, 1236edo has subset edos {{EDOs| 2, 3, 6, 12, 103, 206, 309, and 618 }}. It is divisible by 12, and is an [[atomic]] system.</text>
      <sha1>orv643iltfwa2qzo8y78ouebqvevr58</sha1>
    </revision>
  </page>
  <page>
    <title>125/72</title>
    <ns>0</ns>
    <id>111</id>
    <revision>
      <id>111</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>111</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="242" sha1="tqw0u82qjm6i0yunxxkyr0tk5fc6j59" xml:space="preserve">'''125/72'''
|-3 -2 3&amp;gt;

955.0311 cents

[[File:jid_125_72_pluck_adu_dr220.mp3]] [[:File:jid_125_72_pluck_adu_dr220.mp3|sound sample]]

The classic ([[5-limit|5-limit]]) augmented sixth.
[[Category:neutral_seventh]]
[[Category:todo:expand]]</text>
      <sha1>tqw0u82qjm6i0yunxxkyr0tk5fc6j59</sha1>
    </revision>
    <revision>
      <id>4542</id>
      <timestamp>2014-06-09T22:25:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513417162 - Original comment: **</comment>
      <origin>4542</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2932" sha1="9v7qlr0092mez9x908b9je5778v5haj" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-09 22:25:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513417162&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**125/72**
|-3 -2 3&amp;gt;
955.0311 cents
[[media type="file" key="jid_125_72_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_125_72_pluck_adu_dr220.mp3|sound sample]]

The classic ([[5-limit]]) augmented sixth.
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;125_72&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;125/72&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|-3 -2 3&amp;amp;gt;&amp;lt;br /&amp;gt;
955.0311 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_125_72_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_125_72_pluck_adu_dr220.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_125_72_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_125_72_pluck_adu_dr220.mp3/513415358/jid_125_72_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_125_72_pluck_adu_dr220.mp3/513415358/jid_125_72_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The classic (&amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;) augmented sixth.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9v7qlr0092mez9x908b9je5778v5haj</sha1>
    </revision>
    <revision>
      <id>50309</id>
      <parentid>111</parentid>
      <timestamp>2020-09-26T14:55:43Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>cleanup, infobox, cats, still limited information here...</comment>
      <origin>50309</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="543" sha1="9246stgfpjs3tbjseid8eyji7hm0r73" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 125/72
| Monzo = -3 -2 3
| Cents = 955.03114
| Name = classic augmented sixth
| Color name = 
| FJS name = A6&lt;sup&gt;125&lt;/sup&gt;
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]].

== See also ==

* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Listen]]

[[Category:Todo:expand]]</text>
      <sha1>9246stgfpjs3tbjseid8eyji7hm0r73</sha1>
    </revision>
    <revision>
      <id>50375</id>
      <parentid>50309</parentid>
      <timestamp>2020-09-27T14:49:50Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>+cat</comment>
      <origin>50375</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="570" sha1="cbctquok89vr17l97vq0ufndtjwiai0" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 125/72
| Monzo = -3 -2 3
| Cents = 955.03114
| Name = classic augmented sixth
| Color name = 
| FJS name = A6&lt;sup&gt;125&lt;/sup&gt;
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]].

== See also ==

* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal]]
[[Category:Listen]]

[[Category:Todo:expand]]</text>
      <sha1>cbctquok89vr17l97vq0ufndtjwiai0</sha1>
    </revision>
    <revision>
      <id>62866</id>
      <parentid>50375</parentid>
      <timestamp>2021-03-12T13:38:12Z</timestamp>
      <contributor>
        <username>Nolovoto</username>
        <id>277</id>
      </contributor>
      <comment>added nearness to 49edo in description</comment>
      <origin>62866</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="642" sha1="dzlup363lwhhmjh0gw5movbhkf7p3jh" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 125/72
| Monzo = -3 -2 3
| Cents = 955.03114
| Name = classic augmented sixth
| Color name = 
| FJS name = A6&lt;sup&gt;125&lt;/sup&gt;
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]].

This interval is especially close to the 39th step of [[49edo|49edo.]]

== See also ==

* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal]]
[[Category:Listen]]

[[Category:Todo:expand]]</text>
      <sha1>dzlup363lwhhmjh0gw5movbhkf7p3jh</sha1>
    </revision>
    <revision>
      <id>64582</id>
      <parentid>62866</parentid>
      <timestamp>2021-03-27T17:16:43Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64582</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="690" sha1="p6et4jjupj98xjzew56c2dqf3uv204s" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 125/72
| Monzo = -3 -2 3
| Cents = 955.03114
| Name = classic augmented sixth
| Color name = 
| FJS name = A6&lt;sup&gt;125&lt;/sup&gt;
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]].

This interval is especially close to the 39th step of [[49edo|49edo.]]

== See also ==

* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal]]
[[Category:Listen]]
[[Category:Pages with internal sound examples]]

[[Category:Todo:expand]]</text>
      <sha1>p6et4jjupj98xjzew56c2dqf3uv204s</sha1>
    </revision>
    <revision>
      <id>80415</id>
      <parentid>64582</parentid>
      <timestamp>2021-11-07T22:31:44Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Expansion</comment>
      <origin>80415</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="871" sha1="shivkvu5bjen5mx4gtrrvk2denl82xx" xml:space="preserve">{{Infobox Interval
| Ratio = 125/72
| Monzo = -3 -2 3
| Cents = 955.03114
| Name = classic augmented sixth
| Color name = 
| FJS name = A6&lt;sup&gt;125&lt;/sup&gt;
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]]. It can be obtained by adding [[5/3]], the classic major sixth, by [[25/24]], the classic chroma. In any [[kleismic]] system, it is tuned to an exact semitwelfth, tempered together with [[216/125]].

== Approximation ==
This interval is especially close to the 39th step of [[49edo]]. 

== See also ==
* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal]]
[[Category:Semitwelfth]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>shivkvu5bjen5mx4gtrrvk2denl82xx</sha1>
    </revision>
    <revision>
      <id>86303</id>
      <parentid>80415</parentid>
      <timestamp>2022-02-22T23:46:40Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86303</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="823" sha1="9y9o5wpum6mslkc5tqobsglst27hym8" xml:space="preserve">{{Infobox Interval
| Ratio = 125/72
| Monzo = -3 -2 3
| Cents = 955.03114
| Name = classic augmented sixth
| Color name = 
| FJS name = A6&lt;sup&gt;125&lt;/sup&gt;
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]]. It can be obtained by adding [[5/3]], the classic major sixth, by [[25/24]], the classic chroma. In any [[kleismic]] system, it is tuned to an exact semitwelfth, tempered together with [[216/125]].

== Approximation ==
This interval is especially close to the 39th step of [[49edo]]. 

== See also ==
* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:5-limit]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal]]
[[Category:Semitwelfth]]</text>
      <sha1>9y9o5wpum6mslkc5tqobsglst27hym8</sha1>
    </revision>
    <revision>
      <id>90019</id>
      <parentid>86303</parentid>
      <timestamp>2022-03-21T12:36:39Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Categories</comment>
      <origin>90019</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="795" sha1="jn3vywt7pbpsue4x95vuilexuj5tkfr" xml:space="preserve">{{Infobox Interval
| Ratio = 125/72
| Monzo = -3 -2 3
| Cents = 955.03114
| Name = classic augmented sixth
| Color name = 
| FJS name = A6&lt;sup&gt;125&lt;/sup&gt;
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]]. It can be obtained by adding [[5/3]], the classic major sixth, by [[25/24]], the classic chroma. In any [[kleismic]] system, it is tuned to an exact semitwelfth, tempered together with [[216/125]].

== Approximation ==
This interval is especially close to the 39th step of [[49edo]]. 

== See also ==
* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:5-limit]]
[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal]]
[[Category:Semitwelfth]]</text>
      <sha1>jn3vywt7pbpsue4x95vuilexuj5tkfr</sha1>
    </revision>
    <revision>
      <id>98097</id>
      <parentid>90019</parentid>
      <timestamp>2022-10-25T14:05:51Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>98097</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="673" sha1="d47ubv9vqmlpqzf42q366lwey7iw9zs" xml:space="preserve">{{Infobox Interval
| Name = classic augmented sixth
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]]. It can be obtained by adding [[5/3]], the classic major sixth, by [[25/24]], the classic chroma. In any [[kleismic]] system, it is tuned to an exact semitwelfth, tempered together with [[216/125]].

== Approximation ==
This interval is especially close to the 39th step of [[49edo]]. 

== See also ==
* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal]]
[[Category:Semitwelfth]]</text>
      <sha1>d47ubv9vqmlpqzf42q366lwey7iw9zs</sha1>
    </revision>
    <revision>
      <id>100565</id>
      <parentid>98097</parentid>
      <timestamp>2022-12-29T02:12:51Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>added color name</comment>
      <origin>100565</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="713" sha1="21y46xp2vpx2z906qd4aks9v8aac90q" xml:space="preserve">{{Infobox Interval
| Name = classic augmented sixth
| Color name = y&lt;sup&gt;3&lt;/sup&gt;6,triyo 6th
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]]. It can be obtained by adding [[5/3]], the classic major sixth, by [[25/24]], the classic chroma. In any [[kleismic]] system, it is tuned to an exact semitwelfth, tempered together with [[216/125]].

== Approximation ==
This interval is especially close to the 39th step of [[49edo]]. 

== See also ==
* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal]]
[[Category:Semitwelfth]]</text>
      <sha1>21y46xp2vpx2z906qd4aks9v8aac90q</sha1>
    </revision>
    <revision>
      <id>100566</id>
      <parentid>100565</parentid>
      <timestamp>2022-12-29T02:13:03Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <origin>100566</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="714" sha1="huqeefko0qqgblacrbl0izu4wccbiw4" xml:space="preserve">{{Infobox Interval
| Name = classic augmented sixth
| Color name = y&lt;sup&gt;3&lt;/sup&gt;6, triyo 6th
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]]. It can be obtained by adding [[5/3]], the classic major sixth, by [[25/24]], the classic chroma. In any [[kleismic]] system, it is tuned to an exact semitwelfth, tempered together with [[216/125]].

== Approximation ==
This interval is especially close to the 39th step of [[49edo]]. 

== See also ==
* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal]]
[[Category:Semitwelfth]]</text>
      <sha1>huqeefko0qqgblacrbl0izu4wccbiw4</sha1>
    </revision>
    <revision>
      <id>109848</id>
      <parentid>100566</parentid>
      <timestamp>2023-04-28T03:27:03Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Interseptimal]] to [[Category:Interseptimal intervals]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>109848</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="724" sha1="bt3o448ke1cerg8xzztz6z3t1yj13kv" xml:space="preserve">{{Infobox Interval
| Name = classic augmented sixth
| Color name = y&lt;sup&gt;3&lt;/sup&gt;6, triyo 6th
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]]. It can be obtained by adding [[5/3]], the classic major sixth, by [[25/24]], the classic chroma. In any [[kleismic]] system, it is tuned to an exact semitwelfth, tempered together with [[216/125]].

== Approximation ==
This interval is especially close to the 39th step of [[49edo]]. 

== See also ==
* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal intervals]]
[[Category:Semitwelfth]]</text>
      <sha1>bt3o448ke1cerg8xzztz6z3t1yj13kv</sha1>
    </revision>
    <revision>
      <id>120252</id>
      <parentid>109848</parentid>
      <timestamp>2023-07-13T06:49:07Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+name and explain the idea behind them</comment>
      <origin>120252</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="912" sha1="11zvys4ssbnudne3t7jkerroqordqnn" xml:space="preserve">{{Infobox Interval
| Name = classic(al) augmented sixth, triptolemaic augmented sixth
| Color name = y&lt;sup&gt;3&lt;/sup&gt;6, triyo 6th
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]]. It can be obtained by adding [[5/3]], the classic major sixth, by [[25/24]], the classic chroma. It is also the Pythagorean augmented sixth (59049/32768) sharpened by three [[81/80|syntonic commas]], which lends itself to the term ''triptolemaic''. 

In any [[kleismic]] system, it is tuned to an exact semitwelfth, tempered together with [[216/125]].

== Approximation ==
This interval is especially close to the 39th step of [[49edo]]. 

== See also ==
* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal intervals]]
[[Category:Semitwelfth]]</text>
      <sha1>11zvys4ssbnudne3t7jkerroqordqnn</sha1>
    </revision>
    <revision>
      <id>156425</id>
      <parentid>120252</parentid>
      <timestamp>2024-09-24T18:54:12Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>minor corrections</comment>
      <origin>156425</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="914" sha1="eotr0xg8g3aixbwr7gexuisveqwzff1" xml:space="preserve">{{Infobox Interval
| Name = classic(al) augmented sixth, triptolemaic augmented sixth
| Color name = y&lt;sup&gt;3&lt;/sup&gt;6, triyo 6th
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955 [[cent]]. It can be obtained by widening [[5/3]], the classic major sixth, by [[25/24]], the classic chroma. It is also the Pythagorean augmented sixth (59049/32768) flattened by three [[81/80|syntonic commas]], which lends itself to the term ''triptolemaic''. 

In any [[kleismic]] system, it is tuned to an exact semitwelfth, tempered together with [[216/125]].

== Approximation ==
This interval is especially close to the 39th step of [[49edo]]. 

== See also ==
* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal intervals]]
[[Category:Semitwelfth]]</text>
      <sha1>eotr0xg8g3aixbwr7gexuisveqwzff1</sha1>
    </revision>
    <revision>
      <id>221450</id>
      <parentid>156425</parentid>
      <timestamp>2026-01-13T15:04:02Z</timestamp>
      <contributor>
        <username>Hotcrystal0</username>
        <id>614</id>
      </contributor>
      <comment>cent</comment>
      <origin>221450</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="913" sha1="3a05grzb3txmrmeepxs9qnxzbr6g2he" xml:space="preserve">{{Infobox Interval
| Name = classic(al) augmented sixth, triptolemaic augmented sixth
| Color name = y&lt;sup&gt;3&lt;/sup&gt;6, triyo 6th
| Sound = jid_125_72_pluck_adu_dr220.mp3
}}
'''125/72''', the '''classic augmented sixth''' is [[5-limit]] just interval of about 955{{cent}}. It can be obtained by widening [[5/3]], the classic major sixth, by [[25/24]], the classic chroma. It is also the Pythagorean augmented sixth (59049/32768) flattened by three [[81/80|syntonic commas]], which lends itself to the term ''triptolemaic''. 

In any [[kleismic]] system, it is tuned to an exact semitwelfth, tempered together with [[216/125]].

== Approximation ==
This interval is especially close to the 39th step of [[49edo]]. 

== See also ==
* [[144/125]] – its [[octave complement]]
* [[Gallery of just intervals]]

[[Category:Sixth]]
[[Category:Augmented sixth]]
[[Category:Interseptimal intervals]]
[[Category:Semitwelfth]]</text>
      <sha1>3a05grzb3txmrmeepxs9qnxzbr6g2he</sha1>
    </revision>
  </page>
  <page>
    <title>1ed125c</title>
    <ns>0</ns>
    <id>113</id>
    <redirect title="4ed4/3" />
    <revision>
      <id>113</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>113</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1288" sha1="dq23cqgoz3uluz2qg52xx6gfod2mud1" xml:space="preserve">=125cET=

This [[nonoctave|nonoctave]] tuning uses equal steps of 125 cents each. This could be considered dividing the 500 cent perfect fourth into 4 equal parts.

{| class="wikitable"
|-
| | ordinal number
| | interval name
| | cents
|-
| | 0
| | 
| | 0
|-
| | 1
| | 2/3-tone, trienthird
| | 125
|-
| | 2
| | semifourth
| | 250
|-
| | 3
| | narrow perde segah,

marvelous major third,

near just major third
| | 375
|-
| | 4
| | perfect fourth
| | 500
|-
| | 5
| | pental diminished fifth,

classic diminshed fifth
| | 625
|-
| | 6
| | septendecimal subminor sixth
| | 750
|-
| | 7
| | major sixth
| | 875
|-
| | 8
| | Pythagorean minor seventh
| | 1000
|-
| | 9
| | classic ([[5-limit|5-limit]]) diminished octave.
| | 1125
|-
| | 10
| | 
| | 1250
|-
| | 11
| | 
| | 1375
|-
| | 12
| | 
| | 1500
|-
| | 13
| | 
| | 1625
|-
| | 14
| | 
| | 1750
|-
| | 15
| | 
| | 1875
|-
| | 16
| | 
| | 2000
|}
! E:\cakewalk\scales\125cent.scl
!

125 cent tuning

4
!

125.00000

250.00000

375.00000

500.00000

&lt;u&gt;Example music with this tuning&lt;/u&gt;

[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall]

[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over]      [[Category:31edo]]
[[Category:et]]
[[Category:listen]]
[[Category:nonoctave]]
[[Category:what_is]]
[[Category:wiki]]</text>
      <sha1>dq23cqgoz3uluz2qg52xx6gfod2mud1</sha1>
    </revision>
    <revision>
      <id>4543</id>
      <timestamp>2014-10-11T14:31:34Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 526068290 - Original comment: **</comment>
      <origin>4543</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6722" sha1="8v9r11fxt4qq29f3f3yokp04xvh9f6h" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2014-10-11 14:31:34 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;526068290&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=125cET= 

This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered dividing the 500 cent perfect fourth into 4 equal parts.

|| ordinal number || interval name || cents ||
|| 0 ||   || 0 ||
|| 1 || 2/3-tone, trienthird || 125 ||
|| 2 || semifourth || 250 ||
|| 3 || narrow perde segah,
marvelous major third,
near just major third || 375 ||
|| 4 || perfect fourth || 500 ||
|| 5 || pental diminished fifth,
classic diminshed fifth || 625 ||
|| 6 || septendecimal subminor sixth || 750 ||
|| 7 || major sixth || 875 ||
|| 8 || Pythagorean minor seventh || 1000 ||
|| 9 || classic ([[xenharmonic/5-limit|5-limit]]) diminished octave. || 1125 ||
|| 10 ||   || 1250 ||
|| 11 ||   || 1375 ||
|| 12 ||   || 1500 ||
|| 13 ||   || 1625 ||
|| 14 ||   || 1750 ||
|| 15 ||   || 1875 ||
|| 16 ||   || 2000 ||


! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000

__Example music with this tuning__

[[http://chrisvaisvil.com/canyon-diablo-fall/|Canyon Diablo Fall]]
[[http://chrisvaisvil.com/crossing-over-125cet/|Crossing Over]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;125cET&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x125cET"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;125cET&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This &amp;lt;a class="wiki_link" href="/nonoctave"&amp;gt;nonoctave&amp;lt;/a&amp;gt; tuning uses equal steps of 125 cents each. This could be considered dividing the 500 cent perfect fourth into 4 equal parts.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;ordinal number&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;interval name&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2/3-tone, trienthird&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;semifourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;250&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;narrow perde segah,&amp;lt;br /&amp;gt;
marvelous major third,&amp;lt;br /&amp;gt;
near just major third&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;375&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;perfect fourth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;500&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;pental diminished fifth,&amp;lt;br /&amp;gt;
classic diminshed fifth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;septendecimal subminor sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;major sixth&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;875&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Pythagorean minor seventh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;classic (&amp;lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;) diminished octave.&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1250&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1375&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1500&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1875&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
! E:\cakewalk\scales\125cent.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
125 cent tuning&amp;lt;br /&amp;gt;
4&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
125.00000&amp;lt;br /&amp;gt;
250.00000&amp;lt;br /&amp;gt;
375.00000&amp;lt;br /&amp;gt;
500.00000&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;Example music with this tuning&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/canyon-diablo-fall/" rel="nofollow"&amp;gt;Canyon Diablo Fall&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/crossing-over-125cet/" rel="nofollow"&amp;gt;Crossing Over&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>8v9r11fxt4qq29f3f3yokp04xvh9f6h</sha1>
    </revision>
    <revision>
      <id>4544</id>
      <timestamp>2014-10-11T14:20:34Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 526067474 - Original comment: **</comment>
      <origin>4544</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5899" sha1="jijszviaeqgsatt578ab44g4aedhczd" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2014-10-11 14:20:34 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;526067474&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=125cET= 

This [[nonoctave]] tuning uses equal steps of 125 cents each.

|| ordinal number || interval || cents ||
|| 0 ||   || 0 ||
|| 1 ||   || 125 ||
|| 2 ||   || 250 ||
|| 3 ||   || 375 ||
|| 4 ||   || 500 ||
|| 5 ||   || 625 ||
|| 6 ||   || 750 ||
|| 7 ||   || 875 ||
|| 8 ||   || 1000 ||
|| 9 ||   || 1125 ||
|| 10 ||   || 1250 ||
|| 11 ||   || 1375 ||
|| 12 ||   || 1500 ||
|| 13 ||   || 1625 ||
|| 14 ||   || 1750 ||
|| 15 ||   || 1875 ||
|| 16 ||   || 2000 ||


! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000

__Example music with this tuning__

[[http://chrisvaisvil.com/canyon-diablo-fall/|Canyon Diablo Fall]]
[[http://chrisvaisvil.com/crossing-over-125cet/|Crossing Over]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;125cET&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x125cET"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;125cET&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This &amp;lt;a class="wiki_link" href="/nonoctave"&amp;gt;nonoctave&amp;lt;/a&amp;gt; tuning uses equal steps of 125 cents each.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;ordinal number&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;250&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;375&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;500&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;875&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1250&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1375&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1500&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1875&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
! E:\cakewalk\scales\125cent.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
125 cent tuning&amp;lt;br /&amp;gt;
4&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
125.00000&amp;lt;br /&amp;gt;
250.00000&amp;lt;br /&amp;gt;
375.00000&amp;lt;br /&amp;gt;
500.00000&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;Example music with this tuning&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/canyon-diablo-fall/" rel="nofollow"&amp;gt;Canyon Diablo Fall&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/crossing-over-125cet/" rel="nofollow"&amp;gt;Crossing Over&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>jijszviaeqgsatt578ab44g4aedhczd</sha1>
    </revision>
    <revision>
      <id>4545</id>
      <timestamp>2014-10-11T14:19:46Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 526067418 - Original comment: **</comment>
      <origin>4545</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5700" sha1="l5xp81tpbgpdlaj5c3dg55zf3qb3jtw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2014-10-11 14:19:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;526067418&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=125cET= 

This [[nonoctave]] tuning uses equal steps of 125 cents each.

|| ordinal number || interval || cents ||
|| 0 ||   || 0 ||
|| 1 ||   || 125 ||
|| 2 ||   || 250 ||
|| 3 ||   || 375 ||
|| 4 ||   || 500 ||
|| 5 ||   || 625 ||
|| 6 ||   || 750 ||
|| 7 ||   || 875 ||
|| 8 ||   || 1000 ||
|| 9 ||   || 1125 ||
|| 10 ||   || 1250 ||
|| 11 ||   || 1375 ||
|| 12 ||   || 1500 ||
|| 13 ||   || 1625 ||
|| 14 ||   || 1750 ||
|| 15 ||   || 1875 ||
|| 16 ||   || 2000 ||


! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000

__Example music with this tuning__

[[http://chrisvaisvil.com/canyon-diablo-fall/|Canyon Diablo Fall]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;125cET&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x125cET"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;125cET&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
This &amp;lt;a class="wiki_link" href="/nonoctave"&amp;gt;nonoctave&amp;lt;/a&amp;gt; tuning uses equal steps of 125 cents each.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;ordinal number&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;interval&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;cents&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;250&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;375&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;500&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;875&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;8&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;10&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1250&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1375&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1500&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;14&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1750&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;15&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1875&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;16&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;2000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
! E:\cakewalk\scales\125cent.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
125 cent tuning&amp;lt;br /&amp;gt;
4&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
125.00000&amp;lt;br /&amp;gt;
250.00000&amp;lt;br /&amp;gt;
375.00000&amp;lt;br /&amp;gt;
500.00000&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;u&amp;gt;Example music with this tuning&amp;lt;/u&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://chrisvaisvil.com/canyon-diablo-fall/" rel="nofollow"&amp;gt;Canyon Diablo Fall&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>l5xp81tpbgpdlaj5c3dg55zf3qb3jtw</sha1>
    </revision>
    <revision>
      <id>33810</id>
      <parentid>113</parentid>
      <timestamp>2018-09-28T01:55:25Z</timestamp>
      <contributor>
        <username>Vaisvil</username>
        <id>7</id>
      </contributor>
      <origin>33810</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1432" sha1="62ak3qllpi4y6bmfxem5ma77wku1ti1" xml:space="preserve">=125cET=

This [[nonoctave|nonoctave]] tuning uses equal steps of 125 cents each. This could be considered dividing the 500 cent perfect fourth into 4 equal parts.

{| class="wikitable"
|-
| | ordinal number
| | interval name
| | cents
|-
| | 0
| | 
| | 0
|-
| | 1
| | 2/3-tone, trienthird
| | 125
|-
| | 2
| | semifourth
| | 250
|-
| | 3
| | narrow perde segah,

marvelous major third,

near just major third
| | 375
|-
| | 4
| | perfect fourth
| | 500
|-
| | 5
| | pental diminished fifth,

classic diminshed fifth
| | 625
|-
| | 6
| | septendecimal subminor sixth
| | 750
|-
| | 7
| | major sixth
| | 875
|-
| | 8
| | Pythagorean minor seventh
| | 1000
|-
| | 9
| | classic ([[5-limit|5-limit]]) diminished octave.
| | 1125
|-
| | 10
| | 
| | 1250
|-
| | 11
| | 
| | 1375
|-
| | 12
| | 
| | 1500
|-
| | 13
| | 
| | 1625
|-
| | 14
| | 
| | 1750
|-
| | 15
| | 
| | 1875
|-
| | 16
| | 
| | 2000
|}
! E:\cakewalk\scales\125cent.scl
!

125 cent tuning

4
!

125.00000

250.00000

375.00000

500.00000

&lt;u&gt;Example music with this tuning&lt;/u&gt;

[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall]  
[[File:http://micro.soonlabel.com/125cent/20130317_125cent.mp3]]
[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over]
[[File:http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3]]      
[[Category:31edo]]
[[Category:et]]
[[Category:listen]]
[[Category:nonoctave]]
[[Category:what_is]]
[[Category:wiki]]</text>
      <sha1>62ak3qllpi4y6bmfxem5ma77wku1ti1</sha1>
    </revision>
    <revision>
      <id>33812</id>
      <parentid>33810</parentid>
      <timestamp>2018-09-28T01:58:22Z</timestamp>
      <contributor>
        <username>Vaisvil</username>
        <id>7</id>
      </contributor>
      <origin>33812</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1452" sha1="tqovc4vhd6q58p0d6lx0sza3mldboj6" xml:space="preserve">=125cET=

This [[nonoctave|nonoctave]] tuning uses equal steps of 125 cents each. This could be considered dividing the 500 cent perfect fourth into 4 equal parts.

{| class="wikitable"
|-
| | ordinal number
| | interval name
| | cents
|-
| | 0
| | 
| | 0
|-
| | 1
| | 2/3-tone, trienthird
| | 125
|-
| | 2
| | semifourth
| | 250
|-
| | 3
| | narrow perde segah,

marvelous major third,

near just major third
| | 375
|-
| | 4
| | perfect fourth
| | 500
|-
| | 5
| | pental diminished fifth,

classic diminshed fifth
| | 625
|-
| | 6
| | septendecimal subminor sixth
| | 750
|-
| | 7
| | major sixth
| | 875
|-
| | 8
| | Pythagorean minor seventh
| | 1000
|-
| | 9
| | classic ([[5-limit|5-limit]]) diminished octave.
| | 1125
|-
| | 10
| | 
| | 1250
|-
| | 11
| | 
| | 1375
|-
| | 12
| | 
| | 1500
|-
| | 13
| | 
| | 1625
|-
| | 14
| | 
| | 1750
|-
| | 15
| | 
| | 1875
|-
| | 16
| | 
| | 2000
|}
! E:\cakewalk\scales\125cent.scl
!

125 cent tuning

4
!

125.00000

250.00000

375.00000

500.00000

&lt;u&gt;Example music with this tuning&lt;/u&gt;

[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall]
&lt;br&gt;  
[[File:http://micro.soonlabel.com/125cent/20130317_125cent.mp3]]
&lt;br&gt;
[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over]
&lt;br&gt;
[[File:http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3]]
&lt;br&gt;      
[[Category:31edo]]
[[Category:et]]
[[Category:listen]]
[[Category:nonoctave]]
[[Category:what_is]]
[[Category:wiki]]</text>
      <sha1>tqovc4vhd6q58p0d6lx0sza3mldboj6</sha1>
    </revision>
    <revision>
      <id>38916</id>
      <parentid>33812</parentid>
      <timestamp>2018-12-08T18:46:35Z</timestamp>
      <contributor>
        <username>Spt3125</username>
        <id>16</id>
      </contributor>
      <comment>various cleanup</comment>
      <origin>38916</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1463" sha1="if382x1n9lgtd0hfjfw8em5fzftukiu" xml:space="preserve">This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts.  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step).

{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>if382x1n9lgtd0hfjfw8em5fzftukiu</sha1>
    </revision>
    <revision>
      <id>80812</id>
      <parentid>38916</parentid>
      <timestamp>2021-11-14T21:33:20Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>80812</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1504" sha1="iynro7gd14swg1qm2emfw9o60ftsyki" xml:space="preserve">This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts.  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.

{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>iynro7gd14swg1qm2emfw9o60ftsyki</sha1>
    </revision>
    <revision>
      <id>84121</id>
      <parentid>80812</parentid>
      <timestamp>2022-01-15T20:15:37Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>add harmonics table</comment>
      <origin>84121</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1546" sha1="g1cmxxk2v0lmpeu9hlfc93ec4q0rogd" xml:space="preserve">This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts.  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.

{{harmonics in cet|125}}

== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>g1cmxxk2v0lmpeu9hlfc93ec4q0rogd</sha1>
    </revision>
    <revision>
      <id>84127</id>
      <parentid>84121</parentid>
      <timestamp>2022-01-15T20:19:06Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <minor/>
      <origin>84127</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1544" sha1="2yh4ors3r5ybdt5k72i5ndtou1b6xfp" xml:space="preserve">This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts.  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>2yh4ors3r5ybdt5k72i5ndtou1b6xfp</sha1>
    </revision>
    <revision>
      <id>111062</id>
      <parentid>84127</parentid>
      <timestamp>2023-05-02T03:56:34Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>111062</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1575" sha1="clafhopzqzweacxh04r1ue73ee2dwtb" xml:space="preserve">This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts making it stretched 4ed[[4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>clafhopzqzweacxh04r1ue73ee2dwtb</sha1>
    </revision>
    <revision>
      <id>111063</id>
      <parentid>111062</parentid>
      <timestamp>2023-05-02T03:57:10Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>111063</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1613" sha1="hl1mmirtux1w2t0icx35khzw0m9o7hk" xml:space="preserve">{{Infobox ET|48ed32}}
This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly compressed [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>hl1mmirtux1w2t0icx35khzw0m9o7hk</sha1>
    </revision>
    <revision>
      <id>111064</id>
      <parentid>111063</parentid>
      <timestamp>2023-05-02T03:57:52Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>111064</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1615" sha1="gghejsf9xkdwzl31isg0wepfwll2fos" xml:space="preserve">{{Infobox ET|96ed1024}}
This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly compressed [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>gghejsf9xkdwzl31isg0wepfwll2fos</sha1>
    </revision>
    <revision>
      <id>111065</id>
      <parentid>111064</parentid>
      <timestamp>2023-05-02T04:00:06Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>111065</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1615" sha1="155jd2jztrabhftxkw8bk8w2alco1ps" xml:space="preserve">{{Infobox ET|1ed72/67}}
This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly compressed [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>155jd2jztrabhftxkw8bk8w2alco1ps</sha1>
    </revision>
    <revision>
      <id>111066</id>
      <parentid>111065</parentid>
      <timestamp>2023-05-02T04:00:39Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>111066</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1617" sha1="3x3zc7zts6v3otzii3r7zoytvc24er4" xml:space="preserve">{{Infobox ET|1ed201/187}}
This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly compressed [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>3x3zc7zts6v3otzii3r7zoytvc24er4</sha1>
    </revision>
    <revision>
      <id>111070</id>
      <parentid>111066</parentid>
      <timestamp>2023-05-02T04:02:51Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>111070</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1625" sha1="s30p2trn0t8g77xoi0x3j2u83yfn9iu" xml:space="preserve">{{Infobox ET|1ed2683575/2496643}}
This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly compressed [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>s30p2trn0t8g77xoi0x3j2u83yfn9iu</sha1>
    </revision>
    <revision>
      <id>111075</id>
      <parentid>111070</parentid>
      <timestamp>2023-05-02T04:07:19Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <comment>6-digit continued fraction isn't needed</comment>
      <origin>111075</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1621" sha1="0hiylia6xwj7lbubol3eb8m968kb1kw" xml:space="preserve">{{Infobox ET|1ed33521/31186}}
This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly compressed [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>0hiylia6xwj7lbubol3eb8m968kb1kw</sha1>
    </revision>
    <revision>
      <id>111076</id>
      <parentid>111075</parentid>
      <timestamp>2023-05-02T04:07:34Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>111076</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1679" sha1="73h27iv7p6e2km3a7gqn2krihu39yuv" xml:space="preserve">{{Infobox ET|1ed33521/31186}}
&lt;!-- temporary until infobox ET works with cents value--&gt;
This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly compressed [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>73h27iv7p6e2km3a7gqn2krihu39yuv</sha1>
    </revision>
    <revision>
      <id>111216</id>
      <parentid>111076</parentid>
      <timestamp>2023-05-02T13:44:49Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>The bug has been fixed</comment>
      <origin>111216</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1614" sha1="mtexftn6zii5hl6m0njrg0q989futfn" xml:space="preserve">{{Infobox ET|48ed32}}

This [[nonoctave]] tuning uses equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly compressed [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>mtexftn6zii5hl6m0njrg0q989futfn</sha1>
    </revision>
    <revision>
      <id>127059</id>
      <parentid>111216</parentid>
      <timestamp>2023-11-07T19:37:11Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>make 1ed name primary, add APS/AS name, del refs to cET/equal-step, per https://en.xen.wiki/w/Talk:CET</comment>
      <origin>127059</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1789" sha1="8gqux7129rog5bdddio7wp4mmxzgtrg" xml:space="preserve">{{Infobox ET|48ed32}}

'''1 equal division of 125¢''' ('''1ed125c'''), also known as '''APS125¢''', is a [[nonoctave]] tuning using equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly compressed [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]] (125cET is an outdated name for this tuning system, standing for "125-cent equal temperament")
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>8gqux7129rog5bdddio7wp4mmxzgtrg</sha1>
    </revision>
    <revision>
      <id>127060</id>
      <parentid>127059</parentid>
      <timestamp>2023-11-07T19:37:28Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <minor/>
      <comment>Cmloegcmluin moved page [[125cET]] to [[1ed125c]]: make 1ed name primary per https://en.xen.wiki/w/Talk:CET</comment>
      <origin>127059</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1789" sha1="8gqux7129rog5bdddio7wp4mmxzgtrg" xml:space="preserve">{{Infobox ET|48ed32}}

'''1 equal division of 125¢''' ('''1ed125c'''), also known as '''APS125¢''', is a [[nonoctave]] tuning using equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly compressed [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]] (125cET is an outdated name for this tuning system, standing for "125-cent equal temperament")
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>8gqux7129rog5bdddio7wp4mmxzgtrg</sha1>
    </revision>
    <revision>
      <id>127068</id>
      <parentid>127060</parentid>
      <timestamp>2023-11-07T19:49:38Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>expand APS</comment>
      <origin>127068</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1832" sha1="pwhywhnh6nbt2mxgh8bvq7036nlchob" xml:space="preserve">{{Infobox ET|48ed32}}

'''1 equal division of 125¢''' ('''1ed125c'''), also known as '''arithmetic pitch sequence of 125¢''' ('''APS125¢'''), is a [[nonoctave]] tuning using equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly compressed [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | ordinal number
! | cents
! | interval name
|-
| | 0
| | 0
| | unison
|-
| | 1
| | 125
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | semifourth
|-
| | 3
| | 375
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | perfect fourth
|-
| | 5
| | 625
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | major sixth
|-
| | 8
| | 1000
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | classic ([[5-limit|5-limit]]) diminished octave.
|-
| | 10
| | 1250
| | 
|-
| | 11
| | 1375
| | 
|-
| | 12
| | 1500
| | 
|-
| | 13
| | 1625
| | 
|-
| | 14
| | 1750
| | 
|-
| | 15
| | 1875
| | 
|-
| | 16
| | 2000
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]] (125cET is an outdated name for this tuning system, standing for "125-cent equal temperament")
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>pwhywhnh6nbt2mxgh8bvq7036nlchob</sha1>
    </revision>
    <revision>
      <id>194914</id>
      <parentid>127068</parentid>
      <timestamp>2025-05-02T09:02:43Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <origin>194914</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2073" sha1="mp6mhrlqxbklg7k95sclsbkgs3yl5nu" xml:space="preserve">{{Infobox ET|48ed32}}

'''1 equal division of 125¢''' ('''1ed125c'''), also known as '''arithmetic pitch sequence of 125¢''' ('''APS125¢'''), is a [[nonoctave]] tuning using equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly stretched [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | Degree
! | Cents
! | Associated ratios
! | Interval name
|-
| | 0
| | 0
| | [[1/1]]
| | unison
|-
| | 1
| | 125
| | [[14/13]], [[15/14]]
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | [[15/13]], [[22/19]]
| | semifourth
|-
| | 3
| | 375
| | [[5/4]]
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | [[4/3]]
| | perfect fourth
|-
| | 5
| | 625
| | [[10/7]], [[23/16]]
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | [[17/11]], [[20/13]]
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | [[5/3]]
| | major sixth
|-
| | 8
| | 1000
| | [[16/9]]
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | [[21/11]], [[23/12]]
| | classic ([[5-limit|5-limit]]) diminished octave
|-
| | 10
| | 1250
| | [[33/16]]
| | 
|-
| | 11
| | 1375
| | 
| | 
|-
| | 12
| | 1500
| | 
| | 
|-
| | 13
| | 1625
| | 
| | 
|-
| | 14
| | 1750
| | 
| | 
|-
| | 15
| | 1875
| | 
| | 
|-
| | 16
| | 2000
| | 
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]] (125cET is an outdated name for this tuning system, standing for "125-cent equal temperament")
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>mp6mhrlqxbklg7k95sclsbkgs3yl5nu</sha1>
    </revision>
    <revision>
      <id>205652</id>
      <parentid>194914</parentid>
      <timestamp>2025-08-01T01:35:44Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Propose merge (very similar tunings)</comment>
      <origin>205652</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2145" sha1="mk9iho1d0a4nzqbzh021dzhbg4pcu4r" xml:space="preserve">{{todo|merge articles|inline=1|text=Merge [[4ed4/3]] and [[1ed125c]]?}}
{{Infobox ET|48ed32}}

'''1 equal division of 125¢''' ('''1ed125c'''), also known as '''arithmetic pitch sequence of 125¢''' ('''APS125¢'''), is a [[nonoctave]] tuning using equal steps of 125 cents each. This could be considered as dividing the approximate [[perfect fourth]] of 500 cents into 4 equal parts, making it very slightly stretched [[4ed4/3]].  It is equivalent to 9.6[[edo]], and is a subset of [[48edo]] (every fifth step). Therefore, it can be regarded as 48ed32.
{{harmonics in cet|125}}
== Intervals ==
{| class="wikitable"
|-
! | Degree
! | Cents
! | Associated ratios
! | Interval name
|-
| | 0
| | 0
| | [[1/1]]
| | unison
|-
| | 1
| | 125
| | [[14/13]], [[15/14]]
| | 2/3-tone, trienthird
|-
| | 2
| | 250
| | [[15/13]], [[22/19]]
| | semifourth
|-
| | 3
| | 375
| | [[5/4]]
| | narrow perde segah, marvelous major third, near just major third
|-
| | 4
| | 500
| | [[4/3]]
| | perfect fourth
|-
| | 5
| | 625
| | [[10/7]], [[23/16]]
| | pental diminished fifth, classic diminshed fifth
|-
| | 6
| | 750
| | [[17/11]], [[20/13]]
| | septendecimal subminor sixth
|-
| | 7
| | 875
| | [[5/3]]
| | major sixth
|-
| | 8
| | 1000
| | [[16/9]]
| | Pythagorean minor seventh
|-
| | 9
| | 1125
| | [[21/11]], [[23/12]]
| | classic ([[5-limit|5-limit]]) diminished octave
|-
| | 10
| | 1250
| | [[33/16]]
| | 
|-
| | 11
| | 1375
| | 
| | 
|-
| | 12
| | 1500
| | 
| | 
|-
| | 13
| | 1625
| | 
| | 
|-
| | 14
| | 1750
| | 
| | 
|-
| | 15
| | 1875
| | 
| | 
|-
| | 16
| | 2000
| | 
| | 
|}

==Scala file==
&lt;pre&gt;
! E:\cakewalk\scales\125cent.scl
!
125 cent tuning
4
!
125.00000
250.00000
375.00000
500.00000
&lt;/pre&gt;

==Music==
*[http://chrisvaisvil.com/canyon-diablo-fall/ Canyon Diablo Fall] by [[Chris Vaisvil]]
**http://micro.soonlabel.com/125cent/20130317_125cent.mp3
*[http://chrisvaisvil.com/crossing-over-125cet/ Crossing Over] by [[Chris Vaisvil]] (125cET is an outdated name for this tuning system, standing for "125-cent equal temperament")
**http://micro.soonlabel.com/125cent/20141011_125ct_crossing_over.mp3

[[Category:listen]]
[[Category:nonoctave]]</text>
      <sha1>mk9iho1d0a4nzqbzh021dzhbg4pcu4r</sha1>
    </revision>
    <revision>
      <id>232866</id>
      <parentid>205652</parentid>
      <timestamp>2026-06-28T08:36:47Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Merged</comment>
      <origin>232866</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="27" sha1="h0dqspfqganmibwoah926t7502kgqa0" xml:space="preserve">#REDIRECT [[4ed4/3#125cET]]</text>
      <sha1>h0dqspfqganmibwoah926t7502kgqa0</sha1>
    </revision>
  </page>
  <page>
    <title>125edo</title>
    <ns>0</ns>
    <id>114</id>
    <revision>
      <id>114</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>114</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="390" sha1="dqsfnx7pcfq7k2rvdo9l13737dzx0bn" xml:space="preserve">The ''125 equal temperament'' divides the octave into 125 equal parts of exactly 9.6 cents each. It defines the [[Optimal_patent_val|optimal patent val]] for 7- and 11-limit [[Marvel_temperaments|slender temperament]]. It tempers out 15625/15552 in the 5-limit; 225/224 and 4375/4374 in the 7-limit; 385/384 in the 11-limit; and 275/273 in the 13-limit.
[[Category:edo]]
[[Category:theory]]</text>
      <sha1>dqsfnx7pcfq7k2rvdo9l13737dzx0bn</sha1>
    </revision>
    <revision>
      <id>4546</id>
      <timestamp>2011-03-25T16:02:21Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 214071944 - Original comment: **</comment>
      <origin>4546</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2021" sha1="q2ujkkkuflv7myw06s8n1qhg4jjjz7k" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-25 16:02:21 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214071944&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //125 equal temperament// divides the octave into 125 equal parts of exactly 9.6 cents each. It defines the [[optimal patent val]] for 7- and 11-limit [[Marvel temperaments|slender temperament]]. It tempers out 15625/15552 in the 5-limit; 225/224 and 4375/4374 in the 7-limit; 385/384 in the 11-limit; and 275/273 in the 13-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;125edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;125 equal temperament&amp;lt;/em&amp;gt; divides the octave into 125 equal parts of exactly 9.6 cents each. It defines the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 7- and 11-limit &amp;lt;a class="wiki_link" href="/Marvel%20temperaments"&amp;gt;slender temperament&amp;lt;/a&amp;gt;. It tempers out 15625/15552 in the 5-limit; 225/224 and 4375/4374 in the 7-limit; 385/384 in the 11-limit; and 275/273 in the 13-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>q2ujkkkuflv7myw06s8n1qhg4jjjz7k</sha1>
    </revision>
    <revision>
      <id>40209</id>
      <parentid>114</parentid>
      <timestamp>2019-02-13T01:49:22Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>40209</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="586" sha1="ajb1u65rbrw3c3po6bo4kognav5lglg" xml:space="preserve">The ''125 equal temperament'' divides the octave into 125 equal parts of exactly 9.6 cents each. It defines the [[Optimal_patent_val|optimal patent val]] for 7- and 11-limit [[Marvel_temperaments|slender temperament]]. It tempers out 15625/15552 in the 5-limit; 225/224 and 4375/4374 in the 7-limit; 385/384 in the 11-limit; and 275/273 in the 13-limit. Being the cube closest to division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the cubic (fine) relative cent of [[1edo]].
[[Category:edo]]
[[Category:theory]]</text>
      <sha1>ajb1u65rbrw3c3po6bo4kognav5lglg</sha1>
    </revision>
    <revision>
      <id>54979</id>
      <parentid>40209</parentid>
      <timestamp>2020-12-04T23:13:07Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54979</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="612" sha1="mga3y8tbihcsg1kmg2c509gj5a1epal" xml:space="preserve">The ''125 equal temperament'' divides the octave into 125 equal parts of exactly 9.6 cents each. It defines the [[Optimal_patent_val|optimal patent val]] for 7- and 11-limit [[Marvel_temperaments|slender temperament]]. It tempers out 15625/15552 in the 5-limit; 225/224 and 4375/4374 in the 7-limit; 385/384 in the 11-limit; and 275/273 in the 13-limit. Being the cube closest to division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the cubic (fine) relative cent of [[1edo]].
[[Category:Equal divisions of the octave]]
[[Category:theory]]</text>
      <sha1>mga3y8tbihcsg1kmg2c509gj5a1epal</sha1>
    </revision>
    <revision>
      <id>73742</id>
      <parentid>54979</parentid>
      <timestamp>2021-07-05T07:54:19Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+prime error table, +temperament section</comment>
      <origin>73742</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1196" sha1="rw7r1zs9i3au7741c6vrhjfu4kqxu9o" xml:space="preserve">The '''125 equal temperament''' divides the octave into 125 equal parts of exactly 9.6 cents each. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] in the 11-limit; and [[275/273]] in the 13-limit. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], it has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

=== Prime harmonics ===
{{Primes in edo|125|prec=2}}

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Catakleismic]]</text>
      <sha1>rw7r1zs9i3au7741c6vrhjfu4kqxu9o</sha1>
    </revision>
    <revision>
      <id>73743</id>
      <parentid>73742</parentid>
      <timestamp>2021-07-05T07:59:24Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Expansion on 13-limit interpretation</comment>
      <origin>73743</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1400" sha1="thqfscl5nvfeb292ha50s81zmbh9wce" xml:space="preserve">The '''125 equal temperament''' divides the octave into 125 equal parts of exactly 9.6 cents each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125|prec=2}}

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Catakleismic]]</text>
      <sha1>thqfscl5nvfeb292ha50s81zmbh9wce</sha1>
    </revision>
    <revision>
      <id>73744</id>
      <parentid>73743</parentid>
      <timestamp>2021-07-05T08:05:15Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */</comment>
      <origin>73744</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1416" sha1="86sem0232wib0wnohxwq2ymf34stayh" xml:space="preserve">The '''125 equal temperament''' divides the octave into 125 equal parts of exactly 9.6 cents each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125|prec=2}}

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Catakleismic]]</text>
      <sha1>86sem0232wib0wnohxwq2ymf34stayh</sha1>
    </revision>
    <revision>
      <id>73894</id>
      <parentid>73744</parentid>
      <timestamp>2021-07-08T14:16:06Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ +RTT table</comment>
      <origin>73894</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2254" sha1="c7cdgnrye8iv3iej9x7z7am0tfnw4d0" xml:space="preserve">The '''125 equal temperament''' divides the octave into 125 equal parts of exactly 9.6 cents each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125|prec=2}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Catakleismic]]</text>
      <sha1>c7cdgnrye8iv3iej9x7z7am0tfnw4d0</sha1>
    </revision>
    <revision>
      <id>74131</id>
      <parentid>73894</parentid>
      <timestamp>2021-07-11T12:07:20Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* Prime harmonics */ the same prec is now estimated by EDO magnitude</comment>
      <origin>74131</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2247" sha1="av6cn26l8phv68r4ipfpz1ah6zek5fq" xml:space="preserve">The '''125 equal temperament''' divides the octave into 125 equal parts of exactly 9.6 cents each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Catakleismic]]</text>
      <sha1>av6cn26l8phv68r4ipfpz1ah6zek5fq</sha1>
    </revision>
    <revision>
      <id>74209</id>
      <parentid>74131</parentid>
      <timestamp>2021-07-12T16:04:46Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Rank-2 temperaments */ +link to pental and hemipental</comment>
      <origin>74209</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2402" sha1="aia12koyd0e74ue5e286hzs93xv54u5" xml:space="preserve">The '''125 equal temperament''' divides the octave into 125 equal parts of exactly 9.6 cents each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 24\125&lt;br&gt;(1\125)
| 230.4&lt;br&gt;(9.6)
| 8/7&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Catakleismic]]</text>
      <sha1>aia12koyd0e74ue5e286hzs93xv54u5</sha1>
    </revision>
    <revision>
      <id>75218</id>
      <parentid>74209</parentid>
      <timestamp>2021-07-29T10:31:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Improve intro and +category</comment>
      <origin>75218</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2589" sha1="tht7yv1vw85itmhf34octiez06r8p9o" xml:space="preserve">The '''125 equal divisions of the octave''' ('''125edo'''), or the '''125(-tone) equal temperament''' ('''125tet''', '''125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 125 [[equal]] parts of exactly 9.6 [[cent]]s each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 24\125&lt;br&gt;(1\125)
| 230.4&lt;br&gt;(9.6)
| 8/7&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Catakleismic]]
[[Category:Slender]]</text>
      <sha1>tht7yv1vw85itmhf34octiez06r8p9o</sha1>
    </revision>
    <revision>
      <id>75219</id>
      <parentid>75218</parentid>
      <timestamp>2021-07-29T10:35:19Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Rank-2 temperaments */ adopt (8ve reduced) mapping generator</comment>
      <origin>75219</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2591" sha1="fcqe3q1neo2e9792ccrcvdyqjk41dqx" xml:space="preserve">The '''125 equal divisions of the octave''' ('''125edo'''), or the '''125(-tone) equal temperament''' ('''125tet''', '''125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 125 [[equal]] parts of exactly 9.6 [[cent]]s each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Catakleismic]]
[[Category:Slender]]</text>
      <sha1>fcqe3q1neo2e9792ccrcvdyqjk41dqx</sha1>
    </revision>
    <revision>
      <id>77035</id>
      <parentid>75219</parentid>
      <timestamp>2021-08-28T13:12:19Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+infobox</comment>
      <origin>77035</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2789" sha1="ri1v4hqmo26sk9x0vsntk03uz4c8ygf" xml:space="preserve">{{Infobox ET
| Prime factorization = 5&lt;sup&gt;3&lt;/sup&gt;
| Step size = 9.60000¢
| Fifth = 73\125 (700.80¢)
| Major 2nd = 21\125 (202¢)
| Minor 2nd = 10\125 (96¢)
| Augmented 1sn = 11\125  (106¢)
}}

The '''125 equal divisions of the octave''' ('''125edo'''), or the '''125(-tone) equal temperament''' ('''125tet''', '''125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 125 [[equal]] parts of exactly 9.6 [[cent]]s each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Catakleismic]]
[[Category:Slender]]</text>
      <sha1>ri1v4hqmo26sk9x0vsntk03uz4c8ygf</sha1>
    </revision>
    <revision>
      <id>79777</id>
      <parentid>77035</parentid>
      <timestamp>2021-10-22T10:44:40Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Update infobox</comment>
      <origin>79777</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2779" sha1="l8tkygdepptny7371j6k70yre93y0t8" xml:space="preserve">{{Infobox ET
| Prime factorization = 5&lt;sup&gt;3&lt;/sup&gt;
| Step size = 9.600¢
| Fifth = 73\125 (700.800¢)
| Major 2nd = 21\125 (202¢)
| Semitones = 11:10 (106¢ : 96¢)
| Consistency = 9
}}

The '''125 equal divisions of the octave''' ('''125edo'''), or the '''125(-tone) equal temperament''' ('''125tet''', '''125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 125 [[equal]] parts of exactly 9.6 [[cent]]s each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Catakleismic]]
[[Category:Slender]]</text>
      <sha1>l8tkygdepptny7371j6k70yre93y0t8</sha1>
    </revision>
    <revision>
      <id>87445</id>
      <parentid>79777</parentid>
      <timestamp>2022-02-26T16:48:49Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87445</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2759" sha1="gbg2tjb6zbbt0beghbkm06ip21z0rws" xml:space="preserve">{{Infobox ET
| Prime factorization = 5&lt;sup&gt;3&lt;/sup&gt;
| Step size = 9.600¢
| Fifth = 73\125 (700.800¢)
| Major 2nd = 21\125 (202¢)
| Semitones = 11:10 (106¢ : 96¢)
| Consistency = 9
}}

The '''125 equal divisions of the octave''' ('''125edo'''), or the '''125(-tone) equal temperament''' ('''125tet''', '''125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 125 [[equal]] parts of exactly 9.6 [[cent]]s each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Catakleismic]]
[[Category:Slender]]</text>
      <sha1>gbg2tjb6zbbt0beghbkm06ip21z0rws</sha1>
    </revision>
    <revision>
      <id>87699</id>
      <parentid>87445</parentid>
      <timestamp>2022-02-27T20:06:29Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Rank-2 temperaments */ +semigamera</comment>
      <origin>87699</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2808" sha1="b5z0m6dh9q2whwli102xuoafl6ols5l" xml:space="preserve">{{Infobox ET
| Prime factorization = 5&lt;sup&gt;3&lt;/sup&gt;
| Step size = 9.600¢
| Fifth = 73\125 (700.800¢)
| Major 2nd = 21\125 (202¢)
| Semitones = 11:10 (106¢ : 96¢)
| Consistency = 9
}}

The '''125 equal divisions of the octave''' ('''125edo'''), or the '''125(-tone) equal temperament''' ('''125tet''', '''125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 125 [[equal]] parts of exactly 9.6 [[cent]]s each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}

[[Category:Equal divisions of the octave]]
[[Category:Catakleismic]]
[[Category:Slender]]</text>
      <sha1>b5z0m6dh9q2whwli102xuoafl6ols5l</sha1>
    </revision>
    <revision>
      <id>92935</id>
      <parentid>87699</parentid>
      <timestamp>2022-07-02T16:26:19Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92935</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2836" sha1="tbfgg8fmr697y5s13kfzvnawk6pfexv" xml:space="preserve">{{Infobox ET
| Prime factorization = 5&lt;sup&gt;3&lt;/sup&gt;
| Step size = 9.600¢
| Fifth = 73\125 (700.800¢)
| Major 2nd = 21\125 (202¢)
| Semitones = 11:10 (106¢ : 96¢)
| Consistency = 9
}}

The '''125 equal divisions of the octave''' ('''125edo'''), or the '''125(-tone) equal temperament''' ('''125tet''', '''125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 125 [[equal]] parts of exactly 9.6 [[cent]]s each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Catakleismic]]
[[Category:Slender]]</text>
      <sha1>tbfgg8fmr697y5s13kfzvnawk6pfexv</sha1>
    </revision>
    <revision>
      <id>96087</id>
      <parentid>92935</parentid>
      <timestamp>2022-10-04T16:04:13Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET now computes most parameters automatically</comment>
      <origin>96087</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2664" sha1="qxr0gt8h0x0kw2cijlh8o5qk6u4xd1b" xml:space="preserve">{{Infobox ET}}

The '''125 equal divisions of the octave''' ('''125edo'''), or the '''125(-tone) equal temperament''' ('''125tet''', '''125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 125 [[equal]] parts of exactly 9.6 [[cent]]s each. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. 

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Catakleismic]]
[[Category:Slender]]</text>
      <sha1>qxr0gt8h0x0kw2cijlh8o5qk6u4xd1b</sha1>
    </revision>
    <revision>
      <id>97305</id>
      <parentid>96087</parentid>
      <timestamp>2022-10-12T17:37:40Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>cleanup</comment>
      <origin>97305</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2715" sha1="co7fzqxpzgo910et1r5j8f0dv6cs37u" xml:space="preserve">{{Infobox ET}}

The '''125 equal divisions of the octave''' ('''125edo'''), or the '''125(-tone) equal temperament''' ('''125tet''', '''125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 125 [[equal]] parts of exactly 9.6 [[cent]]s each.  

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125}}

=== Miscellaneous properties ===
125 is 5 cubed. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Catakleismic]]
[[Category:Slender]]</text>
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    <revision>
      <id>109651</id>
      <parentid>97305</parentid>
      <timestamp>2023-04-26T12:24:31Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
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      <text bytes="2732" sha1="fvs8khxrfhuign5xjp81ar8i8v3vw0q" xml:space="preserve">{{Infobox ET}}

The '''125 equal divisions of the octave''' ('''125edo'''), or the '''125(-tone) equal temperament''' ('''125tet''', '''125et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 125 [[equal]] parts of exactly 9.6 [[cent]]s each.  

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Primes in edo|125}}

=== Miscellaneous properties ===
125 is 5 cubed. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per octave
! Generator&lt;br&gt;(reduced)
! Cents&lt;br&gt;(reduced)
! Associated&lt;br&gt;ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Ragismic_microtemperaments#Semigamera|Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Minortone]] / [[mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Hanson]] / [[catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Catakleismic]]</text>
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    <revision>
      <id>125018</id>
      <parentid>109651</parentid>
      <timestamp>2023-10-03T08:53:39Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
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      <text bytes="2361" sha1="hjhqdfshipmw62r7u05go2z9z2x0tlh" xml:space="preserve">{{Infobox ET}}
{{EDO intro|125}}

== Theory ==
125edo defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. It tempers out [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Miscellaneous properties ===
125 is 5 cubed. Being the cube closest to division of the octave by the Germanic [[Wikipedia: Long hundred|long hundred]], 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}

[[Category:Catakleismic]]</text>
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    <revision>
      <id>145212</id>
      <parentid>125018</parentid>
      <timestamp>2024-05-29T05:46:16Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
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      <format>text/x-wiki</format>
      <text bytes="2507" sha1="i5ra6f4j8y4j6spevx24au2u69s6jrq" xml:space="preserve">{{Infobox ET}}
{{EDO intro|125}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Miscellaneous properties ===
125 is 5 cubed. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[Category:Catakleismic]]</text>
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    </revision>
    <revision>
      <id>145213</id>
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      <timestamp>2024-05-29T05:48:19Z</timestamp>
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        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */</comment>
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      <text bytes="2589" sha1="nu5vbihiznnublnotoqtuyrr5dbw5ri" xml:space="preserve">{{Infobox ET}}
{{EDO intro|125}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Subsets and supersets ===
Since 125 factors into {{factorization|125}}, 125edo contains [[5edo]] and [[25edo]] as its subsets. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[Category:Catakleismic]]</text>
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    </revision>
    <revision>
      <id>154409</id>
      <parentid>145213</parentid>
      <timestamp>2024-09-09T01:56:40Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Skip disambiguation</comment>
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      <text bytes="2610" sha1="jqtdt72os1pfzpedt4cknpzu6y7fwsb" xml:space="preserve">{{Infobox ET}}
{{EDO intro|125}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Subsets and supersets ===
Since 125 factors into {{factorization|125}}, 125edo contains [[5edo]] and [[25edo]] as its subsets. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental (temperament)|Pental]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[Category:Catakleismic]]</text>
      <sha1>jqtdt72os1pfzpedt4cknpzu6y7fwsb</sha1>
    </revision>
    <revision>
      <id>165851</id>
      <parentid>154409</parentid>
      <timestamp>2024-11-16T03:21:30Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>165851</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2075" sha1="qxmd99lfzeam7mkgyneza3s2ilw08x6" xml:space="preserve">{{Infobox ET}}
{{EDO intro|125}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Subsets and supersets ===
Since 125 factors into {{factorization|125}}, 125edo contains [[5edo]] and [[25edo]] as its subsets. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

== Regular temperament properties ==
{{comma basis begin}}
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
{{comma basis end}}

=== Rank-2 temperaments ===
{{rank-2 begin}}
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental (temperament)|Pental]]
{{rank-2 end}}
{{orf}}

[[Category:Catakleismic]]</text>
      <sha1>qxmd99lfzeam7mkgyneza3s2ilw08x6</sha1>
    </revision>
    <revision>
      <id>166223</id>
      <parentid>165851</parentid>
      <timestamp>2024-11-16T12:53:07Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Partial undo</comment>
      <origin>166223</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2632" sha1="01r00nayqgf8wk02a5igxgybyylcx5d" xml:space="preserve">{{Infobox ET}}
{{EDO intro|125}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Subsets and supersets ===
Since 125 factors into {{factorization|125}}, 125edo contains [[5edo]] and [[25edo]] as its subsets. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Pental (temperament)|Pental]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[Category:Catakleismic]]</text>
      <sha1>01r00nayqgf8wk02a5igxgybyylcx5d</sha1>
    </revision>
    <revision>
      <id>166224</id>
      <parentid>166223</parentid>
      <timestamp>2024-11-16T12:53:28Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>166224</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2644" sha1="4hhenhzzfnru6ppw2xxzuyr1rxso9ay" xml:space="preserve">{{Infobox ET}}
{{EDO intro|125}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Subsets and supersets ===
Since 125 factors into {{factorization|125}}, 125edo contains [[5edo]] and [[25edo]] as its subsets. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br /&gt;(1\125)
| 249.6&lt;br /&gt;(9.6)
| 81/70&lt;br /&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br /&gt;(2\125)
| 499.2&lt;br /&gt;(19.2)
| 4/3&lt;br /&gt;(81/80)
| [[Pental (temperament)|Pental]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[Category:Catakleismic]]</text>
      <sha1>4hhenhzzfnru6ppw2xxzuyr1rxso9ay</sha1>
    </revision>
    <revision>
      <id>169125</id>
      <parentid>166224</parentid>
      <timestamp>2024-12-05T17:22:41Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <minor/>
      <comment>/* Subsets and supersets */ 1ed86.4c</comment>
      <origin>169125</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2735" sha1="egyago1h5jjntmq02eqqybtk1cf2er1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|125}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. 

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Subsets and supersets ===
Since 125 factors into {{factorization|125}}, 125edo contains [[5edo]] and [[25edo]] as its subsets. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. Using every 9th step, or [[1ed86.4c]] still encapsulates many of its best-tuned harmonics.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br /&gt;(1\125)
| 249.6&lt;br /&gt;(9.6)
| 81/70&lt;br /&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br /&gt;(2\125)
| 499.2&lt;br /&gt;(19.2)
| 4/3&lt;br /&gt;(81/80)
| [[Pental (temperament)|Pental]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[Category:Catakleismic]]</text>
      <sha1>egyago1h5jjntmq02eqqybtk1cf2er1</sha1>
    </revision>
    <revision>
      <id>171357</id>
      <parentid>169125</parentid>
      <timestamp>2024-12-17T13:15:37Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <origin>171357</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2868" sha1="fj8rki07f3jhjnltqx0ui2ah2zk059e" xml:space="preserve">{{Infobox ET}}
{{EDO intro|125}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Subsets and supersets ===
Since 125 factors into {{factorization|125}}, 125edo contains [[5edo]] and [[25edo]] as its subsets. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. Using every 9th step, or [[1ed86.4c]] still encapsulates many of its best-tuned harmonics.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br /&gt;(1\125)
| 249.6&lt;br /&gt;(9.6)
| 81/70&lt;br /&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br /&gt;(2\125)
| 499.2&lt;br /&gt;(19.2)
| 4/3&lt;br /&gt;(81/80)
| [[Pental (temperament)|Pental]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

[[Category:Catakleismic]]</text>
      <sha1>fj8rki07f3jhjnltqx0ui2ah2zk059e</sha1>
    </revision>
    <revision>
      <id>176349</id>
      <parentid>171357</parentid>
      <timestamp>2025-01-15T19:38:57Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176349</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2862" sha1="ccr67ainyzkq8wvc5fktwygdiehe9bq" xml:space="preserve">{{Infobox ET}}
{{EDO intro|125}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Subsets and supersets ===
Since 125 factors into {{factorization|125}}, 125edo contains [[5edo]] and [[25edo]] as its subsets. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. Using every 9th step, or [[1ed86.4c]] still encapsulates many of its best-tuned harmonics.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br /&gt;(1\125)
| 249.6&lt;br /&gt;(9.6)
| 81/70&lt;br /&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br /&gt;(2\125)
| 499.2&lt;br /&gt;(19.2)
| 4/3&lt;br /&gt;(81/80)
| [[Pental (temperament)|Pental]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:Catakleismic]]</text>
      <sha1>ccr67ainyzkq8wvc5fktwygdiehe9bq</sha1>
    </revision>
    <revision>
      <id>181334</id>
      <parentid>176349</parentid>
      <timestamp>2025-02-18T16:59:50Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181334</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2857" sha1="70deb0963jgnahwmtdz44o0dy3tubv6" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Subsets and supersets ===
Since 125 factors into {{factorization|125}}, 125edo contains [[5edo]] and [[25edo]] as its subsets. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. Using every 9th step, or [[1ed86.4c]] still encapsulates many of its best-tuned harmonics.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -198 125 }}
| [{{val| 125 198 }}]
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| [{{val| 125 198 290 }}]
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| [{{val| 125 198 290 351 }}]
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| [{{val| 125 198 290 351 432 }}]
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| [{{val| 125 198 290 351 432 462 }}] (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br /&gt;(1\125)
| 249.6&lt;br /&gt;(9.6)
| 81/70&lt;br /&gt;(176/175)
| [[Hemipental]]
|-
| 5
| 52\125&lt;br /&gt;(2\125)
| 499.2&lt;br /&gt;(19.2)
| 4/3&lt;br /&gt;(81/80)
| [[Pental (temperament)|Pental]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:Catakleismic]]</text>
      <sha1>70deb0963jgnahwmtdz44o0dy3tubv6</sha1>
    </revision>
    <revision>
      <id>181415</id>
      <parentid>181334</parentid>
      <timestamp>2025-02-19T14:28:56Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ cleanup and update</comment>
      <origin>181415</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2831" sha1="4m5hmkgvjq6ibwxn1rzeax7axuk675t" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Subsets and supersets ===
Since 125 factors into {{factorization|125}}, 125edo contains [[5edo]] and [[25edo]] as its subsets. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. Using every 9th step, or [[1ed86.4c]] still encapsulates many of its best-tuned harmonics.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:Catakleismic]]</text>
      <sha1>4m5hmkgvjq6ibwxn1rzeax7axuk675t</sha1>
    </revision>
    <revision>
      <id>192247</id>
      <parentid>181415</parentid>
      <timestamp>2025-04-15T12:36:14Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ +octave stretch</comment>
      <origin>192247</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3230" sha1="8903drrsg94lpqnp5cpv9x0g1hspek8" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. Using every 9th step, or [[1ed86.4c]] still encapsulates many of its best-tuned harmonics.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:Catakleismic]]</text>
      <sha1>8903drrsg94lpqnp5cpv9x0g1hspek8</sha1>
    </revision>
    <revision>
      <id>195529</id>
      <parentid>192247</parentid>
      <timestamp>2025-05-06T17:28:59Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>add thunderclysmic</comment>
      <origin>195529</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3327" sha1="sl0wou49an178ouldb1995r21vtfza5" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. Using every 9th step, or [[1ed86.4c]] still encapsulates many of its best-tuned harmonics.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br/&gt;(76.8)
| 6/5&lt;br/&gt;(24/23~23/22~22/21)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:Catakleismic]]</text>
      <sha1>sl0wou49an178ouldb1995r21vtfza5</sha1>
    </revision>
    <revision>
      <id>195810</id>
      <parentid>195529</parentid>
      <timestamp>2025-05-08T18:21:24Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>/* Prime harmonics */ good approximate (no-13's no-41's) 67-limit system (flat-tending)</comment>
      <origin>195810</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4704" sha1="ijdbag9dh0rsimagcmain0er4nkbtxh" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd-limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

=== Prime harmonics ===
{{Harmonics in equal|125|columns=12}}
{{Harmonics in equal|125|start=13|columns=7}}
{{Harmonics in equal|125|start=20|columns=10|collapsed=true}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. Using every 9th step, or [[1ed86.4c]] still encapsulates many of its best-tuned harmonics.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br/&gt;(76.8)
| 6/5&lt;br/&gt;(24/23~23/22~22/21)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:Catakleismic]]</text>
      <sha1>ijdbag9dh0rsimagcmain0er4nkbtxh</sha1>
    </revision>
    <revision>
      <id>195812</id>
      <parentid>195810</parentid>
      <timestamp>2025-05-08T18:22:04Z</timestamp>
      <contributor>
        <username>Godtone</username>
        <id>261</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ hyphen isnt needed here</comment>
      <origin>195812</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4704" sha1="j5dy1fh1mzqppmlrk9z9xz6y0or1eh8" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

=== Prime harmonics ===
{{Harmonics in equal|125|columns=12}}
{{Harmonics in equal|125|start=13|columns=7}}
{{Harmonics in equal|125|start=20|columns=10|collapsed=true}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]]. Using every 9th step, or [[1ed86.4c]] still encapsulates many of its best-tuned harmonics.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br/&gt;(76.8)
| 6/5&lt;br/&gt;(24/23~23/22~22/21)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

[[Category:Catakleismic]]</text>
      <sha1>j5dy1fh1mzqppmlrk9z9xz6y0or1eh8</sha1>
    </revision>
    <revision>
      <id>205377</id>
      <parentid>195812</parentid>
      <timestamp>2025-07-30T03:41:42Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Merged in relevant content from [[1ed86.4c]], with a few edits for clarity.</comment>
      <origin>205377</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5199" sha1="eoau0bn2hszqz1tizuexfofgnttpo4y" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

=== Prime harmonics ===
{{Harmonics in equal|125|columns=12}}
{{Harmonics in equal|125|start=13|columns=7}}
{{Harmonics in equal|125|start=20|columns=10|collapsed=true}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''') still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{harmonics in cet|86.4|prec=1}}
{{harmonics in cet|86.4|prec=1|start=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br/&gt;(76.8)
| 6/5&lt;br/&gt;(24/23~23/22~22/21)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>eoau0bn2hszqz1tizuexfofgnttpo4y</sha1>
    </revision>
    <revision>
      <id>205381</id>
      <parentid>205377</parentid>
      <timestamp>2025-07-30T03:45:03Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>/* Subsets and supersets */ Mention 13.888edo</comment>
      <origin>205381</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5247" sha1="reu1ge3w75g793tqant2zrl8dx8otwf" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

=== Prime harmonics ===
{{Harmonics in equal|125|columns=12}}
{{Harmonics in equal|125|start=13|columns=7}}
{{Harmonics in equal|125|start=20|columns=10|collapsed=true}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{harmonics in cet|86.4|prec=1}}
{{harmonics in cet|86.4|prec=1|start=11}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br/&gt;(76.8)
| 6/5&lt;br/&gt;(24/23~23/22~22/21)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>reu1ge3w75g793tqant2zrl8dx8otwf</sha1>
    </revision>
    <revision>
      <id>209855</id>
      <parentid>205381</parentid>
      <timestamp>2025-09-12T17:10:05Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Omg pls don't creep up the limit of the harmonics table. This entire paragraph of readability hell has no place in the theory section, either.</comment>
      <origin>209855</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5405" sha1="s6tinvbn1bzeka759bxx3k933035508" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125|columns=9}}
{{Harmonics in equal|125|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>s6tinvbn1bzeka759bxx3k933035508</sha1>
    </revision>
    <revision>
      <id>209870</id>
      <parentid>209855</parentid>
      <timestamp>2025-09-13T14:05:36Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>columns=11 is standard</comment>
      <origin>209870</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5395" sha1="itmkupauzb5yfz5dcgvyqdv92b5jjkg" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}
{{Harmonics in equal|125|columns=9|start=12|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>itmkupauzb5yfz5dcgvyqdv92b5jjkg</sha1>
    </revision>
    <revision>
      <id>217456</id>
      <parentid>209870</parentid>
      <timestamp>2025-11-21T22:47:52Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <minor/>
      <comment>/* Octave stretch */ fixed harmonics</comment>
      <origin>217456</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5394" sha1="c5nwtk2hiy43uow2s6o219h3liorlyk" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}
{{Harmonics in equal|125|columns=9|start=12|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 11 can be improved, and moreover the approximated harmonic 13 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this.

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>c5nwtk2hiy43uow2s6o219h3liorlyk</sha1>
    </revision>
    <revision>
      <id>217501</id>
      <parentid>217456</parentid>
      <timestamp>2025-11-22T14:52:27Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Undo revision [[Special:Diff/217456|217456]] by [[Special:Contributions/Overthink|Overthink]] ([[User talk:Overthink|talk]]). Was never wrong in the first place</comment>
      <origin>217501</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5395" sha1="itmkupauzb5yfz5dcgvyqdv92b5jjkg" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}
{{Harmonics in equal|125|columns=9|start=12|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>itmkupauzb5yfz5dcgvyqdv92b5jjkg</sha1>
    </revision>
    <revision>
      <id>223091</id>
      <parentid>217501</parentid>
      <timestamp>2026-02-03T07:08:14Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>/* Music */ add scales section</comment>
      <origin>223091</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5821" sha1="2sy6050rvm4t3j5c81b5fzcx3odlyo2" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}
{{Harmonics in equal|125|columns=9|start=12|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Catakleismic]][7]: 7 26 7 26 7 26 26
* Catakleismic[11]: 7 19 7 7 19 7 7 19 7 19 7
* Catakleismic[15]: 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7
* Catakleismic[19]: 7 7 5 7 7 7 7 5 7 7 7 5 7 7 7 7 5 7 7
* Catakleismic[34]: 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5
* Catakleismic[53]: 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>2sy6050rvm4t3j5c81b5fzcx3odlyo2</sha1>
    </revision>
    <revision>
      <id>223180</id>
      <parentid>223091</parentid>
      <timestamp>2026-02-04T07:12:22Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Scales */ add 3</comment>
      <origin>223180</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6029" sha1="p17m2z7bvpa25xuvqi7vxi9o6pkx0xb" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}
{{Harmonics in equal|125|columns=9|start=12|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Catakleismic]][7]: 7 26 7 26 7 26 26
* Catakleismic[11]: 7 19 7 7 19 7 7 19 7 19 7
* Catakleismic[15]: 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7
* Catakleismic[19]: 7 7 5 7 7 7 7 5 7 7 7 5 7 7 7 7 5 7 7
* Catakleismic[34]: 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5
** Subsets
*** 21 31 21 14 14 24 (''canopy{{idio}} (approximated from [[16afdo]])'')
*** 21 12 26 14 33 19 (''resembles [[antipental blues]]'')
*** 19 21 21 19 26 19 (''resembles whole tone scale, [[6edo]]'')
* Catakleismic[53]: 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>p17m2z7bvpa25xuvqi7vxi9o6pkx0xb</sha1>
    </revision>
    <revision>
      <id>223181</id>
      <parentid>223180</parentid>
      <timestamp>2026-02-04T07:12:36Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Scales */</comment>
      <origin>223181</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6028" sha1="tphp7z534t13ocldildp7x43kiadokj" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}
{{Harmonics in equal|125|columns=9|start=12|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Catakleismic]][7]: 7 26 7 26 7 26 26
* Catakleismic[11]: 7 19 7 7 19 7 7 19 7 19 7
* Catakleismic[15]: 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7
* Catakleismic[19]: 7 7 5 7 7 7 7 5 7 7 7 5 7 7 7 7 5 7 7
* Catakleismic[34]: 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5
** Subsets
*** 21 31 21 14 14 24 (''canopy{{idio}}, approximated from [[16afdo]]'')
*** 21 12 26 14 33 19 (''resembles [[antipental blues]]'')
*** 19 21 21 19 26 19 (''resembles whole tone scale, [[6edo]]'')
* Catakleismic[53]: 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>tphp7z534t13ocldildp7x43kiadokj</sha1>
    </revision>
    <revision>
      <id>223182</id>
      <parentid>223181</parentid>
      <timestamp>2026-02-04T07:36:07Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Scales */  add 3</comment>
      <origin>223182</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6181" sha1="a771w4gmruirk5qbfdtze697dxkx3gi" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}
{{Harmonics in equal|125|columns=9|start=12|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Catakleismic]][7]: 7 26 7 26 7 26 26
* Catakleismic[11]: 7 19 7 7 19 7 7 19 7 19 7
* Catakleismic[15]: 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7
* Catakleismic[19]: 7 7 5 7 7 7 7 5 7 7 7 5 7 7 7 7 5 7 7
* Catakleismic[34]: 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5
** Subsets
*** 21 38 14 14 14 24 (''city-at-dawn scale''{{idio}})
*** 40 19 14 14 14 24 (''city-at-midday scale''{{idio}})
*** 40 19 14 28 12 12 (''city-at-dusk scale''{{idio}})
*** 40 19 14 14 26 12 (''city-at-midnight scale''{{idio}})
*** 21 12 26 14 33 19 (''resembles [[antipental blues]]'')
*** 19 21 21 19 26 19 (''resembles whole tone scale, [[6edo]]'')
* Catakleismic[53]: 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>a771w4gmruirk5qbfdtze697dxkx3gi</sha1>
    </revision>
    <revision>
      <id>223185</id>
      <parentid>223182</parentid>
      <timestamp>2026-02-04T08:04:12Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Scales */</comment>
      <origin>223185</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6286" sha1="ixp7415usba9hyrmuiqlmnz30g3h284" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}
{{Harmonics in equal|125|columns=9|start=12|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Catakleismic]][7]: 7 26 7 26 7 26 26
* Catakleismic[11]: 7 19 7 7 19 7 7 19 7 19 7
* Catakleismic[15]: 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7
* Catakleismic[19]: 7 7 5 7 7 7 7 5 7 7 7 5 7 7 7 7 5 7 7
* Catakleismic[34]: 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5
* Catakleismic[53]: 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3

; Catakleismic[34] subsets
{{Idiosyncratic terms}}
* City-at-dawn scale: 21 38 14 14 14 24
* City-at-midday scale: 40 19 14 14 14 24
* City-at-dusk scale: 40 19 14 28 12 12
* City-at-midnight scale: 40 19 14 14 26 12
* Port-at-dawn scale: 26 33 14 19 9 24
* Port-at-midday scale: 40 19 14 19 9 24
* Port-at-dusk scale: 40 19 14 28 10 14
* Port-at-midnight scale: 40 19 14 19 19 14
* Pseudo-minor blues: 21 12 26 14 33 19
* Pseudo-whole tone scale 19 21 21 19 26 19

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>ixp7415usba9hyrmuiqlmnz30g3h284</sha1>
    </revision>
    <revision>
      <id>223190</id>
      <parentid>223185</parentid>
      <timestamp>2026-02-04T08:10:19Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Scales */</comment>
      <origin>223190</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6407" sha1="q1vofksky8srcifi6rjdoz57y18p7lx" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}
{{Harmonics in equal|125|columns=9|start=12|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Catakleismic]][7]: 7 26 7 26 7 26 26
* Catakleismic[11]: 7 19 7 7 19 7 7 19 7 19 7
* Catakleismic[15]: 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7
* Catakleismic[19]: 7 7 5 7 7 7 7 5 7 7 7 5 7 7 7 7 5 7 7
* Catakleismic[34]: 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5
* Catakleismic[53]: 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3

; Catakleismic[34] subsets
{{Idiosyncratic terms|Meta-tropolis and its subsets named by [[Budjarn Lambeth]]}}
* Meta-tropolis scale: 14 7 5 14 19 14 14 5 9 10 2 12
** City-at-dawn scale: 21 38 14 14 14 24
** City-at-midday scale: 40 19 14 14 14 24
** City-at-dusk scale: 40 19 14 28 12 12
** City-at-midnight scale: 40 19 14 14 26 12
** Port-at-dawn scale: 26 33 14 19 9 24
** Port-at-midday scale: 40 19 14 19 9 24
** Port-at-dusk scale: 40 19 14 28 10 14
** Port-at-midnight scale: 40 19 14 19 19 14
* Pseudo-minor blues: 21 12 26 14 33 19
* Pseudo-whole tone scale 19 21 21 19 26 19

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>q1vofksky8srcifi6rjdoz57y18p7lx</sha1>
    </revision>
    <revision>
      <id>223193</id>
      <parentid>223190</parentid>
      <timestamp>2026-02-04T08:12:18Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Scales */</comment>
      <origin>223193</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6417" sha1="5s8bpwq6vfteb1or1dlxm3yrc8921c2" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}
{{Harmonics in equal|125|columns=9|start=12|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Catakleismic]][7]: 7 26 7 26 7 26 26
* Catakleismic[11]: 7 19 7 7 19 7 7 19 7 19 7
* Catakleismic[15]: 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7
* Catakleismic[19]: 7 7 5 7 7 7 7 5 7 7 7 5 7 7 7 7 5 7 7
* Catakleismic[34]: 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5
* Catakleismic[53]: 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3

; Catakleismic[34] subsets
{{Idiosyncratic terms|Meta-tropolis and its subsets named by [[Budjarn Lambeth]]}}
* Meta-tropolis scale (12-tone): 14 7 5 14 19 14 14 5 9 10 2 12
** City-at-dawn scale: 21 38 14 14 14 24
** City-at-midday scale: 40 19 14 14 14 24
** City-at-dusk scale: 40 19 14 28 12 12
** City-at-midnight scale: 40 19 14 14 26 12
** Port-at-dawn scale: 26 33 14 19 9 24
** Port-at-midday scale: 40 19 14 19 9 24
** Port-at-dusk scale: 40 19 14 28 10 14
** Port-at-midnight scale: 40 19 14 19 19 14
* Pseudo-minor blues: 21 12 26 14 33 19
* Pseudo-whole tone scale 19 21 21 19 26 19

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>5s8bpwq6vfteb1or1dlxm3yrc8921c2</sha1>
    </revision>
    <revision>
      <id>223713</id>
      <parentid>223193</parentid>
      <timestamp>2026-02-10T00:24:43Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Subsets and supersets */ mark idiosyncratic term</comment>
      <origin>223713</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6425" sha1="tvkc4x7h8kusyrvq8ymweonnni8dcym" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}
{{Harmonics in equal|125|columns=9|start=12|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent{{Idio}} of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Catakleismic]][7]: 7 26 7 26 7 26 26
* Catakleismic[11]: 7 19 7 7 19 7 7 19 7 19 7
* Catakleismic[15]: 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7
* Catakleismic[19]: 7 7 5 7 7 7 7 5 7 7 7 5 7 7 7 7 5 7 7
* Catakleismic[34]: 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5
* Catakleismic[53]: 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3

; Catakleismic[34] subsets
{{Idiosyncratic terms|Meta-tropolis and its subsets named by [[Budjarn Lambeth]]}}
* Meta-tropolis scale (12-tone): 14 7 5 14 19 14 14 5 9 10 2 12
** City-at-dawn scale: 21 38 14 14 14 24
** City-at-midday scale: 40 19 14 14 14 24
** City-at-dusk scale: 40 19 14 28 12 12
** City-at-midnight scale: 40 19 14 14 26 12
** Port-at-dawn scale: 26 33 14 19 9 24
** Port-at-midday scale: 40 19 14 19 9 24
** Port-at-dusk scale: 40 19 14 28 10 14
** Port-at-midnight scale: 40 19 14 19 19 14
* Pseudo-minor blues: 21 12 26 14 33 19
* Pseudo-whole tone scale 19 21 21 19 26 19

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
      <sha1>tvkc4x7h8kusyrvq8ymweonnni8dcym</sha1>
    </revision>
    <revision>
      <id>225735</id>
      <parentid>223713</parentid>
      <timestamp>2026-03-13T13:31:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct" to "[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct"</comment>
      <origin>225735</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6480" sha1="1uym0ztwwhagdohyug47bhfzmscwxdq" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament [[tempering out|tempers out]] [[15625/15552]] in the 5-limit; [[225/224]] and [[4375/4374]] in the 7-limit; [[385/384]] and [[540/539]] in the 11-limit. It defines the [[optimal patent val]] for 7- and 11-limit [[slender]] temperament. In the 13-limit the 125f val {{val| 125 198 290 351 432 462 }} does a better job, where it tempers out [[169/168]], [[325/324]], [[351/350]], [[625/624]] and [[676/675]], providing a good tuning for [[catakleismic]]. Among well-known intervals, the approximation of [[10/9]], as 19 steps, is notable for being a strong convergent, within 0.004 cents.

=== Prime harmonics ===
{{Harmonics in equal|125}}
{{Harmonics in equal|125|columns=9|start=12|collapsed=true|title=Approximation of prime harmonics in 125edo (continued)}}

=== Octave stretch ===
125edo's approximated harmonics 3, 5, and 13 can be improved, and moreover the approximated harmonic 11 can be brought to consistency, by slightly [[stretched and compressed tuning|stretching the octave]], though it comes at the expense of somewhat less accurate approximations of 7, 17, and 19. Tunings such as [[198edt]] and [[323ed6]] are great demonstrations of this. 

=== Subsets and supersets ===
Since 125 factors into primes as 5&lt;sup&gt;3&lt;/sup&gt;, 125edo contains [[5edo]] and [[25edo]] as subset edos. Being the cube closest to division of the octave by the Germanic {{w|long hundred}}, 125edo has a unit step which is the cubic (fine) relative cent{{Idio}} of [[1edo]].

Using every 9th step of 125edo, '''86.4-cET''' (also known as '''1ed86.4{{cent}}''', and sometimes '''13.888edo''' by approximation) still encapsulates many of its best-tuned harmonics, such as the 3rd, 7th, 9th and 11th. It has been voted "monthly tuning" multiple times on the [[Monthly Tunings]] Facebook group. This subset is closely related to [[22edt]], another tuning that closely approximates [[42zpi]].
{{Harmonics in cet|86.4|prec=1|columns=11|collapsed=true}}
{{Harmonics in cet|86.4|prec=1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 1ed86.4c (continued)}}

== Approximation to JI ==
=== Higher-limit JI ===
125edo does well as a flat-tending no-13's no-41's 67-limit system by using error cancellations to achieve frequently accurate approximations of the corresponding [[odd limit]]. Due to the relative simplicity of intervals of 13, harmonies of 13 are usable in practice, but will run into numerous inconsistencies no matter which mapping for 13 you use (the flat one or the sharp one, the latter being used by the [[patent val]]). In the no-13's no-41's 67-odd-limit (so omitting 39 and 65*), there are 31 inconsistent interval pairs out of 362 interval pairs total, meaning less than 9% of intervals are mapped to their second-best mapping rather than their best. In ascending order, these intervals are: 56/55, 50/49, 34/33, 33/32, 57/55, 55/53, 49/46, 55/51, 49/45, 54/49, 49/44, 28/25, 55/49, 63/55, 55/48, 38/33, 64/55, 33/28, 28/23, 60/49, 68/55, 56/45, 61/49, 66/53, 14/11, 55/42, 45/34, 66/49, 23/17, 34/25, 76/55, 55/38, 25/17, and their [[octave complement]]s. (Therefore, all 331 other intervals are mapped with strictly less than 4.8{{cent}} of error.)

&lt;nowiki&gt;*&lt;/nowiki&gt; including odd 39 and/or 65 is possible if you don't mind about a dozen more inconsistent interval pairs so that there's more inconsistencies in total but with more coverage.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -198 125 }}
| {{Mapping| 125 198 }}
| +0.364
| 0.364
| 3.80
|-
| 2.3.5
| 15625/15552, 17433922005/17179869184
| {{Mapping| 125 198 290 }}
| +0.575
| 0.421
| 4.39
|-
| 2.3.5.7
| 225/224, 4375/4374, 589824/588245
| {{Mapping| 125 198 290 351 }}
| +0.362
| 0.519
| 5.40
|-
| 2.3.5.7.11
| 225/224, 385/384, 1331/1323, 4375/4374
| {{Mapping| 125 198 290 351 432 }}
| +0.528
| 0.570
| 5.94
|-
| 2.3.5.7.11.13
| 169/168, 225/224, 325/324, 385/384, 1331/1323
| {{Mapping| 125 198 290 351 432 462 }} (125f)
| +0.680
| 0.622
| 6.47
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;ratio*
! Temperaments
|-
| 1
| 4\125
| 38.4
| 49/48
| [[Slender]]
|-
| 1
| 12\125
| 115.2
| 77/72
| [[Semigamera]]
|-
| 1
| 19\125
| 182.4
| 10/9
| [[Mitonic]]
|-
| 1
| 24\125
| 230.4
| 8/7
| [[Gamera]]
|-
| 1
| 33\125
| 316.8
| 6/5
| [[Catakleismic]]
|-
| 1
| 52\125
| 499.2
| 4/3
| [[Gracecordial]]
|-
| 1
| 61\125
| 585.6
| 7/5
| [[Merman]]
|-
| 5
| 26\125&lt;br&gt;(1\125)
| 249.6&lt;br&gt;(9.6)
| 81/70&lt;br&gt;(176/175)
| [[Hemiquintile]]
|-
| 5
| 52\125&lt;br&gt;(2\125)
| 499.2&lt;br&gt;(19.2)
| 4/3&lt;br&gt;(81/80)
| [[Quintile]]
|-
| 5
| 33\125&lt;br&gt;(8\125)
| 316.8&lt;br&gt;(76.8)
| 6/5&lt;br&gt;(23/22)
| [[Thunderclysmic]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Scales ==
* [[Catakleismic]][7]: 7 26 7 26 7 26 26
* Catakleismic[11]: 7 19 7 7 19 7 7 19 7 19 7
* Catakleismic[15]: 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7
* Catakleismic[19]: 7 7 5 7 7 7 7 5 7 7 7 5 7 7 7 7 5 7 7
* Catakleismic[34]: 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5 5 2 5 2 5 2 5 5 2 5 2 5 2 5 2 5
* Catakleismic[53]: 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 3 2 2 3

; Catakleismic[34] subsets
{{Idiosyncratic terms|Meta-tropolis and its subsets named by [[Budjarn Lambeth]]}}
* Meta-tropolis scale (12-tone): 14 7 5 14 19 14 14 5 9 10 2 12
** City-at-dawn scale: 21 38 14 14 14 24
** City-at-midday scale: 40 19 14 14 14 24
** City-at-dusk scale: 40 19 14 28 12 12
** City-at-midnight scale: 40 19 14 14 26 12
** Port-at-dawn scale: 26 33 14 19 9 24
** Port-at-midday scale: 40 19 14 19 9 24
** Port-at-dusk scale: 40 19 14 28 10 14
** Port-at-midnight scale: 40 19 14 19 19 14
* Pseudo-minor blues: 21 12 26 14 33 19
* Pseudo-whole tone scale 19 21 21 19 26 19

== Music ==
* ''[https://soundcloud.com/morphosyntax-1/the-butterflys-dream The Butterfly’s Dream]'' (in 86.4-cET) by [[Herman Miller]] (2022)

[[Category:Catakleismic]]</text>
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  <page>
    <title>126/125</title>
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    <id>115</id>
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      <id>115</id>
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      <text bytes="1220" sha1="7ew93hndwrhp4lse6a4r6aa554x65px" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[:de:126/125 Deutsch]]&lt;/span&gt;

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the only superparticular [[7-limit|7-limit]] [[Comma|comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12/7|12/7]] falls short of three [[6/5|6/5]] minor thirds. It is also the amount by which two [[5/3|5/3 major sixths]] (octave-reduced) exceed the [[7/5|7/5 tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6|7/6 septimal minor third]]. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

Tempering it out leads to [[Starling_temperaments#Starling comma|starling temperament]], and three minor thirds plus a [[7/6|7/6]] subminor third, when 126/125 is tempered out, gives the [[starling_tetrad|starling tetrad]] or septimal semicomma diminished seventh chord.

[http://en.wikipedia.org/wiki/Septimal_semicomma http://en.wikipedia.org/wiki/Septimal_semicomma]      [[Category:7-limit]]
[[Category:comma]]
[[Category:interval]]
[[Category:septimal]]
[[Category:starling]]</text>
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      <comment>**Imported revision 600277782 - Original comment: **</comment>
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      <text bytes="4077" sha1="am5ywxizs771zfu829d2zczfgx5erw2" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:toddiharrop|toddiharrop]] and made on &lt;tt&gt;2016-11-23 16:06:29 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;600277782&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[xenharmonie/126_125|Deutsch]]
&amp;lt;/span&amp;gt;
The **starling comma** or **septimal semicomma**, 126/125 (about 13.8 cents), is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12_7|12/7]] falls short of three [[6_5|6/5]] minor thirds. It is also the amount by which two [[5_3|5/3 major sixths]] (octave-reduced) exceed the [[7_5|7/5 tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7_6|7/6 septimal minor third]]. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

Tempering it out leads to [[Starling temperaments#Starling%20comma|starling temperament]], and three minor thirds plus a [[7_6|7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.


[[http://en.wikipedia.org/wiki/Septimal_semicomma]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
The &amp;lt;strong&amp;gt;starling comma&amp;lt;/strong&amp;gt; or &amp;lt;strong&amp;gt;septimal semicomma&amp;lt;/strong&amp;gt;, 126/125 (about 13.8 cents), is the only superparticular &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt; which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which &amp;lt;a class="wiki_link" href="/12_7"&amp;gt;12/7&amp;lt;/a&amp;gt; falls short of three &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt; minor thirds. It is also the amount by which two &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3 major sixths&amp;lt;/a&amp;gt; (octave-reduced) exceed the &amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5 tritone&amp;lt;/a&amp;gt;, and the amount by which three 5/3s (octave-reduced) fall short of the &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6 septimal minor third&amp;lt;/a&amp;gt;. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Tempering it out leads to &amp;lt;a class="wiki_link" href="/Starling%20temperaments#Starling%20comma"&amp;gt;starling temperament&amp;lt;/a&amp;gt;, and three minor thirds plus a &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt; subminor third, when 126/125 is tempered out, gives the &amp;lt;a class="wiki_link" href="/starling%20tetrad"&amp;gt;starling tetrad&amp;lt;/a&amp;gt; or septimal semicomma diminished seventh chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_semicomma" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Septimal_semicomma&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 581264315 - Original comment: **</comment>
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      <text bytes="3769" sha1="pwga6blug0ur7ir2lp7x07fj7xcdg3u" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2016-04-26 11:20:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;581264315&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[xenharmonie/126_125|Deutsch]]
&amp;lt;/span&amp;gt;
The **starling comma** or **septimal semicomma**, 126/125, is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12_7|12/7]] falls short of three [[6_5|6/5]] minor thirds. It is also the amount by which two [[5_3|5/3 major sixths]] (octave-reduced) exceed the [[7_5|7/5 tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7_6|7/6 septimal minor third]].

Tempering it out leads to [[Starling temperaments#Starling%20comma|starling temperament]], and three minor thirds plus a [[7_6|7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.


[[http://en.wikipedia.org/wiki/Septimal_semicomma]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
The &amp;lt;strong&amp;gt;starling comma&amp;lt;/strong&amp;gt; or &amp;lt;strong&amp;gt;septimal semicomma&amp;lt;/strong&amp;gt;, 126/125, is the only superparticular &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt; which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which &amp;lt;a class="wiki_link" href="/12_7"&amp;gt;12/7&amp;lt;/a&amp;gt; falls short of three &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt; minor thirds. It is also the amount by which two &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3 major sixths&amp;lt;/a&amp;gt; (octave-reduced) exceed the &amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5 tritone&amp;lt;/a&amp;gt;, and the amount by which three 5/3s (octave-reduced) fall short of the &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6 septimal minor third&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Tempering it out leads to &amp;lt;a class="wiki_link" href="/Starling%20temperaments#Starling%20comma"&amp;gt;starling temperament&amp;lt;/a&amp;gt;, and three minor thirds plus a &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt; subminor third, when 126/125 is tempered out, gives the &amp;lt;a class="wiki_link" href="/starling%20tetrad"&amp;gt;starling tetrad&amp;lt;/a&amp;gt; or septimal semicomma diminished seventh chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_semicomma" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Septimal_semicomma&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 444304260 - Original comment: **</comment>
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      <text bytes="3487" sha1="mnxf9zg3obsheubh1eweyb5g7l1csws" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2013-08-11 13:55:55 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;444304260&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The **starling comma** or **septimal semicomma**, 126/125, is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12_7|12/7]] falls short of three [[6_5|6/5]] minor thirds. It is also the amount by which two [[5_3|5/3 major sixths]] (octave-reduced) exceed the [[7_5|7/5 tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7_6|7/6 septimal minor third]].

Tempering it out leads to [[Starling temperaments#Starling%20comma|starling temperament]], and three minor thirds plus a [[7_6|7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.


[[http://en.wikipedia.org/wiki/Septimal_semicomma]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;126_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;strong&amp;gt;starling comma&amp;lt;/strong&amp;gt; or &amp;lt;strong&amp;gt;septimal semicomma&amp;lt;/strong&amp;gt;, 126/125, is the only superparticular &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt; which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which &amp;lt;a class="wiki_link" href="/12_7"&amp;gt;12/7&amp;lt;/a&amp;gt; falls short of three &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt; minor thirds. It is also the amount by which two &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3 major sixths&amp;lt;/a&amp;gt; (octave-reduced) exceed the &amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5 tritone&amp;lt;/a&amp;gt;, and the amount by which three 5/3s (octave-reduced) fall short of the &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6 septimal minor third&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Tempering it out leads to &amp;lt;a class="wiki_link" href="/Starling%20temperaments#Starling%20comma"&amp;gt;starling temperament&amp;lt;/a&amp;gt;, and three minor thirds plus a &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt; subminor third, when 126/125 is tempered out, gives the &amp;lt;a class="wiki_link" href="/starling%20tetrad"&amp;gt;starling tetrad&amp;lt;/a&amp;gt; or septimal semicomma diminished seventh chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_semicomma" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Septimal_semicomma&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4550</id>
      <timestamp>2013-08-11T12:39:15Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jdfreivald</username>
        <id>0</id>
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      <comment>**Imported revision 444298596 - Original comment: **</comment>
      <origin>4550</origin>
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      <text bytes="3238" sha1="o5z4zbk5rgvcriyz5yxwklf4yk6xzf6" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2013-08-11 12:39:15 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;444298596&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The **starling comma** or **septimal semicomma**, 126/125, is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12_7|12/7]] falls short of three [[6_5|6/5]] minor thirds. It is also the amount by which two [[5_3|5/3 major sixths]] (octave-reduced) exceed the [[7_5|7/5 tritone]].

Tempering it out leads to [[Starling temperaments#Starling%20comma|starling temperament]], and three minor thirds plus a [[7_6|7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.


[[http://en.wikipedia.org/wiki/Septimal_semicomma]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;126_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;strong&amp;gt;starling comma&amp;lt;/strong&amp;gt; or &amp;lt;strong&amp;gt;septimal semicomma&amp;lt;/strong&amp;gt;, 126/125, is the only superparticular &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt; which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which &amp;lt;a class="wiki_link" href="/12_7"&amp;gt;12/7&amp;lt;/a&amp;gt; falls short of three &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt; minor thirds. It is also the amount by which two &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3 major sixths&amp;lt;/a&amp;gt; (octave-reduced) exceed the &amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5 tritone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Tempering it out leads to &amp;lt;a class="wiki_link" href="/Starling%20temperaments#Starling%20comma"&amp;gt;starling temperament&amp;lt;/a&amp;gt;, and three minor thirds plus a &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt; subminor third, when 126/125 is tempered out, gives the &amp;lt;a class="wiki_link" href="/starling%20tetrad"&amp;gt;starling tetrad&amp;lt;/a&amp;gt; or septimal semicomma diminished seventh chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_semicomma" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Septimal_semicomma&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4551</id>
      <timestamp>2013-08-11T12:38:30Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jdfreivald</username>
        <id>0</id>
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      <comment>**Imported revision 444298570 - Original comment: **</comment>
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      <text bytes="3204" sha1="rsdx8l2b2xg2wx9s3din7v2fgjn0wvd" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2013-08-11 12:38:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;444298570&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The **starling comma** or **septimal semicomma**, 126/125, is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12_7|12/7]] falls short of three [[6_5|6/5]] minor thirds. It is also the amount by which two [[5_3|5/3 major sixths]] exceed the [[7_5|7/5 tritone]].

Tempering it out leads to [[Starling temperaments#Starling%20comma|starling temperament]], and three minor thirds plus a [[7_6|7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.


[[http://en.wikipedia.org/wiki/Septimal_semicomma]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;126_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;strong&amp;gt;starling comma&amp;lt;/strong&amp;gt; or &amp;lt;strong&amp;gt;septimal semicomma&amp;lt;/strong&amp;gt;, 126/125, is the only superparticular &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt; which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which &amp;lt;a class="wiki_link" href="/12_7"&amp;gt;12/7&amp;lt;/a&amp;gt; falls short of three &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt; minor thirds. It is also the amount by which two &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3 major sixths&amp;lt;/a&amp;gt; exceed the &amp;lt;a class="wiki_link" href="/7_5"&amp;gt;7/5 tritone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Tempering it out leads to &amp;lt;a class="wiki_link" href="/Starling%20temperaments#Starling%20comma"&amp;gt;starling temperament&amp;lt;/a&amp;gt;, and three minor thirds plus a &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt; subminor third, when 126/125 is tempered out, gives the &amp;lt;a class="wiki_link" href="/starling%20tetrad"&amp;gt;starling tetrad&amp;lt;/a&amp;gt; or septimal semicomma diminished seventh chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_semicomma" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Septimal_semicomma&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <timestamp>2011-08-08T17:27:55Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
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      <comment>**Imported revision 244906905 - Original comment: **</comment>
      <origin>4552</origin>
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      <text bytes="2904" sha1="0muu45bc3jkq2ywwomjbuxf2v1zjlx6" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-08-08 17:27:55 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;244906905&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The **starling comma** or **septimal semicomma**, 126/125, is the only superparticular [[7-limit]] [[comma]] which is not the difference between two  7-limit superparticular ratios. Instead, it is the amount by which [[12_7|12/7]] falls short of three [[6_5|6/5]] minor thirds. Tempering it out leads to [[Starling temperaments#Starling comma|starling temperament]], and three minor thirds plus a [[7_6|7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.


[[http://en.wikipedia.org/wiki/Septimal_semicomma]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;126_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;strong&amp;gt;starling comma&amp;lt;/strong&amp;gt; or &amp;lt;strong&amp;gt;septimal semicomma&amp;lt;/strong&amp;gt;, 126/125, is the only superparticular &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt; which is not the difference between two  7-limit superparticular ratios. Instead, it is the amount by which &amp;lt;a class="wiki_link" href="/12_7"&amp;gt;12/7&amp;lt;/a&amp;gt; falls short of three &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt; minor thirds. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Starling%20temperaments#Starling comma"&amp;gt;starling temperament&amp;lt;/a&amp;gt;, and three minor thirds plus a &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt; subminor third, when 126/125 is tempered out, gives the &amp;lt;a class="wiki_link" href="/starling%20tetrad"&amp;gt;starling tetrad&amp;lt;/a&amp;gt; or septimal semicomma diminished seventh chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_semicomma" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Septimal_semicomma&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <comment>**Imported revision 244893613 - Original comment: **</comment>
      <origin>4553</origin>
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      <text bytes="2738" sha1="7ec2p6eu93tt9pvupq8vwi12o5e7sse" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-08 16:06:19 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;244893613&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The starling comma or septimal semicomma, 126/125, is the only superparticular 7-limit comma which is not the difference between two  7-limit superparticular ratios. Instead, it is the amount by which [[12_7|12/7]] falls short of three [[6_5|6/5]] minor thirds. Tempering it out leads to [[Starling temperaments#Starling comma|starling temperament]], and three minor thirds plus a [[7_6|7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.


[[http://en.wikipedia.org/wiki/Septimal_semicomma]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;126_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The starling comma or septimal semicomma, 126/125, is the only superparticular 7-limit comma which is not the difference between two  7-limit superparticular ratios. Instead, it is the amount by which &amp;lt;a class="wiki_link" href="/12_7"&amp;gt;12/7&amp;lt;/a&amp;gt; falls short of three &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt; minor thirds. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Starling%20temperaments#Starling comma"&amp;gt;starling temperament&amp;lt;/a&amp;gt;, and three minor thirds plus a &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt; subminor third, when 126/125 is tempered out, gives the &amp;lt;a class="wiki_link" href="/starling%20tetrad"&amp;gt;starling tetrad&amp;lt;/a&amp;gt; or septimal semicomma diminished seventh chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_semicomma" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Septimal_semicomma&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 244893409 - Original comment: **</comment>
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      <text bytes="2730" sha1="5agrlhjbgnszzy7cxki7ki5vkcs3abz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-08 16:04:56 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;244893409&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The starling comma or septimal semicomma, 126/125, is the only superparticular 7-limit comma which is not the difference between two  7 (or 9) limit consonances. Instead, it is the amount by which [[12_7|12/7]] falls short of three [[6_5|6/5]] minor thirds. Tempering it out leads to [[Starling temperaments#Starling comma|starling temperament]], and three minor thirds plus a [[7_6|7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.


[[http://en.wikipedia.org/wiki/Septimal_semicomma]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;126_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The starling comma or septimal semicomma, 126/125, is the only superparticular 7-limit comma which is not the difference between two  7 (or 9) limit consonances. Instead, it is the amount by which &amp;lt;a class="wiki_link" href="/12_7"&amp;gt;12/7&amp;lt;/a&amp;gt; falls short of three &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt; minor thirds. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Starling%20temperaments#Starling comma"&amp;gt;starling temperament&amp;lt;/a&amp;gt;, and three minor thirds plus a &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt; subminor third, when 126/125 is tempered out, gives the &amp;lt;a class="wiki_link" href="/starling%20tetrad"&amp;gt;starling tetrad&amp;lt;/a&amp;gt; or septimal semicomma diminished seventh chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_semicomma" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Septimal_semicomma&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>5agrlhjbgnszzy7cxki7ki5vkcs3abz</sha1>
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    <revision>
      <id>4555</id>
      <timestamp>2011-08-08T15:36:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 244889129 - Original comment: **</comment>
      <origin>4555</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1439" sha1="n1ww99tsxlz6zu6sg7gpydfi47oo9ls" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-08 15:36:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;244889129&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[http://en.wikipedia.org/wiki/Septimal_semicomma]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;126_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Septimal_semicomma" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Septimal_semicomma&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>n1ww99tsxlz6zu6sg7gpydfi47oo9ls</sha1>
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    <revision>
      <id>32312</id>
      <parentid>115</parentid>
      <timestamp>2018-09-17T00:09:16Z</timestamp>
      <contributor>
        <username>Mike Battaglia</username>
        <id>3</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "(\[\[\:(dev|es|purdal|de|en):[^[:space:]\|]*) (.*])" to "$1|$3"</comment>
      <origin>32312</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1220" sha1="nut98l19dhmx9zlq0n4k1xoz2u7xykg" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[:de:126/125|Deutsch]]&lt;/span&gt;

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the only superparticular [[7-limit|7-limit]] [[Comma|comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12/7|12/7]] falls short of three [[6/5|6/5]] minor thirds. It is also the amount by which two [[5/3|5/3 major sixths]] (octave-reduced) exceed the [[7/5|7/5 tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6|7/6 septimal minor third]]. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

Tempering it out leads to [[Starling_temperaments#Starling comma|starling temperament]], and three minor thirds plus a [[7/6|7/6]] subminor third, when 126/125 is tempered out, gives the [[starling_tetrad|starling tetrad]] or septimal semicomma diminished seventh chord.

[http://en.wikipedia.org/wiki/Septimal_semicomma http://en.wikipedia.org/wiki/Septimal_semicomma]      [[Category:7-limit]]
[[Category:comma]]
[[Category:interval]]
[[Category:septimal]]
[[Category:starling]]</text>
      <sha1>nut98l19dhmx9zlq0n4k1xoz2u7xykg</sha1>
    </revision>
    <revision>
      <id>47673</id>
      <parentid>32312</parentid>
      <timestamp>2020-07-14T08:48:14Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>added infobox, simplified wiki markup</comment>
      <origin>47673</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1338" sha1="jj7tjm66mwv0r83lu25wbfyrmx7ufs4" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Icon = 
| Ratio = 126/125
| Monzo = 2 3 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name =
| Sound = 
}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) exceed the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

Tempering it out leads to [[Starling_temperaments#Starling comma|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

[http://en.wikipedia.org/wiki/Septimal_semicomma http://en.wikipedia.org/wiki/Septimal_semicomma]

[[Category:7-limit]]
[[Category:Comma]]
[[Category:Interval]]
[[Category:Septimal]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>jj7tjm66mwv0r83lu25wbfyrmx7ufs4</sha1>
    </revision>
    <revision>
      <id>50176</id>
      <parentid>47673</parentid>
      <timestamp>2020-09-25T17:15:16Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>recat, added: Color name, FJS name, See also</comment>
      <origin>50176</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1467" sha1="ly4llg8kr1shmwtot2ofkgy8zx7jujc" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Icon = 
| Ratio = 126/125
| Monzo = 2 3 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) exceed the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

Tempering it out leads to [[Starling_temperaments#Starling comma|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==

* [[Gallery of just intervals]]
* [http://en.wikipedia.org/wiki/Septimal_semicomma http://en.wikipedia.org/wiki/Septimal_semicomma]

[[Category:7-limit]]
[[Category:Small comma]]
[[Category:Interval]]
[[Category:Septimal]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>ly4llg8kr1shmwtot2ofkgy8zx7jujc</sha1>
    </revision>
    <revision>
      <id>51832</id>
      <parentid>50176</parentid>
      <timestamp>2020-10-23T17:53:39Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>septimal is synonymous to 7-limit</comment>
      <origin>51832</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1445" sha1="2zzom8mzlptq76irze7ug56a8ta6yks" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Icon = 
| Ratio = 126/125
| Monzo = 2 3 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) exceed the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

Tempering it out leads to [[Starling_temperaments#Starling comma|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==

* [[Gallery of just intervals]]
* [http://en.wikipedia.org/wiki/Septimal_semicomma http://en.wikipedia.org/wiki/Septimal_semicomma]

[[Category:7-limit]]
[[Category:Small comma]]
[[Category:Interval]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>2zzom8mzlptq76irze7ug56a8ta6yks</sha1>
    </revision>
    <revision>
      <id>58959</id>
      <parentid>51832</parentid>
      <timestamp>2021-01-14T06:41:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+links</comment>
      <origin>58959</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1493" sha1="oi8ehy4spi1onzev4l6qiko05o10mfy" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Icon = 
| Ratio = 126/125
| Monzo = 2 3 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) exceed the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

Tempering it out leads to [[Starling temperaments #Starling comma|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==
* [[Starling family]]
* [[Starling temperaments]]
* [[Starling chords]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]
* [[Wikipedia: Septimal semicomma]]

[[Category:7-limit]]
[[Category:Small comma]]
[[Category:Interval]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>oi8ehy4spi1onzev4l6qiko05o10mfy</sha1>
    </revision>
    <revision>
      <id>58961</id>
      <parentid>58959</parentid>
      <timestamp>2021-01-14T06:42:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+1</comment>
      <origin>58961</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1511" sha1="5r95749urrx6w1nutwqvpdyquf8ve3t" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Icon = 
| Ratio = 126/125
| Monzo = 2 3 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) exceed the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

Tempering it out leads to [[Starling temperaments #Starling comma|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==
* [[Starling family]]
* [[Starling temperaments]]
* [[Starling chords]]
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]
* [[Wikipedia: Septimal semicomma]]

[[Category:7-limit]]
[[Category:Small comma]]
[[Category:Interval]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>5r95749urrx6w1nutwqvpdyquf8ve3t</sha1>
    </revision>
    <revision>
      <id>65683</id>
      <parentid>58961</parentid>
      <timestamp>2021-04-05T05:41:00Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>65683</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1511" sha1="lhipfjbe0vdt30227duzoobooj10bqf" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Icon = 
| Ratio = 126/125
| Monzo = 2 2 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) exceed the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

Tempering it out leads to [[Starling temperaments #Starling comma|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==
* [[Starling family]]
* [[Starling temperaments]]
* [[Starling chords]]
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]
* [[Wikipedia: Septimal semicomma]]

[[Category:7-limit]]
[[Category:Small comma]]
[[Category:Interval]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>lhipfjbe0vdt30227duzoobooj10bqf</sha1>
    </revision>
    <revision>
      <id>68459</id>
      <parentid>65683</parentid>
      <timestamp>2021-04-23T22:38:12Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>correct typo in monzo</comment>
      <origin>68459</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1511" sha1="1xkln4eamjzsfnuq5i71y16udkpd8ac" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Icon = 
| Ratio = 126/125
| Monzo = 1 2 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios. Instead, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) exceed the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

Tempering it out leads to [[Starling temperaments #Starling comma|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==
* [[Starling family]]
* [[Starling temperaments]]
* [[Starling chords]]
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]
* [[Wikipedia: Septimal semicomma]]

[[Category:7-limit]]
[[Category:Small comma]]
[[Category:Interval]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>1xkln4eamjzsfnuq5i71y16udkpd8ac</sha1>
    </revision>
    <revision>
      <id>72549</id>
      <parentid>68459</parentid>
      <timestamp>2021-06-13T09:27:58Z</timestamp>
      <contributor>
        <username>Henrik Ljungstrand</username>
        <id>185</id>
      </contributor>
      <comment>Corrected factually incorrect information, the text needs to be reworked a bit though.</comment>
      <origin>72549</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1638" sha1="suyq0cupcuz4kq14i3kizn4hpoecunr" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Icon = 
| Ratio = 126/125
| Monzo = 1 2 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the only superparticular [[7-limit]] [[comma]] which is not the difference between two 7-limit superparticular ratios (actually it is the difference between the [[36/35]] septimal quartertone and the [[50/49]] jubilisma, so this is not correct). Instead, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) exceed the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

Tempering it out leads to [[Starling temperaments #Starling comma|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==
* [[Starling family]]
* [[Starling temperaments]]
* [[Starling chords]]
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]
* [[Wikipedia: Septimal semicomma]]

[[Category:7-limit]]
[[Category:Small comma]]
[[Category:Interval]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>suyq0cupcuz4kq14i3kizn4hpoecunr</sha1>
    </revision>
    <revision>
      <id>72553</id>
      <parentid>72549</parentid>
      <timestamp>2021-06-13T10:03:14Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>General improvement</comment>
      <origin>72553</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1678" sha1="15xiuapi2vzebma77151eqis2zk06b8" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Icon = 
| Ratio = 126/125
| Monzo = 1 2 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the superparticular [[7-limit]] [[comma]] which is the difference between the [[36/35]] septimal quartertone and the [[50/49]] jubilisma. In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) exceed the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

== Temperaments ==
Tempering it out leads to the [[Starling family #Starling|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==
* [[Starling family]], the rank-3 temperament family where it is tempered out
* [[Starling temperaments]], a collection of rank-2 temperaments where it is tempered out
* [[Starling chords]]
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]
* [[Wikipedia: Septimal semicomma]]

[[Category:7-limit]]
[[Category:Small comma]]
[[Category:Interval]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>15xiuapi2vzebma77151eqis2zk06b8</sha1>
    </revision>
    <revision>
      <id>79109</id>
      <parentid>72553</parentid>
      <timestamp>2021-10-06T21:50:42Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <minor/>
      <comment>add an equated interval pair</comment>
      <origin>79109</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1743" sha1="3347lzr9unvu4ixkeolovyozi9vjujf" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Icon = 
| Ratio = 126/125
| Monzo = 1 2 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the superparticular [[7-limit]] [[comma]] which is the difference between the [[36/35]] septimal quartertone and the [[50/49]] jubilisma. In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) exceed the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can be found as the difference between [[25/18]] and [[7/5]]. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

== Temperaments ==
Tempering it out leads to the [[Starling family #Starling|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==
* [[Starling family]], the rank-3 temperament family where it is tempered out
* [[Starling temperaments]], a collection of rank-2 temperaments where it is tempered out
* [[Starling chords]]
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]
* [[Wikipedia: Septimal semicomma]]

[[Category:7-limit]]
[[Category:Small comma]]
[[Category:Interval]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>3347lzr9unvu4ixkeolovyozi9vjujf</sha1>
    </revision>
    <revision>
      <id>81841</id>
      <parentid>79109</parentid>
      <timestamp>2021-12-12T05:52:34Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Added Wikipedia box, misc. edits, categories</comment>
      <origin>81841</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1708" sha1="dmey82c97motcrlm7wgq016s887nv9q" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Ratio = 126/125
| Monzo = 1 2 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}
{{Wikipedia|Septimal semicomma}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the superparticular [[7-limit]] [[comma]] which is the difference between the [[36/35]] septimal quartertone and the [[50/49]] jubilisma. In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) exceed the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can be found as the difference between [[25/18]] and [[7/5]]. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

== Temperaments ==
Tempering it out leads to the [[Starling family #Starling|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==
* [[Starling family]], the rank-3 temperament family where it is tempered out
* [[Starling temperaments]], a collection of rank-2 temperaments where it is tempered out
* [[Starling chords]]
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:7-limit]]
[[Category:Small comma]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>dmey82c97motcrlm7wgq016s887nv9q</sha1>
    </revision>
    <revision>
      <id>89835</id>
      <parentid>81841</parentid>
      <timestamp>2022-03-20T21:59:54Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Small comma]] to [[Category:Small commas]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>89835</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1709" sha1="o8snrf9g97wnpw8kahqxqds3998vm8h" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Ratio = 126/125
| Monzo = 1 2 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}
{{Wikipedia|Septimal semicomma}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the superparticular [[7-limit]] [[comma]] which is the difference between the [[36/35]] septimal quartertone and the [[50/49]] jubilisma. In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) exceed the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can be found as the difference between [[25/18]] and [[7/5]]. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

== Temperaments ==
Tempering it out leads to the [[Starling family #Starling|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==
* [[Starling family]], the rank-3 temperament family where it is tempered out
* [[Starling temperaments]], a collection of rank-2 temperaments where it is tempered out
* [[Starling chords]]
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:7-limit]]
[[Category:Small commas]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>o8snrf9g97wnpw8kahqxqds3998vm8h</sha1>
    </revision>
    <revision>
      <id>95112</id>
      <parentid>89835</parentid>
      <timestamp>2022-08-29T09:58:59Z</timestamp>
      <contributor>
        <username>Stavats</username>
        <id>339</id>
      </contributor>
      <comment>Fixed content – 25/18 is flatter than 7/5</comment>
      <origin>95112</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1716" sha1="dr0v4dc39t4fne11xejmt1e81fwjhev" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Ratio = 126/125
| Monzo = 1 2 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}
{{Wikipedia|Septimal semicomma}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the superparticular [[7-limit]] [[comma]] which is the difference between the [[36/35]] septimal quartertone and the [[50/49]] jubilisma. In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced) fall short of the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can be found as the difference between [[25/18]] and [[7/5]]. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

== Temperaments ==
Tempering it out leads to the [[Starling family #Starling|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==
* [[Starling family]], the rank-3 temperament family where it is tempered out
* [[Starling temperaments]], a collection of rank-2 temperaments where it is tempered out
* [[Starling chords]]
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:7-limit]]
[[Category:Small commas]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>dr0v4dc39t4fne11xejmt1e81fwjhev</sha1>
    </revision>
    <revision>
      <id>95113</id>
      <parentid>95112</parentid>
      <timestamp>2022-08-29T10:02:28Z</timestamp>
      <contributor>
        <username>Stavats</username>
        <id>339</id>
      </contributor>
      <comment>Remove duplicate</comment>
      <origin>95113</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1662" sha1="lxpdq3t9rmdyrv98ql3xnr711i192ag" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Ratio = 126/125
| Monzo = 1 2 -3 1
| Cents = 13.79477
| Name = Starling comma, &lt;br&gt; septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}
{{Wikipedia|Septimal semicomma}}

The '''starling comma''' or '''septimal semicomma''', 126/125 (about 13.8 cents), is the superparticular [[7-limit]] [[comma]] which is the difference between the [[36/35]] septimal quartertone and the [[50/49]] jubilisma. In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced, [[25/18]]) fall short of the [[7/5]] [[tritone]], and the amount by which three 5/3s (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest Bohlen–Pierce intervals.

== Temperaments ==
Tempering it out leads to the [[Starling family #Starling|starling temperament]], and three minor thirds plus a [[7/6]] subminor third, when 126/125 is tempered out, gives the [[starling tetrad]] or septimal semicomma diminished seventh chord.

== See also ==
* [[Starling family]], the rank-3 temperament family where it is tempered out
* [[Starling temperaments]], a collection of rank-2 temperaments where it is tempered out
* [[Starling chords]]
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:7-limit]]
[[Category:Small commas]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>lxpdq3t9rmdyrv98ql3xnr711i192ag</sha1>
    </revision>
    <revision>
      <id>97818</id>
      <parentid>95113</parentid>
      <timestamp>2022-10-22T13:42:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Merge some see-also links to the temperament section; misc cleanup</comment>
      <origin>97818</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1533" sha1="r1ylrsa1v8v3e5irq8fguw45gdr4iwq" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Ratio = 126/125
| Monzo = 1 2 -3 1
| Cents = 13.79477
| Name = starling comma, &lt;br&gt;septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}
{{Wikipedia|Septimal semicomma}}

The '''starling comma''' or '''septimal semicomma''', '''126/125''' (about 13.8 [[cent]]s), is the [[superparticular]] [[7-limit]] [[comma]] which is the difference between [[36/35]] (septimal quartertone) and [[50/49]] (jubilisma). In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced, [[25/18]]) fall short of the [[7/5]] [[tritone]], and the amount by which three 5/3's (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest [[Bohlen–Pierce]] intervals.

== Temperaments ==
Tempering it out alone in the 7-limit leads to the [[starling]] temperament, and enables [[starling chords]]. See [[Starling family]] for the rank-3 temperament family where it is tempered out. See [[starling temperaments]] for a collection of rank-2 temperaments where it is tempered out. 

== See also ==
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:7-limit]]
[[Category:Small commas]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>r1ylrsa1v8v3e5irq8fguw45gdr4iwq</sha1>
    </revision>
    <revision>
      <id>97819</id>
      <parentid>97818</parentid>
      <timestamp>2022-10-22T13:42:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Fix a link</comment>
      <origin>97819</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1547" sha1="oonx19up1qz29dmy41b0gonbwhpmidh" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Ratio = 126/125
| Monzo = 1 2 -3 1
| Cents = 13.79477
| Name = starling comma, &lt;br&gt;septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| FJS name = d2&lt;sup&gt;7&lt;/sup&gt;&lt;sub&gt;125&lt;/sub&gt;
| Sound = 
}}
{{Wikipedia|Septimal semicomma}}

The '''starling comma''' or '''septimal semicomma''', '''126/125''' (about 13.8 [[cent]]s), is the [[superparticular]] [[7-limit]] [[comma]] which is the difference between [[36/35]] (septimal quartertone) and [[50/49]] (jubilisma). In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced, [[25/18]]) fall short of the [[7/5]] [[tritone]], and the amount by which three 5/3's (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest [[Bohlen-Pierce|Bohlen–Pierce]] intervals.

== Temperaments ==
Tempering it out alone in the 7-limit leads to the [[starling]] temperament, and enables [[starling chords]]. See [[Starling family]] for the rank-3 temperament family where it is tempered out. See [[starling temperaments]] for a collection of rank-2 temperaments where it is tempered out. 

== See also ==
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:7-limit]]
[[Category:Small commas]]
[[Category:Starling]]
[[Category:Superparticular]]</text>
      <sha1>oonx19up1qz29dmy41b0gonbwhpmidh</sha1>
    </revision>
    <revision>
      <id>97945</id>
      <parentid>97819</parentid>
      <timestamp>2022-10-25T11:31:41Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97945</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1370" sha1="t4gwnmwn4ao8ifga6km79ty1ve6n8tj" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Name = starling comma, septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| Comma = 1
}}
{{Wikipedia|Septimal semicomma}}

The '''starling comma''' or '''septimal semicomma''', '''126/125''' (about 13.8 [[cent]]s), is the [[superparticular]] [[7-limit]] [[comma]] which is the difference between [[36/35]] (septimal quartertone) and [[50/49]] (jubilisma). In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced, [[25/18]]) fall short of the [[7/5]] [[tritone]], and the amount by which three 5/3's (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest [[Bohlen-Pierce|Bohlen–Pierce]] intervals.

== Temperaments ==
Tempering it out alone in the 7-limit leads to the [[starling]] temperament, and enables [[starling chords]]. See [[Starling family]] for the rank-3 temperament family where it is tempered out. See [[starling temperaments]] for a collection of rank-2 temperaments where it is tempered out. 

== See also ==
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Starling]]</text>
      <sha1>t4gwnmwn4ao8ifga6km79ty1ve6n8tj</sha1>
    </revision>
    <revision>
      <id>98482</id>
      <parentid>97945</parentid>
      <timestamp>2022-10-25T18:12:26Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <origin>98482</origin>
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      <format>text/x-wiki</format>
      <text bytes="1372" sha1="7a1199fcgll12d0kpztkmgx0pyx3s3o" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Name = starling comma, septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu comma
| Comma = yes
}}
{{Wikipedia|Septimal semicomma}}

The '''starling comma''' or '''septimal semicomma''', '''126/125''' (about 13.8 [[cent]]s), is the [[superparticular]] [[7-limit]] [[comma]] which is the difference between [[36/35]] (septimal quartertone) and [[50/49]] (jubilisma). In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced, [[25/18]]) fall short of the [[7/5]] [[tritone]], and the amount by which three 5/3's (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest [[Bohlen-Pierce|Bohlen–Pierce]] intervals.

== Temperaments ==
Tempering it out alone in the 7-limit leads to the [[starling]] temperament, and enables [[starling chords]]. See [[Starling family]] for the rank-3 temperament family where it is tempered out. See [[starling temperaments]] for a collection of rank-2 temperaments where it is tempered out. 

== See also ==
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Starling]]</text>
      <sha1>7a1199fcgll12d0kpztkmgx0pyx3s3o</sha1>
    </revision>
    <revision>
      <id>101164</id>
      <parentid>98482</parentid>
      <timestamp>2023-01-10T02:33:52Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
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      <minor/>
      <origin>101164</origin>
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      <text bytes="1388" sha1="rctnlq60qpq2g94yb7ussx18kbtzmju" xml:space="preserve">{{interwiki
| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Name = starling comma, septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu 2nd,&lt;br&gt;Zotrigu comma
| Comma = yes
}}
{{Wikipedia|Septimal semicomma}}

The '''starling comma''' or '''septimal semicomma''', '''126/125''' (about 13.8 [[cent]]s), is the [[superparticular]] [[7-limit]] [[comma]] which is the difference between [[36/35]] (septimal quartertone) and [[50/49]] (jubilisma). In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced, [[25/18]]) fall short of the [[7/5]] [[tritone]], and the amount by which three 5/3's (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest [[Bohlen-Pierce|Bohlen–Pierce]] intervals.

== Temperaments ==
Tempering it out alone in the 7-limit leads to the [[starling]] temperament, and enables [[starling chords]]. See [[Starling family]] for the rank-3 temperament family where it is tempered out. See [[starling temperaments]] for a collection of rank-2 temperaments where it is tempered out. 

== See also ==
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Starling]]</text>
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| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Name = starling comma, septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu 2nd,&lt;br&gt;Zotrigu comma
| Comma = yes
}}
{{Wikipedia|Septimal semicomma}}

The '''starling comma''' or '''septimal semicomma''', '''126/125''' (about 13.8 [[cent]]s), is the [[superparticular]] [[7-limit]] [[comma]] which is the difference between [[36/35]] (septimal quartertone) and [[50/49]] (jubilisma). In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced, [[25/18]]) fall short of the [[7/5]] [[tritone]], and the amount by which three 5/3's (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest [[Bohlen-Pierce|Bohlen–Pierce]] intervals.

== Temperaments ==
Tempering it out alone in the 7-limit leads to the [[starling]] temperament, and enables [[starling chords]]. See [[Starling family]] for the rank-3 temperament family where it is tempered out. See [[starling temperaments]] for a collection of rank-2 temperaments where it is tempered out. 

== See also ==
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Starling]]
[[Category:Commas named after musical traditions]]</text>
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    <revision>
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| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Name = starling comma, septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu 2nd,&lt;br&gt;Zotrigu comma
| Comma = yes
}}
{{Wikipedia|Septimal semicomma}}

The '''starling comma''' or '''septimal semicomma''', '''126/125''' (about 13.8 [[cent]]s), is the [[superparticular]] [[7-limit]] [[comma]] which is the difference between [[36/35]] (septimal quartertone) and [[50/49]] (jubilisma). In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced, [[25/18]]) fall short of the [[7/5]] [[tritone]], and the amount by which three 5/3's (octave-reduced) fall short of the [[7/6]] septimal minor third. It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest [[Bohlen–Pierce]] intervals.

== Temperaments ==
Tempering it out alone in the 7-limit leads to the [[starling]] temperament, and enables [[starling chords]]. See [[Starling family]] for the rank-3 temperament family where it is tempered out. See [[starling temperaments]] for a collection of rank-2 temperaments where it is tempered out. 

== See also ==
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Starling]]
[[Category:Commas named after musical traditions]]</text>
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      <timestamp>2026-05-18T11:16:02Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
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      <comment>Equates the 7th harmonic with [[7/1]] with ([[5/3]])^3 * ([[3/2]])</comment>
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| de = 126/125
| en = 126/125
| es = 
| ja = 
}}
{{Infobox Interval
| Name = starling comma, septimal semicomma
| Color name = zg&lt;sup&gt;3&lt;/sup&gt;2, zotrigu 2nd,&lt;br&gt;Zotrigu comma
| Comma = yes
}}
{{Wikipedia|Septimal semicomma}}

The '''starling comma''' or '''septimal semicomma''', '''126/125''' (about 13.8 [[cent]]s), is the [[superparticular]] [[7-limit]] [[comma]] which is the difference between [[36/35]] (septimal quartertone) and [[50/49]] (jubilisma). In terms of just intervals, it is the amount by which [[12/7]] falls short of three [[6/5]] minor thirds. It is also the amount by which two [[5/3]] major sixths (octave-reduced, [[25/18]]) fall short of the [[7/5]] [[tritone]], and the amount by which three 5/3's (octave-reduced) fall short of the [[7/6]] septimal minor third. It equates the [[7/1|7th harmonic]] with ([[5/3]])&lt;sup&gt;3&lt;/sup&gt;([[3/2]]). It can also be found when comparing the conventional 5-limit minor third and major tenth to the nearest [[Bohlen–Pierce]] intervals.

== Temperaments ==
Tempering it out alone in the 7-limit leads to the [[starling]] temperament, and enables [[starling chords]]. See [[Starling family]] for the rank-3 temperament family where it is tempered out. See [[starling temperaments]] for a collection of rank-2 temperaments where it is tempered out. 

== See also ==
* [[Small comma]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Starling]]
[[Category:Commas named after musical traditions]]</text>
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      <text bytes="21" sha1="s58oe66ppug4vep0374ndovi5m4xyd8" xml:space="preserve">#REDIRECT [[126/125]]</text>
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    <title>126edo</title>
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      <timestamp>2018-07-17T00:00:00Z</timestamp>
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      <text bytes="459" sha1="kmxo76rtda380hgjn17gvxqpfqhinat" xml:space="preserve">The 126 equal temperament divides the octave into 126 equal parts of 9.524 cents each. It has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It tempers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit [[Diaschismic_family#Srutal-11-limit|srutal temperament]]. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.</text>
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      <comment>**Imported revision 339008238 - Original comment: **</comment>
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      <text bytes="2190" sha1="at1mr1z7vozltgyte4kjp8pre8wyqfn" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-05-24 00:55:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;339008238&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 126 equal temperament divides the octave into 126 equal parts of 9.524 cents each. It has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It tempers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit [[Diaschismic family#Srutal-11-limit|srutal temperament]]. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;126edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 126 equal temperament divides the octave into 126 equal parts of 9.524 cents each. It has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It tempers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit &amp;lt;a class="wiki_link" href="/Diaschismic%20family#Srutal-11-limit"&amp;gt;srutal temperament&amp;lt;/a&amp;gt;. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 339008192 - Original comment: **</comment>
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      <text bytes="2192" sha1="qmhyhka0nqbqixv1gpllkak08lg6t0j" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-05-24 00:55:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;339008192&lt;/tt&gt;.&lt;br&gt;
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The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 126 equal temperament divides the octave into 126 equal parts of 9.524 cents each. It has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It temdpers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit [[Diaschismic family#Srutal-11-limit|srutal temperament]]. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;126edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 126 equal temperament divides the octave into 126 equal parts of 9.524 cents each. It has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It temdpers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit &amp;lt;a class="wiki_link" href="/Diaschismic%20family#Srutal-11-limit"&amp;gt;srutal temperament&amp;lt;/a&amp;gt;. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2020-06-24T10:07:55Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
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      <comment>Talk on why this edo is particularly well-suited to Porcupine scale.</comment>
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      <text bytes="693" sha1="4uuyb6m4qo7rprkkwxvpfof4pzs48x0" xml:space="preserve">The 126 equal temperament divides the octave into 126 equal parts of 9.524 cents each. It has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It tempers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit [[Diaschismic_family#Srutal-11-limit|srutal temperament]]. It also creates an excellent Porcupine [8] scale, mapping the large quills to 17 steps, and the small to 7, which is the precise amount of tempering needed to make the 3rds and 4ths equally consonant within a few fractions of a cent. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.</text>
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      <comment>Categories</comment>
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      <text bytes="765" sha1="az5hsk2e6jtp19dc0kv5idk4nierk0q" xml:space="preserve">The 126 equal temperament divides the octave into 126 equal parts of 9.524 cents each. It has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It tempers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit [[Diaschismic_family#Srutal-11-limit|srutal temperament]]. It also creates an excellent Porcupine [8] scale, mapping the large quills to 17 steps, and the small to 7, which is the precise amount of tempering needed to make the 3rds and 4ths equally consonant within a few fractions of a cent. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
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      <text bytes="780" sha1="qwlik1skf12271qdx4gloliisl87izl" xml:space="preserve">{{Infobox ET}}
The 126 equal temperament divides the octave into 126 equal parts of 9.524 cents each. It has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It tempers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit [[Diaschismic_family#Srutal-11-limit|srutal temperament]]. It also creates an excellent Porcupine [8] scale, mapping the large quills to 17 steps, and the small to 7, which is the precise amount of tempering needed to make the 3rds and 4ths equally consonant within a few fractions of a cent. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
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      <comment>Added "harmonics in equal" table</comment>
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      <text bytes="806" sha1="isvemk1reg2o96r3peuaagrwblz3mxc" xml:space="preserve">{{Infobox ET}}
The 126 equal temperament divides the octave into 126 equal parts of 9.524 cents each. It has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It tempers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit [[Diaschismic_family#Srutal-11-limit|srutal temperament]]. It also creates an excellent Porcupine [8] scale, mapping the large quills to 17 steps, and the small to 7, which is the precise amount of tempering needed to make the 3rds and 4ths equally consonant within a few fractions of a cent. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.
{{Harmonics in equal|126}}
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
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      <comment>Add link to porcupine</comment>
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      <text bytes="809" sha1="ns2i47mzjbns7xeg0ars28q02x62qf2" xml:space="preserve">{{Infobox ET}}
The 126 equal temperament divides the octave into 126 equal parts of 9.524 cents each. It has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It tempers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit [[Diaschismic_family#Srutal-11-limit|srutal temperament]]. It also creates an excellent [[Porcupine]][8] scale, mapping the large quills to 17 steps, and the small to 7, which is the precise amount of tempering needed to make the 3rds and 4ths equally consonant within a few fractions of a cent. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.
{{Harmonics in equal|126}}
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
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      <comment>Add link to octave</comment>
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      <text bytes="813" sha1="tbx9u9mb1h9f2kfk4adbqj39ppyoysz" xml:space="preserve">{{Infobox ET}}
The 126 equal temperament divides the [[octave]] into 126 equal parts of 9.524 cents each. It has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It tempers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit [[Diaschismic_family#Srutal-11-limit|srutal temperament]]. It also creates an excellent [[Porcupine]][8] scale, mapping the large quills to 17 steps, and the small to 7, which is the precise amount of tempering needed to make the 3rds and 4ths equally consonant within a few fractions of a cent. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.
{{Harmonics in equal|126}}
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
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    <revision>
      <id>132901</id>
      <parentid>130631</parentid>
      <timestamp>2024-01-25T09:51:46Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Edo intro</comment>
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      <text bytes="745" sha1="ihq1g3110go4imy32e2dtjo8vq5droz" xml:space="preserve">{{Infobox ET}}
{{EDO intro|126}}

126edo has a distinctly sharp tendency, with the 3, 5, 7 and 11 all sharp. It tempers out 2048/2025 in the 5-limit, 2401/2400 and 4375/4374 in the 7-limit, and 176/175, 1331/1323 and 896/891 in the 11-limit. It provides the optimal patent val for 7- and 11-limit [[Diaschismic_family#Srutal-11-limit|srutal temperament]]. It also creates an excellent [[Porcupine]][8] scale, mapping the large quills to 17 steps, and the small to 7, which is the precise amount of tempering needed to make the 3rds and 4ths equally consonant within a few fractions of a cent. It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.
{{Harmonics in equal|126}}
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;</text>
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    <revision>
      <id>145210</id>
      <parentid>132901</parentid>
      <timestamp>2024-05-29T05:36:36Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
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      <text bytes="819" sha1="aix3zlhtw7wslmk45jiljoq1j7fe2ge" xml:space="preserve">{{Infobox ET}}
{{EDO intro|126}}

126edo has a distinctly sharp tendency, with the [[3/1|3]], [[5/1|5]], [[7/1|7]] and [[11/1|11]] all sharp. The equal temperament [[tempering out|tempers out]] [[2048/2025]] in the 5-limit, [[2401/2400]] and [[4375/4374]] in the 7-limit, and [[176/175]], [[896/891]], and 1331/1323 in the 11-limit. It provides the [[optimal patent val]] for 7- and 11-limit [[srutal]] temperament. It also creates an excellent [[Porcupine]][8] scale, mapping the large [[quill]]s to 17 steps, and the small to 7, which is the precise amount of tempering needed to make the thirds and fourths equally consonant within a few fractions of a cent. 

=== Odd harmonics ===
{{Harmonics in equal|126}}

=== Subsets and supersets ===
It has divisors 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63.

[[Category:Srutal]]</text>
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    <revision>
      <id>145211</id>
      <parentid>145210</parentid>
      <timestamp>2024-05-29T05:38:20Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
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      <minor/>
      <comment>Rework the subsets and supersets section</comment>
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{{EDO intro|126}}

126edo has a distinctly sharp tendency, with the [[3/1|3]], [[5/1|5]], [[7/1|7]] and [[11/1|11]] all sharp. The equal temperament [[tempering out|tempers out]] [[2048/2025]] in the 5-limit, [[2401/2400]] and [[4375/4374]] in the 7-limit, and [[176/175]], [[896/891]], and 1331/1323 in the 11-limit. It provides the [[optimal patent val]] for 7- and 11-limit [[srutal]] temperament. It also creates an excellent [[Porcupine]][8] scale, mapping the large [[quill]]s to 17 steps, and the small to 7, which is the precise amount of tempering needed to make the thirds and fourths equally consonant within a few fractions of a cent. 

=== Odd harmonics ===
{{Harmonics in equal|126}}

=== Subsets and supersets ===
Since 126 factors into {{factorization|126}}, 126edo has subset edos {{EDOs| 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63 }}.

[[Category:Srutal]]</text>
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      <id>181335</id>
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      <timestamp>2025-02-18T17:00:00Z</timestamp>
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        <id>448</id>
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{{ED intro}}

126edo has a distinctly sharp tendency, with the [[3/1|3]], [[5/1|5]], [[7/1|7]] and [[11/1|11]] all sharp. The equal temperament [[tempering out|tempers out]] [[2048/2025]] in the 5-limit, [[2401/2400]] and [[4375/4374]] in the 7-limit, and [[176/175]], [[896/891]], and 1331/1323 in the 11-limit. It provides the [[optimal patent val]] for 7- and 11-limit [[srutal]] temperament. It also creates an excellent [[Porcupine]][8] scale, mapping the large [[quill]]s to 17 steps, and the small to 7, which is the precise amount of tempering needed to make the thirds and fourths equally consonant within a few fractions of a cent. 

=== Odd harmonics ===
{{Harmonics in equal|126}}

=== Subsets and supersets ===
Since 126 factors into {{factorization|126}}, 126edo has subset edos {{EDOs| 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63 }}.

[[Category:Srutal]]</text>
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    <revision>
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{{ED intro}}

126edo has a distinctly sharp tendency, with the [[3/1|3]], [[5/1|5]], [[7/1|7]] and [[11/1|11]] all sharp. The equal temperament [[tempering out|tempers out]] [[2048/2025]] in the 5-limit, [[2401/2400]] and [[4375/4374]] in the 7-limit, and [[176/175]], [[896/891]], and 1331/1323 in the 11-limit. It provides the [[optimal patent val]] for 7- and 11-limit [[srutal]] temperament. It also creates an excellent [[Porcupine]][8] scale, mapping the generators to 17 steps, and the smaller interval to 7, which is the precise amount of tempering needed to make the thirds and fourths equally consonant within a few fractions of a cent. 

=== Odd harmonics ===
{{Harmonics in equal|126}}

=== Subsets and supersets ===
Since 126 factors into {{factorization|126}}, 126edo has subset edos {{EDOs| 2, 3, 6, 7, 9, 14, 18, 21, 42, and 63 }}.

[[Category:Srutal]]</text>
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  <page>
    <title>127edo</title>
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      <id>118</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
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        <username>Wikispaces&gt;FREEZE</username>
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      <text bytes="1175" sha1="inxk0ayvyjshxiyb1bccgvniaihvrl7" xml:space="preserve">'''127edo''', which divides the [[Octave|octave]] into 127 parts of 9.45 [[cents|cents]] each, is another equal division interesting because of its approximations, defined by the [[Comma|comma]]s it [[tempering_out|tempers out]]. In the [[5-limit|5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence supports [[Würschmidt_family|würschmidt temperament]]. In the [[7-limit|7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the [[11-limit|11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[Optimal_patent_val|optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

127edo is the 31st [[prime_numbers|prime]] edo.

[[MOS_Scales_of_127edo|MOS Scales of 127edo]]      [[Category:edo]]
[[Category:hemiwuerschmidt]]
[[Category:minerva]]
[[Category:prime_edo]]
[[Category:theory]]
[[Category:wuerschmidt]]
[[Category:wurschmidt]]</text>
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    <revision>
      <id>4558</id>
      <timestamp>2011-12-31T02:11:10Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 288887301 - Original comment: **</comment>
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      <text bytes="3715" sha1="ia06xd3107ih0nyws1kfzabg2mnwp0r" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-12-31 02:11:10 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288887301&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**127edo**, which divides the [[octave]] into 127 parts of 9.45 [[cents]] each, is another equal division interesting because of its approximations, defined by the [[comma]]s it [[tempering out|tempers out]]. In the [[5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence supports [[Würschmidt family|würschmidt temperament]]. In the [[7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the [[11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

127edo is the 31st [[prime numbers|prime]] edo.

[[MOS Scales of 127edo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;127edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;127edo&amp;lt;/strong&amp;gt;, which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 127 parts of 9.45 &amp;lt;a class="wiki_link" href="/cents"&amp;gt;cents&amp;lt;/a&amp;gt; each, is another equal division interesting because of its approximations, defined by the &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, it tempers out the würschmidt comma, 393216/390625 and hence supports &amp;lt;a class="wiki_link" href="/W%C3%BCrschmidt%20family"&amp;gt;würschmidt temperament&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
127edo is the 31st &amp;lt;a class="wiki_link" href="/prime%20numbers"&amp;gt;prime&amp;lt;/a&amp;gt; edo.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/MOS%20Scales%20of%20127edo"&amp;gt;MOS Scales of 127edo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4559</id>
      <timestamp>2011-12-21T16:01:27Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 288013272 - Original comment: **</comment>
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      <text bytes="3542" sha1="0rud0at1snrgq7dfs9w6jfauwr26lns" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-12-21 16:01:27 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288013272&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**127edo**, which divides the [[octave]] into 127 parts of 9.45 [[cents]] each, is another equal division interesting because of its approximations, defined by the [[comma]]s it [[tempering out|tempers out]]. In the [[5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence supports [[Würschmidt family|würschmidt temperament]]. In the [[7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the [[11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

[[MOS Scales of 127edo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;127edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;127edo&amp;lt;/strong&amp;gt;, which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 127 parts of 9.45 &amp;lt;a class="wiki_link" href="/cents"&amp;gt;cents&amp;lt;/a&amp;gt; each, is another equal division interesting because of its approximations, defined by the &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, it tempers out the würschmidt comma, 393216/390625 and hence supports &amp;lt;a class="wiki_link" href="/W%C3%BCrschmidt%20family"&amp;gt;würschmidt temperament&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/MOS%20Scales%20of%20127edo"&amp;gt;MOS Scales of 127edo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4560</id>
      <timestamp>2011-12-21T16:01:10Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 288013240 - Original comment: **</comment>
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      <text bytes="3542" sha1="d7r0001gc5j7taka89gj4peu3a7pd7g" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-12-21 16:01:10 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288013240&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**127edo**, which divides the [[octave]] into 127 parts of 9.45 [[cents]] each, is another equal division interesting because of its approximations, defined by the [[comma]]s it [[tempering out|tempers out]]. In the [[5-limit]], it tempers out the wuerschmidt comma, 393216/390625 and hence supports [[Würschmidt family|würschmidt temperament]]. In the [[7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the [[11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

[[MOS Scales of 127edo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;127edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;127edo&amp;lt;/strong&amp;gt;, which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 127 parts of 9.45 &amp;lt;a class="wiki_link" href="/cents"&amp;gt;cents&amp;lt;/a&amp;gt; each, is another equal division interesting because of its approximations, defined by the &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, it tempers out the wuerschmidt comma, 393216/390625 and hence supports &amp;lt;a class="wiki_link" href="/W%C3%BCrschmidt%20family"&amp;gt;würschmidt temperament&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/MOS%20Scales%20of%20127edo"&amp;gt;MOS Scales of 127edo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4561</id>
      <timestamp>2011-12-17T02:21:35Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 287019768 - Original comment: **</comment>
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      <text bytes="3592" sha1="ry3zw10q7i0eaw61mmucmnaxq7tysh1" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-12-17 02:21:35 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;287019768&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**127edo**, which divides the [[octave]] into 127 parts of 9.45 [[cents]] each, is another equal division interesting because of its approximations, defined by the [[comma]]s it [[tempering out|tempers out]]. In the [[5-limit]], it tempers out the wuerschmidt comma, 393216/390625 and hence supports [[Wuerschmidt family|wuerschmidt temperament]]. In the [[7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension ("wurschmidt") of wuerschmidt which tempers this out also. In the [[11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of wurschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

[[MOS Scales of 127edo]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;127edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;127edo&amp;lt;/strong&amp;gt;, which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 127 parts of 9.45 &amp;lt;a class="wiki_link" href="/cents"&amp;gt;cents&amp;lt;/a&amp;gt; each, is another equal division interesting because of its approximations, defined by the &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, it tempers out the wuerschmidt comma, 393216/390625 and hence supports &amp;lt;a class="wiki_link" href="/Wuerschmidt%20family"&amp;gt;wuerschmidt temperament&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, it also tempers out 225/224, and is an excellent tuning for the 7-limit extension (&amp;amp;quot;wurschmidt&amp;amp;quot;) of wuerschmidt which tempers this out also. In the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of wurschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/MOS%20Scales%20of%20127edo"&amp;gt;MOS Scales of 127edo&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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        <id>0</id>
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      <comment>**Imported revision 239839467 - Original comment: **</comment>
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      <text bytes="3440" sha1="e0xc0tbco4azugfvcgx3j7h4agovbx5" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-07-03 15:04:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239839467&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**127edo**, which divides the [[octave]] into 127 parts of 9.45 [[cents]] each, is another equal division interesting because of its approximations, defined by the [[comma]]s it [[tempering out|tempers out]]. In the [[5-limit]], it tempers out the wuerschmidt comma, 393216/390625 and hence supports [[Wuerschmidt family|wuerschmidt temperament]]. In the [[7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension ("wurschmidt") of wuerschmidt which tempers this out also. In the [[11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of wurschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;127edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;127edo&amp;lt;/strong&amp;gt;, which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 127 parts of 9.45 &amp;lt;a class="wiki_link" href="/cents"&amp;gt;cents&amp;lt;/a&amp;gt; each, is another equal division interesting because of its approximations, defined by the &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, it tempers out the wuerschmidt comma, 393216/390625 and hence supports &amp;lt;a class="wiki_link" href="/Wuerschmidt%20family"&amp;gt;wuerschmidt temperament&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, it also tempers out 225/224, and is an excellent tuning for the 7-limit extension (&amp;amp;quot;wurschmidt&amp;amp;quot;) of wuerschmidt which tempers this out also. In the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of wurschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 239315103 - Original comment: **</comment>
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      <text bytes="3176" sha1="9cqj8ewzqqv3l6u7ygxqiukrvedjhi9" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 10:01:40 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239315103&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**127edo**, which divides the [[octave]] into 127 parts of 9.45 [[cents]] each, is another equal division interesting because of its approximations, defined by the [[comma]]s it [[tempering out|tempers out]]. In the [[5-limit]], it tempers out the wuerschmidt comma, 393216/390625 and hence supports [[Wuerschmidt family|wuerschmidt temperament]]. In the [[7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension ("wurschmidt") of wuerschmidt which tempers this out also. In the [[11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of wurschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, and the rank four temperament tempering out 99/98.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;127edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;127edo&amp;lt;/strong&amp;gt;, which divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 127 parts of 9.45 &amp;lt;a class="wiki_link" href="/cents"&amp;gt;cents&amp;lt;/a&amp;gt; each, is another equal division interesting because of its approximations, defined by the &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s it &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;, it tempers out the wuerschmidt comma, 393216/390625 and hence supports &amp;lt;a class="wiki_link" href="/Wuerschmidt%20family"&amp;gt;wuerschmidt temperament&amp;lt;/a&amp;gt;. In the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;, it also tempers out 225/224, and is an excellent tuning for the 7-limit extension (&amp;amp;quot;wurschmidt&amp;amp;quot;) of wuerschmidt which tempers this out also. In the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;, it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of wurschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, and the rank four temperament tempering out 99/98.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 213333640 - Original comment: **</comment>
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      <text bytes="2763" sha1="m939dvmu5zko4kz77zgme3ebk56v9gc" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-23 15:58:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;213333640&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//127edo//, which divides the octave into 127 parts of 9.45 cents each, is another equal division interesting because of its approximations, defined by the commas it tempers out. In the 5-limit, it tempers out the wuerschmidt comma, 393216/390625 and hence supports [[Wuerschmidt family|wuerschmidt temperament]]. In the 7-limit, it also tempers out 225/224, and is an excellent tuning for the 7-limit extension ("wurschmidt") of wuerschmidt which tempers this out also. In the 11-limit, it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of wurschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, and the rank four temperament tempering out 99/98.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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      <comment>**Imported revision 155968181 - Original comment: **</comment>
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      <text bytes="2661" sha1="sbu80fgmt30scklyu093ars1e1pjfko" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2010-08-10 15:34:47 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;155968181&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;//127edo//, which divides the octave into 127 parts of 9.45 cents each, is another equal division interesting because of its approximations, defined by the commas it tempers out. In the 5-limit, it tempers out the wuerschmidt comma, 393216/390625 and hence supports [[Wuerschmidt family|wuerschmidt temperament]]. In the 7-limit, it also tempers out 225/224, and is an excellent tuning for the 7-limit extension ("wurschmidt") of wuerschmidt which tempers this out also. In the 11-limit, it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of wurschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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      <timestamp>2018-11-02T10:20:52Z</timestamp>
      <contributor>
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        <id>6</id>
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      <text bytes="1171" sha1="6p5562a6u7zwm8wjd20bhmvnzp89yz0" xml:space="preserve">'''127edo''', which divides the [[Octave|octave]] into 127 parts of 9.45 [[cents|cents]] each, is another equal division interesting because of its approximations, defined by the [[Comma|comma]]s it [[tempering_out|tempers out]]. In the [[5-limit|5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence supports [[Würschmidt_family|würschmidt temperament]]. In the [[7-limit|7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the [[11-limit|11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[Optimal_patent_val|optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

127edo is the 31st [[prime_numbers|prime]] edo.

[[MOS_Scales_of_127edo|MOS Scales of 127edo]]

[[Category:Edo]]
[[Category:Hemiwuerschmidt]]
[[Category:Minerva]]
[[Category:Prime EDO]]
[[Category:Theory]]
[[Category:Wuerschmidt]]
[[Category:Wurschmidt]]</text>
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        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1197" sha1="t0dc3voje3ynpqfih7flwxgl5eyzr2p" xml:space="preserve">'''127edo''', which divides the [[Octave|octave]] into 127 parts of 9.45 [[cents|cents]] each, is another equal division interesting because of its approximations, defined by the [[Comma|comma]]s it [[tempering_out|tempers out]]. In the [[5-limit|5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence supports [[Würschmidt_family|würschmidt temperament]]. In the [[7-limit|7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the [[11-limit|11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[Optimal_patent_val|optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

127edo is the 31st [[prime_numbers|prime]] edo.

[[MOS_Scales_of_127edo|MOS Scales of 127edo]]

[[Category:Equal divisions of the octave]]
[[Category:Hemiwuerschmidt]]
[[Category:Minerva]]
[[Category:Prime EDO]]
[[Category:Theory]]
[[Category:Wuerschmidt]]
[[Category:Wurschmidt]]</text>
      <sha1>t0dc3voje3ynpqfih7flwxgl5eyzr2p</sha1>
    </revision>
    <revision>
      <id>57108</id>
      <parentid>54978</parentid>
      <timestamp>2020-12-20T12:33:29Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>correct cat name</comment>
      <origin>57108</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1173" sha1="nrc5qw0pfjn4wtk8w5arjy7ne1wqj9k" xml:space="preserve">'''127edo''', which divides the [[Octave|octave]] into 127 parts of 9.45 [[cents|cents]] each, is another equal division interesting because of its approximations, defined by the [[Comma|comma]]s it [[tempering_out|tempers out]]. In the [[5-limit|5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence supports [[Würschmidt_family|würschmidt temperament]]. In the [[7-limit|7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the [[11-limit|11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[Optimal_patent_val|optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

127edo is the 31st [[prime_numbers|prime]] edo.

[[MOS_Scales_of_127edo|MOS Scales of 127edo]]

[[Category:Equal divisions of the octave]]
[[Category:Hemiwuerschmidt]]
[[Category:Minerva]]
[[Category:Prime EDO]]
[[Category:Theory]]
[[Category:Würschmidt]]</text>
      <sha1>nrc5qw0pfjn4wtk8w5arjy7ne1wqj9k</sha1>
    </revision>
    <revision>
      <id>57119</id>
      <parentid>57108</parentid>
      <timestamp>2020-12-20T12:44:01Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>recat</comment>
      <origin>57119</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1173" sha1="9by67e9262wwqjj7g8n8kxdl5xin363" xml:space="preserve">'''127edo''', which divides the [[Octave|octave]] into 127 parts of 9.45 [[cents|cents]] each, is another equal division interesting because of its approximations, defined by the [[Comma|comma]]s it [[tempering_out|tempers out]]. In the [[5-limit|5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence supports [[Würschmidt_family|würschmidt temperament]]. In the [[7-limit|7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the [[11-limit|11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[Optimal_patent_val|optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

127edo is the 31st [[prime_numbers|prime]] edo.

[[MOS_Scales_of_127edo|MOS Scales of 127edo]]

[[Category:Equal divisions of the octave]]
[[Category:Hemiwürschmidt]]
[[Category:Minerva]]
[[Category:Prime EDO]]
[[Category:Theory]]
[[Category:Würschmidt]]</text>
      <sha1>9by67e9262wwqjj7g8n8kxdl5xin363</sha1>
    </revision>
    <revision>
      <id>84775</id>
      <parentid>57119</parentid>
      <timestamp>2022-01-25T18:35:09Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>link to new page for Supports</comment>
      <origin>84775</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1177" sha1="kjuwabmrqgk4m8p4zqjcgom7wikogzx" xml:space="preserve">'''127edo''', which divides the [[Octave|octave]] into 127 parts of 9.45 [[cents|cents]] each, is another equal division interesting because of its approximations, defined by the [[Comma|comma]]s it [[tempering_out|tempers out]]. In the [[5-limit|5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence [[support]]s [[Würschmidt_family|würschmidt temperament]]. In the [[7-limit|7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the [[11-limit|11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[Optimal_patent_val|optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

127edo is the 31st [[prime_numbers|prime]] edo.

[[MOS_Scales_of_127edo|MOS Scales of 127edo]]

[[Category:Equal divisions of the octave]]
[[Category:Hemiwürschmidt]]
[[Category:Minerva]]
[[Category:Prime EDO]]
[[Category:Theory]]
[[Category:Würschmidt]]</text>
      <sha1>kjuwabmrqgk4m8p4zqjcgom7wikogzx</sha1>
    </revision>
    <revision>
      <id>87444</id>
      <parentid>84775</parentid>
      <timestamp>2022-02-26T16:48:49Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87444</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1157" sha1="7wouys3uo2ez85k4hvcwoikk62rvzyj" xml:space="preserve">'''127edo''', which divides the [[Octave|octave]] into 127 parts of 9.45 [[cents|cents]] each, is another equal division interesting because of its approximations, defined by the [[Comma|comma]]s it [[tempering_out|tempers out]]. In the [[5-limit|5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence [[support]]s [[Würschmidt_family|würschmidt temperament]]. In the [[7-limit|7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the [[11-limit|11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[Optimal_patent_val|optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

127edo is the 31st [[prime_numbers|prime]] edo.

[[MOS_Scales_of_127edo|MOS Scales of 127edo]]

[[Category:Equal divisions of the octave]]
[[Category:Hemiwürschmidt]]
[[Category:Minerva]]
[[Category:Prime EDO]]
[[Category:Würschmidt]]</text>
      <sha1>7wouys3uo2ez85k4hvcwoikk62rvzyj</sha1>
    </revision>
    <revision>
      <id>92937</id>
      <parentid>87444</parentid>
      <timestamp>2022-07-02T16:26:32Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92937</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1185" sha1="nhtincapc1eux7hqndpa03fpltbcgno" xml:space="preserve">'''127edo''', which divides the [[Octave|octave]] into 127 parts of 9.45 [[cents|cents]] each, is another equal division interesting because of its approximations, defined by the [[Comma|comma]]s it [[tempering_out|tempers out]]. In the [[5-limit|5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence [[support]]s [[Würschmidt_family|würschmidt temperament]]. In the [[7-limit|7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the [[11-limit|11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[Optimal_patent_val|optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

127edo is the 31st [[prime_numbers|prime]] edo.

[[MOS_Scales_of_127edo|MOS Scales of 127edo]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Hemiwürschmidt]]
[[Category:Minerva]]
[[Category:Prime EDO]]
[[Category:Würschmidt]]</text>
      <sha1>nhtincapc1eux7hqndpa03fpltbcgno</sha1>
    </revision>
    <revision>
      <id>96204</id>
      <parentid>92937</parentid>
      <timestamp>2022-10-04T18:54:32Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96204</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1200" sha1="0wm6b6uww98txuiimklnhfv6tpcv5nz" xml:space="preserve">{{Infobox ET}}
'''127edo''', which divides the [[Octave|octave]] into 127 parts of 9.45 [[cents|cents]] each, is another equal division interesting because of its approximations, defined by the [[Comma|comma]]s it [[tempering_out|tempers out]]. In the [[5-limit|5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence [[support]]s [[Würschmidt_family|würschmidt temperament]]. In the [[7-limit|7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. In the [[11-limit|11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[Optimal_patent_val|optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

127edo is the 31st [[prime_numbers|prime]] edo.

[[MOS_Scales_of_127edo|MOS Scales of 127edo]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Hemiwürschmidt]]
[[Category:Minerva]]
[[Category:Prime EDO]]
[[Category:Würschmidt]]</text>
      <sha1>0wm6b6uww98txuiimklnhfv6tpcv5nz</sha1>
    </revision>
    <revision>
      <id>117482</id>
      <parentid>96204</parentid>
      <timestamp>2023-06-24T02:58:23Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added subheadings, turned paragraph into dot point list, added table of harmonics</comment>
      <origin>117482</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1219" sha1="apuu2286alogihx1w53s2klb3mndgdi" xml:space="preserve">{{Infobox ET}}
'''127edo''', which divides the [[Octave|octave]] into 127 parts of 9.45 [[cents|cents]] each, is an equal division interesting because of its approximations, defined by the [[comma]]s it [[tempering_out|tempers out]]:

* In the [[5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence [[support]]s [[Würschmidt_family|würschmidt temperament]]. 
* In the [[7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. 
* In the [[11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

127edo is the 31st [[prime_numbers|prime]] edo.

=== Harmonics ===
{{Harmonics in equal|127}}

=== MOS Scales ===
[[MOS_Scales_of_127edo|MOS Scales of 127edo]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Hemiwürschmidt]]
[[Category:Minerva]]
[[Category:Prime EDO]]
[[Category:Würschmidt]]</text>
      <sha1>apuu2286alogihx1w53s2klb3mndgdi</sha1>
    </revision>
    <revision>
      <id>132902</id>
      <parentid>117482</parentid>
      <timestamp>2024-01-25T09:53:09Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>EDO intro</comment>
      <origin>132902</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1132" sha1="hrmss438avpelsh8twkfagfuvw9bc54" xml:space="preserve">{{Infobox ET}}
{{EDO intro|127}}

127edo is interesting because of its approximations, defined by the [[comma]]s it [[tempering_out|tempers out]]:

* In the [[5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence [[support]]s [[Würschmidt_family|würschmidt temperament]]. 
* In the [[7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. 
* In the [[11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

127edo is the 31st [[prime_numbers|prime]] edo.

=== Harmonics ===
{{Harmonics in equal|127}}

=== MOS Scales ===
[[MOS_Scales_of_127edo|MOS Scales of 127edo]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Hemiwürschmidt]]
[[Category:Minerva]]
[[Category:Prime EDO]]
[[Category:Würschmidt]]</text>
      <sha1>hrmss438avpelsh8twkfagfuvw9bc54</sha1>
    </revision>
    <revision>
      <id>145208</id>
      <parentid>132902</parentid>
      <timestamp>2024-05-29T05:31:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>145208</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1077" sha1="q78zgzf15u4wzpi5m11qs9v4neendau" xml:space="preserve">{{Infobox ET}}
{{EDO intro|127}}

== Theory ==
127edo is interesting because of its approximations, defined by the [[comma]]s it [[tempering out|tempers out]]:

* In the [[5-limit]], it tempers out the würschmidt comma, 393216/390625 and hence [[support]]s [[Würschmidt_family|würschmidt temperament]]. 
* In the [[7-limit]], it also tempers out 225/224, and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. 
* In the [[11-limit]], it tempers out 99/98, 176/175 and 243/242, and is an excellent tuning for the 11-limit version of würschmidt, as well as minerva, the rank three temperament tempering out 99/98 and 176/175, for which it is the [[optimal patent val]] and the rank four temperament tempering out 99/98, for which it also provides the optimal patent val.

=== Odd harmonics ===
{{Harmonics in equal|127}}

=== Subsets and supersets ===
127edo is the 31st [[prime edo]].

== Scales ==
=== MOS scales ===
See [[List of MOS scales in 127edo]]. 

[[Category:Würschmidt]]
[[Category:Hemiwürschmidt]]
[[Category:Minerva]]</text>
      <sha1>q78zgzf15u4wzpi5m11qs9v4neendau</sha1>
    </revision>
    <revision>
      <id>145209</id>
      <parentid>145208</parentid>
      <timestamp>2024-05-29T05:33:35Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ cleanup and link to previous and next prime edos</comment>
      <origin>145209</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1123" sha1="0zvmuaj6td7bxtigmw932j5fau6a95x" xml:space="preserve">{{Infobox ET}}
{{EDO intro|127}}

== Theory ==
127edo is interesting because of its approximations, defined by the [[comma]]s it [[tempering out|tempers out]]:
* In the [[5-limit]], it tempers out 393216/390625 ([[würschmidt comma]]) and hence [[support]]s the [[würschmidt]] temperament. 
* In the [[7-limit]], it also tempers out [[225/224]], and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. 
* In the [[11-limit]], it tempers out [[99/98]], [[176/175]] and [[243/242]], and is an excellent tuning for the 11-limit version of würschmidt, as well as [[minerva]], the [[rank-3 temperament]] tempering out 99/98 and 176/175, for which it is the [[optimal patent val]] and the rank-4 temperament tempering out 99/98, for which it also provides the optimal patent val.

=== Odd harmonics ===
{{Harmonics in equal|127}}

=== Subsets and supersets ===
127edo is the 31st [[prime edo]], following [[113edo]] and before [[131edo]].

== Scales ==
=== MOS scales ===
See [[List of MOS scales in 127edo]]. 

[[Category:Würschmidt]]
[[Category:Hemiwürschmidt]]
[[Category:Minerva]]</text>
      <sha1>0zvmuaj6td7bxtigmw932j5fau6a95x</sha1>
    </revision>
    <revision>
      <id>181336</id>
      <parentid>145209</parentid>
      <timestamp>2025-02-18T17:00:09Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181336</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1118" sha1="9qfvqk6e0xcnjy50jmgypbyuoimduwq" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
127edo is interesting because of its approximations, defined by the [[comma]]s it [[tempering out|tempers out]]:
* In the [[5-limit]], it tempers out 393216/390625 ([[würschmidt comma]]) and hence [[support]]s the [[würschmidt]] temperament. 
* In the [[7-limit]], it also tempers out [[225/224]], and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. 
* In the [[11-limit]], it tempers out [[99/98]], [[176/175]] and [[243/242]], and is an excellent tuning for the 11-limit version of würschmidt, as well as [[minerva]], the [[rank-3 temperament]] tempering out 99/98 and 176/175, for which it is the [[optimal patent val]] and the rank-4 temperament tempering out 99/98, for which it also provides the optimal patent val.

=== Odd harmonics ===
{{Harmonics in equal|127}}

=== Subsets and supersets ===
127edo is the 31st [[prime edo]], following [[113edo]] and before [[131edo]].

== Scales ==
=== MOS scales ===
See [[List of MOS scales in 127edo]]. 

[[Category:Würschmidt]]
[[Category:Hemiwürschmidt]]
[[Category:Minerva]]</text>
      <sha1>9qfvqk6e0xcnjy50jmgypbyuoimduwq</sha1>
    </revision>
    <revision>
      <id>205441</id>
      <parentid>181336</parentid>
      <timestamp>2025-07-30T06:51:04Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <comment>Add lumatone mapping link.</comment>
      <origin>205441</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1171" sha1="3wxeo6iaitrj4i8qa9hiaxdt3uop00g" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

== Theory ==
127edo is interesting because of its approximations, defined by the [[comma]]s it [[tempering out|tempers out]]:
* In the [[5-limit]], it tempers out 393216/390625 ([[würschmidt comma]]) and hence [[support]]s the [[würschmidt]] temperament. 
* In the [[7-limit]], it also tempers out [[225/224]], and is an excellent tuning for the 7-limit extension of würschmidt which tempers this out also. 
* In the [[11-limit]], it tempers out [[99/98]], [[176/175]] and [[243/242]], and is an excellent tuning for the 11-limit version of würschmidt, as well as [[minerva]], the [[rank-3 temperament]] tempering out 99/98 and 176/175, for which it is the [[optimal patent val]] and the rank-4 temperament tempering out 99/98, for which it also provides the optimal patent val.

=== Odd harmonics ===
{{Harmonics in equal|127}}

=== Subsets and supersets ===
127edo is the 31st [[prime edo]], following [[113edo]] and before [[131edo]].

== Scales ==
=== MOS scales ===
See [[List of MOS scales in 127edo]]. 

== Instruments ==
* [[Lumatone mapping for 127edo]]

[[Category:Würschmidt]]
[[Category:Hemiwürschmidt]]
[[Category:Minerva]]</text>
      <sha1>3wxeo6iaitrj4i8qa9hiaxdt3uop00g</sha1>
    </revision>
  </page>
  <page>
    <title>128/125</title>
    <ns>0</ns>
    <id>119</id>
    <revision>
      <id>119</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>119</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1025" sha1="8xslycgw0cx2ibfctfh0warzyc5hbjh" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;Other languages: [[:de:128/125 Deutsch]]&lt;/span&gt;&lt;span style="display: block; text-align: left;"&gt;

&lt;/span&gt;&lt;span style="display: block; text-align: left;"&gt;The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[Comma|comma]]'''; it represents the gap between three [[5/4|5/4]] just major thirds and the [[Octave|octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[Quarter-comma_meantone|quarter comma meantone]], it will be exact. Furthermore, in quarter-comma.meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic comma''' for this reason. Tempering it out leads to [[Augmented_family|augmented temperament]].&lt;/span&gt;      [[Category:5-limit]]
[[Category:comma]]
[[Category:diesis]]</text>
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    </revision>
    <revision>
      <id>4566</id>
      <timestamp>2014-02-05T17:47:43Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 487593550 - Original comment: **</comment>
      <origin>4566</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3824" sha1="hr1668d9rofrux3rkdx9tb3g10zi758" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2014-02-05 17:47:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;487593550&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;Other languages: [[xenharmonie/128_125|Deutsch]]&amp;lt;/span&amp;gt;&amp;lt;span style="display: block; text-align: left;"&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;span style="display: block; text-align: left;"&amp;gt;The 41.059 cent interval of **128/125** is called the **diesis** or **augmented [[comma]]**; it represents the gap between three [[5_4|5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[Quarter-comma meantone|quarter comma meantone]], it will be exact. Furthermore, in quarter-comma.meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called **enharmonic comma** for this reason. Tempering it out leads to [[Augmented family|augmented temperament]].&amp;lt;/span&amp;gt;&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;Other languages: &amp;lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/128_125"&amp;gt;Deutsch&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;span style="display: block; text-align: left;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;span style="display: block; text-align: left;"&amp;gt;The 41.059 cent interval of &amp;lt;strong&amp;gt;128/125&amp;lt;/strong&amp;gt; is called the &amp;lt;strong&amp;gt;diesis&amp;lt;/strong&amp;gt; or &amp;lt;strong&amp;gt;augmented &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;; it represents the gap between three &amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt; just major thirds and the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt;, or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in &amp;lt;a class="wiki_link" href="/28edo"&amp;gt;28&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34&amp;lt;/a&amp;gt; EDO, and by two steps of &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/59edo"&amp;gt;59&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/65edo"&amp;gt;65&amp;lt;/a&amp;gt;. In any tuning with just major, thirds, such as &amp;lt;a class="wiki_link" href="/Quarter-comma%20meantone"&amp;gt;quarter comma meantone&amp;lt;/a&amp;gt;, it will be exact. Furthermore, in quarter-comma.meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called &amp;lt;strong&amp;gt;enharmonic comma&amp;lt;/strong&amp;gt; for this reason. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Augmented%20family"&amp;gt;augmented temperament&amp;lt;/a&amp;gt;.&amp;lt;/span&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4567</id>
      <timestamp>2014-02-05T17:46:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 487593392 - Original comment: **</comment>
      <origin>4567</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3676" sha1="aqghfptrlmewf81qquxro4leeinsxun" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2014-02-05 17:46:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;487593392&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;Other languages: [[xenharmonie/128_125|Deutsch]]&amp;lt;/span&amp;gt;&amp;lt;span style="display: block; text-align: left;"&amp;gt;The 41.059 cent interval of **128/125** is called the **diesis** or **augmented [[comma]]**; it represents the gap between three [[5_4|5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[Quarter-comma meantone|quarter comma meantone]], it will be exact. Furthermore, in quarter-comma.meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called **enharmonic comma** for this reason. Tempering it out leads to [[Augmented family|augmented temperament]].&amp;lt;/span&amp;gt;&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;Other languages: &amp;lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/128_125"&amp;gt;Deutsch&amp;lt;/a&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;span style="display: block; text-align: left;"&amp;gt;The 41.059 cent interval of &amp;lt;strong&amp;gt;128/125&amp;lt;/strong&amp;gt; is called the &amp;lt;strong&amp;gt;diesis&amp;lt;/strong&amp;gt; or &amp;lt;strong&amp;gt;augmented &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;; it represents the gap between three &amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt; just major thirds and the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt;, or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in &amp;lt;a class="wiki_link" href="/28edo"&amp;gt;28&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34&amp;lt;/a&amp;gt; EDO, and by two steps of &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/59edo"&amp;gt;59&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/65edo"&amp;gt;65&amp;lt;/a&amp;gt;. In any tuning with just major, thirds, such as &amp;lt;a class="wiki_link" href="/Quarter-comma%20meantone"&amp;gt;quarter comma meantone&amp;lt;/a&amp;gt;, it will be exact. Furthermore, in quarter-comma.meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called &amp;lt;strong&amp;gt;enharmonic comma&amp;lt;/strong&amp;gt; for this reason. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Augmented%20family"&amp;gt;augmented temperament&amp;lt;/a&amp;gt;.&amp;lt;/span&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    </revision>
    <revision>
      <id>4568</id>
      <timestamp>2013-05-24T07:36:12Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 434012608 - Original comment: **</comment>
      <origin>4568</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3248" sha1="dz4j2jfhwt7nx51hyks11m9irrfh9ou" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2013-05-24 07:36:12 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;434012608&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 41.059 cent interval of **128/125** is called the **diesis** or **augmented [[comma]]**; it represents the gap between three [[5_4|5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[Quarter-comma meantone|quarter comma meantone]], it will be exact. Furthermore, in quarter-comma.meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called **enharmonic comma** for this reason. Tempering it out leads to [[Augmented family|augmented temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 41.059 cent interval of &amp;lt;strong&amp;gt;128/125&amp;lt;/strong&amp;gt; is called the &amp;lt;strong&amp;gt;diesis&amp;lt;/strong&amp;gt; or &amp;lt;strong&amp;gt;augmented &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;; it represents the gap between three &amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt; just major thirds and the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt;, or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in &amp;lt;a class="wiki_link" href="/28edo"&amp;gt;28&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34&amp;lt;/a&amp;gt; EDO, and by two steps of &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/59edo"&amp;gt;59&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/65edo"&amp;gt;65&amp;lt;/a&amp;gt;. In any tuning with just major, thirds, such as &amp;lt;a class="wiki_link" href="/Quarter-comma%20meantone"&amp;gt;quarter comma meantone&amp;lt;/a&amp;gt;, it will be exact. Furthermore, in quarter-comma.meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called &amp;lt;strong&amp;gt;enharmonic comma&amp;lt;/strong&amp;gt; for this reason. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Augmented%20family"&amp;gt;augmented temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    </revision>
    <revision>
      <id>4569</id>
      <timestamp>2013-05-24T07:35:57Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 434012576 - Original comment: **</comment>
      <origin>4569</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3215" sha1="boq7pufcl5w9c9zw4iejgq9t4tob0gf" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2013-05-24 07:35:57 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;434012576&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 41.059 cent interval of **128/125** is called the **diesis or augmented [[comma]]**; it represents the gap between three [[5_4|5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[Quarter-comma meantone|quarter comma meantone]], it will be exact. Furthermore, in quarter-comma.meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called **enharmonic comma** for this reason. Tempering it out leads to [[Augmented family|augmented temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 41.059 cent interval of &amp;lt;strong&amp;gt;128/125&amp;lt;/strong&amp;gt; is called the &amp;lt;strong&amp;gt;diesis or augmented &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;; it represents the gap between three &amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt; just major thirds and the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt;, or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in &amp;lt;a class="wiki_link" href="/28edo"&amp;gt;28&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34&amp;lt;/a&amp;gt; EDO, and by two steps of &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/59edo"&amp;gt;59&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/65edo"&amp;gt;65&amp;lt;/a&amp;gt;. In any tuning with just major, thirds, such as &amp;lt;a class="wiki_link" href="/Quarter-comma%20meantone"&amp;gt;quarter comma meantone&amp;lt;/a&amp;gt;, it will be exact. Furthermore, in quarter-comma.meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called &amp;lt;strong&amp;gt;enharmonic comma&amp;lt;/strong&amp;gt; for this reason. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Augmented%20family"&amp;gt;augmented temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4570</id>
      <timestamp>2013-05-24T06:54:18Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 434006950 - Original comment: typographic changes only**</comment>
      <origin>4570</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2907" sha1="1v9owyid0vw56v4sqsezubnuk47jb2f" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2013-05-24 06:54:18 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;434006950&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;typographic changes only&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 41.059 cent interval of **128/125** is called the **diesis** or **augmented [[comma]]**; it represents the gap between three [[5_4|5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[Quarter-comma meantone|quarter comma meantone]], it will be exact. Tempering it out leads to [[Augmented family|augmented temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 41.059 cent interval of &amp;lt;strong&amp;gt;128/125&amp;lt;/strong&amp;gt; is called the &amp;lt;strong&amp;gt;diesis&amp;lt;/strong&amp;gt; or &amp;lt;strong&amp;gt;augmented &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;&amp;lt;/strong&amp;gt;; it represents the gap between three &amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt; just major thirds and the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt;, or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in &amp;lt;a class="wiki_link" href="/28edo"&amp;gt;28&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34&amp;lt;/a&amp;gt; EDO, and by two steps of &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/59edo"&amp;gt;59&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/65edo"&amp;gt;65&amp;lt;/a&amp;gt;. In any tuning with just major, thirds, such as &amp;lt;a class="wiki_link" href="/Quarter-comma%20meantone"&amp;gt;quarter comma meantone&amp;lt;/a&amp;gt;, it will be exact. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Augmented%20family"&amp;gt;augmented temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4571</id>
      <timestamp>2011-12-06T06:38:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 282758034 - Original comment: **</comment>
      <origin>4571</origin>
      <model>wikitext</model>
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      <text bytes="2796" sha1="5uk2rl9ts9iw1bgcfacuujce6ygwxzt" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-12-06 06:38:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;282758034&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 41.059 cent interval of 128/125 is called the diesis or augmented [[comma]]; it represents the gap between three [[5_4|5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[Quarter-comma meantone|quarter comma meantone]], it will be exact. Tempering it out leads to [[Augmented family|augmented temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 41.059 cent interval of 128/125 is called the diesis or augmented &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;; it represents the gap between three &amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt; just major thirds and the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt;, or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in &amp;lt;a class="wiki_link" href="/28edo"&amp;gt;28&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34&amp;lt;/a&amp;gt; EDO, and by two steps of &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/59edo"&amp;gt;59&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/65edo"&amp;gt;65&amp;lt;/a&amp;gt;. In any tuning with just major, thirds, such as &amp;lt;a class="wiki_link" href="/Quarter-comma%20meantone"&amp;gt;quarter comma meantone&amp;lt;/a&amp;gt;, it will be exact. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Augmented%20family"&amp;gt;augmented temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4572</id>
      <timestamp>2011-12-05T21:47:54Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 282668710 - Original comment: **</comment>
      <origin>4572</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2812" sha1="391cqgu0ds3eko6gaoaxosq6g9ql8e9" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-12-05 21:47:54 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;282668710&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 41.059 cent interval of 128/125 is called the diesis or augmented [[comma]]; it represents the gap between three [[5_4|5/4]] just major thirds and the [[octave]], or in other words 2/(128/125)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[Quarter-comma meantone|quarter comma meantone]], it will be exact. Tempering it out leads to [[Augmented family|augmented temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 41.059 cent interval of 128/125 is called the diesis or augmented &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;; it represents the gap between three &amp;lt;a class="wiki_link" href="/5_4"&amp;gt;5/4&amp;lt;/a&amp;gt; just major thirds and the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt;, or in other words 2/(128/125)^3. It is fairly accurately represented by a single step in &amp;lt;a class="wiki_link" href="/28edo"&amp;gt;28&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34&amp;lt;/a&amp;gt; EDO, and by two steps of &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/59edo"&amp;gt;59&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/65edo"&amp;gt;65&amp;lt;/a&amp;gt;. In any tuning with just major, thirds, such as &amp;lt;a class="wiki_link" href="/Quarter-comma%20meantone"&amp;gt;quarter comma meantone&amp;lt;/a&amp;gt;, it will be exact. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Augmented%20family"&amp;gt;augmented temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4573</id>
      <timestamp>2011-08-26T12:20:27Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 248654139 - Original comment: **</comment>
      <origin>4573</origin>
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      <format>text/x-wiki</format>
      <text bytes="2636" sha1="02kpostorc7w0jra8gpdha8rhz30vph" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-26 12:20:27 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;248654139&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 41.059 cent interval of 128/125 is called the diesis or augmented comma; it represents the gap between three 5/4 just major thirds and the octave, or in other words 2/(128/125)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[Quarter-comma meantone|quarter comma meantone]], it will be exact. Tempering it out leads to [[Augmented family|augmented temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 41.059 cent interval of 128/125 is called the diesis or augmented comma; it represents the gap between three 5/4 just major thirds and the octave, or in other words 2/(128/125)^3. It is fairly accurately represented by a single step in &amp;lt;a class="wiki_link" href="/28edo"&amp;gt;28&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34&amp;lt;/a&amp;gt; EDO, and by two steps of &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/59edo"&amp;gt;59&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/65edo"&amp;gt;65&amp;lt;/a&amp;gt;. In any tuning with just major, thirds, such as &amp;lt;a class="wiki_link" href="/Quarter-comma%20meantone"&amp;gt;quarter comma meantone&amp;lt;/a&amp;gt;, it will be exact. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Augmented%20family"&amp;gt;augmented temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>02kpostorc7w0jra8gpdha8rhz30vph</sha1>
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    <revision>
      <id>4574</id>
      <timestamp>2011-08-26T11:48:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 248647945 - Original comment: **</comment>
      <origin>4574</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2612" sha1="9timx56xdam7hnifjusur2fm166t91j" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-26 11:48:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;248647945&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The interval of 128/125 is called the diesis or augmented comma; it represents the gap between three 5/4 just major thirds and the octave, or in other words 2/(128/125)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[Quarter-comma meantone|quarter comma meantone]], it will be exact. Tempering it out leads to [[Augmented family|augmented temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The interval of 128/125 is called the diesis or augmented comma; it represents the gap between three 5/4 just major thirds and the octave, or in other words 2/(128/125)^3. It is fairly accurately represented by a single step in &amp;lt;a class="wiki_link" href="/28edo"&amp;gt;28&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34&amp;lt;/a&amp;gt; EDO, and by two steps of &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/59edo"&amp;gt;59&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/65edo"&amp;gt;65&amp;lt;/a&amp;gt;. In any tuning with just major, thirds, such as &amp;lt;a class="wiki_link" href="/Quarter-comma%20meantone"&amp;gt;quarter comma meantone&amp;lt;/a&amp;gt;, it will be exact. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Augmented%20family"&amp;gt;augmented temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9timx56xdam7hnifjusur2fm166t91j</sha1>
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    <revision>
      <id>4575</id>
      <timestamp>2011-08-26T11:46:50Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 248647581 - Original comment: **</comment>
      <origin>4575</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2591" sha1="e2grmuf0zr72odfhla001a3oeqiq05t" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-26 11:46:50 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;248647581&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The interval of 128/125 is called the diesis or augmented comma; it represents the gap between three 5/4 just major thirds and the octave, or in other words 2/(128/125)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[quarter comma meantone]], it will be exact. Tempering it out leads to [[Augmented family|augmented temperament]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128_125&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The interval of 128/125 is called the diesis or augmented comma; it represents the gap between three 5/4 just major thirds and the octave, or in other words 2/(128/125)^3. It is fairly accurately represented by a single step in &amp;lt;a class="wiki_link" href="/28edo"&amp;gt;28&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/34edo"&amp;gt;34&amp;lt;/a&amp;gt; EDO, and by two steps of &amp;lt;a class="wiki_link" href="/53edo"&amp;gt;53&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/59edo"&amp;gt;59&amp;lt;/a&amp;gt; or &amp;lt;a class="wiki_link" href="/65edo"&amp;gt;65&amp;lt;/a&amp;gt;. In any tuning with just major, thirds, such as &amp;lt;a class="wiki_link" href="/quarter%20comma%20meantone"&amp;gt;quarter comma meantone&amp;lt;/a&amp;gt;, it will be exact. Tempering it out leads to &amp;lt;a class="wiki_link" href="/Augmented%20family"&amp;gt;augmented temperament&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>32355</id>
      <parentid>119</parentid>
      <timestamp>2018-09-17T00:09:22Z</timestamp>
      <contributor>
        <username>Mike Battaglia</username>
        <id>3</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "(\[\[\:(dev|es|purdal|de|en):[^[:space:]\|]*) (.*])" to "$1|$3"</comment>
      <origin>32355</origin>
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      <format>text/x-wiki</format>
      <text bytes="1025" sha1="ok90q4bp6soe6sr2e27f2l11i6cz6tb" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;Other languages: [[:de:128/125|Deutsch]]&lt;/span&gt;&lt;span style="display: block; text-align: left;"&gt;

&lt;/span&gt;&lt;span style="display: block; text-align: left;"&gt;The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[Comma|comma]]'''; it represents the gap between three [[5/4|5/4]] just major thirds and the [[Octave|octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[Quarter-comma_meantone|quarter comma meantone]], it will be exact. Furthermore, in quarter-comma.meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic comma''' for this reason. Tempering it out leads to [[Augmented_family|augmented temperament]].&lt;/span&gt;      [[Category:5-limit]]
[[Category:comma]]
[[Category:diesis]]</text>
      <sha1>ok90q4bp6soe6sr2e27f2l11i6cz6tb</sha1>
    </revision>
    <revision>
      <id>33893</id>
      <parentid>32355</parentid>
      <timestamp>2018-09-29T11:48:06Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <origin>33893</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1007" sha1="74r0p5rce4y06w9q71fomwy0kgqgjp2" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[de:128/125|Deutsch]]&lt;/span&gt;&lt;span style="display: block; text-align: left;"&gt;

&lt;/span&gt;&lt;span style="display: block; text-align: left;"&gt;The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[Comma|comma]]'''; it represents the gap between three [[5/4|5/4]] just major thirds and the [[Octave|octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]]. In any tuning with just major, thirds, such as [[Quarter-comma_meantone|quarter comma meantone]], it will be exact. Furthermore, in quarter-comma.meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic comma''' for this reason. Tempering it out leads to [[Augmented_family|augmented temperament]].&lt;/span&gt;      [[Category:5-limit]]
[[Category:comma]]
[[Category:diesis]]</text>
      <sha1>74r0p5rce4y06w9q71fomwy0kgqgjp2</sha1>
    </revision>
    <revision>
      <id>34288</id>
      <parentid>33893</parentid>
      <timestamp>2018-10-01T12:01:31Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>reworked, simplified wiki links, placed interwiki below categories</comment>
      <origin>34288</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="850" sha1="7k8wz0pfqmgyk15m1kl1jow0lfu8int" xml:space="preserve">The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]] EDO. In any tuning with just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic comma''' for this reason. Tempering it out leads to [[Augmented_family|augmented temperament]].

[[Category:5-limit]]
[[Category:comma]]
[[Category:diesis]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]</text>
      <sha1>7k8wz0pfqmgyk15m1kl1jow0lfu8int</sha1>
    </revision>
    <revision>
      <id>49365</id>
      <parentid>34288</parentid>
      <timestamp>2020-09-13T12:14:09Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>cats</comment>
      <origin>49365</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="891" sha1="1s9dt80twfg0r6898cei8p81wlldc9t" xml:space="preserve">The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]] EDO. In any tuning with just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic comma''' for this reason. Tempering it out leads to [[Augmented_family|augmented temperament]].

[[Category:5-limit]]
[[Category:Medium comma]]
[[Category:Diesis]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>1s9dt80twfg0r6898cei8p81wlldc9t</sha1>
    </revision>
    <revision>
      <id>63143</id>
      <parentid>49365</parentid>
      <timestamp>2021-03-15T09:30:23Z</timestamp>
      <contributor>
        <username>Flirora</username>
        <id>236</id>
      </contributor>
      <comment>+infobox</comment>
      <origin>63143</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1070" sha1="rh7d1ad4b6m9zemyn0u2illrw44v4j8" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 128/125
| Monzo = 7 0 -3
| Cents = 41.05886
| Name = diesis, augmented comma
| Color name = 
| FJS name = d2&lt;sub&gt;5,5,5&lt;/sub&gt;
| Sound = 
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]] EDO. In any tuning with just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic comma''' for this reason. Tempering it out leads to [[Augmented_family|augmented temperament]].

[[Category:5-limit]]
[[Category:Medium comma]]
[[Category:Diesis]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>rh7d1ad4b6m9zemyn0u2illrw44v4j8</sha1>
    </revision>
    <revision>
      <id>71882</id>
      <parentid>63143</parentid>
      <timestamp>2021-06-05T10:35:35Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>71882</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1271" sha1="e542figjrhoertm7jm9k0rx1dnajf2g" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 128/125
| Monzo = 7 0 -3
| Cents = 41.05886
| Name = diesis, augmented comma
| Color name = 
| FJS name = d2&lt;sub&gt;5,5,5&lt;/sub&gt;
| Sound = 
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]] EDO. In any tuning with just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic comma''' for this reason. Tempering it out leads to [[Augmented_family|augmented temperament]].

This interval can also be called a '''kilobyte comma''', since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

[[Category:5-limit]]
[[Category:Medium comma]]
[[Category:Diesis]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>e542figjrhoertm7jm9k0rx1dnajf2g</sha1>
    </revision>
    <revision>
      <id>84627</id>
      <parentid>71882</parentid>
      <timestamp>2022-01-23T20:56:38Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>84627</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1308" sha1="ghslz5ehimqazxyy9yxqnji7u6nf6r0" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 128/125
| Monzo = 7 0 -3
| Cents = 41.05886
| Name = diesis, augmented comma
| Color name = 
| FJS name = d2&lt;sub&gt;5,5,5&lt;/sub&gt;
| Sound = 
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)^3. It is fairly accurately represented by a single step in [[28edo|28]], [[31edo|31]] or [[34edo|34]] EDO, and by two steps of [[53edo|53]], [[59edo|59]] or [[65edo|65]] EDO. In any tuning with just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic comma''' for this reason. Tempering it out leads to [[Augmented_family|augmented temperament]].

This interval can also be called a '''kilobyte comma''', since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

[[Category:5-limit]]
[[Category:Medium comma]]
[[Category:Diesis]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>ghslz5ehimqazxyy9yxqnji7u6nf6r0</sha1>
    </revision>
    <revision>
      <id>85780</id>
      <parentid>84627</parentid>
      <timestamp>2022-02-12T22:24:07Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Clarify and improve links</comment>
      <origin>85780</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1306" sha1="n0qaebjbin7oo7whkxbz52k4fdqfxau" xml:space="preserve">{{Infobox Interval
| Ratio = 128/125
| Monzo = 7 0 -3
| Cents = 41.05886
| Name = diesis, &lt;br&gt;augmented comma
| Color name = 
| FJS name = d2&lt;sub&gt;5,5,5&lt;/sub&gt;
| Sound = 
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic comma''' for this reason. Tempering it out leads to [[augmented]] temperament.

This interval can also be called a '''kilobyte comma''', since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

[[Category:5-limit]]
[[Category:Medium comma]]
[[Category:Diesis]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>n0qaebjbin7oo7whkxbz52k4fdqfxau</sha1>
    </revision>
    <revision>
      <id>85861</id>
      <parentid>85780</parentid>
      <timestamp>2022-02-13T05:42:41Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>added the color name</comment>
      <origin>85861</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1336" sha1="66dt369zy447bsn4eqq915ityi1e0t8" xml:space="preserve">{{Infobox Interval
| Ratio = 128/125
| Monzo = 7 0 -3
| Cents = 41.05886
| Name = diesis, &lt;br&gt;augmented comma
| Color name = g32, trigu unison, Trigu comma
| FJS name = d2&lt;sub&gt;5,5,5&lt;/sub&gt;
| Sound = 
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic comma''' for this reason. Tempering it out leads to [[augmented]] temperament.

This interval can also be called a '''kilobyte comma''', since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

[[Category:5-limit]]
[[Category:Medium comma]]
[[Category:Diesis]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>66dt369zy447bsn4eqq915ityi1e0t8</sha1>
    </revision>
    <revision>
      <id>89785</id>
      <parentid>85861</parentid>
      <timestamp>2022-03-20T21:53:41Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Medium comma]] to [[Category:Medium commas]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>89785</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1337" sha1="kptg47hz2ajjoab81dlqqk818bcxxfv" xml:space="preserve">{{Infobox Interval
| Ratio = 128/125
| Monzo = 7 0 -3
| Cents = 41.05886
| Name = diesis, &lt;br&gt;augmented comma
| Color name = g32, trigu unison, Trigu comma
| FJS name = d2&lt;sub&gt;5,5,5&lt;/sub&gt;
| Sound = 
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic comma''' for this reason. Tempering it out leads to [[augmented]] temperament.

This interval can also be called a '''kilobyte comma''', since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

[[Category:5-limit]]
[[Category:Medium commas]]
[[Category:Diesis]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>kptg47hz2ajjoab81dlqqk818bcxxfv</sha1>
    </revision>
    <revision>
      <id>97757</id>
      <parentid>89785</parentid>
      <timestamp>2022-10-20T00:50:31Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>De-bold the anecdotal name. The original names are pretty established and not subject to changes. +"enharmonic diesis"</comment>
      <origin>97757</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1358" sha1="1n3y56orh9pz872w0mnwazn8zrnmkfr" xml:space="preserve">{{Infobox Interval
| Ratio = 128/125
| Monzo = 7 0 -3
| Cents = 41.05886
| Name = diesis, &lt;br&gt;augmented comma
| Color name = g32, trigu unison, Trigu comma
| FJS name = d2&lt;sub&gt;5,5,5&lt;/sub&gt;
| Sound = 
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. Tempering it out leads to [[augmented]] temperament.

This interval can also be called a kilobyte comma, since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

[[Category:5-limit]]
[[Category:Medium commas]]
[[Category:Diesis]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>1n3y56orh9pz872w0mnwazn8zrnmkfr</sha1>
    </revision>
    <revision>
      <id>97758</id>
      <parentid>97757</parentid>
      <timestamp>2022-10-20T00:51:37Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Improve categories</comment>
      <origin>97758</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1361" sha1="if1j93ix7y82svgsggin0bxpixd9gaq" xml:space="preserve">{{Infobox Interval
| Ratio = 128/125
| Monzo = 7 0 -3
| Cents = 41.05886
| Name = diesis, &lt;br&gt;augmented comma
| Color name = g32, trigu unison, Trigu comma
| FJS name = d2&lt;sub&gt;5,5,5&lt;/sub&gt;
| Sound = 
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. Tempering it out leads to [[augmented]] temperament.

This interval can also be called a kilobyte comma, since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

[[Category:5-limit]]
[[Category:Medium commas]]
[[Category:Augmented]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>if1j93ix7y82svgsggin0bxpixd9gaq</sha1>
    </revision>
    <revision>
      <id>97762</id>
      <parentid>97758</parentid>
      <timestamp>2022-10-20T01:12:27Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+see also</comment>
      <origin>97762</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1412" sha1="acsp7w5mbgincf3nnc1b1m8ul4gunlf" xml:space="preserve">{{Infobox Interval
| Ratio = 128/125
| Monzo = 7 0 -3
| Cents = 41.05886
| Name = diesis, &lt;br&gt;augmented comma
| Color name = g32, trigu unison, Trigu comma
| FJS name = d2&lt;sub&gt;5,5,5&lt;/sub&gt;
| Sound = 
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. Tempering it out leads to [[augmented]] temperament.

This interval can also be called a kilobyte comma, since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:5-limit]]
[[Category:Medium commas]]
[[Category:Augmented]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>acsp7w5mbgincf3nnc1b1m8ul4gunlf</sha1>
    </revision>
    <revision>
      <id>97994</id>
      <parentid>97762</parentid>
      <timestamp>2022-10-25T12:00:35Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97994</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1275" sha1="ioks74m3ffv3amljwgi2b637ryn3ylb" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma
| Color name = g32, trigu unison, Trigu comma
| Comma = 1
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. Tempering it out leads to [[augmented]] temperament.

This interval can also be called a kilobyte comma, since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>ioks74m3ffv3amljwgi2b637ryn3ylb</sha1>
    </revision>
    <revision>
      <id>98485</id>
      <parentid>97994</parentid>
      <timestamp>2022-10-25T18:15:03Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <origin>98485</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1277" sha1="9oohicvzewww4au6iskk63hck0ca57f" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma
| Color name = g32, trigu unison, Trigu comma
| Comma = yes
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. Tempering it out leads to [[augmented]] temperament.

This interval can also be called a kilobyte comma, since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>9oohicvzewww4au6iskk63hck0ca57f</sha1>
    </revision>
    <revision>
      <id>100545</id>
      <parentid>98485</parentid>
      <timestamp>2022-12-29T01:43:08Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>corrected color name</comment>
      <origin>100545</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1274" sha1="fbunm77ptrtd4tphjieiq6v40dry3ts" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, Trigu comma
| Comma = yes
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. Tempering it out leads to [[augmented]] temperament.

This interval can also be called a kilobyte comma, since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>fbunm77ptrtd4tphjieiq6v40dry3ts</sha1>
    </revision>
    <revision>
      <id>101165</id>
      <parentid>100545</parentid>
      <timestamp>2023-01-10T02:35:41Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <origin>101165</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1288" sha1="0hm8ab0qmoklzku4rwmt2bwe60z2mr6" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. Tempering it out leads to [[augmented]] temperament.

This interval can also be called a kilobyte comma, since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>0hm8ab0qmoklzku4rwmt2bwe60z2mr6</sha1>
    </revision>
    <revision>
      <id>105466</id>
      <parentid>101165</parentid>
      <timestamp>2023-03-13T03:54:19Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>105466</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1499" sha1="fxe4y93pjqoe59ud37vumgecqlqa00s" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. Tempering it out leads to [[augmented]] temperament.

This interval can also be called a kilobyte comma, since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce 7-limit and 11-limit harmony into the 5-limit.

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>fxe4y93pjqoe59ud37vumgecqlqa00s</sha1>
    </revision>
    <revision>
      <id>105467</id>
      <parentid>105466</parentid>
      <timestamp>2023-03-13T03:55:16Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>105467</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1510" sha1="9uxmjlcz1klgpv3yvj5xxwz59n9ev14" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented [[comma]]'''; it represents the gap between three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in quarter-comma meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. Tempering it out leads to [[augmented]] temperament.

This interval can also be called a kilobyte comma, since it represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales.

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>9uxmjlcz1klgpv3yvj5xxwz59n9ev14</sha1>
    </revision>
    <revision>
      <id>105492</id>
      <parentid>105467</parentid>
      <timestamp>2023-03-13T07:16:57Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Sectioning</comment>
      <origin>105492</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1589" sha1="damwu8vtx758xd0g142sy03ou6bqbkp" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. 

== Approximation ==
If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. 

It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. 

== Temperaments ==
Tempering out this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out. 

== Trivia ==
This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>damwu8vtx758xd0g142sy03ou6bqbkp</sha1>
    </revision>
    <revision>
      <id>117758</id>
      <parentid>105492</parentid>
      <timestamp>2023-06-29T13:33:49Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added enharmonic diesis and enharmonic comma to the list of names in the infobox</comment>
      <origin>117758</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1626" sha1="ni78a1j1l9vr9z50e7p3t6szq8zl1zj" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma, enharmonic diesis, enharmonic comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. 

== Approximation ==
If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. 

It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. 

== Temperaments ==
Tempering out this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out. 

== Trivia ==
This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Real-life sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>ni78a1j1l9vr9z50e7p3t6szq8zl1zj</sha1>
    </revision>
    <revision>
      <id>121730</id>
      <parentid>117758</parentid>
      <timestamp>2023-08-08T07:31:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Recategorize</comment>
      <origin>121730</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1616" sha1="mhxh406nrud4y2tqu9zusmaz4s9ey4r" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma, enharmonic diesis, enharmonic comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. 

== Approximation ==
If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. 

It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. 

== Temperaments ==
Tempering out this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out. 

== Trivia ==
This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>mhxh406nrud4y2tqu9zusmaz4s9ey4r</sha1>
    </revision>
    <revision>
      <id>129096</id>
      <parentid>121730</parentid>
      <timestamp>2023-12-15T07:51:37Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>129096</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2131" sha1="12jzm105co4h09pznnmbuh8a9z582wd" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma, enharmonic diesis, enharmonic comma, cyberisma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. 

== Approximation ==
If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. 

It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. 

== Temperaments ==
Tempering out this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out.  

=== Walker temperament ===
Walker temperament flattens the octave in order to temper out the difference between 2^10 and 10^3, which is an occurrence of the diesis. [[40ed10]] is an example of Walker temperament. It was created by Vector Graphics (Jay Walker) before he even knew what temperament was, as an attempt to align powers of 2 and powers of 10. In the context of Walker temperament, the augmented comma can be called the '''cyberisma,''' after its appearance in computing.

== Trivia ==
This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[https://de.xen.wiki/w/128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>12jzm105co4h09pznnmbuh8a9z582wd</sha1>
    </revision>
    <revision>
      <id>129097</id>
      <parentid>129096</parentid>
      <timestamp>2023-12-15T08:05:25Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Undo. More names not accepted for this comma and its corresponding temperament</comment>
      <origin>129097</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1616" sha1="mhxh406nrud4y2tqu9zusmaz4s9ey4r" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma, enharmonic diesis, enharmonic comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. 

== Approximation ==
If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. 

It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. 

== Temperaments ==
Tempering out this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out. 

== Trivia ==
This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]

[[Category:Todo:improve layout]]</text>
      <sha1>mhxh406nrud4y2tqu9zusmaz4s9ey4r</sha1>
    </revision>
    <revision>
      <id>130655</id>
      <parentid>129097</parentid>
      <timestamp>2024-01-04T02:53:50Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Improved layout</comment>
      <origin>130655</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1614" sha1="ntyre79dojrubjdrkq46cs2yeo8v8co" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma, enharmonic diesis, enharmonic comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. 

== As a comma ==
Tempering out this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out. 

== As an interval ==
If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. 

== Approximations ==
It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. 

== Relation to binary ==
This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[de:128/125]]</text>
      <sha1>ntyre79dojrubjdrkq46cs2yeo8v8co</sha1>
    </revision>
    <revision>
      <id>130656</id>
      <parentid>130655</parentid>
      <timestamp>2024-01-04T02:55:33Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Add to category augmented family</comment>
      <origin>130656</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1663" sha1="sc07s62o3zqa8mqct0wvknsrsotznkm" xml:space="preserve">{{Infobox Interval
| Name = diesis, augmented comma, enharmonic diesis, enharmonic comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059 cent interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. 

== As a comma ==
Tempering out this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out. 

== As an interval ==
If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. 

== Approximations ==
It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. 

== Relation to binary ==
This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Sonifications]]

&lt;!-- interwiki: links are being placed in the sidebar --&gt;
[[https://de.xen.wiki/w/128/125]]
[[Category:Augmented family]]</text>
      <sha1>sc07s62o3zqa8mqct0wvknsrsotznkm</sha1>
    </revision>
    <revision>
      <id>130713</id>
      <parentid>130656</parentid>
      <timestamp>2024-01-04T07:26:37Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Add back original section titles; adopt template: Interwiki</comment>
      <origin>130713</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1604" sha1="fuf5uek16tbfhaby9zafq7kt0htud5y" xml:space="preserve">{{Interwiki
| en = 128/125
| de = 128/125
}}
{{Infobox Interval
| Name = diesis, augmented comma, enharmonic diesis, enharmonic comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059-[[cent]] interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. 

== Temperaments ==
=== As a comma ===
[[Tempering out]] this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out. 

=== As an interval ===
If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. 

== Approximations ==
It is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. 

== Trivia ==
This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Sonifications]]</text>
      <sha1>fuf5uek16tbfhaby9zafq7kt0htud5y</sha1>
    </revision>
    <revision>
      <id>130717</id>
      <parentid>130713</parentid>
      <timestamp>2024-01-04T07:39:09Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Revert to original section order (I don't see anything wrong with this)</comment>
      <origin>130717</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1615" sha1="72jsk29nt1nudi079fc18us403qdvkh" xml:space="preserve">{{Interwiki
| en = 128/125
| de = 128/125
}}
{{Infobox Interval
| Name = diesis, augmented comma, enharmonic diesis, enharmonic comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059-[[cent]] interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. 

== Approximations ==
This interval is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. 

== Temperaments ==
=== As a comma ===
[[Tempering out]] this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out. 

=== As an interval ===
If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. 

== Trivia ==
This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Sonifications]]</text>
      <sha1>72jsk29nt1nudi079fc18us403qdvkh</sha1>
    </revision>
    <revision>
      <id>163509</id>
      <parentid>130717</parentid>
      <timestamp>2024-11-03T07:02:05Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <origin>163509</origin>
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      <text bytes="1667" sha1="d1kiwq4whpblkex0h96ud1lxsvq5a5d" xml:space="preserve">{{Interwiki
| en = 128/125
| de = 128/125
}}
{{Infobox Interval
| Name = diesis, augmented comma, enharmonic diesis, enharmonic comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059-[[cent]] interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. 

== Approximations ==
This interval is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. 

== Temperaments ==
=== As a comma ===
[[Tempering out]] this comma leads to [[augmented]] temperament. See [[augmented family]] for the family where it is tempered out. 

=== As an interval ===
If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales. 

== Trivia ==
This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Sonifications]]
[[Category:Commas named after their interval size]]</text>
      <sha1>d1kiwq4whpblkex0h96ud1lxsvq5a5d</sha1>
    </revision>
    <revision>
      <id>209725</id>
      <parentid>163509</parentid>
      <timestamp>2025-09-11T04:24:40Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Temperaments */ link</comment>
      <origin>209725</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1690" sha1="k0cuu4mqpkzhiy9kfje72vytj0wolhl" xml:space="preserve">{{Interwiki
| en = 128/125
| de = 128/125
}}
{{Infobox Interval
| Name = diesis, augmented comma, enharmonic diesis, enharmonic comma
| Color name = g&lt;sup&gt;3&lt;/sup&gt;2, trigu 2nd,&lt;br&gt;Trigu comma
| Comma = yes
}}

The 41.059-[[cent]] interval of '''128/125''' is called the '''diesis''' or '''augmented comma'''; it represents the gap between a stack of three [[5/4]] just major thirds and the [[octave]], or in other words 2/(5/4)&lt;sup&gt;3&lt;/sup&gt;. 

== Approximations ==
This interval is fairly accurately represented by a single step in [[28edo|28-]], [[31edo|31-]] or [[34edo]], and by two steps of [[53edo|53-]], [[59edo|59-]] or [[65edo]]. In any tuning with pure octaves and just major thirds, such as [[quarter-comma meantone]], it will be exact. Furthermore, in meantone it appears as the difference between sharps and flats, e.g. between D# and Eb. It is also called '''enharmonic diesis''' or '''enharmonic comma''' for this reason. 

== Temperaments ==
=== As a comma ===
[[Tempering out]] this comma leads to [[augmented (temperament)|augmented]] temperament. See [[Augmented family]] for the family where it is tempered out. 

=== As an interval ===
If the diesis is treated as a musical interval in its own right as opposed to tempering it out, it is approximately a quarter-tone and so can be used to introduce [[7-limit]] and [[11-limit]] harmony into 5-limit scales.

== Trivia ==
This interval represents the amount by which the binary definition of kilobyte, 1024 bytes, exceeds the nominal definition of "kilo" prefix, 1000. 

== See also ==
* [[Diesis]] (disambiguation page)

[[Category:Augmented]]
[[Category:Sonifications]]
[[Category:Commas named after their interval size]]</text>
      <sha1>k0cuu4mqpkzhiy9kfje72vytj0wolhl</sha1>
    </revision>
  </page>
  <page>
    <title>128 125</title>
    <ns>0</ns>
    <id>120</id>
    <redirect title="128/125" />
    <revision>
      <id>120</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>120</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21" sha1="64p87iyfpkoqpcyi9serhmoo15x0pp7" xml:space="preserve">#REDIRECT [[128/125]]</text>
      <sha1>64p87iyfpkoqpcyi9serhmoo15x0pp7</sha1>
    </revision>
  </page>
  <page>
    <title>128/81</title>
    <ns>0</ns>
    <id>121</id>
    <revision>
      <id>121</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>121</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="269" sha1="eo8h94nwqdllc95a56wl0g5pq6ka9b0" xml:space="preserve">'''128/81'''
|7 -4&amp;gt;

792.1800 cents

[[File:jid_128_81_pluck_adu_dr220.mp3]] [[:File:jid_128_81_pluck_adu_dr220.mp3|sound sample]]

128/81 is the Pythagorean minor sixth, at -4 steps on the chain of fifths.  Its inversion is [[81/64|81/64]].
[[Category:todo:expand]]</text>
      <sha1>eo8h94nwqdllc95a56wl0g5pq6ka9b0</sha1>
    </revision>
    <revision>
      <id>4576</id>
      <timestamp>2014-06-09T22:12:26Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513415858 - Original comment: **</comment>
      <origin>4576</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3052" sha1="p09ghlzq5ijoxllkp6lo392v791h1qd" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-09 22:12:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513415858&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**128/81**
|7 -4&amp;gt;
792.1800 cents
[[media type="file" key="jid_128_81_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_128_81_pluck_adu_dr220.mp3|sound sample]]

128/81 is the Pythagorean minor sixth, at -4 steps on the chain of fifths.  Its inversion is [[81_64|81/64]].
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128_81&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;128/81&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|7 -4&amp;amp;gt;&amp;lt;br /&amp;gt;
792.1800 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_128_81_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_128_81_pluck_adu_dr220.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_128_81_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_128_81_pluck_adu_dr220.mp3/513415368/jid_128_81_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_128_81_pluck_adu_dr220.mp3/513415368/jid_128_81_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
128/81 is the Pythagorean minor sixth, at -4 steps on the chain of fifths.  Its inversion is &amp;lt;a class="wiki_link" href="/81_64"&amp;gt;81/64&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>p09ghlzq5ijoxllkp6lo392v791h1qd</sha1>
    </revision>
    <revision>
      <id>62868</id>
      <parentid>121</parentid>
      <timestamp>2021-03-12T13:43:22Z</timestamp>
      <contributor>
        <username>Nolovoto</username>
        <id>277</id>
      </contributor>
      <comment>Created some changes toward making this an updated interval page.</comment>
      <origin>62868</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="403" sha1="ecqsc9cskpg5j3391ywyveouw8zcqy8" xml:space="preserve">'''128/81'''
|7 -4&amp;gt;

792.1800 cents

[[File:jid_128_81_pluck_adu_dr220.mp3]] [[:File:jid_128_81_pluck_adu_dr220.mp3|sound sample]]

128/81 is the '''Pythagorean minor sixth''', at -4 steps on the chain of fifths.  Its octave complement is [[81/64|81/64]].
[[Category:todo:expand]]
[[Category:3-limit]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Pythagorean]]
[[Category:Sound example]]</text>
      <sha1>ecqsc9cskpg5j3391ywyveouw8zcqy8</sha1>
    </revision>
    <revision>
      <id>62869</id>
      <parentid>62868</parentid>
      <timestamp>2021-03-12T13:44:37Z</timestamp>
      <contributor>
        <username>Nolovoto</username>
        <id>277</id>
      </contributor>
      <minor/>
      <comment>added listen category</comment>
      <origin>62869</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="396" sha1="ge0z2pneb57hevbe5gclamxtqdlgrvl" xml:space="preserve">'''128/81'''
|7 -4&amp;gt;

792.1800 cents

[[File:jid_128_81_pluck_adu_dr220.mp3]] [[:File:jid_128_81_pluck_adu_dr220.mp3|sound sample]]

128/81 is the '''Pythagorean minor sixth''', at -4 steps on the chain of fifths.  Its octave complement is [[81/64|81/64]].
[[Category:todo:expand]]
[[Category:3-limit]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Pythagorean]]
[[Category:Listen]]</text>
      <sha1>ge0z2pneb57hevbe5gclamxtqdlgrvl</sha1>
    </revision>
    <revision>
      <id>62922</id>
      <parentid>62869</parentid>
      <timestamp>2021-03-13T04:22:33Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Infoboxified</comment>
      <origin>62922</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="590" sha1="tkrmqqha10owhew7l7pk6b918wmwwsv" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 128/81
| Monzo = 7 -4
| Cents = 792.18000
| Name = Pythagorean minor sixth
| Color name = sw6, small wa 6th
| FJS name = m6
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', at -4 steps on the chain of fifths. 

== See also ==
* [[81/64]] – its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:3-limit]]
[[Category:Interval]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Pythagorean]]
[[Category:Listen]]

{{todo| expand }}</text>
      <sha1>tkrmqqha10owhew7l7pk6b918wmwwsv</sha1>
    </revision>
    <revision>
      <id>64590</id>
      <parentid>62922</parentid>
      <timestamp>2021-03-27T17:39:59Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64590</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="638" sha1="q1uf33mihj5rscxgbikm8pk2bf17t48" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 128/81
| Monzo = 7 -4
| Cents = 792.18000
| Name = Pythagorean minor sixth
| Color name = sw6, small wa 6th
| FJS name = m6
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', at -4 steps on the chain of fifths. 

== See also ==
* [[81/64]] – its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:3-limit]]
[[Category:Interval]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Pythagorean]]
[[Category:Listen]]
[[Category:Pages with internal sound examples]]

{{todo| expand }}</text>
      <sha1>q1uf33mihj5rscxgbikm8pk2bf17t48</sha1>
    </revision>
    <revision>
      <id>74265</id>
      <parentid>64590</parentid>
      <timestamp>2021-07-13T02:00:48Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <origin>74265</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="634" sha1="65nkhms53u3x2317qq5rg76q9gjovgg" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 128/81
| Monzo = 7 -4
| Cents = 792.18000
| Name = Pythagorean minor sixth
| Color name = sw6, sawa 6th
| FJS name = m6
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', at -4 steps on the chain of fifths. 

== See also ==
* [[81/64]] – its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:3-limit]]
[[Category:Interval]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Pythagorean]]
[[Category:Listen]]
[[Category:Pages with internal sound examples]]

{{todo| expand }}</text>
      <sha1>65nkhms53u3x2317qq5rg76q9gjovgg</sha1>
    </revision>
    <revision>
      <id>74293</id>
      <parentid>74265</parentid>
      <timestamp>2021-07-13T16:31:54Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>74293</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="836" sha1="io3pjer90o2t1h1xnxfzjbp7gn1m2d3" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 128/81
| Monzo = 7 -4
| Cents = 792.18000
| Name = Pythagorean minor sixth
| Color name = sw6, sawa 6th
| FJS name = m6
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', created by stacking for instances of [[4/3]] and [[Octave reduction|octave-reducing]].  It has a [[harmonic entropy]] level roughly on par with that of [[12/11]], and thus, some would argue that it is functionally an imperfect dissonance.

== See also ==
* [[81/64]] – its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:3-limit]]
[[Category:Interval]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Pythagorean]]
[[Category:Listen]]
[[Category:Pages with internal sound examples]]

{{todo| expand }}</text>
      <sha1>io3pjer90o2t1h1xnxfzjbp7gn1m2d3</sha1>
    </revision>
    <revision>
      <id>74294</id>
      <parentid>74293</parentid>
      <timestamp>2021-07-13T16:34:07Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>74294</origin>
      <model>wikitext</model>
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      <text bytes="929" sha1="1jp3jxphyafyimd4x3vml13585nxpyj" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 128/81
| Monzo = 7 -4
| Cents = 792.18000
| Name = Pythagorean minor sixth
| Color name = sw6, sawa 6th
| FJS name = m6
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', created by stacking for instances of [[4/3]] and [[Octave reduction|octave-reducing]].  In contrast to the more typical [[8/5]]- with which it is conflated in [[meantone]]- this interval has a [[harmonic entropy]] level roughly on par with that of [[12/11]].  Thus, some would argue that it is functionally an imperfect dissonance.

== See also ==
* [[81/64]] – its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:3-limit]]
[[Category:Interval]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Pythagorean]]
[[Category:Listen]]
[[Category:Pages with internal sound examples]]

{{todo| expand }}</text>
      <sha1>1jp3jxphyafyimd4x3vml13585nxpyj</sha1>
    </revision>
    <revision>
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      <parentid>74294</parentid>
      <timestamp>2021-11-15T15:22:09Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
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| Icon = 
| Ratio = 128/81
| Monzo = 7 -4
| Cents = 792.18000
| Name = Pythagorean minor sixth
| Color name = sw6, sawa 6th
| FJS name = m6
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', created by stacking four instances of [[4/3]] and [[Octave reduction|octave-reducing]].  In contrast to the more typical [[8/5]]- with which it is conflated in [[meantone]]- this interval has a [[harmonic entropy]] level roughly on par with that of [[12/11]].  Thus, some would argue that it is functionally an imperfect dissonance.

== See also ==
* [[81/64]] – its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:3-limit]]
[[Category:Interval]]
[[Category:Interval ratio]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Pythagorean]]
[[Category:Listen]]
[[Category:Pages with internal sound examples]]

{{todo| expand }}</text>
      <sha1>aqzhz66x6z9w8nidz0w7c22gaqz3d5i</sha1>
    </revision>
    <revision>
      <id>81846</id>
      <parentid>80873</parentid>
      <timestamp>2021-12-12T06:04:48Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Misc. edits, categories</comment>
      <origin>81846</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="837" sha1="ddjd0em1uuwb51r491dfid75wyveh4m" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 128/81
| Monzo = 7 -4
| Cents = 792.18000
| Name = Pythagorean minor sixth
| Color name = sw6, sawa 6th
| FJS name = m6
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', created by stacking four instances of [[4/3]] and [[Octave reduction|octave-reducing]]. In contrast to the more typical [[8/5]]— with which it is conflated in [[meantone]]— this interval has a [[harmonic entropy]] level roughly on par with that of [[12/11]]. Thus, some would argue that it is functionally an imperfect dissonance.

== See also ==
* [[81/64]] – its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:3-limit]]
[[Category:Sixth]]
[[Category:Minor sixth]]
[[Category:Pages with internal sound examples]]

{{todo| expand }}</text>
      <sha1>ddjd0em1uuwb51r491dfid75wyveh4m</sha1>
    </revision>
    <revision>
      <id>86304</id>
      <parentid>81846</parentid>
      <timestamp>2022-02-22T23:46:40Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86304</origin>
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      <text bytes="788" sha1="8er439e6jd1da23rsgm83qxttqk34mz" xml:space="preserve">{{Infobox Interval
| Icon = 
| Ratio = 128/81
| Monzo = 7 -4
| Cents = 792.18000
| Name = Pythagorean minor sixth
| Color name = sw6, sawa 6th
| FJS name = m6
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', created by stacking four instances of [[4/3]] and [[Octave reduction|octave-reducing]]. In contrast to the more typical [[8/5]]— with which it is conflated in [[meantone]]— this interval has a [[harmonic entropy]] level roughly on par with that of [[12/11]]. Thus, some would argue that it is functionally an imperfect dissonance.

== See also ==
* [[81/64]] – its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:3-limit]]
[[Category:Sixth]]
[[Category:Minor sixth]]
{{todo| expand }}</text>
      <sha1>8er439e6jd1da23rsgm83qxttqk34mz</sha1>
    </revision>
    <revision>
      <id>97960</id>
      <parentid>86304</parentid>
      <timestamp>2022-10-25T11:41:33Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97960</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="689" sha1="r75xhzvb7ud13ut756wf409fq9fqy5w" xml:space="preserve">{{Infobox Interval
| Name = Pythagorean minor sixth
| Color name = sw6, sawa 6th
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', created by stacking four instances of [[4/3]] and [[Octave reduction|octave-reducing]]. In contrast to the more typical [[8/5]]— with which it is conflated in [[meantone]]— this interval has a [[harmonic entropy]] level roughly on par with that of [[12/11]]. Thus, some would argue that it is functionally an imperfect dissonance.

== See also ==
* [[81/64]] – its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:Sixth]]
[[Category:Minor sixth]]
{{todo| expand }}</text>
      <sha1>r75xhzvb7ud13ut756wf409fq9fqy5w</sha1>
    </revision>
    <revision>
      <id>144727</id>
      <parentid>97960</parentid>
      <timestamp>2024-05-23T13:33:40Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>144727</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="703" sha1="edsdkq002ignhoss7ji6t4hs2kuhejr" xml:space="preserve">{{Infobox Interval
| Name = Pythagorean minor sixth
| Color name = sw6, sawa 6th
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', created by stacking four instances of [[4/3]] and [[Octave reduction|octave-reducing]]. In contrast to the more typical [[8/5]]{{mdash}}with which it is conflated in [[meantone]]{{mdash}}this interval has a [[harmonic entropy]] level roughly on par with that of [[12/11]]. Thus, some would argue that it is functionally an imperfect dissonance.

== See also ==
* [[81/64]] &amp;ndash; Its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:Sixth]]
[[Category:Minor sixth]]
{{todo| expand }}</text>
      <sha1>edsdkq002ignhoss7ji6t4hs2kuhejr</sha1>
    </revision>
    <revision>
      <id>144801</id>
      <parentid>144727</parentid>
      <timestamp>2024-05-23T21:51:50Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>144801</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="699" sha1="2dewvu4scznhefrjm43f7yqk58hkb5q" xml:space="preserve">{{Infobox Interval
| Name = Pythagorean minor sixth
| Color name = sw6, sawa 6th
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', created by stacking four instances of [[4/3]] and [[Octave reduction|octave-reducing]]. In contrast to the more typical [[8/5]]&amp;mdash;with which it is conflated in [[meantone]]&amp;mdash;this interval has a [[harmonic entropy]] level roughly on par with that of [[12/11]]. Thus, some would argue that it is functionally an imperfect dissonance.

== See also ==
* [[81/64]] &amp;ndash; Its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:Sixth]]
[[Category:Minor sixth]]
{{todo| expand }}</text>
      <sha1>2dewvu4scznhefrjm43f7yqk58hkb5q</sha1>
    </revision>
    <revision>
      <id>174332</id>
      <parentid>144801</parentid>
      <timestamp>2025-01-03T07:26:25Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Change/remove expand/stub in accordance with XW:IntTodo</comment>
      <origin>174332</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="681" sha1="0pa9w71i91wq3dkcwp0nwfarrospt2i" xml:space="preserve">{{Infobox Interval
| Name = Pythagorean minor sixth
| Color name = sw6, sawa 6th
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', created by stacking four instances of [[4/3]] and [[Octave reduction|octave-reducing]]. In contrast to the more typical [[8/5]]&amp;mdash;with which it is conflated in [[meantone]]&amp;mdash;this interval has a [[harmonic entropy]] level roughly on par with that of [[12/11]]. Thus, some would argue that it is functionally an imperfect dissonance.

== See also ==
* [[81/64]] &amp;ndash; Its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:Sixth]]
[[Category:Minor sixth]]</text>
      <sha1>0pa9w71i91wq3dkcwp0nwfarrospt2i</sha1>
    </revision>
    <revision>
      <id>211129</id>
      <parentid>174332</parentid>
      <timestamp>2025-09-27T10:43:35Z</timestamp>
      <contributor>
        <username>M1k1 H2tsun2</username>
        <id>569</id>
      </contributor>
      <comment>add it in the first, rather than the other place.</comment>
      <origin>211129</origin>
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      <text bytes="704" sha1="p5tfg45grou9x8ief9fco024zdqosh8" xml:space="preserve">{{Infobox Interval
| Name = Threerish minor sixth, Pythagorean minor sixth
| Color name = sw6, sawa 6th
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', created by stacking four instances of [[4/3]] and [[Octave reduction|octave-reducing]]. In contrast to the more typical [[8/5]]&amp;mdash;with which it is conflated in [[meantone]]&amp;mdash;this interval has a [[harmonic entropy]] level roughly on par with that of [[12/11]]. Thus, some would argue that it is functionally an imperfect dissonance.

== See also ==
* [[81/64]] &amp;ndash; Its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:Sixth]]
[[Category:Minor sixth]]</text>
      <sha1>p5tfg45grou9x8ief9fco024zdqosh8</sha1>
    </revision>
    <revision>
      <id>211155</id>
      <parentid>211129</parentid>
      <timestamp>2025-09-27T11:48:47Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Undo revision [[Special:Diff/211129|211129]] by [[Special:Contributions/M1k1 H2tsun2|M1k1 H2tsun2]] ([[User talk:M1k1 H2tsun2|talk]])</comment>
      <origin>211155</origin>
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      <text bytes="681" sha1="0pa9w71i91wq3dkcwp0nwfarrospt2i" xml:space="preserve">{{Infobox Interval
| Name = Pythagorean minor sixth
| Color name = sw6, sawa 6th
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', created by stacking four instances of [[4/3]] and [[Octave reduction|octave-reducing]]. In contrast to the more typical [[8/5]]&amp;mdash;with which it is conflated in [[meantone]]&amp;mdash;this interval has a [[harmonic entropy]] level roughly on par with that of [[12/11]]. Thus, some would argue that it is functionally an imperfect dissonance.

== See also ==
* [[81/64]] &amp;ndash; Its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:Sixth]]
[[Category:Minor sixth]]</text>
      <sha1>0pa9w71i91wq3dkcwp0nwfarrospt2i</sha1>
    </revision>
    <revision>
      <id>212062</id>
      <parentid>211155</parentid>
      <timestamp>2025-10-05T14:06:11Z</timestamp>
      <contributor>
        <username>Squib</username>
        <id>562</id>
      </contributor>
      <origin>212062</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="737" sha1="9ywpb4rapf4gzvwsjiiklto8rrzrq6a" xml:space="preserve">{{Infobox Interval
| Name = Pythagorean minor sixth
| Color name = sw6, sawa 6th
| Sound = jid_128_81_pluck_adu_dr220.mp3
}}

'''128/81''' is the '''Pythagorean minor sixth''', created by stacking four instances of [[4/3]] and [[Octave reduction|octave-reducing]]. In contrast to the more typical [[8/5]]&amp;mdash;with which it is conflated in [[meantone]]&amp;mdash;this interval has a [[harmonic entropy]] level roughly on par with that of [[12/11]]. Thus, some would argue that it is functionally an imperfect dissonance.

It differs from [[11/7]] by the [[pentacircle comma]].

== See also ==
* [[81/64]] &amp;ndash; Its [[octave complement]]
* [[Gallery of just intervals]]
* [[Pythagorean tuning]]

[[Category:Sixth]]
[[Category:Minor sixth]]</text>
      <sha1>9ywpb4rapf4gzvwsjiiklto8rrzrq6a</sha1>
    </revision>
  </page>
  <page>
    <title>128 81</title>
    <ns>0</ns>
    <id>122</id>
    <redirect title="128/81" />
    <revision>
      <id>122</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <text bytes="20" sha1="iuqd47iox7d7rjdt43depvq295wot66" xml:space="preserve">#REDIRECT [[128/81]]</text>
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  </page>
  <page>
    <title>128edo</title>
    <ns>0</ns>
    <id>123</id>
    <revision>
      <id>123</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>123</origin>
      <model>wikitext</model>
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      <text bytes="639" sha1="fhmgspbpgtmknf1uw3fnw9576iiwlhj" xml:space="preserve">The 128 equal division divides the [[Octave|octave]] into 128 equal parts of exactly 9.375 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for [[7-limit|7-limit]] [[Gamelismic_clan|rodan temperament]]. It [[tempering_out|tempers out]] 2109375/2097152 in the [[5-limit|5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.

=Scales=
[[radon5|radon5]]

[[radon11|radon11]]

[[radon16|radon16]]

[https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophon] - Philipp Gerschlauer
[[Category:128edo]]
[[Category:edo]]
[[Category:rodan]]
[[Category:theory]]</text>
      <sha1>fhmgspbpgtmknf1uw3fnw9576iiwlhj</sha1>
    </revision>
    <revision>
      <id>4577</id>
      <timestamp>2016-01-06T01:37:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Osmiorisbendi</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 571226595 - Original comment: **</comment>
      <origin>4577</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3005" sha1="mxrwstsnbkk4905gq1epyo5ueqybygc" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Osmiorisbendi|Osmiorisbendi]] and made on &lt;tt&gt;2016-01-06 01:37:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;571226595&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 128 equal division divides the [[octave]] into 128 equal parts of exactly 9.375 [[cent]]s each. It is the [[optimal patent val]] for [[7-limit]] [[Gamelismic clan|rodan temperament]]. It [[tempering out|tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.

=Scales= 
[[radon5]]
[[radon11]]
[[radon16]]

[[https://www.youtube.com/watch?v=lGa66qHzKME|128 notes per octave on Alto Saxophon]] - Philipp Gerschlauer&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 128 equal division divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 128 equal parts of exactly 9.375 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;rodan temperament&amp;lt;/a&amp;gt;. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 2109375/2097152 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/radon5"&amp;gt;radon5&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/radon11"&amp;gt;radon11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/radon16"&amp;gt;radon16&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="https://www.youtube.com/watch?v=lGa66qHzKME" rel="nofollow"&amp;gt;128 notes per octave on Alto Saxophon&amp;lt;/a&amp;gt; - Philipp Gerschlauer&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <id>4578</id>
      <timestamp>2011-06-29T10:04:01Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 239315431 - Original comment: **</comment>
      <origin>4578</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2692" sha1="7fujvfmgju3ktsdypj39uefmfla4qpb" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 10:04:01 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239315431&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 128 equal division divides the [[octave]] into 128 equal parts of exactly 9.375 [[cent]]s each. It is the [[optimal patent val]] for [[7-limit]] [[Gamelismic clan|rodan temperament]]. It [[tempering out|tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.

=Scales= 
[[radon5]]
[[radon11]]
[[radon16]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 128 equal division divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 128 equal parts of exactly 9.375 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;rodan temperament&amp;lt;/a&amp;gt;. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 2109375/2097152 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link" href="/radon5"&amp;gt;radon5&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/radon11"&amp;gt;radon11&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/radon16"&amp;gt;radon16&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <timestamp>2011-06-22T07:26:29Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 238144747 - Original comment: **</comment>
      <origin>4579</origin>
      <model>wikitext</model>
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      <text bytes="2613" sha1="atj94yd09ipo3r6lu6o5nwu2pime76m" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2011-06-22 07:26:29 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;238144747&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 128 equal division divides the [[octave]] into 128 equal parts of exactly 9.375 [[cent]]s each. It is the [[optimal patent val]] for [[7-limit]] [[Gamelismic clan|rodan temperament]]. It tempers out 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.

=Scales= 
[[radon5]]
[[radon11]]
[[radon16]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 128 equal division divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 128 equal parts of exactly 9.375 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;rodan temperament&amp;lt;/a&amp;gt;. It tempers out 2109375/2097152 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
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&amp;lt;a class="wiki_link" href="/radon16"&amp;gt;radon16&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2011-03-28T03:14:32Z</timestamp>
      <contributor>
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        <id>0</id>
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      <comment>**Imported revision 214558306 - Original comment: **</comment>
      <origin>4580</origin>
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      <text bytes="2158" sha1="3ibtt6q7i8smr2kukvs0ch6gyoj0a5f" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-03-28 03:14:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214558306&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 128 equal division divides the [[octave]] into 128 equal parts of exactly 9.375 [[cent]]s each. It is the [[optimal patent val]] for [[7-limit]] [[Gamelismic clan|rodan temperament]]. It tempers out 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 128 equal division divides the &amp;lt;a class="wiki_link" href="/octave"&amp;gt;octave&amp;lt;/a&amp;gt; into 128 equal parts of exactly 9.375 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;rodan temperament&amp;lt;/a&amp;gt;. It tempers out 2109375/2097152 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4581</id>
      <timestamp>2011-03-25T16:55:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 214083274 - Original comment: **</comment>
      <origin>4581</origin>
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      <text bytes="1946" sha1="coowc28jbq40vxsrtwz21128iwv26dr" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-25 16:55:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214083274&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The 128 equal division divides the octave into 128 equal parts of exactly 9.375 cents each. It is the [[optimal patent val]] for 7-limit [[Gamelismic clan|rodan temperament]]. It tempers out 2109375/2097152 in the 5-limit; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;128edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The 128 equal division divides the octave into 128 equal parts of exactly 9.375 cents each. It is the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for 7-limit &amp;lt;a class="wiki_link" href="/Gamelismic%20clan"&amp;gt;rodan temperament&amp;lt;/a&amp;gt;. It tempers out 2109375/2097152 in the 5-limit; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>coowc28jbq40vxsrtwz21128iwv26dr</sha1>
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    <revision>
      <id>40206</id>
      <parentid>123</parentid>
      <timestamp>2019-02-13T01:37:23Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
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      <model>wikitext</model>
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      <text bytes="844" sha1="qdeodu6x474tepcspvaxmuqz868tmbe" xml:space="preserve">The 128 equal division divides the [[Octave|octave]] into 128 equal parts of exactly 9.375 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for [[7-limit|7-limit]] [[Gamelismic_clan|rodan temperament]]. It [[tempering_out|tempers out]] 2109375/2097152 in the [[5-limit|5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. Being the power of two closest to division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the binary (fine) relative cent of [[1edo]].

=Scales=
[[radon5|radon5]]

[[radon11|radon11]]

[[radon16|radon16]]

[https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophon] - Philipp Gerschlauer
[[Category:128edo]]
[[Category:edo]]
[[Category:rodan]]
[[Category:theory]]</text>
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    <revision>
      <id>40207</id>
      <parentid>40206</parentid>
      <timestamp>2019-02-13T01:41:43Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
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      <text bytes="880" sha1="2hifv4qx0zaff045ov9jygtqqu9ow3z" xml:space="preserve">The 128 equal division divides the [[Octave|octave]] into 128 equal parts of exactly 9.375 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for [[7-limit|7-limit]] [[Gamelismic_clan|rodan temperament]]. It [[tempering_out|tempers out]] 2109375/2097152 in the [[5-limit|5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. Being the power of two closest to division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].

=Scales=
[[radon5|radon5]]

[[radon11|radon11]]

[[radon16|radon16]]

[https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophon] - Philipp Gerschlauer
[[Category:128edo]]
[[Category:edo]]
[[Category:rodan]]
[[Category:theory]]</text>
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    <revision>
      <id>54981</id>
      <parentid>40207</parentid>
      <timestamp>2020-12-04T23:13:07Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54981</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="906" sha1="86346gjzbkjb52hnv1qyc6sgi2uqk1l" xml:space="preserve">The 128 equal division divides the [[Octave|octave]] into 128 equal parts of exactly 9.375 [[cent|cent]]s each. It is the [[Optimal_patent_val|optimal patent val]] for [[7-limit|7-limit]] [[Gamelismic_clan|rodan temperament]]. It [[tempering_out|tempers out]] 2109375/2097152 in the [[5-limit|5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. Being the power of two closest to division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].

=Scales=
[[radon5|radon5]]

[[radon11|radon11]]

[[radon16|radon16]]

[https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophon] - Philipp Gerschlauer
[[Category:128edo]]
[[Category:Equal divisions of the octave]]
[[Category:rodan]]
[[Category:theory]]</text>
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    <revision>
      <id>57138</id>
      <parentid>54981</parentid>
      <timestamp>2020-12-20T13:43:42Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>some cleanup</comment>
      <origin>57138</origin>
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      <text bytes="849" sha1="hcxrtwa4j2s9r3ukqixmfypm8mjls83" xml:space="preserve">The 128 equal division divides the [[octave]] into 128 equal parts of exactly 9.375 [[cent]]s each. It is the [[optimal patent val]] for [[7-limit]] [[Gamelismic clan|rodan temperament]]. It [[tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. Being the power of two closest to division of the octave by the Germanic [https://en.wikipedia.org/wiki/Long_hundred long hundred], it has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophon] (Demo by Philipp Gerschlauer)

== Scales ==

* [[radon5]]
* [[radon11]]
* [[radon16]]

[[Category:128edo]]
[[Category:Equal divisions of the octave]]
[[Category:Rodan]]
[[Category:Theory]]</text>
      <sha1>hcxrtwa4j2s9r3ukqixmfypm8mjls83</sha1>
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    <revision>
      <id>62223</id>
      <parentid>57138</parentid>
      <timestamp>2021-02-23T18:27:50Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>adjusted sorting on category page, simplified links</comment>
      <origin>62223</origin>
      <model>wikitext</model>
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      <text bytes="841" sha1="752kip8n52vji7alwgy1713i7hb86l0" xml:space="preserve">The 128 equal division divides the [[octave]] into 128 equal parts of exactly 9.375 [[cent]]s each. It is the [[optimal patent val]] for [[7-limit]] [[Rodan]] temperament. It [[tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. Being the power of two closest to division of the octave by the Germanic [[Wikipedia: long hundred| long hundred]], it has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophon] (Demo by Philipp Gerschlauer)

== Scales ==

* [[radon5]]
* [[radon11]]
* [[radon16]]

[[Category:128edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Rodan]]
[[Category:Theory]]</text>
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      <id>87447</id>
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      <timestamp>2022-02-26T16:48:50Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87447</origin>
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      <format>text/x-wiki</format>
      <text bytes="821" sha1="42yqzj1j99mlgx9rv3bjp5hjh5rknwy" xml:space="preserve">The 128 equal division divides the [[octave]] into 128 equal parts of exactly 9.375 [[cent]]s each. It is the [[optimal patent val]] for [[7-limit]] [[Rodan]] temperament. It [[tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. Being the power of two closest to division of the octave by the Germanic [[Wikipedia: long hundred| long hundred]], it has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophon] (Demo by Philipp Gerschlauer)

== Scales ==

* [[radon5]]
* [[radon11]]
* [[radon16]]

[[Category:128edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Rodan]]</text>
      <sha1>42yqzj1j99mlgx9rv3bjp5hjh5rknwy</sha1>
    </revision>
    <revision>
      <id>91981</id>
      <parentid>87447</parentid>
      <timestamp>2022-05-18T14:27:41Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>91981</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="884" sha1="0tlhjz92c0rsgg67lpk3dgatcb9blas" xml:space="preserve">{{EDO intro|128}}It is notable because it is the equal division corresponding to a standard MIDI piano roll of 128 notes.

== Theory ==
{{Harmonics in equal|128}}
It is the [[optimal patent val]] for [[7-limit]] [[Rodan]] temperament. It [[tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. Being the power of two closest to division of the octave by the Germanic [[Wikipedia: long hundred| long hundred]], it has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophon] (Demo by Philipp Gerschlauer)

== Scales ==

* [[radon5]]
* [[radon11]]
* [[radon16]]

[[Category:128edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave]]
[[Category:Rodan]]</text>
      <sha1>0tlhjz92c0rsgg67lpk3dgatcb9blas</sha1>
    </revision>
    <revision>
      <id>92938</id>
      <parentid>91981</parentid>
      <timestamp>2022-07-02T16:27:21Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92938</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="913" sha1="2hdqaeo495jfh47dt7nbmdmtdid8djr" xml:space="preserve">{{EDO intro|128}}It is notable because it is the equal division corresponding to a standard MIDI piano roll of 128 notes.

== Theory ==
{{Harmonics in equal|128}}
It is the [[optimal patent val]] for [[7-limit]] [[Rodan]] temperament. It [[tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. Being the power of two closest to division of the octave by the Germanic [[Wikipedia: long hundred| long hundred]], it has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

== Scales ==

* [[radon5]]
* [[radon11]]
* [[radon16]]

[[Category:128edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Rodan]]</text>
      <sha1>2hdqaeo495jfh47dt7nbmdmtdid8djr</sha1>
    </revision>
    <revision>
      <id>96205</id>
      <parentid>92938</parentid>
      <timestamp>2022-10-04T18:55:18Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96205</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="928" sha1="5l9sr8w39rmc2qdif6dk4k3mbms2dd4" xml:space="preserve">{{Infobox ET}}
{{EDO intro|128}}It is notable because it is the equal division corresponding to a standard MIDI piano roll of 128 notes.

== Theory ==
{{Harmonics in equal|128}}
It is the [[optimal patent val]] for [[7-limit]] [[Rodan]] temperament. It [[tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. Being the power of two closest to division of the octave by the Germanic [[Wikipedia: long hundred| long hundred]], it has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

== Scales ==

* [[radon5]]
* [[radon11]]
* [[radon16]]

[[Category:128edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Rodan]]</text>
      <sha1>5l9sr8w39rmc2qdif6dk4k3mbms2dd4</sha1>
    </revision>
    <revision>
      <id>98701</id>
      <parentid>96205</parentid>
      <timestamp>2022-10-30T02:42:07Z</timestamp>
      <contributor>
        <username>Royalmilktea</username>
        <id>285</id>
      </contributor>
      <origin>98701</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1543" sha1="74ho4ua3e3bld3vgzc4t7wdta7dbygy" xml:space="preserve">{{Infobox ET}}
{{EDO intro|128}}It is notable because it is the equal division corresponding to a standard MIDI piano roll of 128 notes.

== Theory ==
{{Harmonics in equal|128}}
It is the [[optimal patent val]] for [[7-limit]] [[Rodan]] temperament. It [[tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. Being the power of two closest to division of the octave by the Germanic [[Wikipedia: long hundred| long hundred]], it has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

== Regular temperament properties ==
{| class="wikitable center-1 center-2 center-3"
|+Rank-2 temperaments
!Periods
per octave
!Generator
(reduced)
!Cents
(reduced)
!Associated
Ratio
!Temperaments
|-
|1
|25\128
|234.375
|8/7
|[[Rodan]]
|-
|1
|29\128
|271.875
|75/64
|[[Orson]]
|-
|1
|33\128
|309.375
|448/375
|[[Triwell]]
|-
|1
|53\128
|496.875
|4/3
|[[Undecental]]
|-
|2
|13\128
|121.875
|15/14
|[[Gamelismic clan#Lagaca|Lagaca]]
|-
|2
|15\128
|140.625
|27/25
|[[Fifive]]
|-
|4
|15\128
|140.625
|27/25
|[[Fifive family#Fourfives|Fourfives]]
|-
|4
|53\128
(11\128)
|496.875

(103.125)
|4/3
|[[Undim]] (7-limit)
|}

== Scales ==

* [[radon5]]
* [[radon11]]
* [[radon16]]

[[Category:128edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Rodan]]</text>
      <sha1>74ho4ua3e3bld3vgzc4t7wdta7dbygy</sha1>
    </revision>
    <revision>
      <id>98702</id>
      <parentid>98701</parentid>
      <timestamp>2022-10-30T02:42:45Z</timestamp>
      <contributor>
        <username>Royalmilktea</username>
        <id>285</id>
      </contributor>
      <minor/>
      <comment>(forgot edit summary) rank-2 temperaments</comment>
      <origin>98702</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1544" sha1="62xa43btyf80zu3vyl3mojlmqhpcb2t" xml:space="preserve">{{Infobox ET}}
{{EDO intro|128}}It is notable because it is the equal division corresponding to a standard MIDI piano roll of 128 notes. 

== Theory ==
{{Harmonics in equal|128}}
It is the [[optimal patent val]] for [[7-limit]] [[Rodan]] temperament. It [[tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. Being the power of two closest to division of the octave by the Germanic [[Wikipedia: long hundred| long hundred]], it has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

== Regular temperament properties ==
{| class="wikitable center-1 center-2 center-3"
|+Rank-2 temperaments
!Periods
per octave
!Generator
(reduced)
!Cents
(reduced)
!Associated
Ratio
!Temperaments
|-
|1
|25\128
|234.375
|8/7
|[[Rodan]]
|-
|1
|29\128
|271.875
|75/64
|[[Orson]]
|-
|1
|33\128
|309.375
|448/375
|[[Triwell]]
|-
|1
|53\128
|496.875
|4/3
|[[Undecental]]
|-
|2
|13\128
|121.875
|15/14
|[[Gamelismic clan#Lagaca|Lagaca]]
|-
|2
|15\128
|140.625
|27/25
|[[Fifive]]
|-
|4
|15\128
|140.625
|27/25
|[[Fifive family#Fourfives|Fourfives]]
|-
|4
|53\128
(11\128)
|496.875

(103.125)
|4/3
|[[Undim]] (7-limit)
|}

== Scales ==

* [[radon5]]
* [[radon11]]
* [[radon16]]

[[Category:128edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Rodan]]</text>
      <sha1>62xa43btyf80zu3vyl3mojlmqhpcb2t</sha1>
    </revision>
    <revision>
      <id>98752</id>
      <parentid>98702</parentid>
      <timestamp>2022-11-01T14:17:00Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Fix unclear pronoun; sectioning; style</comment>
      <origin>98752</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1633" sha1="1dve33kd3me0wglne5u5dzgfo80tv59" xml:space="preserve">{{Infobox ET}}
{{EDO intro|128}}It is notable because it is the equal division corresponding to a standard MIDI piano roll of 128 notes. 

== Theory ==
128edo is the [[optimal patent val]] for [[7-limit]] [[Rodan]] temperament. It [[tempers out]] 2109375/2097152 in the [[5-limit]]; 245/243, 1029/1024 and 5120/5103 in the 7-limit; 385/384 and 441/440 in the limit. 

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

=== Prime harmonics ===
{{Harmonics in equal|128|columns=11}}

=== Miscellaneous properties ===
Being the power of two closest to division of the octave by the Germanic [[Wikipedia: long hundred|long hundred]], 128edo has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].

== Regular temperament properties ==
{| class="wikitable center-1 center-2 center-3"
|+Rank-2 temperaments
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 25\128
| 234.375
| 8/7
| [[Rodan]]
|-
| 1
| 29\128
| 271.875
| 75/64
| [[Orson]]
|-
| 1
| 33\128
| 309.375
| 448/375
| [[Triwell]]
|-
| 1
| 53\128
| 496.875
| 4/3
| [[Undecental]]
|-
| 2
| 13\128
| 121.875
| 15/14
| [[Lagaca]]
|-
| 2
| 15\128
| 140.625
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128&lt;br&gt;(11\128)
| 496.875&lt;br&gt;(103.125)
| 4/3
| [[Undim]] (7-limit)
|}

== Scales ==
* [[radon5]]
* [[radon11]]
* [[radon16]]

[[Category:128edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Rodan]]</text>
      <sha1>1dve33kd3me0wglne5u5dzgfo80tv59</sha1>
    </revision>
    <revision>
      <id>117727</id>
      <parentid>98752</parentid>
      <timestamp>2023-06-29T07:37:34Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup and expansion</comment>
      <origin>117727</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1777" sha1="tp2z4ytrfi6q1cuhu46sgiib4uli0r1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|128}} It is notable for being the equal division corresponding to a standard MIDI piano roll of 128 notes. 

== Theory ==
128edo [[tempers out]] 2109375/2097152 in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the 41 &amp; 87 temperament, as well as for 7-limit [[fourfives]], the 60 &amp; 68 temperament. 

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

=== Prime harmonics ===
{{Harmonics in equal|128|columns=11}}

=== Subsets and supersets ===
Since 128 factors into 2&lt;sup&gt;7&lt;/sup&gt;, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}. 

=== Miscellaneous properties ===
Being the power of two closest to division of the octave by the Germanic [[Wikipedia: long hundred|long hundred]], 128edo has a unit step which is the binary (fine) relative cent (or relative heptamu in MIDI terms) of [[1edo]].

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+Rank-2 temperaments by generators
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 25\128
| 234.375
| 8/7
| [[Rodan]]
|-
| 1
| 29\128
| 271.875
| 75/64
| [[Orson]]
|-
| 1
| 33\128
| 309.375
| 448/375
| [[Triwell]]
|-
| 1
| 53\128
| 496.875
| 4/3
| [[Undecental]]
|-
| 2
| 13\128
| 121.875
| 15/14
| [[Lagaca]]
|-
| 2
| 15\128
| 140.625
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128&lt;br&gt;(11\128)
| 496.875&lt;br&gt;(103.125)
| 4/3
| [[Undim]] (7-limit)
|}

== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]

[[Category:Rodan]]
[[Category:Fourfives]]</text>
      <sha1>tp2z4ytrfi6q1cuhu46sgiib4uli0r1</sha1>
    </revision>
    <revision>
      <id>117812</id>
      <parentid>117727</parentid>
      <timestamp>2023-06-30T14:02:58Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Subsets and supersets */ remove the statement since that is already said in the intro</comment>
      <origin>117812</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1515" sha1="e79d2w44hdxjw6n73ln15f3zre4l1m9" xml:space="preserve">{{Infobox ET}}
{{EDO intro|128}} It is notable for being the equal division corresponding to a standard MIDI piano roll of 128 notes. 

== Theory ==
128edo [[tempers out]] 2109375/2097152 in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the 41 &amp; 87 temperament, as well as for 7-limit [[fourfives]], the 60 &amp; 68 temperament. 

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

=== Prime harmonics ===
{{Harmonics in equal|128|columns=11}}

=== Subsets and supersets ===
Since 128 factors into 2&lt;sup&gt;7&lt;/sup&gt;, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}. 

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+Rank-2 temperaments by generators
! Periods&lt;br&gt;per 8ve
! Generator&lt;br&gt;(Reduced)
! Cents&lt;br&gt;(Reduced)
! Associated&lt;br&gt;Ratio
! Temperaments
|-
| 1
| 25\128
| 234.375
| 8/7
| [[Rodan]]
|-
| 1
| 29\128
| 271.875
| 75/64
| [[Orson]]
|-
| 1
| 33\128
| 309.375
| 448/375
| [[Triwell]]
|-
| 1
| 53\128
| 496.875
| 4/3
| [[Undecental]]
|-
| 2
| 13\128
| 121.875
| 15/14
| [[Lagaca]]
|-
| 2
| 15\128
| 140.625
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128&lt;br&gt;(11\128)
| 496.875&lt;br&gt;(103.125)
| 4/3
| [[Undim]] (7-limit)
|}

== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]

[[Category:Rodan]]
[[Category:Fourfives]]</text>
      <sha1>e79d2w44hdxjw6n73ln15f3zre4l1m9</sha1>
    </revision>
    <revision>
      <id>145130</id>
      <parentid>117812</parentid>
      <timestamp>2024-05-27T17:30:03Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Cleanup</comment>
      <origin>145130</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1688" sha1="pq82cdqze8chnbxufjq1637q9yf5afg" xml:space="preserve">{{Infobox ET}}
{{EDO intro|128}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes. 

== Theory ==
The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the 41 &amp; 87 temperament, as well as for 7-limit [[fourfives]], the 60 &amp; 68 temperament. 

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

=== Prime harmonics ===
{{Harmonics in equal|128}}

=== Subsets and supersets ===
Since 128 factors into 2&lt;sup&gt;7&lt;/sup&gt;, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}. 

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+Rank-2 temperaments by generators
! Periods&lt;br&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br&gt;Ratio*
! Temperaments
|-
| 1
| 25\128
| 234.375
| 8/7
| [[Rodan]]
|-
| 1
| 29\128
| 271.875
| 75/64
| [[Orson]]
|-
| 1
| 33\128
| 309.375
| 448/375
| [[Triwell]]
|-
| 1
| 53\128
| 496.875
| 4/3
| [[Undecental]]
|-
| 2
| 13\128
| 121.875
| 15/14
| [[Lagaca]]
|-
| 2
| 15\128
| 140.625
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128&lt;br&gt;(11\128)
| 496.875&lt;br&gt;(103.125)
| 4/3
| [[Undim]] (7-limit)
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]

[[Category:Rodan]]
[[Category:Fourfives]]</text>
      <sha1>pq82cdqze8chnbxufjq1637q9yf5afg</sha1>
    </revision>
    <revision>
      <id>165567</id>
      <parentid>145130</parentid>
      <timestamp>2024-11-16T00:53:55Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>165567</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1585" sha1="la47vk16gjohk5vc2oediqs58h94iwh" xml:space="preserve">{{Infobox ET}}
{{EDO intro|128}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes. 

== Theory ==
The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the 41 &amp; 87 temperament, as well as for 7-limit [[fourfives]], the 60 &amp; 68 temperament. 

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

=== Prime harmonics ===
{{Harmonics in equal|128}}

=== Subsets and supersets ===
Since 128 factors into 2&lt;sup&gt;7&lt;/sup&gt;, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}. 

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;Ratio*
! Temperaments
|-
| 1
| 25\128
| 234.375
| 8/7
| [[Rodan]]
|-
| 1
| 29\128
| 271.875
| 75/64
| [[Orson]]
|-
| 1
| 33\128
| 309.375
| 448/375
| [[Triwell]]
|-
| 1
| 53\128
| 496.875
| 4/3
| [[Undecental]]
|-
| 2
| 13\128
| 121.875
| 15/14
| [[Lagaca]]
|-
| 2
| 15\128
| 140.625
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128&lt;br /&gt;(11\128)
| 496.875&lt;br /&gt;(103.125)
| 4/3
| [[Undim]] (7-limit)
|}
{{orf}}

== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]

[[Category:Rodan]]
[[Category:Fourfives]]</text>
      <sha1>la47vk16gjohk5vc2oediqs58h94iwh</sha1>
    </revision>
    <revision>
      <id>165641</id>
      <parentid>165567</parentid>
      <timestamp>2024-11-16T01:35:52Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>165641</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1415" sha1="jrdu07bqplp8tt06f4w1hu820idh1p1" xml:space="preserve">{{Infobox ET}}
{{EDO intro|128}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes. 

== Theory ==
The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the 41 &amp; 87 temperament, as well as for 7-limit [[fourfives]], the 60 &amp; 68 temperament. 

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

=== Prime harmonics ===
{{Harmonics in equal|128}}

=== Subsets and supersets ===
Since 128 factors into 2&lt;sup&gt;7&lt;/sup&gt;, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}. 

== Regular temperament properties ==
{{rank-2 begin}}
|-
| 1
| 25\128
| 234.375
| 8/7
| [[Rodan]]
|-
| 1
| 29\128
| 271.875
| 75/64
| [[Orson]]
|-
| 1
| 33\128
| 309.375
| 448/375
| [[Triwell]]
|-
| 1
| 53\128
| 496.875
| 4/3
| [[Undecental]]
|-
| 2
| 13\128
| 121.875
| 15/14
| [[Lagaca]]
|-
| 2
| 15\128
| 140.625
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128&lt;br /&gt;(11\128)
| 496.875&lt;br /&gt;(103.125)
| 4/3
| [[Undim]] (7-limit)
{{rank-2 end}}
{{orf}}

== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]

[[Category:Rodan]]
[[Category:Fourfives]]</text>
      <sha1>jrdu07bqplp8tt06f4w1hu820idh1p1</sha1>
    </revision>
    <revision>
      <id>166323</id>
      <parentid>165641</parentid>
      <timestamp>2024-11-16T13:20:00Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Partial undo</comment>
      <origin>166323</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1728" sha1="fqwht4i4yga9u3ug8zvpcc4n4ha8yxo" xml:space="preserve">{{Infobox ET}}
{{EDO intro|128}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes. 

== Theory ==
The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the 41 &amp; 87 temperament, as well as for 7-limit [[fourfives]], the 60 &amp; 68 temperament. 

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

=== Prime harmonics ===
{{Harmonics in equal|128}}

=== Subsets and supersets ===
Since 128 factors into 2&lt;sup&gt;7&lt;/sup&gt;, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}. 

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 25\128
| 234.375
| 8/7
| [[Rodan]]
|-
| 1
| 29\128
| 271.875
| 75/64
| [[Orson]]
|-
| 1
| 33\128
| 309.375
| 448/375
| [[Triwell]]
|-
| 1
| 53\128
| 496.875
| 4/3
| [[Undecental]]
|-
| 2
| 13\128
| 121.875
| 15/14
| [[Lagaca]]
|-
| 2
| 15\128
| 140.625
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128&lt;br /&gt;(11\128)
| 496.875&lt;br /&gt;(103.125)
| 4/3
| [[Undim]] (7-limit)
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]

[[Category:Rodan]]
[[Category:Fourfives]]</text>
      <sha1>fqwht4i4yga9u3ug8zvpcc4n4ha8yxo</sha1>
    </revision>
    <revision>
      <id>176182</id>
      <parentid>166323</parentid>
      <timestamp>2025-01-15T16:47:12Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>176182</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1752" sha1="sipe6jmqook7v2ah7ko69qkx4rlrh1e" xml:space="preserve">{{Infobox ET}}
{{EDO intro|128}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes. 

== Theory ==
The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the {{nowrap|41 &amp;amp; 87}} temperament, as well as for 7-limit [[fourfives]], the {{nowrap|60 &amp;amp; 68}} temperament. 

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

=== Prime harmonics ===
{{Harmonics in equal|128}}

=== Subsets and supersets ===
Since 128 factors into 2&lt;sup&gt;7&lt;/sup&gt;, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}. 

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 25\128
| 234.375
| 8/7
| [[Rodan]]
|-
| 1
| 29\128
| 271.875
| 75/64
| [[Orson]]
|-
| 1
| 33\128
| 309.375
| 448/375
| [[Triwell]]
|-
| 1
| 53\128
| 496.875
| 4/3
| [[Undecental]]
|-
| 2
| 13\128
| 121.875
| 15/14
| [[Lagaca]]
|-
| 2
| 15\128
| 140.625
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128&lt;br /&gt;(11\128)
| 496.875&lt;br /&gt;(103.125)
| 4/3
| [[Undim]] (7-limit)
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]

[[Category:Rodan]]
[[Category:Fourfives]]</text>
      <sha1>sipe6jmqook7v2ah7ko69qkx4rlrh1e</sha1>
    </revision>
    <revision>
      <id>181337</id>
      <parentid>176182</parentid>
      <timestamp>2025-02-18T17:00:20Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181337</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1747" sha1="ixlwhwqzk5t0yf4v7yuiqcj3um7rxrp" xml:space="preserve">{{Infobox ET}}
{{ED intro}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes. 

== Theory ==
The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the {{nowrap|41 &amp;amp; 87}} temperament, as well as for 7-limit [[fourfives]], the {{nowrap|60 &amp;amp; 68}} temperament. 

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

=== Prime harmonics ===
{{Harmonics in equal|128}}

=== Subsets and supersets ===
Since 128 factors into 2&lt;sup&gt;7&lt;/sup&gt;, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}. 

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 25\128
| 234.375
| 8/7
| [[Rodan]]
|-
| 1
| 29\128
| 271.875
| 75/64
| [[Orson]]
|-
| 1
| 33\128
| 309.375
| 448/375
| [[Triwell]]
|-
| 1
| 53\128
| 496.875
| 4/3
| [[Undecental]]
|-
| 2
| 13\128
| 121.875
| 15/14
| [[Lagaca]]
|-
| 2
| 15\128
| 140.625
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128&lt;br /&gt;(11\128)
| 496.875&lt;br /&gt;(103.125)
| 4/3
| [[Undim]] (7-limit)
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]

[[Category:Rodan]]
[[Category:Fourfives]]</text>
      <sha1>ixlwhwqzk5t0yf4v7yuiqcj3um7rxrp</sha1>
    </revision>
    <revision>
      <id>191038</id>
      <parentid>181337</parentid>
      <timestamp>2025-04-11T07:42:18Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Theory */ (''See [[regular temperament]] for more about what all this means and how to use it.'')</comment>
      <origin>191038</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1833" sha1="9y21vzklrrhimyigvtpm9mwpf9j6mat" xml:space="preserve">{{Infobox ET}}
{{ED intro}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes. 

== Theory ==
See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the {{nowrap|41 &amp;amp; 87}} temperament, as well as for 7-limit [[fourfives]], the {{nowrap|60 &amp;amp; 68}} temperament. (''See [[regular temperament]] for more about what all this means and how to use it.'')

=== Prime harmonics ===
{{Harmonics in equal|128}}

=== Subsets and supersets ===
Since 128 factors into 2&lt;sup&gt;7&lt;/sup&gt;, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}.

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 25\128
| 234.375
| 8/7
| [[Rodan]]
|-
| 1
| 29\128
| 271.875
| 75/64
| [[Orson]]
|-
| 1
| 33\128
| 309.375
| 448/375
| [[Triwell]]
|-
| 1
| 53\128
| 496.875
| 4/3
| [[Undecental]]
|-
| 2
| 13\128
| 121.875
| 15/14
| [[Lagaca]]
|-
| 2
| 15\128
| 140.625
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128&lt;br /&gt;(11\128)
| 496.875&lt;br /&gt;(103.125)
| 4/3
| [[Undim]] (7-limit)
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]

[[Category:Rodan]]
[[Category:Fourfives]]</text>
      <sha1>9y21vzklrrhimyigvtpm9mwpf9j6mat</sha1>
    </revision>
    <revision>
      <id>191153</id>
      <parentid>191038</parentid>
      <timestamp>2025-04-11T11:18:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>No need to remind readers of what a regular temperament is everywhere</comment>
      <origin>191153</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1747" sha1="ixlwhwqzk5t0yf4v7yuiqcj3um7rxrp" xml:space="preserve">{{Infobox ET}}
{{ED intro}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes. 

== Theory ==
The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the {{nowrap|41 &amp;amp; 87}} temperament, as well as for 7-limit [[fourfives]], the {{nowrap|60 &amp;amp; 68}} temperament. 

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

=== Prime harmonics ===
{{Harmonics in equal|128}}

=== Subsets and supersets ===
Since 128 factors into 2&lt;sup&gt;7&lt;/sup&gt;, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}. 

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 25\128
| 234.375
| 8/7
| [[Rodan]]
|-
| 1
| 29\128
| 271.875
| 75/64
| [[Orson]]
|-
| 1
| 33\128
| 309.375
| 448/375
| [[Triwell]]
|-
| 1
| 53\128
| 496.875
| 4/3
| [[Undecental]]
|-
| 2
| 13\128
| 121.875
| 15/14
| [[Lagaca]]
|-
| 2
| 15\128
| 140.625
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128&lt;br /&gt;(11\128)
| 496.875&lt;br /&gt;(103.125)
| 4/3
| [[Undim]] (7-limit)
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]

[[Category:Rodan]]
[[Category:Fourfives]]</text>
      <sha1>ixlwhwqzk5t0yf4v7yuiqcj3um7rxrp</sha1>
    </revision>
    <revision>
      <id>225766</id>
      <parentid>191153</parentid>
      <timestamp>2026-03-13T13:31:48Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct" to "[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct"</comment>
      <origin>225766</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1802" sha1="c7pix51n8hj55ec7odc9q4dw29h4j47" xml:space="preserve">{{Infobox ET}}
{{ED intro}} It is notable for being the equal division corresponding to a standard [[MIDI]] piano roll of 128 notes. 

== Theory ==
The equal temperament [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) in the [[5-limit]]; [[245/243]], [[1029/1024]] and [[5120/5103]] in the 7-limit; [[385/384]] and [[441/440]] in the 11-limit. It provides the [[optimal patent val]] for [[7-limit]] [[rodan]], the {{nowrap|41 &amp;amp; 87}} temperament, as well as for 7-limit [[fourfives]], the {{nowrap|60 &amp;amp; 68}} temperament. 

See also [https://www.youtube.com/watch?v=lGa66qHzKME 128 notes per octave on Alto Saxophone] (Demo by Philipp Gerschlauer)

=== Prime harmonics ===
{{Harmonics in equal|128}}

=== Subsets and supersets ===
Since 128 factors into 2&lt;sup&gt;7&lt;/sup&gt;, 128edo has subset edos {{EDOs| 2, 4, 8, 16, 32, and 64 }}. 

== Regular temperament properties ==
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Cents*
! Associated&lt;br /&gt;ratio*
! Temperaments
|-
| 1
| 25\128
| 234.375
| 8/7
| [[Rodan]]
|-
| 1
| 29\128
| 271.875
| 75/64
| [[Orson]]
|-
| 1
| 33\128
| 309.375
| 448/375
| [[Triwell]]
|-
| 1
| 53\128
| 496.875
| 4/3
| [[Undecental]]
|-
| 2
| 13\128
| 121.875
| 15/14
| [[Lagaca]]
|-
| 2
| 15\128
| 140.625
| 27/25
| [[Fifive]]
|-
| 4
| 15\128
| 140.625
| 27/25
| [[Fourfives]]
|-
| 4
| 53\128&lt;br /&gt;(11\128)
| 496.875&lt;br /&gt;(103.125)
| 4/3
| [[Undim]] (7-limit)
|}
&lt;nowiki /&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Scales ==
* [[Radon5]]
* [[Radon11]]
* [[Radon16]]

[[Category:Rodan]]
[[Category:Fourfives]]</text>
      <sha1>c7pix51n8hj55ec7odc9q4dw29h4j47</sha1>
    </revision>
  </page>
  <page>
    <title>129edo</title>
    <ns>0</ns>
    <id>124</id>
    <revision>
      <id>124</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>124</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="694" sha1="9nj811vy6utcur5ie26gsvx26z7lydz" xml:space="preserve">'''129edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 129 parts of 9.302 [[cent|cent]]s each. It provides the [[Optimal_patent_val|optimal patent val]] for the 11-limit rank three [[Didymus_rank_three_family|clio temperament]]. It [[tempering_out|tempers out]] 81/80 in the [[5-limit|5-limit]]; 1029/1024 and 1728/1715 in the [[7-limit|7-limit]]; 176/175 and 540/539 in the [[11-limit|11-limit]]; 507/500, 676/675 and 847/845 in the [[13-limit|13-limit]]; 221/220 in the [[17-limit|17-limit]]; 171/170 and 286/285 in the [[19-limit|19-limit]]. 

The factorization of 129 is [[3edo|3]] and [[43edo|43]]      [[Category:clio]]
[[Category:edo]]
[[Category:theory]]</text>
      <sha1>9nj811vy6utcur5ie26gsvx26z7lydz</sha1>
    </revision>
    <revision>
      <id>4582</id>
      <timestamp>2011-06-29T10:08:59Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 239316131 - Original comment: **</comment>
      <origin>4582</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2978" sha1="5xf5z8rxc10raalv1modccwb93shzr4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 10:08:59 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239316131&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**129edo** is the [[equal division of the octave]] into 129 parts of 9.302 [[cent]]s each. It provides the [[optimal patent val]] for the 11-limit rank three [[Didymus rank three family|clio temperament]]. It [[tempering out|tempers out]] 81/80 in the [[5-limit]]; 1029/1024 and 1728/1715 in the [[7-limit]]; 176/175 and 540/539 in the [[11-limit]]; 507/500, 676/675 and 847/845 in the [[13-limit]]; 221/220 in the [[17-limit]]; 171/170 and 286/285 in the [[19-limit]]. 

The factorization of 129 is [[3edo|3]] and [[43edo|43]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;129edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;129edo&amp;lt;/strong&amp;gt; is the &amp;lt;a class="wiki_link" href="/equal%20division%20of%20the%20octave"&amp;gt;equal division of the octave&amp;lt;/a&amp;gt; into 129 parts of 9.302 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cent&amp;lt;/a&amp;gt;s each. It provides the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for the 11-limit rank three &amp;lt;a class="wiki_link" href="/Didymus%20rank%20three%20family"&amp;gt;clio temperament&amp;lt;/a&amp;gt;. It &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; 81/80 in the &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt;; 1029/1024 and 1728/1715 in the &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt;; 176/175 and 540/539 in the &amp;lt;a class="wiki_link" href="/11-limit"&amp;gt;11-limit&amp;lt;/a&amp;gt;; 507/500, 676/675 and 847/845 in the &amp;lt;a class="wiki_link" href="/13-limit"&amp;gt;13-limit&amp;lt;/a&amp;gt;; 221/220 in the &amp;lt;a class="wiki_link" href="/17-limit"&amp;gt;17-limit&amp;lt;/a&amp;gt;; 171/170 and 286/285 in the &amp;lt;a class="wiki_link" href="/19-limit"&amp;gt;19-limit&amp;lt;/a&amp;gt;. &amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The factorization of 129 is &amp;lt;a class="wiki_link" href="/3edo"&amp;gt;3&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/43edo"&amp;gt;43&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>5xf5z8rxc10raalv1modccwb93shzr4</sha1>
    </revision>
    <revision>
      <id>4583</id>
      <timestamp>2011-03-25T21:41:24Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 214124472 - Original comment: **</comment>
      <origin>4583</origin>
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      <text bytes="2205" sha1="gzdf74ti7vrqg3jcmz110cit8xz73zm" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-03-25 21:41:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;214124472&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The //129 equal division// divides the octave into 129 equal parts of 9.302 cents each. It provides the [[optimal patent val]] for the 11-limit rank three [[Didymus rank three family|clio temperament]]. It tempers out 81/80 in the 5-limit; 1029/1024 and 1728/1715 in the 7-limit; 176/175 and 540/539 in the 11-limit; 507/500, 676/675 and 847/845 in the 13-limit; 221/220 in the 17-limit; 171/170 and 286/285 in the 19-limit.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;129edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The &amp;lt;em&amp;gt;129 equal division&amp;lt;/em&amp;gt; divides the octave into 129 equal parts of 9.302 cents each. It provides the &amp;lt;a class="wiki_link" href="/optimal%20patent%20val"&amp;gt;optimal patent val&amp;lt;/a&amp;gt; for the 11-limit rank three &amp;lt;a class="wiki_link" href="/Didymus%20rank%20three%20family"&amp;gt;clio temperament&amp;lt;/a&amp;gt;. It tempers out 81/80 in the 5-limit; 1029/1024 and 1728/1715 in the 7-limit; 176/175 and 540/539 in the 11-limit; 507/500, 676/675 and 847/845 in the 13-limit; 221/220 in the 17-limit; 171/170 and 286/285 in the 19-limit.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    </revision>
    <revision>
      <id>54980</id>
      <parentid>124</parentid>
      <timestamp>2020-12-04T23:13:07Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54980</origin>
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      <text bytes="720" sha1="0j4y1y8l3rtrrvsaj6cb5zd3xw77nog" xml:space="preserve">'''129edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 129 parts of 9.302 [[cent|cent]]s each. It provides the [[Optimal_patent_val|optimal patent val]] for the 11-limit rank three [[Didymus_rank_three_family|clio temperament]]. It [[tempering_out|tempers out]] 81/80 in the [[5-limit|5-limit]]; 1029/1024 and 1728/1715 in the [[7-limit|7-limit]]; 176/175 and 540/539 in the [[11-limit|11-limit]]; 507/500, 676/675 and 847/845 in the [[13-limit|13-limit]]; 221/220 in the [[17-limit|17-limit]]; 171/170 and 286/285 in the [[19-limit|19-limit]]. 

The factorization of 129 is [[3edo|3]] and [[43edo|43]]      [[Category:clio]]
[[Category:Equal divisions of the octave]]
[[Category:theory]]</text>
      <sha1>0j4y1y8l3rtrrvsaj6cb5zd3xw77nog</sha1>
    </revision>
    <revision>
      <id>87443</id>
      <parentid>54980</parentid>
      <timestamp>2022-02-26T16:48:49Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Theory]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>87443</origin>
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      <format>text/x-wiki</format>
      <text bytes="700" sha1="ajs8xljd5z28qzbv5iqb9soy1q91hvp" xml:space="preserve">'''129edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 129 parts of 9.302 [[cent|cent]]s each. It provides the [[Optimal_patent_val|optimal patent val]] for the 11-limit rank three [[Didymus_rank_three_family|clio temperament]]. It [[tempering_out|tempers out]] 81/80 in the [[5-limit|5-limit]]; 1029/1024 and 1728/1715 in the [[7-limit|7-limit]]; 176/175 and 540/539 in the [[11-limit|11-limit]]; 507/500, 676/675 and 847/845 in the [[13-limit|13-limit]]; 221/220 in the [[17-limit|17-limit]]; 171/170 and 286/285 in the [[19-limit|19-limit]]. 

The factorization of 129 is [[3edo|3]] and [[43edo|43]]      [[Category:clio]]
[[Category:Equal divisions of the octave]]</text>
      <sha1>ajs8xljd5z28qzbv5iqb9soy1q91hvp</sha1>
    </revision>
    <revision>
      <id>92939</id>
      <parentid>87443</parentid>
      <timestamp>2022-07-02T16:27:38Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92939</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="724" sha1="mu06l9156hbioit49pn3j0q2dz28vqf" xml:space="preserve">'''129edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 129 parts of 9.302 [[cent|cent]]s each. It provides the [[Optimal_patent_val|optimal patent val]] for the 11-limit rank three [[Didymus_rank_three_family|clio temperament]]. It [[tempering_out|tempers out]] 81/80 in the [[5-limit|5-limit]]; 1029/1024 and 1728/1715 in the [[7-limit|7-limit]]; 176/175 and 540/539 in the [[11-limit|11-limit]]; 507/500, 676/675 and 847/845 in the [[13-limit|13-limit]]; 221/220 in the [[17-limit|17-limit]]; 171/170 and 286/285 in the [[19-limit|19-limit]]. 

The factorization of 129 is [[3edo|3]] and [[43edo|43]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Clio]]</text>
      <sha1>mu06l9156hbioit49pn3j0q2dz28vqf</sha1>
    </revision>
    <revision>
      <id>96206</id>
      <parentid>92939</parentid>
      <timestamp>2022-10-04T18:55:29Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>96206</origin>
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      <text bytes="739" sha1="1wym33alwk3p3neacr4hnwic5mgkwi8" xml:space="preserve">{{Infobox ET}}
'''129edo''' is the [[Equal_division_of_the_octave|equal division of the octave]] into 129 parts of 9.302 [[cent|cent]]s each. It provides the [[Optimal_patent_val|optimal patent val]] for the 11-limit rank three [[Didymus_rank_three_family|clio temperament]]. It [[tempering_out|tempers out]] 81/80 in the [[5-limit|5-limit]]; 1029/1024 and 1728/1715 in the [[7-limit|7-limit]]; 176/175 and 540/539 in the [[11-limit|11-limit]]; 507/500, 676/675 and 847/845 in the [[13-limit|13-limit]]; 221/220 in the [[17-limit|17-limit]]; 171/170 and 286/285 in the [[19-limit|19-limit]]. 

The factorization of 129 is [[3edo|3]] and [[43edo|43]]

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Clio]]</text>
      <sha1>1wym33alwk3p3neacr4hnwic5mgkwi8</sha1>
    </revision>
    <revision>
      <id>99321</id>
      <parentid>96206</parentid>
      <timestamp>2022-11-27T13:01:05Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>This is the last meantone pval edo, not 105edo</comment>
      <origin>99321</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="810" sha1="9eel9rslynt22xh3uya92879ma1my7y" xml:space="preserve">{{Infobox ET}}
'''129edo''' is the [[equal division of the octave]] into 129 parts of 9.302 [[cent]]s each. It provides the [[optimal patent val]] for the 11-limit rank-3 [[clio]] temperament. It is the last [[patent val]] that [[tempering out|tempers out]] 81/80 so as to support [[meantone]] and its higher-limit expansions. It also tempers out [[1029/1024]] and [[1728/1715]] in the [[7-limit]]; [[176/175]] and [[540/539]] in the [[11-limit]]; [[507/500]], [[676/675]] and [[847/845]] in the [[13-limit]]; [[221/220]] in the [[17-limit]]; [[171/170]] and [[286/285]] in the [[19-limit]]. 

=== Odd harmonics ===
{{Harmonics in equal|129}}

=== Miscellany ===
The factorization of 129 is [[3edo|3]] and [[43edo|43]]. 

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Clio]]</text>
      <sha1>9eel9rslynt22xh3uya92879ma1my7y</sha1>
    </revision>
    <revision>
      <id>132903</id>
      <parentid>99321</parentid>
      <timestamp>2024-01-25T09:55:57Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Edo intro</comment>
      <origin>132903</origin>
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      <text bytes="720" sha1="8s15ifg3hinijl4gc5uejh550qhx3ns" xml:space="preserve">{{Infobox ET}}
{{EDO intro|129}}

129edo provides the [[optimal patent val]] for the 11-limit rank-3 [[clio]] temperament. It is the last [[patent val]] that [[tempering out|tempers out]] 81/80 so as to support [[meantone]] and its higher-limit expansions. It also tempers out [[1029/1024]] and [[1728/1715]] in the [[7-limit]]; [[176/175]] and [[540/539]] in the [[11-limit]]; [[507/500]], [[676/675]] and [[847/845]] in the [[13-limit]]; [[221/220]] in the [[17-limit]]; [[171/170]] and [[286/285]] in the [[19-limit]]. 

The factorization of 129 is [[3edo|3]] and [[43edo|43]].

=== Odd harmonics ===
{{Harmonics in equal|129}}

[[Category:Equal divisions of the octave|###]] &lt;!-- 3-digit number --&gt;
[[Category:Clio]]</text>
      <sha1>8s15ifg3hinijl4gc5uejh550qhx3ns</sha1>
    </revision>
    <revision>
      <id>145128</id>
      <parentid>132903</parentid>
      <timestamp>2024-05-27T17:23:02Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+overview to its harmonic quality, before going on to talk about the patent val. Misc. cleanup</comment>
      <origin>145128</origin>
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      <text bytes="861" sha1="h0ythcglfu5lvu0d9f7zfg7kc9jz67q" xml:space="preserve">{{Infobox ET}}
{{EDO intro|129}}

129edo is in[[consistent]] to the [[5-odd-limit]] and both [[harmonic]]s [[3/1|3]] and [[5/1|5]] are about halfway between its steps. It is the last [[patent val]] that [[tempering out|tempers out]] [[81/80]] so as to [[support]] [[meantone]] and its higher-limit expansions. It also tempers out [[1029/1024]] and [[1728/1715]] in the [[7-limit]]; [[176/175]] and [[540/539]] in the [[11-limit]]; [[507/500]], [[676/675]] and [[847/845]] in the [[13-limit]]; [[221/220]] in the [[17-limit]]; [[171/170]] and [[286/285]] in the [[19-limit]]. It provides the [[optimal patent val]] for the 11-limit rank-3 [[clio]] temperament. 

=== Odd harmonics ===
{{Harmonics in equal|129}}

=== Subsets and supersets ===
Since 129 factors into {{factorization|129}}, 129edo contains [[3edo]] and [[43edo]] as its subsets.

[[Category:Clio]]</text>
      <sha1>h0ythcglfu5lvu0d9f7zfg7kc9jz67q</sha1>
    </revision>
    <revision>
      <id>145129</id>
      <parentid>145128</parentid>
      <timestamp>2024-05-27T17:26:29Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Note its patent val is the same as 43et</comment>
      <origin>145129</origin>
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      <text bytes="957" sha1="ktzuzij4uqqei83w6815ol0lmt9n5uj" xml:space="preserve">{{Infobox ET}}
{{EDO intro|129}}

129edo is in[[consistent]] to the [[5-odd-limit]] and both [[harmonic]]s [[3/1|3]] and [[5/1|5]] are about halfway between its steps. The [[patent val]] is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[43edo]]. It is the last patent val that [[tempering out|tempers out]] [[81/80]] so as to [[support]] [[meantone]] and its higher-limit expansions. It also tempers out [[1029/1024]] and [[1728/1715]] in the [[7-limit]]; [[176/175]] and [[540/539]] in the [[11-limit]]; [[507/500]], [[676/675]] and [[847/845]] in the [[13-limit]]; [[221/220]] in the [[17-limit]]; [[171/170]] and [[286/285]] in the [[19-limit]]. It provides the [[optimal patent val]] for the 11-limit rank-3 [[clio]] temperament. 

=== Odd harmonics ===
{{Harmonics in equal|129}}

=== Subsets and supersets ===
Since 129 factors into {{factorization|129}}, 129edo contains [[3edo]] and [[43edo]] as its subsets.

[[Category:Clio]]</text>
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    </revision>
    <revision>
      <id>181338</id>
      <parentid>145129</parentid>
      <timestamp>2025-02-18T17:00:48Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>181338</origin>
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      <text bytes="952" sha1="hx8c6wi7loulcmfy0li23raos5zgjvh" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

129edo is in[[consistent]] to the [[5-odd-limit]] and both [[harmonic]]s [[3/1|3]] and [[5/1|5]] are about halfway between its steps. The [[patent val]] is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[43edo]]. It is the last patent val that [[tempering out|tempers out]] [[81/80]] so as to [[support]] [[meantone]] and its higher-limit expansions. It also tempers out [[1029/1024]] and [[1728/1715]] in the [[7-limit]]; [[176/175]] and [[540/539]] in the [[11-limit]]; [[507/500]], [[676/675]] and [[847/845]] in the [[13-limit]]; [[221/220]] in the [[17-limit]]; [[171/170]] and [[286/285]] in the [[19-limit]]. It provides the [[optimal patent val]] for the 11-limit rank-3 [[clio]] temperament. 

=== Odd harmonics ===
{{Harmonics in equal|129}}

=== Subsets and supersets ===
Since 129 factors into {{factorization|129}}, 129edo contains [[3edo]] and [[43edo]] as its subsets.

[[Category:Clio]]</text>
      <sha1>hx8c6wi7loulcmfy0li23raos5zgjvh</sha1>
    </revision>
    <revision>
      <id>204487</id>
      <parentid>181338</parentid>
      <timestamp>2025-07-13T17:21:26Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <minor/>
      <comment>Add lumatone mapping link.</comment>
      <origin>204487</origin>
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      <format>text/x-wiki</format>
      <text bytes="1005" sha1="p50ywdoudzpauqsyay59y861f24kh94" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

129edo is in[[consistent]] to the [[5-odd-limit]] and both [[harmonic]]s [[3/1|3]] and [[5/1|5]] are about halfway between its steps. The [[patent val]] is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[43edo]]. It is the last patent val that [[tempering out|tempers out]] [[81/80]] so as to [[support]] [[meantone]] and its higher-limit expansions. It also tempers out [[1029/1024]] and [[1728/1715]] in the [[7-limit]]; [[176/175]] and [[540/539]] in the [[11-limit]]; [[507/500]], [[676/675]] and [[847/845]] in the [[13-limit]]; [[221/220]] in the [[17-limit]]; [[171/170]] and [[286/285]] in the [[19-limit]]. It provides the [[optimal patent val]] for the 11-limit rank-3 [[clio]] temperament. 

=== Odd harmonics ===
{{Harmonics in equal|129}}

=== Subsets and supersets ===
Since 129 factors into {{factorization|129}}, 129edo contains [[3edo]] and [[43edo]] as its subsets.

== Instruments ==
* [[Lumatone mapping for 129edo]]

[[Category:Clio]]</text>
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    </revision>
    <revision>
      <id>204498</id>
      <parentid>204487</parentid>
      <timestamp>2025-07-13T22:57:23Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>/* Subsets and supersets */</comment>
      <origin>204498</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1093" sha1="ilcsan3ed9e1wzfy6s933ysijdnpj0o" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

129edo is in[[consistent]] to the [[5-odd-limit]] and both [[harmonic]]s [[3/1|3]] and [[5/1|5]] are about halfway between its steps. The [[patent val]] is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[43edo]]. It is the last patent val that [[tempering out|tempers out]] [[81/80]] so as to [[support]] [[meantone]] and its higher-limit expansions. It also tempers out [[1029/1024]] and [[1728/1715]] in the [[7-limit]]; [[176/175]] and [[540/539]] in the [[11-limit]]; [[507/500]], [[676/675]] and [[847/845]] in the [[13-limit]]; [[221/220]] in the [[17-limit]]; [[171/170]] and [[286/285]] in the [[19-limit]]. It provides the [[optimal patent val]] for the 11-limit rank-3 [[clio]] temperament. 

=== Odd harmonics ===
{{Harmonics in equal|129}}

=== Subsets and supersets ===
Since 129 factors into {{factorization|129}}, 129edo contains [[3edo]] and [[43edo]] as its subsets. [[258edo]], which doubles it, provides a good correction for the 3rd and 5th harmonics.

== Instruments ==
* [[Lumatone mapping for 129edo]]

[[Category:Clio]]</text>
      <sha1>ilcsan3ed9e1wzfy6s933ysijdnpj0o</sha1>
    </revision>
  </page>
  <page>
    <title>12/1</title>
    <ns>0</ns>
    <id>125</id>
    <revision>
      <id>125</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>125</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="178" sha1="d6cp5hc0y4k8if1z2wb9uzk94w650pm" xml:space="preserve">'''12/1'''
|2 1&amp;gt;

4301.9550 cents

[[File:jid_12_1_pluck_adu_dr55.mp3]] [[:File:jid_12_1_pluck_adu_dr55.mp3|sound sample]]

The 12th harmonic; three octaves above [[3/2|3/2]].</text>
      <sha1>d6cp5hc0y4k8if1z2wb9uzk94w650pm</sha1>
    </revision>
    <revision>
      <id>4584</id>
      <timestamp>2014-06-08T14:04:19Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513249394 - Original comment: **</comment>
      <origin>4584</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2903" sha1="5dgfupbh7ttn6ttfbvlxsh6uf7kz6hl" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-08 14:04:19 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513249394&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**12/1**
|2 1&amp;gt;
4301.9550 cents
[[media type="file" key="jid_12_1_pluck_adu_dr55.mp3"]] [[file:xenharmonic/jid_12_1_pluck_adu_dr55.mp3|sound sample]]

The 12th harmonic; three octaves above [[3_2|3/2]].
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12_1&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;12/1&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|2 1&amp;amp;gt;&amp;lt;br /&amp;gt;
4301.9550 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_12_1_pluck_adu_dr55.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_12_1_pluck_adu_dr55.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_12_1_pluck_adu_dr55.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_12_1_pluck_adu_dr55.mp3/513246156/jid_12_1_pluck_adu_dr55.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_12_1_pluck_adu_dr55.mp3/513246156/jid_12_1_pluck_adu_dr55.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The 12th harmonic; three octaves above &amp;lt;a class="wiki_link" href="/3_2"&amp;gt;3/2&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>5dgfupbh7ttn6ttfbvlxsh6uf7kz6hl</sha1>
    </revision>
    <revision>
      <id>64577</id>
      <parentid>125</parentid>
      <timestamp>2021-03-27T17:14:09Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
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      <origin>64577</origin>
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      <text bytes="227" sha1="8twv207metcdqu72ve04rjfnfc0isif" xml:space="preserve">'''12/1'''
|2 1&amp;gt;

4301.9550 cents

[[File:jid_12_1_pluck_adu_dr55.mp3]] [[:File:jid_12_1_pluck_adu_dr55.mp3|sound sample]]

The 12th harmonic; three octaves above [[3/2|3/2]].

[[Category:Pages with internal sound examples]]</text>
      <sha1>8twv207metcdqu72ve04rjfnfc0isif</sha1>
    </revision>
    <revision>
      <id>82512</id>
      <parentid>64577</parentid>
      <timestamp>2021-12-23T12:46:45Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+infobox</comment>
      <origin>82512</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="396" sha1="q1e44zozhlgxcnuxqkty6a3eptzbsl3" xml:space="preserve">{{Infobox Interval
| Ratio = 12/1
| Monzo = 2 1
| Cents = 4301.95500
| Name = 12th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;w5, tricowa 5th
| FJS name = P26
| Sound = jid_12_1_pluck_adu_dr55.mp3
}}
'''12/1''' is the 12th harmonic; three octaves above [[3/2]].

== See also ==
* [[Gallery of just intervals]]

[[Category:3-limit]]
[[Category:Harmonics]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>q1e44zozhlgxcnuxqkty6a3eptzbsl3</sha1>
    </revision>
    <revision>
      <id>86297</id>
      <parentid>82512</parentid>
      <timestamp>2022-02-22T23:46:39Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86297</origin>
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      <text bytes="348" sha1="p2lb0malif9yqsl6wj7uvev2fm5lsbx" xml:space="preserve">{{Infobox Interval
| Ratio = 12/1
| Monzo = 2 1
| Cents = 4301.95500
| Name = 12th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;w5, tricowa 5th
| FJS name = P26
| Sound = jid_12_1_pluck_adu_dr55.mp3
}}
'''12/1''' is the 12th harmonic; three octaves above [[3/2]].

== See also ==
* [[Gallery of just intervals]]

[[Category:3-limit]]
[[Category:Harmonics]]</text>
      <sha1>p2lb0malif9yqsl6wj7uvev2fm5lsbx</sha1>
    </revision>
    <revision>
      <id>98017</id>
      <parentid>86297</parentid>
      <timestamp>2022-10-25T12:20:30Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>98017</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="236" sha1="7usnx4iv1dod780bdbysu9rmkbcw63h" xml:space="preserve">{{Infobox Interval
| Name = 12th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;w5, tricowa 5th
| Sound = jid_12_1_pluck_adu_dr55.mp3
}}
'''12/1''' is the 12th harmonic; three octaves above [[3/2]].

== See also ==
* [[Gallery of just intervals]]</text>
      <sha1>7usnx4iv1dod780bdbysu9rmkbcw63h</sha1>
    </revision>
    <revision>
      <id>118788</id>
      <parentid>98017</parentid>
      <timestamp>2023-07-06T02:27:00Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added to category “equivalence interval”</comment>
      <origin>118788</origin>
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      <text bytes="270" sha1="d46htgkojhnt5oyd098jr3lb24fsu3k" xml:space="preserve">{{Infobox Interval
| Name = 12th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;w5, tricowa 5th
| Sound = jid_12_1_pluck_adu_dr55.mp3
}}
'''12/1''' is the 12th harmonic; three octaves above [[3/2]].

== See also ==
* [[Gallery of just intervals]]
[[Category:Equivalence interval]]</text>
      <sha1>d46htgkojhnt5oyd098jr3lb24fsu3k</sha1>
    </revision>
    <revision>
      <id>118818</id>
      <parentid>118788</parentid>
      <timestamp>2023-07-06T02:56:13Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Equivalence interval]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>118818</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="236" sha1="7usnx4iv1dod780bdbysu9rmkbcw63h" xml:space="preserve">{{Infobox Interval
| Name = 12th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;w5, tricowa 5th
| Sound = jid_12_1_pluck_adu_dr55.mp3
}}
'''12/1''' is the 12th harmonic; three octaves above [[3/2]].

== See also ==
* [[Gallery of just intervals]]</text>
      <sha1>7usnx4iv1dod780bdbysu9rmkbcw63h</sha1>
    </revision>
    <revision>
      <id>129179</id>
      <parentid>118818</parentid>
      <timestamp>2023-12-16T06:56:37Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Expansion</comment>
      <origin>129179</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="703" sha1="cj8hv7500kcuw85e0jpg4my0hk3eqi9" xml:space="preserve">{{Infobox Interval
| Name = 12th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;w5, tricowa 5th
| Sound = jid_12_1_pluck_adu_dr55.mp3
}}
'''12/1''', the '''12th harmonic''', is the [[harmonic]] past [[11/1]] and before [[13/1]]. It is three [[octave]]s above [[3/2]]. Since 12 is a {{w|highly composite number}}, this harmonic can be approached in various ways of stacking, all the components being [[Pythagorean tuning|Pythagorean]] intervals. Alternating [[3/2|fifth]]s and [[4/3|fourth]]s gives a [[consonant]] but simplistic skeleton across multiple registers: 1-2-3-4-6-12, on which [[harmonic limit|higher-limit]] intervals can be added to enrich its colors. 

== See also ==
* [[Gallery of just intervals]]</text>
      <sha1>cj8hv7500kcuw85e0jpg4my0hk3eqi9</sha1>
    </revision>
    <revision>
      <id>129184</id>
      <parentid>129179</parentid>
      <timestamp>2023-12-16T07:09:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Wording</comment>
      <origin>129184</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="727" sha1="nxe25e63chxfjrrvd995iwc9trovlk7" xml:space="preserve">{{Infobox Interval
| Name = 12th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;w5, tricowa 5th
| Sound = jid_12_1_pluck_adu_dr55.mp3
}}
'''12/1''', the '''12th harmonic''', is the [[harmonic]] past [[11/1]] and before [[13/1]]. It is three [[octave]]s above [[3/2]]. Since 12 is a {{w|highly composite number}}, this harmonic can be approached in various ways of stacking, all the components being [[Pythagorean tuning|Pythagorean]] intervals. For example, stacking with octaves, [[3/2|fifth]]s and [[4/3|fourth]]s gives a [[consonant]] but simplistic skeleton across multiple registers: 1-2-3-4-6-12, on which [[harmonic limit|higher-limit]] intervals can be added to enrich its colors. 

== See also ==
* [[Gallery of just intervals]]</text>
      <sha1>nxe25e63chxfjrrvd995iwc9trovlk7</sha1>
    </revision>
    <revision>
      <id>162435</id>
      <parentid>129184</parentid>
      <timestamp>2024-10-27T00:07:54Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* See also */</comment>
      <origin>162435</origin>
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      <text bytes="738" sha1="oiky1syuxisoh4is9az1jl3njrbq90f" xml:space="preserve">{{Infobox Interval
| Name = 12th harmonic
| Color name = c&lt;sup&gt;3&lt;/sup&gt;w5, tricowa 5th
| Sound = jid_12_1_pluck_adu_dr55.mp3
}}
'''12/1''', the '''12th harmonic''', is the [[harmonic]] past [[11/1]] and before [[13/1]]. It is three [[octave]]s above [[3/2]]. Since 12 is a {{w|highly composite number}}, this harmonic can be approached in various ways of stacking, all the components being [[Pythagorean tuning|Pythagorean]] intervals. For example, stacking with octaves, [[3/2|fifth]]s and [[4/3|fourth]]s gives a [[consonant]] but simplistic skeleton across multiple registers: 1-2-3-4-6-12, on which [[harmonic limit|higher-limit]] intervals can be added to enrich its colors. 

== See also ==
* [[Gallery of just intervals]]
* [[Ed12]]</text>
      <sha1>oiky1syuxisoh4is9az1jl3njrbq90f</sha1>
    </revision>
    <revision>
      <id>189102</id>
      <parentid>162435</parentid>
      <timestamp>2025-03-31T10:33:33Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+"duodecuple"</comment>
      <origin>189102</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="770" sha1="avmnmynjhhc4kmy5200ttesg3dwe93a" xml:space="preserve">{{Infobox Interval
| Name = 12th harmonic, duodecuple
| Color name = c&lt;sup&gt;3&lt;/sup&gt;w5, tricowa 5th
| Sound = jid_12_1_pluck_adu_dr55.mp3
}}
'''12/1''', the '''12th harmonic''' or '''duodecuple''', is the [[harmonic]] past [[11/1]] and before [[13/1]]. It is three [[octave]]s above [[3/2]]. Since 12 is a {{w|highly composite number}}, this harmonic can be approached in various ways of stacking, all the components being [[Pythagorean tuning|Pythagorean]] intervals. For example, stacking with octaves, [[3/2|fifth]]s and [[4/3|fourth]]s gives a [[consonant]] but simplistic skeleton across multiple registers: 1-2-3-4-6-12, on which [[harmonic limit|higher-limit]] intervals can be added to enrich its colors. 

== See also ==
* [[Gallery of just intervals]]
* [[Ed12]]</text>
      <sha1>avmnmynjhhc4kmy5200ttesg3dwe93a</sha1>
    </revision>
    <revision>
      <id>189408</id>
      <parentid>189102</parentid>
      <timestamp>2025-04-01T16:00:22Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Link to internal link</comment>
      <origin>189408</origin>
      <model>wikitext</model>
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      <text bytes="768" sha1="p14de5yeak7ogpbotvfhonivk3beet9" xml:space="preserve">{{Infobox Interval
| Name = 12th harmonic, duodecuple
| Color name = c&lt;sup&gt;3&lt;/sup&gt;w5, tricowa 5th
| Sound = jid_12_1_pluck_adu_dr55.mp3
}}
'''12/1''', the '''12th harmonic''' or '''duodecuple''', is the [[harmonic]] past [[11/1]] and before [[13/1]]. It is three [[octave]]s above [[3/2]]. Since 12 is a [[highly composite number]], this harmonic can be approached in various ways of stacking, all the components being [[Pythagorean tuning|Pythagorean]] intervals. For example, stacking with octaves, [[3/2|fifth]]s and [[4/3|fourth]]s gives a [[consonant]] but simplistic skeleton across multiple registers: 1-2-3-4-6-12, on which [[harmonic limit|higher-limit]] intervals can be added to enrich its colors. 

== See also ==
* [[Gallery of just intervals]]
* [[Ed12]]</text>
      <sha1>p14de5yeak7ogpbotvfhonivk3beet9</sha1>
    </revision>
    <revision>
      <id>189595</id>
      <parentid>189408</parentid>
      <timestamp>2025-04-02T10:59:35Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Style</comment>
      <origin>189595</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="778" sha1="281qeuiziogsrwkw9i5487gx90ws7ch" xml:space="preserve">{{Infobox Interval
| Name = 12th harmonic, duodecuple
| Color name = c&lt;sup&gt;3&lt;/sup&gt;w5, tricowa 5th
| Sound = jid_12_1_pluck_adu_dr55.mp3
}}
'''12/1''', the '''12th harmonic''' or '''duodecuple''', is the [[harmonic]] past [[11/1]] and before [[13/1]]. It is three [[octave]]s above [[3/2]]. Since 12 is a [[highly composite number]], this harmonic can be approached in various ways of stacking, all the components being [[Pythagorean tuning|Pythagorean]] intervals. For example, stacking with octaves, [[3/2|fifth]]s and [[4/3|fourth]]s gives a [[consonant]] but simplistic skeleton across multiple registers: 1–2–3–4–6–12, on which [[harmonic limit|higher-limit]] intervals can be added to enrich its colors. 

== See also ==
* [[Gallery of just intervals]]
* [[Ed12]]</text>
      <sha1>281qeuiziogsrwkw9i5487gx90ws7ch</sha1>
    </revision>
  </page>
  <page>
    <title>12 1</title>
    <ns>0</ns>
    <id>126</id>
    <redirect title="12/1" />
    <revision>
      <id>126</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <text bytes="18" sha1="oalg8fq3bl0mafl4627x85jk58ikkz9" xml:space="preserve">#REDIRECT [[12/1]]</text>
      <sha1>oalg8fq3bl0mafl4627x85jk58ikkz9</sha1>
    </revision>
  </page>
  <page>
    <title>12/11</title>
    <ns>0</ns>
    <id>127</id>
    <revision>
      <id>127</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>127</origin>
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      <text bytes="1148" sha1="lyi2cpee91vdk8w29n6r4dmul0zcfv0" xml:space="preserve">'''12/11'''
|2 1 0 0 -1&amp;gt;

150.63706 cents

[[File:jid_12_11_pluck_adu_dr220.mp3]] [[:File:jid_12_11_pluck_adu_dr220.mp3|sound sample]]

The (lesser) neutral second is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular|superparticular]] ratio 12/11, and is about 150.6 [[cent|cents]] large. One step of [[8edo|8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = |15 8 0 0 -8&amp;gt;. It follows that EDOs which are multiples of 8, such as [[16edo|16edo]] and [[24edo|24edo]], will also represent this interval well.

12/11 differs from the larger undecimal neutral second 11/10 (~165 cents) by 121/120 (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo|15edo]], [[22edo|22edo]], [[31edo|31edo]], [[Orwell|orwell]], [[Porcupine|porcupine]], [[Mohajira|mohajira]] and [[Valentine|valentine]].
[[Category:11-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:neutral_2nd]]
[[Category:ratio]]
[[Category:second]]</text>
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    </revision>
    <revision>
      <id>4585</id>
      <timestamp>2014-06-07T16:01:40Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513195536 - Original comment: **</comment>
      <origin>4585</origin>
      <model>wikitext</model>
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      <text bytes="5075" sha1="95p29nkfo2wlm0p2ynmxzujlabiwtle" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-07 16:01:40 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513195536&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**12/11**
|2 1 0 0 -1&amp;gt;
150.63706 cents
[[media type="file" key="jid_12_11_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_12_11_pluck_adu_dr220.mp3|sound sample]]

The (lesser) neutral second is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent|cents]] large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = |15 8 0 0 -8&amp;gt;. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

12/11 differs from the larger undecimal neutral second 11/10 (~165 cents) by 121/120 (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]] and [[valentine]].&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12_11&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;12/11&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|2 1 0 0 -1&amp;amp;gt;&amp;lt;br /&amp;gt;
150.63706 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_12_11_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_12_11_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_12_11_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_12_11_pluck_adu_dr220.mp3/513195200/jid_12_11_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_12_11_pluck_adu_dr220.mp3/513195200/jid_12_11_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The (lesser) neutral second is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; ratio 12/11, and is about 150.6 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cents&amp;lt;/a&amp;gt; large. One step of &amp;lt;a class="wiki_link" href="/8edo"&amp;gt;8edo&amp;lt;/a&amp;gt; is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = |15 8 0 0 -8&amp;amp;gt;. It follows that EDOs which are multiples of 8, such as &amp;lt;a class="wiki_link" href="/16edo"&amp;gt;16edo&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/24edo"&amp;gt;24edo&amp;lt;/a&amp;gt;, will also represent this interval well.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
12/11 differs from the larger undecimal neutral second 11/10 (~165 cents) by 121/120 (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include &amp;lt;a class="wiki_link" href="/15edo"&amp;gt;15edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/22edo"&amp;gt;22edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/orwell"&amp;gt;orwell&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/porcupine"&amp;gt;porcupine&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/mohajira"&amp;gt;mohajira&amp;lt;/a&amp;gt; and &amp;lt;a class="wiki_link" href="/valentine"&amp;gt;valentine&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
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      <timestamp>2011-08-25T16:16:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 248497683 - Original comment: **</comment>
      <origin>4586</origin>
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      <text bytes="2187" sha1="hu3xis8bhvhoykt41i2smcytc1ss7hc" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-08-25 16:16:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;248497683&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The (lesser) neutral second is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.64 [[cent|cents]] large. One step of [[8edo|8-edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = |15 8 0 0 -8&amp;gt;.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12_11&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The (lesser) neutral second is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; ratio 12/11, and is about 150.64 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cents&amp;lt;/a&amp;gt; large. One step of &amp;lt;a class="wiki_link" href="/8edo"&amp;gt;8-edo&amp;lt;/a&amp;gt; is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = |15 8 0 0 -8&amp;amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4587</id>
      <timestamp>2011-08-09T15:17:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Sarzadoce</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 245080117 - Original comment: **</comment>
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      <text bytes="2013" sha1="eu1miqmada20nsxh8l40auycvgib1ms" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on &lt;tt&gt;2011-08-09 15:17:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;245080117&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The (lesser) neutral second is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.64 [[cent|cents]] large. One step of [[8edo|8-edo]] is an excellent approximation of the just neutral second.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12_11&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The (lesser) neutral second is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; ratio 12/11, and is about 150.64 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cents&amp;lt;/a&amp;gt; large. One step of &amp;lt;a class="wiki_link" href="/8edo"&amp;gt;8-edo&amp;lt;/a&amp;gt; is an excellent approximation of the just neutral second.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4588</id>
      <timestamp>2011-08-09T00:51:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Sarzadoce</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 244975507 - Original comment: **</comment>
      <origin>4588</origin>
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      <text bytes="1916" sha1="5xhzzdlcjogs0whdheguzvi2tuqzgul" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Sarzadoce|Sarzadoce]] and made on &lt;tt&gt;2011-08-09 00:51:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;244975507&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The (lesser) neutral second is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the ratio 12/11, and is about 150.64 [[cent|cents]] large. One step of [[8edo|8-edo]] is an excellent approximation of the just neutral second.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12_11&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The (lesser) neutral second is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the ratio 12/11, and is about 150.64 &amp;lt;a class="wiki_link" href="/cent"&amp;gt;cents&amp;lt;/a&amp;gt; large. One step of &amp;lt;a class="wiki_link" href="/8edo"&amp;gt;8-edo&amp;lt;/a&amp;gt; is an excellent approximation of the just neutral second.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>36590</id>
      <parentid>127</parentid>
      <timestamp>2018-10-11T22:54:06Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
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      <comment>switched to infobox interval; simplified links</comment>
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      <text bytes="1102" sha1="m6ml350rrj0jcc6403kug3uv4n61el7" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 12/11
| Monzo = 2 1 0 0 -1
| Cents = 150.63706
| Name = undecimal neutral second
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}

The (lesser) neutral second is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent|cents]] large. One step of [[8edo|8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

'''12/11''' differs from the larger undecimal neutral second 11/10 (~165 cents) by 121/120 (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]] and [[valentine]].

[[Category:11-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:neutral_2nd]]
[[Category:ratio]]
[[Category:second]]</text>
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    <revision>
      <id>37586</id>
      <parentid>36590</parentid>
      <timestamp>2018-10-23T03:26:27Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
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      <origin>37586</origin>
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      <text bytes="1129" sha1="is9ny5uhwdc5zx3mr4wlg6ft5yl9px5" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 12/11
| Monzo = 2 1 0 0 -1
| Cents = 150.63706
| Name = undecimal neutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}

The (lesser) neutral second is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent|cents]] large. One step of [[8edo|8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

'''12/11''' differs from the larger undecimal neutral second 11/10 (~165 cents) by 121/120 (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]] and [[valentine]].

[[Category:11-limit]]
[[Category:interval]]
[[Category:just_interval]]
[[Category:neutral_2nd]]
[[Category:ratio]]
[[Category:second]]</text>
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    <revision>
      <id>38223</id>
      <parentid>37586</parentid>
      <timestamp>2018-10-31T22:20:13Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>added name in synopsis, simplified links, added category, added see-also section</comment>
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      <text bytes="1329" sha1="nlgugp3cn4l3hvl8bndzxiri5x126xx" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 12/11
| Monzo = 2 1 0 0 -1
| Cents = 150.63706
| Name = undecimal neutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

'''12/11''' differs from the larger undecimal neutral second 11/10 (~165 cents) by 121/120 (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]] and [[valentine]].

== See also ==
* [[11/6]] - its [[octave complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval]]
[[Category:Just interval]]
[[Category:Neutral 2nd]]
[[Category:Ratio]]
[[Category:Second]]
[[Category:Superparticular]]</text>
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    <revision>
      <id>45984</id>
      <parentid>38223</parentid>
      <timestamp>2020-06-07T14:29:41Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>recat</comment>
      <origin>45984</origin>
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      <text bytes="1332" sha1="cpdlbgq4q4b0j6wxntlss0cbawdxjou" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 12/11
| Monzo = 2 1 0 0 -1
| Cents = 150.63706
| Name = undecimal neutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

'''12/11''' differs from the larger undecimal neutral second 11/10 (~165 cents) by 121/120 (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]] and [[valentine]].

== See also ==
* [[11/6]] - its [[octave complement]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval]]
[[Category:Just interval]]
[[Category:Neutral second]]
[[Category:Ratio]]
[[Category:Second]]
[[Category:Superparticular]]</text>
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    <revision>
      <id>45989</id>
      <parentid>45984</parentid>
      <timestamp>2020-06-07T14:44:43Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* See also */ +1</comment>
      <origin>45989</origin>
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      <format>text/x-wiki</format>
      <text bytes="1353" sha1="92vcia0khx1mgm4or5csfi7opawil0w" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 12/11
| Monzo = 2 1 0 0 -1
| Cents = 150.63706
| Name = undecimal neutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

'''12/11''' differs from the larger undecimal neutral second 11/10 (~165 cents) by 121/120 (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]] and [[valentine]].

== See also ==
* [[11/6]] - its [[octave complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval]]
[[Category:Just interval]]
[[Category:Neutral second]]
[[Category:Ratio]]
[[Category:Second]]
[[Category:Superparticular]]</text>
      <sha1>92vcia0khx1mgm4or5csfi7opawil0w</sha1>
    </revision>
    <revision>
      <id>46309</id>
      <parentid>45989</parentid>
      <timestamp>2020-06-10T07:08:37Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>cats</comment>
      <origin>46309</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1340" sha1="euidm236fllbe0nskrf5atjmfgkvavk" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 12/11
| Monzo = 2 1 0 0 -1
| Cents = 150.63706
| Name = undecimal neutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

'''12/11''' differs from the larger undecimal neutral second 11/10 (~165 cents) by 121/120 (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]] and [[valentine]].

== See also ==
* [[11/6]] - its [[octave complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Neutral second]]
[[Category:Second]]
[[Category:Superparticular]]</text>
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    </revision>
    <revision>
      <id>48563</id>
      <parentid>46309</parentid>
      <timestamp>2020-09-05T03:50:25Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* See also */</comment>
      <origin>48563</origin>
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      <text bytes="1361" sha1="qaqk1dutzh1n1a0b4tutgzztfzcynqy" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 12/11
| Monzo = 2 1 0 0 -1
| Cents = 150.63706
| Name = undecimal neutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)^8/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

'''12/11''' differs from the larger undecimal neutral second 11/10 (~165 cents) by 121/120 (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]] and [[valentine]].

== See also ==
* [[11/6]] - its [[octave complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Neutral second]]
[[Category:Second]]
[[Category:Superparticular]]
[[Category:Over-11]]</text>
      <sha1>qaqk1dutzh1n1a0b4tutgzztfzcynqy</sha1>
    </revision>
    <revision>
      <id>49877</id>
      <parentid>48563</parentid>
      <timestamp>2020-09-20T06:38:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+FJS name; cleanup</comment>
      <origin>49877</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1446" sha1="ba4h5ibvifep1licfmtc7snbm521o7p" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 12/11
| Monzo = 2 1 0 0 -1
| Cents = 150.63706
| Name = undecimal neutral second
| Color name = 1u2, lu 2nd
| FJS name = M2&lt;sub&gt;11&lt;/sub&gt;
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)&lt;sup&gt;8&lt;/sup&gt;/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

12/11 differs from the larger undecimal neutral second [[11/10]] (~165 cents) by [[121/120]] (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]], [[valentine]], etc.

== See also ==
* [[11/6]] – its [[octave complement]]
* [[11/8]] – its [[fifth complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Neutral second]]
[[Category:Second]]
[[Category:Superparticular]]
[[Category:Over-11]]</text>
      <sha1>ba4h5ibvifep1licfmtc7snbm521o7p</sha1>
    </revision>
    <revision>
      <id>64578</id>
      <parentid>49877</parentid>
      <timestamp>2021-03-27T17:14:28Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64578</origin>
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      <text bytes="1494" sha1="g2f5grj7mwxjhau6tprgojgvthud6yy" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 12/11
| Monzo = 2 1 0 0 -1
| Cents = 150.63706
| Name = undecimal neutral second
| Color name = 1u2, lu 2nd
| FJS name = M2&lt;sub&gt;11&lt;/sub&gt;
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)&lt;sup&gt;8&lt;/sup&gt;/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

12/11 differs from the larger undecimal neutral second [[11/10]] (~165 cents) by [[121/120]] (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]], [[valentine]], etc.

== See also ==
* [[11/6]] – its [[octave complement]]
* [[11/8]] – its [[fifth complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Neutral second]]
[[Category:Second]]
[[Category:Superparticular]]
[[Category:Over-11]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>g2f5grj7mwxjhau6tprgojgvthud6yy</sha1>
    </revision>
    <revision>
      <id>72998</id>
      <parentid>64578</parentid>
      <timestamp>2021-06-23T17:42:29Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>72998</origin>
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      <text bytes="1535" sha1="54w0cpr2sekkld3f5vhlk21z0ci6w1c" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 12/11
| Monzo = 2 1 0 0 -1
| Cents = 150.63706
| Name = undecimal neutral second
| Color name = 1u2, lu 2nd
| FJS name = M2&lt;sub&gt;11&lt;/sub&gt;
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)&lt;sup&gt;8&lt;/sup&gt;/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

12/11 differs from the larger undecimal neutral second [[11/10]] (~165 cents) by [[121/120]] (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]], [[valentine]], etc.

== See also ==
* [[11/6]] – its [[octave complement]]
* [[11/8]] – its [[fifth complement]]
* [[11/9]] – its [[fourth complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Neutral second]]
[[Category:Second]]
[[Category:Superparticular]]
[[Category:Over-11]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>54w0cpr2sekkld3f5vhlk21z0ci6w1c</sha1>
    </revision>
    <revision>
      <id>86298</id>
      <parentid>72998</parentid>
      <timestamp>2022-02-22T23:46:39Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86298</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1487" sha1="gz7kw0zmo6j7r66bwk3k1eqeq8p9c6u" xml:space="preserve">{{Infobox Interval
| Icon =
| Ratio = 12/11
| Monzo = 2 1 0 0 -1
| Cents = 150.63706
| Name = undecimal neutral second
| Color name = 1u2, lu 2nd
| FJS name = M2&lt;sub&gt;11&lt;/sub&gt;
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the harmonic series. In Just Intonation it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)&lt;sup&gt;8&lt;/sup&gt;/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

12/11 differs from the larger undecimal neutral second [[11/10]] (~165 cents) by [[121/120]] (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]], [[valentine]], etc.

== See also ==
* [[11/6]] – its [[octave complement]]
* [[11/8]] – its [[fifth complement]]
* [[11/9]] – its [[fourth complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Interval ratio]]
[[Category:Just interval]]
[[Category:Neutral second]]
[[Category:Second]]
[[Category:Superparticular]]
[[Category:Over-11]]</text>
      <sha1>gz7kw0zmo6j7r66bwk3k1eqeq8p9c6u</sha1>
    </revision>
    <revision>
      <id>89346</id>
      <parentid>86298</parentid>
      <timestamp>2022-03-11T17:06:06Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Added Wikipedia box, misc. edits, categories</comment>
      <origin>89346</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1460" sha1="cj1hepit3fp48e9hkuk883v15vubzri" xml:space="preserve">{{Infobox Interval
| Ratio = 12/11
| Monzo = 2 1 0 0 -1
| Cents = 150.63706
| Name = undecimal neutral second
| Color name = 1u2, lu 2nd
| FJS name = M2&lt;sub&gt;11&lt;/sub&gt;
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral second}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the [[harmonic series]]. In [[just intonation]] it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)&lt;sup&gt;8&lt;/sup&gt;/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

12/11 differs from the larger undecimal neutral second [[11/10]] (~165 cents) by [[121/120]] (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]], [[valentine]], etc.

== See also ==
* [[11/6]] – its [[octave complement]]
* [[11/8]] – its [[fifth complement]]
* [[11/9]] – its [[fourth complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:11-limit]]
[[Category:Second]]
[[Category:Neutral second]]
[[Category:Superparticular]]
[[Category:Over-11]]</text>
      <sha1>cj1hepit3fp48e9hkuk883v15vubzri</sha1>
    </revision>
    <revision>
      <id>97984</id>
      <parentid>89346</parentid>
      <timestamp>2022-10-25T11:55:21Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>97984</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1323" sha1="h0dy6583jitm3d5ujy2vg8wld3mlnyx" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral second}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the [[harmonic series]]. In [[just intonation]] it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)&lt;sup&gt;8&lt;/sup&gt;/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.

12/11 differs from the larger undecimal neutral second [[11/10]] (~165 cents) by [[121/120]] (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]], [[valentine]], etc.

== See also ==
* [[11/6]] – its [[octave complement]]
* [[11/8]] – its [[fifth complement]]
* [[11/9]] – its [[fourth complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Over-11]]</text>
      <sha1>h0dy6583jitm3d5ujy2vg8wld3mlnyx</sha1>
    </revision>
    <revision>
      <id>112372</id>
      <parentid>97984</parentid>
      <timestamp>2023-05-06T23:43:07Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>112372</origin>
      <model>wikitext</model>
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      <text bytes="1441" sha1="k2b20fdu7z85ykfpzinafh155v3i4d9" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral second, Alpharabian tendoneutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral second}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the [[harmonic series]]. In [[just intonation]] it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)&lt;sup&gt;8&lt;/sup&gt;/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.  In [[Alpharabian tuning]] it is known as the '''Alpharabian tendoneutral second'''.

12/11 differs from the larger undecimal neutral second [[11/10]] (~165 cents) by [[121/120]] (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]], [[valentine]], etc.

== See also ==
* [[11/6]] – its [[octave complement]]
* [[11/8]] – its [[fifth complement]]
* [[11/9]] – its [[fourth complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Over-11]]</text>
      <sha1>k2b20fdu7z85ykfpzinafh155v3i4d9</sha1>
    </revision>
    <revision>
      <id>157039</id>
      <parentid>112372</parentid>
      <timestamp>2024-09-30T08:45:07Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Recategorize</comment>
      <origin>157039</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1451" sha1="88gkdhokc8tbxtbp77ian20su3ml2bd" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral second, Alpharabian tendoneutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral second}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the [[harmonic series]]. In [[just intonation]] it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma (12/11)&lt;sup&gt;8&lt;/sup&gt;/2 = {{Monzo|15 8 0 0 -8}}. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.  In [[Alpharabian tuning]] it is known as the '''Alpharabian tendoneutral second'''.

12/11 differs from the larger undecimal neutral second [[11/10]] (~165 cents) by [[121/120]] (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]], [[valentine]], etc.

== See also ==
* [[11/6]] – its [[octave complement]]
* [[11/8]] – its [[fifth complement]]
* [[11/9]] – its [[fourth complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Over-11 intervals]]</text>
      <sha1>88gkdhokc8tbxtbp77ian20su3ml2bd</sha1>
    </revision>
    <revision>
      <id>175609</id>
      <parentid>157039</parentid>
      <timestamp>2025-01-10T18:35:07Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <minor/>
      <comment>Link to referenced comma.</comment>
      <origin>175609</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1481" sha1="oji105eankfop3owvt8ba1haspn0bm6" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral second, Alpharabian tendoneutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral second}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is a strangely exotic interval found between the 11th and 12th partials of the [[harmonic series]]. In [[just intonation]] it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma [[Undecimal_octatonic_comma|(12/11)&lt;sup&gt;8&lt;/sup&gt;/2 = {{Monzo|15 8 0 0 -8}}]]. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.  In [[Alpharabian tuning]] it is known as the '''Alpharabian tendoneutral second'''.

12/11 differs from the larger undecimal neutral second [[11/10]] (~165 cents) by [[121/120]] (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]], [[valentine]], etc.

== See also ==
* [[11/6]] – its [[octave complement]]
* [[11/8]] – its [[fifth complement]]
* [[11/9]] – its [[fourth complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Over-11 intervals]]</text>
      <sha1>oji105eankfop3owvt8ba1haspn0bm6</sha1>
    </revision>
    <revision>
      <id>207211</id>
      <parentid>175609</parentid>
      <timestamp>2025-08-16T23:53:23Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>Neutrality</comment>
      <origin>207211</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1532" sha1="93cniihdfqah7riecnxe2et4c6umvyi" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral second, Alpharabian tendoneutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral second}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is an interval found between the 11th and 12th partials of the [[harmonic series]]. In [[just intonation]] it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma [[Undecimal_octatonic_comma|(12/11)&lt;sup&gt;8&lt;/sup&gt;/2 = {{Monzo|15 8 0 0 -8}}]]. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.  In [[Alpharabian tuning]] it is known as the '''Alpharabian tendoneutral second'''.

12/11 differs from the larger undecimal neutral second [[11/10]] (~165 cents) by [[121/120]] (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]], [[valentine]], etc.

Many Western listeners might describe 12/11 as sounding "exotic".

== See also ==
* [[11/6]] – its [[octave complement]]
* [[11/8]] – its [[fifth complement]]
* [[11/9]] – its [[fourth complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Over-11 intervals]]</text>
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    <revision>
      <id>215773</id>
      <parentid>207211</parentid>
      <timestamp>2025-11-03T06:49:11Z</timestamp>
      <contributor>
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        <id>507</id>
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      <text bytes="1591" sha1="kfdi70rts6j5i5l9pm0kq32e6z1wijw" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral second, Alpharabian tendoneutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral second}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is an interval found between the 11th and 12th partials of the [[harmonic series]]. In [[just intonation]] it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma [[Undecimal_octatonic_comma|(12/11)&lt;sup&gt;8&lt;/sup&gt;/2 = {{Monzo|15 8 0 0 -8}}]]. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.  In [[Alpharabian tuning]] it is known as the '''Alpharabian tendoneutral second'''.

12/11 differs from the larger undecimal neutral second [[11/10]] (~165 cents) by [[121/120]] (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]], [[valentine]], etc.

Many Western listeners might describe 12/11 as sounding "exotic".
== Approximation ==
 {{Interval_Edo_Approximation | 12/11}}
== See also ==
* [[11/6]] – its [[octave complement]]
* [[11/8]] – its [[fifth complement]]
* [[11/9]] – its [[fourth complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Over-11 intervals]]</text>
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    <revision>
      <id>215967</id>
      <parentid>215773</parentid>
      <timestamp>2025-11-03T13:01:53Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"</comment>
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      <text bytes="1588" sha1="hz048xum95i216j1gvhaod9crodmenx" xml:space="preserve">{{Infobox Interval
| Name = undecimal neutral second, Alpharabian tendoneutral second
| Color name = 1u2, lu 2nd
| Sound = jid_12_11_pluck_adu_dr220.mp3
}}
{{Wikipedia|Neutral second}}

'''12/11''', the '''undecimal neutral second''' or '''(lesser) neutral second''', is an interval found between the 11th and 12th partials of the [[harmonic series]]. In [[just intonation]] it is represented by the [[superparticular]] ratio 12/11, and is about 150.6 [[cent]]s large. One step of [[8edo]] is an excellent approximation of the just neutral second, and eight of them exceed the octave by the comma [[Undecimal_octatonic_comma|(12/11)&lt;sup&gt;8&lt;/sup&gt;/2 = {{Monzo|15 8 0 0 -8}}]]. It follows that EDOs which are multiples of 8, such as [[16edo]] and [[24edo]], will also represent this interval well.  In [[Alpharabian tuning]] it is known as the '''Alpharabian tendoneutral second'''.

12/11 differs from the larger undecimal neutral second [[11/10]] (~165 cents) by [[121/120]] (~14.4 cents). Temperaments which conflate the two (thus tempering out 121/120) include [[15edo]], [[22edo]], [[31edo]], [[orwell]], [[porcupine]], [[mohajira]], [[valentine]], etc.

Many Western listeners might describe 12/11 as sounding "exotic".
== Approximation ==
{{Interval edo approximation|12/11}}
== See also ==
* [[11/6]] – its [[octave complement]]
* [[11/8]] – its [[fifth complement]]
* [[11/9]] – its [[fourth complement]]
* [[Iceface Tuning]]
* [[Gallery of just intervals]]
* [[List of superparticular intervals]]

[[Category:Second]]
[[Category:Neutral second]]
[[Category:Over-11 intervals]]</text>
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  </page>
  <page>
    <title>12 11</title>
    <ns>0</ns>
    <id>128</id>
    <redirect title="12/11" />
    <revision>
      <id>128</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <origin>128</origin>
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      <text bytes="19" sha1="6g1148agvrj0nvu5nba8cpdu5lb0jyj" xml:space="preserve">#REDIRECT [[12/11]]</text>
      <sha1>6g1148agvrj0nvu5nba8cpdu5lb0jyj</sha1>
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  </page>
  <page>
    <title>12/5</title>
    <ns>0</ns>
    <id>129</id>
    <revision>
      <id>129</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>129</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="207" sha1="te3njci8dvob09ivw35pqf2egksu6sf" xml:space="preserve">'''12/5'''
|2 1 -1&amp;gt;

1515.6413 cents

[[File:jid_12_5_pluck_adu_dr220.mp3]] [[:File:jid_12_5_pluck_adu_dr220.mp3|sound sample]]

The ratio between the 5th and 12th harmonics; one octave above [[6/5|6/5]].</text>
      <sha1>te3njci8dvob09ivw35pqf2egksu6sf</sha1>
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    <revision>
      <id>4589</id>
      <timestamp>2014-06-08T13:49:48Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513248274 - Original comment: **</comment>
      <origin>4589</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2966" sha1="m6p4ts5zdkffq9i0guc75h9h49gzzy8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-08 13:49:48 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513248274&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**12/5**
|2 1 -1&amp;gt;
1515.6413 cents
[[media type="file" key="jid_12_5_pluck_adu_dr220.mp3"]] [[file:xenharmonic/jid_12_5_pluck_adu_dr220.mp3|sound sample]]

The ratio between the 5th and 12th harmonics; one octave above [[6_5|6/5]].
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12_5&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;12/5&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|2 1 -1&amp;amp;gt;&amp;lt;br /&amp;gt;
1515.6413 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_12_5_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_12_5_pluck_adu_dr220.mp3&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_12_5_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_12_5_pluck_adu_dr220.mp3/513246166/jid_12_5_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_12_5_pluck_adu_dr220.mp3/513246166/jid_12_5_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The ratio between the 5th and 12th harmonics; one octave above &amp;lt;a class="wiki_link" href="/6_5"&amp;gt;6/5&amp;lt;/a&amp;gt;.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>38986</id>
      <parentid>129</parentid>
      <timestamp>2018-12-10T19:48:46Z</timestamp>
      <contributor>
        <username>Spt3125</username>
        <id>16</id>
      </contributor>
      <minor/>
      <comment>category maintenance</comment>
      <origin>38986</origin>
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      <text bytes="276" sha1="68yryrh8dqw6n2qjaii6ktqlq4e2h13" xml:space="preserve">'''12/5'''
|2 1 -1&amp;gt;

1515.6413 cents

[[File:jid_12_5_pluck_adu_dr220.mp3]] [[:File:jid_12_5_pluck_adu_dr220.mp3|sound sample]]

The ratio between the 5th and 12th harmonics; one octave above [[6/5|6/5]].

[[category:interval]]
[[category:5-limit]]
[[category:todo:expand]]</text>
      <sha1>68yryrh8dqw6n2qjaii6ktqlq4e2h13</sha1>
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    <revision>
      <id>64579</id>
      <parentid>38986</parentid>
      <timestamp>2021-03-27T17:14:42Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
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      <text bytes="324" sha1="6ggvtqz0ktwwr3bt99fepjli3ct574f" xml:space="preserve">'''12/5'''
|2 1 -1&amp;gt;

1515.6413 cents

[[File:jid_12_5_pluck_adu_dr220.mp3]] [[:File:jid_12_5_pluck_adu_dr220.mp3|sound sample]]

The ratio between the 5th and 12th harmonics; one octave above [[6/5|6/5]].

[[category:interval]]
[[category:5-limit]]
[[category:todo:expand]]
[[Category:Pages with internal sound examples]]</text>
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    <revision>
      <id>81833</id>
      <parentid>64579</parentid>
      <timestamp>2021-12-12T05:13:33Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
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      <comment>Added infobox interval, cleanup</comment>
      <origin>81833</origin>
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      <text bytes="415" sha1="3hb5lw05xsucrl4vyhv4modqeqp5se2" xml:space="preserve">{{Infobox Interval
| Ratio = 12/5
| Monzo = 2 1 -1
| Cents = 1515.64129
| Name = classic minor tenth
| Color name = gu 10th, g10
| FJS name = m10&lt;sub&gt;5&lt;/sub&gt;
| Sound = jid_12_5_pluck_adu_dr220.mp3
}}

'''12/5''' is the '''classic minor tenth'''. It is one octave above the classic minor third [[6/5]].

[[Category:5-limit]]
[[Category:Tenth]]
[[Category:todo:expand]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>3hb5lw05xsucrl4vyhv4modqeqp5se2</sha1>
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    <revision>
      <id>81839</id>
      <parentid>81833</parentid>
      <timestamp>2021-12-12T05:44:04Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Order of color names</comment>
      <origin>81839</origin>
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      <text bytes="415" sha1="lvgomvtur7ux5y31oette6855qvi2my" xml:space="preserve">{{Infobox Interval
| Ratio = 12/5
| Monzo = 2 1 -1
| Cents = 1515.64129
| Name = classic minor tenth
| Color name = g10, gu 10th
| FJS name = m10&lt;sub&gt;5&lt;/sub&gt;
| Sound = jid_12_5_pluck_adu_dr220.mp3
}}

'''12/5''' is the '''classic minor tenth'''. It is one octave above the classic minor third [[6/5]].

[[Category:5-limit]]
[[Category:Tenth]]
[[Category:todo:expand]]
[[Category:Pages with internal sound examples]]</text>
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      <parentid>81839</parentid>
      <timestamp>2022-02-22T23:46:39Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
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      <text bytes="367" sha1="q11fniyi5yu26rci1l1oluhm1kudaga" xml:space="preserve">{{Infobox Interval
| Ratio = 12/5
| Monzo = 2 1 -1
| Cents = 1515.64129
| Name = classic minor tenth
| Color name = g10, gu 10th
| FJS name = m10&lt;sub&gt;5&lt;/sub&gt;
| Sound = jid_12_5_pluck_adu_dr220.mp3
}}

'''12/5''' is the '''classic minor tenth'''. It is one octave above the classic minor third [[6/5]].

[[Category:5-limit]]
[[Category:Tenth]]
[[Category:todo:expand]]</text>
      <sha1>q11fniyi5yu26rci1l1oluhm1kudaga</sha1>
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    <revision>
      <id>98093</id>
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      <timestamp>2022-10-25T14:02:38Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
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      <comment>Normalising usage of Infobox Interval</comment>
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      <text bytes="264" sha1="luxwc427sqa88ewvu2epqbga1oykff8" xml:space="preserve">{{Infobox Interval
| Name = classic minor tenth
| Color name = g10, gu 10th
| Sound = jid_12_5_pluck_adu_dr220.mp3
}}

'''12/5''' is the '''classic minor tenth'''. It is one octave above the classic minor third [[6/5]].

[[Category:Tenth]]
[[Category:todo:expand]]</text>
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    <revision>
      <id>170419</id>
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      <timestamp>2024-12-15T00:46:46Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Stub</comment>
      <origin>170419</origin>
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      <text bytes="249" sha1="gevc457yihlsfk6v6v2ladh8efgcnpr" xml:space="preserve">{{Infobox Interval
| Name = classic minor tenth
| Color name = g10, gu 10th
| Sound = jid_12_5_pluck_adu_dr220.mp3
}}

'''12/5''' is the '''classic minor tenth'''. It is one octave above the classic minor third [[6/5]].


{{stub}}
[[Category:Tenth]]</text>
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  <page>
    <title>12 5</title>
    <ns>0</ns>
    <id>130</id>
    <redirect title="12/5" />
    <revision>
      <id>130</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
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      <text bytes="18" sha1="p1n1cmuverchnw0xwngppx38xlx056s" xml:space="preserve">#REDIRECT [[12/5]]</text>
      <sha1>p1n1cmuverchnw0xwngppx38xlx056s</sha1>
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  </page>
  <page>
    <title>12/7</title>
    <ns>0</ns>
    <id>131</id>
    <revision>
      <id>131</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
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      <origin>131</origin>
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      <text bytes="1072" sha1="hvcwg3xggdkwt4s4d2wfax88w81pyby" xml:space="preserve">{| class="wikitable"
|-
| | [[File:ji_glyph_12_7.png|alt=ji glyph 12 7.png|136x120px|ji glyph 12 7.png]]
|-
| | JI Glyph for 12/7
|}

'''12/7'''
|-2 -1 0 1&amp;gt;

933.12909 cents

[[File:jid_12_7_pluck_adu_dr220.mp3]] [[:File:jid_12_7_pluck_adu_dr220.mp3|sound sample]]

In [[7-limit|7-limit]] [[Just_intonation|Just Intonation]], 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3|5/3]] by [[36/35|36/25]] -- the septimal quartertone -- a [[superparticular|superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16|27/16]] by [[64/63|64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7/4|7/4]], about 968.8¢ -- by [[49/48|49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6|7/6]].

See: [[Gallery_of_Just_Intervals|Gallery of Just Intervals]]</text>
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    <revision>
      <id>4590</id>
      <timestamp>2015-03-24T20:59:54Z</timestamp>
      <contributor>
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        <id>0</id>
      </contributor>
      <comment>**Imported revision 545216376 - Original comment: **</comment>
      <origin>4590</origin>
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      <format>text/x-wiki</format>
      <text bytes="5750" sha1="nk0obx7qsw618l7t8ptltytp04y8zxr" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:k9assassin|k9assassin]] and made on &lt;tt&gt;2015-03-24 20:59:54 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;545216376&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[image:ji glyph 12 7.png width="136" height="120" caption="JI Glyph for 12/7
"]]

**12/7**
|-2 -1 0 1&amp;gt;
933.12909 cents
[[media type="file" key="jid_12_7_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_12_7_pluck_adu_dr220.mp3|sound sample]]

In [[7-limit]] [[Just Intonation]], 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5_3|5/3]] by [[36_35|36/25]] -- the septimal quartertone -- a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27_16|27/16]] by [[64_63|64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7_4|7/4]], about 968.8¢ -- by [[49_48|49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7_6|7/6]].

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12_7&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:1:&amp;amp;lt;img src=&amp;amp;quot;/file/view/ji%20glyph%2012%207.png/545216352/136x120/ji%20glyph%2012%207.png&amp;amp;quot; alt=&amp;amp;quot;JI Glyph for 12/7&amp;amp;quot; title=&amp;amp;quot;JI Glyph for 12/7&amp;amp;quot; style=&amp;amp;quot;height: 120px; width: 136px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;table class="captionBox"&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="captionedImage"&amp;gt;&amp;lt;img src="/file/view/ji%20glyph%2012%207.png/545216352/136x120/ji%20glyph%2012%207.png" alt="ji glyph 12 7.png" title="ji glyph 12 7.png" style="height: 120px; width: 136px;" /&amp;gt;&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;tr&amp;gt;&amp;lt;td class="imageCaption"&amp;gt;JI Glyph for 12/7&amp;lt;/td&amp;gt;&amp;lt;/tr&amp;gt;&amp;lt;/table&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;12/7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|-2 -1 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
933.12909 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_12_7_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_12_7_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_12_7_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_12_7_pluck_adu_dr220.mp3/513194652/jid_12_7_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_12_7_pluck_adu_dr220.mp3/513194652/jid_12_7_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/36_35"&amp;gt;36/25&amp;lt;/a&amp;gt; -- the septimal quartertone -- a &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; interval of about 48.8¢. It differs from the Pythagorean major sixth of &amp;lt;a class="wiki_link" href="/27_16"&amp;gt;27/16&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/64_63"&amp;gt;64/63&amp;lt;/a&amp;gt; -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- &amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;, about 968.8¢ -- by &amp;lt;a class="wiki_link" href="/49_48"&amp;gt;49/48&amp;lt;/a&amp;gt; -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>nk0obx7qsw618l7t8ptltytp04y8zxr</sha1>
    </revision>
    <revision>
      <id>4591</id>
      <timestamp>2015-03-24T20:58:43Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;k9assassin</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 545216282 - Original comment: **</comment>
      <origin>4591</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5496" sha1="5pjtjulkr7p6bp17cn2pv1xtjao3zsm" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:k9assassin|k9assassin]] and made on &lt;tt&gt;2015-03-24 20:58:43 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;545216282&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;[[image:ji glyph 12 7.png width="113" height="100"]]

**12/7**
|-2 -1 0 1&amp;gt;
933.12909 cents
[[media type="file" key="jid_12_7_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_12_7_pluck_adu_dr220.mp3|sound sample]]

In [[7-limit]] [[Just Intonation]], 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5_3|5/3]] by [[36_35|36/25]] -- the septimal quartertone -- a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27_16|27/16]] by [[64_63|64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7_4|7/4]], about 968.8¢ -- by [[49_48|49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7_6|7/6]].

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12_7&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextLocalImageRule:1:&amp;amp;lt;img src=&amp;amp;quot;/file/view/ji%20glyph%2012%207.png/545216352/113x100/ji%20glyph%2012%207.png&amp;amp;quot; alt=&amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;&amp;amp;quot; style=&amp;amp;quot;height: 100px; width: 113px;&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;img src="/file/view/ji%20glyph%2012%207.png/545216352/113x100/ji%20glyph%2012%207.png" alt="ji glyph 12 7.png" title="ji glyph 12 7.png" style="height: 100px; width: 113px;" /&amp;gt;&amp;lt;!-- ws:end:WikiTextLocalImageRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;strong&amp;gt;12/7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|-2 -1 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
933.12909 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_12_7_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_12_7_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_12_7_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_12_7_pluck_adu_dr220.mp3/513194652/jid_12_7_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_12_7_pluck_adu_dr220.mp3/513194652/jid_12_7_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/36_35"&amp;gt;36/25&amp;lt;/a&amp;gt; -- the septimal quartertone -- a &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; interval of about 48.8¢. It differs from the Pythagorean major sixth of &amp;lt;a class="wiki_link" href="/27_16"&amp;gt;27/16&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/64_63"&amp;gt;64/63&amp;lt;/a&amp;gt; -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- &amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;, about 968.8¢ -- by &amp;lt;a class="wiki_link" href="/49_48"&amp;gt;49/48&amp;lt;/a&amp;gt; -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>5pjtjulkr7p6bp17cn2pv1xtjao3zsm</sha1>
    </revision>
    <revision>
      <id>4592</id>
      <timestamp>2014-06-07T15:41:45Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 513194716 - Original comment: **</comment>
      <origin>4592</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4903" sha1="c215grapd6m8awbis33r2osdjky1yg3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-06-07 15:41:45 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;513194716&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;**12/7**
|-2 -1 0 1&amp;gt;
933.12909 cents
[[media type="file" key="jid_12_7_pluck_adu_dr220.mp3" width="240" height="20"]] [[file:xenharmonic/jid_12_7_pluck_adu_dr220.mp3|sound sample]]

In [[7-limit]] [[Just Intonation]], 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5_3|5/3]] by [[36_35|36/25]] -- the septimal quartertone -- a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27_16|27/16]] by [[64_63|64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7_4|7/4]], about 968.8¢ -- by [[49_48|49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7_6|7/6]].

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12_7&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;strong&amp;gt;12/7&amp;lt;/strong&amp;gt;&amp;lt;br /&amp;gt;
|-2 -1 0 1&amp;amp;gt;&amp;lt;br /&amp;gt;
933.12909 cents&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_12_7_pluck_adu_dr220.mp3?h=20&amp;amp;amp;w=240&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;file&amp;amp;amp;quot; key=&amp;amp;amp;quot;jid_12_7_pluck_adu_dr220.mp3&amp;amp;amp;quot; width=&amp;amp;amp;quot;240&amp;amp;amp;quot; height=&amp;amp;amp;quot;20&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Local Media File&amp;amp;quot;height=&amp;amp;quot;20&amp;amp;quot; width=&amp;amp;quot;240&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_12_7_pluck_adu_dr220.mp3?file_extension=mp3&amp;amp;autostart=false&amp;amp;repeat=false&amp;amp;showdigits=true&amp;amp;showfsbutton=false&amp;amp;width=240&amp;amp;height=20"&amp;gt;&amp;lt;/embed&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt; &amp;lt;a href="http://xenharmonic.wikispaces.com/file/view/jid_12_7_pluck_adu_dr220.mp3/513194652/jid_12_7_pluck_adu_dr220.mp3" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/jid_12_7_pluck_adu_dr220.mp3/513194652/jid_12_7_pluck_adu_dr220.mp3');"&amp;gt;sound sample&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/36_35"&amp;gt;36/25&amp;lt;/a&amp;gt; -- the septimal quartertone -- a &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; interval of about 48.8¢. It differs from the Pythagorean major sixth of &amp;lt;a class="wiki_link" href="/27_16"&amp;gt;27/16&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/64_63"&amp;gt;64/63&amp;lt;/a&amp;gt; -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- &amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;, about 968.8¢ -- by &amp;lt;a class="wiki_link" href="/49_48"&amp;gt;49/48&amp;lt;/a&amp;gt; -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>c215grapd6m8awbis33r2osdjky1yg3</sha1>
    </revision>
    <revision>
      <id>4593</id>
      <timestamp>2011-09-29T18:25:05Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Andrew_Heathwaite</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 259800020 - Original comment: **</comment>
      <origin>4593</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3258" sha1="98yd74h50b7t2t2dkh4q8yqxn443vml" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on &lt;tt&gt;2011-09-29 18:25:05 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;259800020&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;In [[7-limit]] [[Just Intonation]], 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5_3|5/3]] by [[36_35|36/25]] -- the septimal quartertone -- a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27_16|27/16]] by [[64_63|64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7_4|7/4]], about 968.8¢ -- by [[49_48|49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7_6|7/6]].

See: [[Gallery of Just Intervals]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12_7&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;In &amp;lt;a class="wiki_link" href="/7-limit"&amp;gt;7-limit&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link" href="/Just%20Intonation"&amp;gt;Just Intonation&amp;lt;/a&amp;gt;, 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of &amp;lt;a class="wiki_link" href="/5_3"&amp;gt;5/3&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/36_35"&amp;gt;36/25&amp;lt;/a&amp;gt; -- the septimal quartertone -- a &amp;lt;a class="wiki_link" href="/superparticular"&amp;gt;superparticular&amp;lt;/a&amp;gt; interval of about 48.8¢. It differs from the Pythagorean major sixth of &amp;lt;a class="wiki_link" href="/27_16"&amp;gt;27/16&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/64_63"&amp;gt;64/63&amp;lt;/a&amp;gt; -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- &amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;, about 968.8¢ -- by &amp;lt;a class="wiki_link" href="/49_48"&amp;gt;49/48&amp;lt;/a&amp;gt; -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of &amp;lt;a class="wiki_link" href="/7_6"&amp;gt;7/6&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
See: &amp;lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&amp;gt;Gallery of Just Intervals&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>98yd74h50b7t2t2dkh4q8yqxn443vml</sha1>
    </revision>
    <revision>
      <id>37354</id>
      <parentid>131</parentid>
      <timestamp>2018-10-17T22:31:30Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>reworked: simplified links; introduces infobox interval</comment>
      <origin>37354</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="986" sha1="fx97g3py1gs1f3z7aqlvy50aqqt5nou" xml:space="preserve">{{Infobox Interval
| Icon = [[File:ji_glyph_12_7.png|124px]] &lt;small&gt;&lt;br/&gt;[[JI glyphs|JI glyph]]&lt;/small&gt;
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = septimal supermajor sixth
| Sound = jid_12_7_pluck_adu_dr220.mp3
| Color name =
}}

In [[7-limit]] [[Just intonation|Just Intonation]], 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/25]] -- the septimal quartertone -- a [[superparticular|superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7/4]], about 968.8¢ -- by [[49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

See: [[Gallery of Just Intervals]]</text>
      <sha1>fx97g3py1gs1f3z7aqlvy50aqqt5nou</sha1>
    </revision>
    <revision>
      <id>37372</id>
      <parentid>37354</parentid>
      <timestamp>2018-10-18T07:15:22Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>+cat</comment>
      <origin>37372</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1077" sha1="qam2mbww7x0efox44b6tuugp5gi20jy" xml:space="preserve">{{Infobox Interval
| Icon = [[File:ji_glyph_12_7.png|124px]] &lt;small&gt;&lt;br/&gt;[[JI glyphs|JI glyph]]&lt;/small&gt;
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = septimal supermajor sixth
| Sound = jid_12_7_pluck_adu_dr220.mp3
| Color name =
}}

In [[7-limit]] [[Just intonation|Just Intonation]], 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/25]] -- the septimal quartertone -- a [[superparticular|superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7/4]], about 968.8¢ -- by [[49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

See: [[Gallery of Just Intervals]]

[[Category:7-limit]]]]
[[Category:interval]]
[[Category:just]]
[[Category:just interval]]</text>
      <sha1>qam2mbww7x0efox44b6tuugp5gi20jy</sha1>
    </revision>
    <revision>
      <id>37517</id>
      <parentid>37372</parentid>
      <timestamp>2018-10-22T14:12:54Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>updated interval infobox</comment>
      <origin>37517</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1038" sha1="dphtlzf7a7x8si5m6r1verxaj42xfms" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_12_7.png|x48px]]
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = septimal supermajor sixth
| Sound = jid_12_7_pluck_adu_dr220.mp3
| Color name =
}}

In [[7-limit]] [[Just intonation|Just Intonation]], 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/25]] -- the septimal quartertone -- a [[superparticular|superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7/4]], about 968.8¢ -- by [[49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

See: [[Gallery of Just Intervals]]

[[Category:7-limit]]]]
[[Category:interval]]
[[Category:just]]
[[Category:just interval]]</text>
      <sha1>dphtlzf7a7x8si5m6r1verxaj42xfms</sha1>
    </revision>
    <revision>
      <id>37587</id>
      <parentid>37517</parentid>
      <timestamp>2018-10-23T03:27:11Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <origin>37587</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1049" sha1="24lsqcvxk4gm2x0bkt6vjsgqqgv0hh4" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_12_7.png|x48px]]
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = septimal supermajor sixth
| Color name = r6, ru 6th
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just intonation|Just Intonation]], 12/7 is the septimal supermajor sixth of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/25]] -- the septimal quartertone -- a [[superparticular|superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7/4]], about 968.8¢ -- by [[49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

See: [[Gallery of Just Intervals]]

[[Category:7-limit]]]]
[[Category:interval]]
[[Category:just]]
[[Category:just interval]]</text>
      <sha1>24lsqcvxk4gm2x0bkt6vjsgqqgv0hh4</sha1>
    </revision>
    <revision>
      <id>37645</id>
      <parentid>37587</parentid>
      <timestamp>2018-10-23T17:10:40Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>lemma bold</comment>
      <origin>37645</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1036" sha1="p2pcf3is6mkwvxbw4thkj0slmovb0li" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_12_7.png|x48px]]
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = septimal supermajor sixth
| Color name = r6, ru 6th
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''12/7''' is the '''septimal supermajor sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/25]] -- the septimal quartertone -- a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7/4]], about 968.8¢ -- by [[49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

:''See also [[Gallery of Just Intervals]]''

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Just]]
[[Category:Just interval]]</text>
      <sha1>p2pcf3is6mkwvxbw4thkj0slmovb0li</sha1>
    </revision>
    <revision>
      <id>45994</id>
      <parentid>37645</parentid>
      <timestamp>2020-06-07T14:56:45Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>-cat</comment>
      <origin>45994</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1018" sha1="o38twkcvyeb3tv2tpda5unw04l5r9mq" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_12_7.png|x48px]]
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = septimal supermajor sixth
| Color name = r6, ru 6th
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''12/7''' is the '''septimal supermajor sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/25]] -- the septimal quartertone -- a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7/4]], about 968.8¢ -- by [[49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

:''See also [[Gallery of Just Intervals]]''

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Just interval]]</text>
      <sha1>o38twkcvyeb3tv2tpda5unw04l5r9mq</sha1>
    </revision>
    <revision>
      <id>47083</id>
      <parentid>45994</parentid>
      <timestamp>2020-06-18T06:47:23Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>exchanged sound example for better comparison with [[42/25]]</comment>
      <origin>47083</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1192" sha1="fz8em8b521ramdl18qd5nizsrpocqsn" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_12_7.png|x48px]]
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = septimal supermajor sixth
| Color name = r6, ru 6th
| Sound = Ji-12-7-csound-foscil-220hz.mp3
}}

In [[7-limit]] [[Just Intonation]], '''12/7''' is the '''septimal supermajor sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/25]] -- the septimal quartertone -- a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7/4]], about 968.8¢ -- by [[49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

[[File:jid_12_7_pluck_adu_dr220.mp3]]

== See also == 

* [[7/6]] -- its [[octave complement]]
* [[42/25]] -- a lower septimal major sixth
* [[Gallery of Just Intervals]]

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Just interval]]</text>
      <sha1>fz8em8b521ramdl18qd5nizsrpocqsn</sha1>
    </revision>
    <revision>
      <id>47104</id>
      <parentid>47083</parentid>
      <timestamp>2020-06-18T12:06:46Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Correction</comment>
      <origin>47104</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1192" sha1="oreyz7hdgacw7a29o8zkd7kco4zv0cv" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_12_7.png|x48px]]
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = septimal supermajor sixth
| Color name = r6, ru 6th
| Sound = Ji-12-7-csound-foscil-220hz.mp3
}}

In [[7-limit]] [[Just Intonation]], '''12/7''' is the '''septimal supermajor sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] -- the septimal quartertone -- a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7/4]], about 968.8¢ -- by [[49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

[[File:jid_12_7_pluck_adu_dr220.mp3]]

== See also == 

* [[7/6]] -- its [[octave complement]]
* [[42/25]] -- a lower septimal major sixth
* [[Gallery of Just Intervals]]

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Just interval]]</text>
      <sha1>oreyz7hdgacw7a29o8zkd7kco4zv0cv</sha1>
    </revision>
    <revision>
      <id>48389</id>
      <parentid>47104</parentid>
      <timestamp>2020-09-01T07:25:32Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Now that 42/25 is given a more appropriate name, the "septimal major sixth" name can be used for this</comment>
      <origin>48389</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1343" sha1="do01da4kznezlqqjhfvkztli8lne01n" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_12_7.png|x48px]]
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = supermajor sixth, &lt;br&gt;septimal major sixth
| Color name = r6, ru 6th
| FJS name = M6&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] -- the septimal quartertone -- a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7/4]], about 968.8¢ -- by [[49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 

* [[7/6]] -- its [[octave complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal major sixth|Septimal major sixth - Wikipedia]]
* [[File:Ji-12-7-csound-foscil-220hz.mp3]] an alternative sound sample

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Just interval]]</text>
      <sha1>do01da4kznezlqqjhfvkztli8lne01n</sha1>
    </revision>
    <revision>
      <id>48561</id>
      <parentid>48389</parentid>
      <timestamp>2020-09-05T03:48:06Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* See also */</comment>
      <origin>48561</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1363" sha1="5t91so1kge1s3ys49bbmnl0nvsogitl" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_12_7.png|x48px]]
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = supermajor sixth, &lt;br&gt;septimal major sixth
| Color name = r6, ru 6th
| FJS name = M6&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] -- the septimal quartertone -- a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7/4]], about 968.8¢ -- by [[49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 

* [[7/6]] -- its [[octave complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal major sixth|Septimal major sixth - Wikipedia]]
* [[File:Ji-12-7-csound-foscil-220hz.mp3]] an alternative sound sample

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Just interval]]
[[Category:Over-7]]</text>
      <sha1>5t91so1kge1s3ys49bbmnl0nvsogitl</sha1>
    </revision>
    <revision>
      <id>49345</id>
      <parentid>48561</parentid>
      <timestamp>2020-09-13T11:20:57Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* See also */ transformed included file into a file link, the alternative would have been to add a file link info</comment>
      <origin>49345</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1364" sha1="2zcsvow13ziqfl18y6i6pvwrwgli1fn" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_12_7.png|x48px]]
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = supermajor sixth, &lt;br&gt;septimal major sixth
| Color name = r6, ru 6th
| FJS name = M6&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] -- the septimal quartertone -- a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] -- Archytas' comma -- about 27.3¢. And finally, it differs from the harmonic seventh -- [[7/4]], about 968.8¢ -- by [[49/48]] -- the large septimal diesis or slendro diesis -- about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 

* [[7/6]] -- its [[octave complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal major sixth|Septimal major sixth - Wikipedia]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] an alternative sound sample

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Just interval]]
[[Category:Over-7]]</text>
      <sha1>2zcsvow13ziqfl18y6i6pvwrwgli1fn</sha1>
    </revision>
    <revision>
      <id>49854</id>
      <parentid>49345</parentid>
      <timestamp>2020-09-19T14:13:18Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Cleanup</comment>
      <origin>49854</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1357" sha1="diczsmis2j6svuqhhvjx7s751q8kphx" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_12_7.png|x48px]]
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = supermajor sixth, &lt;br&gt;septimal major sixth
| Color name = r6, ru 6th
| FJS name = M6&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 

* [[7/6]] – its [[octave complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal major sixth|Septimal major sixth - Wikipedia]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Just interval]]
[[Category:Over-7]]</text>
      <sha1>diczsmis2j6svuqhhvjx7s751q8kphx</sha1>
    </revision>
    <revision>
      <id>64580</id>
      <parentid>49854</parentid>
      <timestamp>2021-03-27T17:14:57Z</timestamp>
      <contributor>
        <username>Userminusone</username>
        <id>273</id>
      </contributor>
      <origin>64580</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1405" sha1="61d9eeiejajlbf1hzoaqvjlxnjcsrz4" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_12_7.png|x48px]]
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = supermajor sixth, &lt;br&gt;septimal major sixth
| Color name = r6, ru 6th
| FJS name = M6&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 

* [[7/6]] – its [[octave complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal major sixth|Septimal major sixth - Wikipedia]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Just interval]]
[[Category:Over-7]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>61d9eeiejajlbf1hzoaqvjlxnjcsrz4</sha1>
    </revision>
    <revision>
      <id>66275</id>
      <parentid>64580</parentid>
      <timestamp>2021-04-12T02:40:37Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>66275</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1359" sha1="6fv59wmwmyhicyiofh7zjnbc8iialhb" xml:space="preserve">{{Infobox Interval
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = supermajor sixth, &lt;br&gt;septimal major sixth
| Color name = r6, ru 6th
| FJS name = M6&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 

* [[7/6]] – its [[octave complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal major sixth|Septimal major sixth - Wikipedia]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Just interval]]
[[Category:Over-7]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>6fv59wmwmyhicyiofh7zjnbc8iialhb</sha1>
    </revision>
    <revision>
      <id>66333</id>
      <parentid>66275</parentid>
      <timestamp>2021-04-12T06:53:19Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>Undo revision 66275 by [[Special:Contributions/IlL|IlL]] ([[User talk:IlL|talk]])</comment>
      <origin>66333</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1405" sha1="61d9eeiejajlbf1hzoaqvjlxnjcsrz4" xml:space="preserve">{{Infobox Interval
| JI glyph = [[File:ji_glyph_12_7.png|x48px]]
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = supermajor sixth, &lt;br&gt;septimal major sixth
| Color name = r6, ru 6th
| FJS name = M6&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 

* [[7/6]] – its [[octave complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal major sixth|Septimal major sixth - Wikipedia]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Just interval]]
[[Category:Over-7]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>61d9eeiejajlbf1hzoaqvjlxnjcsrz4</sha1>
    </revision>
    <revision>
      <id>66362</id>
      <parentid>66333</parentid>
      <timestamp>2021-04-12T09:11:27Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Undo revision 66333 by [[Special:Contributions/Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]])</comment>
      <origin>66362</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1359" sha1="6fv59wmwmyhicyiofh7zjnbc8iialhb" xml:space="preserve">{{Infobox Interval
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = supermajor sixth, &lt;br&gt;septimal major sixth
| Color name = r6, ru 6th
| FJS name = M6&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}

In [[7-limit]] [[Just Intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 

* [[7/6]] – its [[octave complement]]
* [[Gallery of Just Intervals]]
* [[Wikipedia:Septimal major sixth|Septimal major sixth - Wikipedia]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:7-limit]]
[[Category:Interval]]
[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Just interval]]
[[Category:Over-7]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>6fv59wmwmyhicyiofh7zjnbc8iialhb</sha1>
    </revision>
    <revision>
      <id>81834</id>
      <parentid>66362</parentid>
      <timestamp>2021-12-12T05:20:02Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Added Wikipedia box, misc. edits, categories</comment>
      <origin>81834</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1275" sha1="5ifshds3ez4iz8utpfrhi6t7desfwhn" xml:space="preserve">{{Infobox Interval
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = supermajor sixth, &lt;br&gt;septimal major sixth
| Color name = r6, ru 6th
| FJS name = M6&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}
{{Wikipedia|Septimal major sixth}}

In [[7-limit]] [[just intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 

* [[7/6]] – its [[octave complement]]
* [[Gallery of Just Intervals]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:7-limit]]
[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Over-7]]
[[Category:Pages with internal sound examples]]</text>
      <sha1>5ifshds3ez4iz8utpfrhi6t7desfwhn</sha1>
    </revision>
    <revision>
      <id>86313</id>
      <parentid>81834</parentid>
      <timestamp>2022-02-22T23:46:39Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Pages with internal sound examples]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>86313</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1227" sha1="g0br3ayf9ehhnetgl2xvvl718snpc7t" xml:space="preserve">{{Infobox Interval
| Ratio = 12/7
| Monzo = -2 -1 0 1
| Cents = 933.12909 
| Name = supermajor sixth, &lt;br&gt;septimal major sixth
| Color name = r6, ru 6th
| FJS name = M6&lt;sub&gt;7&lt;/sub&gt;
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}
{{Wikipedia|Septimal major sixth}}

In [[7-limit]] [[just intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 

* [[7/6]] – its [[octave complement]]
* [[Gallery of Just Intervals]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:7-limit]]
[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Over-7]]</text>
      <sha1>g0br3ayf9ehhnetgl2xvvl718snpc7t</sha1>
    </revision>
    <revision>
      <id>98031</id>
      <parentid>86313</parentid>
      <timestamp>2022-10-25T12:30:46Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Normalising usage of Infobox Interval</comment>
      <origin>98031</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1118" sha1="roprtilbgv4ssmhozj4gxi8ymilods2" xml:space="preserve">{{Infobox Interval
| Name = supermajor sixth, septimal major sixth
| Color name = r6, ru 6th
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}
{{Wikipedia|Septimal major sixth}}

In [[7-limit]] [[just intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 

* [[7/6]] – its [[octave complement]]
* [[Gallery of Just Intervals]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Over-7]]</text>
      <sha1>roprtilbgv4ssmhozj4gxi8ymilods2</sha1>
    </revision>
    <revision>
      <id>122001</id>
      <parentid>98031</parentid>
      <timestamp>2023-08-14T11:27:02Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+twelfth complement</comment>
      <origin>122001</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1161" sha1="dj0dg4d58q1ocbyj4wqrc4d4wke11rh" xml:space="preserve">{{Infobox Interval
| Name = supermajor sixth, septimal major sixth
| Color name = r6, ru 6th
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}
{{Wikipedia| Septimal major sixth }}

In [[7-limit]] [[just intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 
* [[7/6]] – its [[octave complement]]
* [[12/7]] – its [[twelfth complement]]
* [[Gallery of Just Intervals]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Over-7]]</text>
      <sha1>dj0dg4d58q1ocbyj4wqrc4d4wke11rh</sha1>
    </revision>
    <revision>
      <id>122002</id>
      <parentid>122001</parentid>
      <timestamp>2023-08-14T11:27:20Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Fix links</comment>
      <origin>122002</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1160" sha1="64qpem4rg732w8j4tvi89kc5tshxlmv" xml:space="preserve">{{Infobox Interval
| Name = supermajor sixth, septimal major sixth
| Color name = r6, ru 6th
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}
{{Wikipedia| Septimal major sixth }}

In [[7-limit]] [[just intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 
* [[7/6]] – its [[octave complement]]
* [[7/4]] – its [[twelfth complement]]
* [[Gallery of just intervals]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Over-7]]</text>
      <sha1>64qpem4rg732w8j4tvi89kc5tshxlmv</sha1>
    </revision>
    <revision>
      <id>157035</id>
      <parentid>122002</parentid>
      <timestamp>2024-09-30T08:43:09Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Recategorize</comment>
      <origin>157035</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1170" sha1="2ifjmf3hu5m5l6go19fiugkyuxeolw0" xml:space="preserve">{{Infobox Interval
| Name = supermajor sixth, septimal major sixth
| Color name = r6, ru 6th
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}
{{Wikipedia| Septimal major sixth }}

In [[7-limit]] [[just intonation]], '''12/7''' is the '''supermajor sixth''' or '''septimal major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 
* [[7/6]] – its [[octave complement]]
* [[7/4]] – its [[twelfth complement]]
* [[Gallery of just intervals]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Over-7 intervals]]</text>
      <sha1>2ifjmf3hu5m5l6go19fiugkyuxeolw0</sha1>
    </revision>
    <revision>
      <id>192134</id>
      <parentid>157035</parentid>
      <timestamp>2025-04-14T16:59:17Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>192134</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1151" sha1="klg3y2jrj88f4jcykffa0n82mqmdns5" xml:space="preserve">{{Infobox Interval
| Name = supermajor sixth, septimal major sixth
| Color name = r6, ru 6th
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}
{{Wikipedia| Septimal major sixth }}

In [[7-limit]] [[just intonation]], '''12/7''' is the '''septimal (super)major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].

== See also == 
* [[7/6]] – its [[octave complement]]
* [[7/4]] – its [[twelfth complement]]
* [[Gallery of just intervals]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Over-7 intervals]]</text>
      <sha1>klg3y2jrj88f4jcykffa0n82mqmdns5</sha1>
    </revision>
    <revision>
      <id>215891</id>
      <parentid>192134</parentid>
      <timestamp>2025-11-03T07:42:53Z</timestamp>
      <contributor>
        <username>Pailiaq</username>
        <id>507</id>
      </contributor>
      <origin>215891</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1210" sha1="i3ewb7sh9krr1ullerxacprvt7x67b1" xml:space="preserve">{{Infobox Interval
| Name = supermajor sixth, septimal major sixth
| Color name = r6, ru 6th
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}
{{Wikipedia| Septimal major sixth }}

In [[7-limit]] [[just intonation]], '''12/7''' is the '''septimal (super)major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].
== Approximation ==
 {{Interval_Edo_Approximation | 12/7}}

== See also == 
* [[7/6]] – its [[octave complement]]
* [[7/4]] – its [[twelfth complement]]
* [[Gallery of just intervals]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Over-7 intervals]]</text>
      <sha1>i3ewb7sh9krr1ullerxacprvt7x67b1</sha1>
    </revision>
    <revision>
      <id>216095</id>
      <parentid>215891</parentid>
      <timestamp>2025-11-03T13:14:43Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Text replacement - " {{Interval_Edo_Approximation | " to "{{Interval edo approximation|"</comment>
      <origin>216095</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1207" sha1="0f3d1g5o0i8nhasxc8spqtgl6rq88jb" xml:space="preserve">{{Infobox Interval
| Name = supermajor sixth, septimal major sixth
| Color name = r6, ru 6th
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}
{{Wikipedia| Septimal major sixth }}

In [[7-limit]] [[just intonation]], '''12/7''' is the '''septimal (super)major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as 4:5:7:9:12. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].
== Approximation ==
{{Interval edo approximation|12/7}}

== See also == 
* [[7/6]] – its [[octave complement]]
* [[7/4]] – its [[twelfth complement]]
* [[Gallery of just intervals]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Over-7 intervals]]</text>
      <sha1>0f3d1g5o0i8nhasxc8spqtgl6rq88jb</sha1>
    </revision>
    <revision>
      <id>219709</id>
      <parentid>216095</parentid>
      <timestamp>2025-12-25T08:39:45Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>less obscure chord example; 4:5:7:9:12 is more often written as 4:5:6:7:9</comment>
      <origin>219709</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1228" sha1="sf7wn4jsg67jfi4he8cgwp3zl6qx100" xml:space="preserve">{{Infobox Interval
| Name = supermajor sixth, septimal major sixth
| Color name = r6, ru 6th
| Sound = jid_12_7_pluck_adu_dr220.mp3
}}
{{Wikipedia| Septimal major sixth }}

In [[7-limit]] [[just intonation]], '''12/7''' is the '''septimal (super)major sixth''' of about 933.1¢. It represents the interval between the 12th and 7th harmonics and appears in chords such as [[70:84:105:120|1/(12:10:8:7)]]. It differs from the 5-limit major sixth of [[5/3]] by [[36/35]] – the septimal quartertone – a [[superparticular]] interval of about 48.8¢. It differs from the Pythagorean major sixth of [[27/16]] by [[64/63]] – Archytas' comma – about 27.3¢. And finally, it differs from the harmonic seventh of [[7/4]] by [[49/48]] – the large septimal diesis or slendro diesis – about 35.7¢. 12/7 is the inversion of the septimal subminor third of [[7/6]].
== Approximation ==
{{Interval edo approximation|12/7}}

== See also == 
* [[7/6]] – its [[octave complement]]
* [[7/4]] – its [[twelfth complement]]
* [[Gallery of just intervals]]
* [[:File:Ji-12-7-csound-foscil-220hz.mp3]] – an alternative sound sample

[[Category:Sixth]]
[[Category:Major sixth]]
[[Category:Supermajor sixth]]
[[Category:Over-7 intervals]]</text>
      <sha1>sf7wn4jsg67jfi4he8cgwp3zl6qx100</sha1>
    </revision>
  </page>
  <page>
    <title>12 7</title>
    <ns>0</ns>
    <id>132</id>
    <redirect title="12/7" />
    <revision>
      <id>132</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <text bytes="18" sha1="8l8l0oui9a216or0c40bk78qh5yy4d1" xml:space="preserve">#REDIRECT [[12/7]]</text>
      <sha1>8l8l0oui9a216or0c40bk78qh5yy4d1</sha1>
    </revision>
  </page>
  <page>
    <title>12edf</title>
    <ns>0</ns>
    <id>133</id>
    <revision>
      <id>133</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>133</origin>
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      <text bytes="379" sha1="pns0rlaob0m8cprs2iefubz4mbb5jz2" xml:space="preserve">=12 ED 3/2=
|

0: 1/1 0.000 unison, perfect prime

1: 58.496 cents 58.496

2: 116.993 cents 116.993

3: 175.489 cents 175.489

4: 233.985 cents 233.985

5: 292.481 cents 292.481

6: 350.978 cents 350.978

7: 409.474 cents 409.474

8: 467.970 cents 467.970

9: 526.466 cents 526.466

10: 584.963 cents 584.963

11: 643.459 cents 643.459

12: 3/2 701.955 perfect fifth

Lookalikes:</text>
      <sha1>pns0rlaob0m8cprs2iefubz4mbb5jz2</sha1>
    </revision>
    <revision>
      <id>4594</id>
      <timestamp>2011-12-26T16:14:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;Kosmorsky</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 288516036 - Original comment: **</comment>
      <origin>4594</origin>
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      <text bytes="2321" sha1="ccuo8291ql962gazmt4wp97g7p6t4w0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:Kosmorsky|Kosmorsky]] and made on &lt;tt&gt;2011-12-26 16:14:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288516036&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=12 ED 3/2= 

|
0: 1/1 0.000 unison, perfect prime
1: 58.496 cents 58.496
2: 116.993 cents 116.993
3: 175.489 cents 175.489
4: 233.985 cents 233.985
5: 292.481 cents 292.481
6: 350.978 cents 350.978
7: 409.474 cents 409.474
8: 467.970 cents 467.970
9: 526.466 cents 526.466
10: 584.963 cents 584.963
11: 643.459 cents 643.459
12: 3/2 701.955 perfect fifth

Lookalikes:&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edf&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="x12 ED 3/2"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;12 ED 3/2&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
|&amp;lt;br /&amp;gt;
0: 1/1 0.000 unison, perfect prime&amp;lt;br /&amp;gt;
1: 58.496 cents 58.496&amp;lt;br /&amp;gt;
2: 116.993 cents 116.993&amp;lt;br /&amp;gt;
3: 175.489 cents 175.489&amp;lt;br /&amp;gt;
4: 233.985 cents 233.985&amp;lt;br /&amp;gt;
5: 292.481 cents 292.481&amp;lt;br /&amp;gt;
6: 350.978 cents 350.978&amp;lt;br /&amp;gt;
7: 409.474 cents 409.474&amp;lt;br /&amp;gt;
8: 467.970 cents 467.970&amp;lt;br /&amp;gt;
9: 526.466 cents 526.466&amp;lt;br /&amp;gt;
10: 584.963 cents 584.963&amp;lt;br /&amp;gt;
11: 643.459 cents 643.459&amp;lt;br /&amp;gt;
12: 3/2 701.955 perfect fifth&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Lookalikes:&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>40045</id>
      <parentid>133</parentid>
      <timestamp>2019-02-06T00:12:39Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
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      <text bytes="1037" sha1="ka8e6yph3k54ex4jbazmmwyocwf25f0" xml:space="preserve">'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.4963 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
| | 0
| | 0.0000
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.4963
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.4888
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.4813
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.4738
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.4663
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| | 
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|}

[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>ka8e6yph3k54ex4jbazmmwyocwf25f0</sha1>
    </revision>
    <revision>
      <id>40721</id>
      <parentid>40045</parentid>
      <timestamp>2019-03-08T21:03:57Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>40721</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1383" sha1="o748qbgv0brc7dcqzxtzmd0vf9olf22" xml:space="preserve">'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.4963 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
| | 0
| | 0.0000
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.4963
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.4888
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.4813
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.4738
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.4663
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| |13/9
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}

[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>o748qbgv0brc7dcqzxtzmd0vf9olf22</sha1>
    </revision>
    <revision>
      <id>42275</id>
      <parentid>40721</parentid>
      <timestamp>2019-08-19T22:32:04Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>42275</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1384" sha1="gsq4r84oq0wnvl2nqdwevhqkldzm6jg" xml:space="preserve">'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.4963 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
|  colspan="2"| 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.4963
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.4888
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.4813
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.4738
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.4663
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| |13/9
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}

[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>gsq4r84oq0wnvl2nqdwevhqkldzm6jg</sha1>
    </revision>
    <revision>
      <id>71157</id>
      <parentid>42275</parentid>
      <timestamp>2021-05-31T02:21:51Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>71157</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1384" sha1="fydyfjnijryx8asexbslle63tcjl6wk" xml:space="preserve">'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.4963 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.4963
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.4888
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.4813
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.4738
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.4663
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| |13/9
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}

[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>fydyfjnijryx8asexbslle63tcjl6wk</sha1>
    </revision>
    <revision>
      <id>96597</id>
      <parentid>71157</parentid>
      <timestamp>2022-10-05T18:35:30Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added</comment>
      <origin>96597</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1399" sha1="lcu6jhq7dp5qon8f0z6zsqwffcujd7w" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.4963 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.4963
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.4888
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.4813
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.4738
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.4663
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| |13/9
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}

[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>lcu6jhq7dp5qon8f0z6zsqwffcujd7w</sha1>
    </revision>
    <revision>
      <id>105843</id>
      <parentid>96597</parentid>
      <timestamp>2023-03-20T00:50:06Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>105843</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5892" sha1="86b8n67mckzwx6dz5vk4ttngbv7mcca" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| |13/9
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 57.14286 cents (4\7/12 = 1\21) to 60 cents (3\5/12 = 1\20)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7 || || || || || || ||57.1429||
|-
| || ||  || || || ||27\47||57.4468 ||
|-
|  || ||  || || ||23\40|| ||57.5000||
|-
| || ||  || || ||  ||42\73||57.53425||
|-
| ||  || || || 19\33|| || || 57.{{Overline|57}}||
|-
| || || || || || ||53\92||57.6087||
|-
| || || || || ||34\59|| ||57.6271||
|-
| || || || || || ||49\85||57.6471||
|-
| || || ||15\26|| || || ||57.6923||
|-
| || || || || || ||56\97||57.7320||
|-
| || || || || ||41\71|| ||57.7465||
|-
| || || || || || ||67\116||57.7586||
|-
| || || || ||26\45|| || ||57.{{Overline|7}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||57.7982||
|-
| || || || || ||37\64|| ||57.8125||
|-
| || || || || || ||48\83||57.8313||
|-
| || ||11\19|| || || || ||57.8947||
|-
| || || || || || ||51\88||57.9{{Overline|54}}||
|-
| || || || || ||40\69|| ||57.9710||
|-
| || || || || || ||69\119||57.9832||
|-
| || || || ||29\50|| || ||58.000||
|-
| || || || || || ||76\131||58.0153||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||58.0247||
|-
| || || || || || ||65\112||58.0357||
|-
| || || ||18\31|| || || ||58.0645||[[Meantone]] is in this region
|-
| || || || || || ||61\105||58.0952||
|-
| || || || || ||43\74|| ||58.{{Overline|108}}||
|-
| || || || || || ||68\117||58.1967||
|-
| || || || ||25\43|| || ||58.1935||
|-
| || || || || || ||57\98||58.1633||
|-
| || || || || ||32\55|| ||58.{{Overline|18}}||
|-
| || || || || || ||39\67||58.2090||The generator closest to a just [[7/5]] for EDOs less than 2400
|-
| ||7\12|| || || || || ||58.{{Overline|3}}||
|-
| || || || || || ||38\65||58.4615||
|-
| || || || || ||31\53|| ||58.4906||The generator closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||58,5106||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||58.5366||
|-
| || || || || || ||65\111||58.{{Overline|558}}||
|-
| || || || || ||41\70|| ||58.5714||
|-
| || || || || || ||58\99||58.{{Overline|58}}||
|-
| || || ||17\29|| || || ||58.6207||
|-
| || || || || || ||61\104||58.65385||
|-
| || || || || ||44\75|| ||58.{{Overline|6}}||
|-
| || || || || || ||71\121||58.6777||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||58.69565||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||58.7155||
|-
| || || || || ||37\63|| ||58.7302||
|-
| || || || || || ||47\80||58.7500||
|-
| || ||10\17|| || || || ||58.8235||
|-
| || || || || || ||43\73||58.9041||
|-
| || || || || ||33\56|| ||58.9286||
|-
| || || || || || ||56\95||58.9474||
|-
| || || || ||23\39|| || ||58.9744||
|-
| || || || || || ||59\100||59.0000||
|-
| || || || || ||36\61|| ||59.0164||
|-
| || || || || || ||49\83||59.0361||
|-
| || || ||13\22|| || || ||59.{{Overline|09}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||59.1549||
|-
| || || || || ||29\49|| ||59.1837||
|-
| || || || || || ||45\76||59.2105||
|-
| || || || ||16\27|| || ||59.{{Overline|259}}||
|-
| || || || || || ||35\59||59.3220||
|-
| || || || || ||19\32|| ||59.3750||
|-
| || || || || || ||22\37||59.{{Overline|459}}||
|-
|3\5|| || || || || || ||60.0000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>86b8n67mckzwx6dz5vk4ttngbv7mcca</sha1>
    </revision>
    <revision>
      <id>106034</id>
      <parentid>105843</parentid>
      <timestamp>2023-03-22T18:02:24Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Scale tree */</comment>
      <origin>106034</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5888" sha1="slch8i0qhv3hpzhqgdpafsfgdicndt2" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| |13/9
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 57.14286 cents (4\7/12 = 1\21) to 60 cents (3\5/12 = 1\20)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7 || || || || || || ||57.1429||
|-
| || ||  || || || ||27\47||57.4468 ||
|-
|  || ||  || || ||23\40|| ||57.5000||
|-
| || ||  || || ||  ||42\73||57.53425||
|-
| ||  || || || 19\33|| || || 57.{{Overline|57}}||
|-
| || || || || || ||53\92||57.6087||
|-
| || || || || ||34\59|| ||57.6271||
|-
| || || || || || ||49\85||57.6471||
|-
| || || ||15\26|| || || ||57.6923||
|-
| || || || || || ||56\97||57.7320||
|-
| || || || || ||41\71|| ||57.7465||
|-
| || || || || || ||67\116||57.7586||
|-
| || || || ||26\45|| || ||57.{{Overline|7}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||57.7982||
|-
| || || || || ||37\64|| ||57.8125||
|-
| || || || || || ||48\83||57.8313||
|-
| || ||11\19|| || || || ||57.8947||
|-
| || || || || || ||51\88||57.9{{Overline|54}}||
|-
| || || || || ||40\69|| ||57.9710||
|-
| || || || || || ||69\119||57.9832||
|-
| || || || ||29\50|| || ||58.000||
|-
| || || || || || ||76\131||58.0153||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||58.0247||
|-
| || || || || || ||65\112||58.0357||
|-
| || || ||18\31|| || || ||58.0645||[[Meantone]] is in this region
|-
| || || || || || ||61\105||58.0952||
|-
| || || || || ||43\74|| ||58.{{Overline|108}}||
|-
| || || || || || ||68\117||58.1967||
|-
| || || || ||25\43|| || ||58.1935||
|-
| || || || || || ||57\98||58.1633||
|-
| || || || || ||32\55|| ||58.{{Overline|18}}||
|-
| || || || || || ||39\67||58.2090||The generator closest to a just [[7/5]] for EDOs less than 2400
|-
| ||7\12|| || || || || ||58.{{Overline|3}}||
|-
| || || || || || ||38\65||58.4615||
|-
| || || || || ||31\53|| ||58.4906||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||58,5106||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||58.5366||
|-
| || || || || || ||65\111||58.{{Overline|558}}||
|-
| || || || || ||41\70|| ||58.5714||
|-
| || || || || || ||58\99||58.{{Overline|58}}||
|-
| || || ||17\29|| || || ||58.6207||
|-
| || || || || || ||61\104||58.65385||
|-
| || || || || ||44\75|| ||58.{{Overline|6}}||
|-
| || || || || || ||71\121||58.6777||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||58.69565||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||58.7155||
|-
| || || || || ||37\63|| ||58.7302||
|-
| || || || || || ||47\80||58.7500||
|-
| || ||10\17|| || || || ||58.8235||
|-
| || || || || || ||43\73||58.9041||
|-
| || || || || ||33\56|| ||58.9286||
|-
| || || || || || ||56\95||58.9474||
|-
| || || || ||23\39|| || ||58.9744||
|-
| || || || || || ||59\100||59.0000||
|-
| || || || || ||36\61|| ||59.0164||
|-
| || || || || || ||49\83||59.0361||
|-
| || || ||13\22|| || || ||59.{{Overline|09}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||59.1549||
|-
| || || || || ||29\49|| ||59.1837||
|-
| || || || || || ||45\76||59.2105||
|-
| || || || ||16\27|| || ||59.{{Overline|259}}||
|-
| || || || || || ||35\59||59.3220||
|-
| || || || || ||19\32|| ||59.3750||
|-
| || || || || || ||22\37||59.{{Overline|459}}||
|-
|3\5|| || || || || || ||60.0000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>slch8i0qhv3hpzhqgdpafsfgdicndt2</sha1>
    </revision>
    <revision>
      <id>118497</id>
      <parentid>106034</parentid>
      <timestamp>2023-07-05T08:27:51Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>118497</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6014" sha1="fan1ou2t52ztj101x9jjyjc71gbegxi" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as either [[20edo]] or [[21edo]]. tIt is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| |13/9
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 57.14286 cents (4\7/12 = 1\21) to 60 cents (3\5/12 = 1\20)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7 || || || || || || ||57.1429||
|-
| || ||  || || || ||27\47||57.4468 ||
|-
|  || ||  || || ||23\40|| ||57.5000||
|-
| || ||  || || ||  ||42\73||57.53425||
|-
| ||  || || || 19\33|| || || 57.{{Overline|57}}||
|-
| || || || || || ||53\92||57.6087||
|-
| || || || || ||34\59|| ||57.6271||
|-
| || || || || || ||49\85||57.6471||
|-
| || || ||15\26|| || || ||57.6923||
|-
| || || || || || ||56\97||57.7320||
|-
| || || || || ||41\71|| ||57.7465||
|-
| || || || || || ||67\116||57.7586||
|-
| || || || ||26\45|| || ||57.{{Overline|7}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||57.7982||
|-
| || || || || ||37\64|| ||57.8125||
|-
| || || || || || ||48\83||57.8313||
|-
| || ||11\19|| || || || ||57.8947||
|-
| || || || || || ||51\88||57.9{{Overline|54}}||
|-
| || || || || ||40\69|| ||57.9710||
|-
| || || || || || ||69\119||57.9832||
|-
| || || || ||29\50|| || ||58.000||
|-
| || || || || || ||76\131||58.0153||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||58.0247||
|-
| || || || || || ||65\112||58.0357||
|-
| || || ||18\31|| || || ||58.0645||[[Meantone]] is in this region
|-
| || || || || || ||61\105||58.0952||
|-
| || || || || ||43\74|| ||58.{{Overline|108}}||
|-
| || || || || || ||68\117||58.1967||
|-
| || || || ||25\43|| || ||58.1935||
|-
| || || || || || ||57\98||58.1633||
|-
| || || || || ||32\55|| ||58.{{Overline|18}}||
|-
| || || || || || ||39\67||58.2090||The generator closest to a just [[7/5]] for EDOs less than 2400
|-
| ||7\12|| || || || || ||58.{{Overline|3}}||
|-
| || || || || || ||38\65||58.4615||
|-
| || || || || ||31\53|| ||58.4906||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||58,5106||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||58.5366||
|-
| || || || || || ||65\111||58.{{Overline|558}}||
|-
| || || || || ||41\70|| ||58.5714||
|-
| || || || || || ||58\99||58.{{Overline|58}}||
|-
| || || ||17\29|| || || ||58.6207||
|-
| || || || || || ||61\104||58.65385||
|-
| || || || || ||44\75|| ||58.{{Overline|6}}||
|-
| || || || || || ||71\121||58.6777||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||58.69565||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||58.7155||
|-
| || || || || ||37\63|| ||58.7302||
|-
| || || || || || ||47\80||58.7500||
|-
| || ||10\17|| || || || ||58.8235||
|-
| || || || || || ||43\73||58.9041||
|-
| || || || || ||33\56|| ||58.9286||
|-
| || || || || || ||56\95||58.9474||
|-
| || || || ||23\39|| || ||58.9744||
|-
| || || || || || ||59\100||59.0000||
|-
| || || || || ||36\61|| ||59.0164||
|-
| || || || || || ||49\83||59.0361||
|-
| || || ||13\22|| || || ||59.{{Overline|09}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||59.1549||
|-
| || || || || ||29\49|| ||59.1837||
|-
| || || || || || ||45\76||59.2105||
|-
| || || || ||16\27|| || ||59.{{Overline|259}}||
|-
| || || || || || ||35\59||59.3220||
|-
| || || || || ||19\32|| ||59.3750||
|-
| || || || || || ||22\37||59.{{Overline|459}}||
|-
|3\5|| || || || || || ||60.0000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>fan1ou2t52ztj101x9jjyjc71gbegxi</sha1>
    </revision>
    <revision>
      <id>118498</id>
      <parentid>118497</parentid>
      <timestamp>2023-07-05T08:28:18Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>118498</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6068" sha1="ebk3w20erqad9um9wp3mknnpx7gp3yi" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this piont. It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| |13/9
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 57.14286 cents (4\7/12 = 1\21) to 60 cents (3\5/12 = 1\20)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7 || || || || || || ||57.1429||
|-
| || ||  || || || ||27\47||57.4468 ||
|-
|  || ||  || || ||23\40|| ||57.5000||
|-
| || ||  || || ||  ||42\73||57.53425||
|-
| ||  || || || 19\33|| || || 57.{{Overline|57}}||
|-
| || || || || || ||53\92||57.6087||
|-
| || || || || ||34\59|| ||57.6271||
|-
| || || || || || ||49\85||57.6471||
|-
| || || ||15\26|| || || ||57.6923||
|-
| || || || || || ||56\97||57.7320||
|-
| || || || || ||41\71|| ||57.7465||
|-
| || || || || || ||67\116||57.7586||
|-
| || || || ||26\45|| || ||57.{{Overline|7}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||57.7982||
|-
| || || || || ||37\64|| ||57.8125||
|-
| || || || || || ||48\83||57.8313||
|-
| || ||11\19|| || || || ||57.8947||
|-
| || || || || || ||51\88||57.9{{Overline|54}}||
|-
| || || || || ||40\69|| ||57.9710||
|-
| || || || || || ||69\119||57.9832||
|-
| || || || ||29\50|| || ||58.000||
|-
| || || || || || ||76\131||58.0153||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||58.0247||
|-
| || || || || || ||65\112||58.0357||
|-
| || || ||18\31|| || || ||58.0645||[[Meantone]] is in this region
|-
| || || || || || ||61\105||58.0952||
|-
| || || || || ||43\74|| ||58.{{Overline|108}}||
|-
| || || || || || ||68\117||58.1967||
|-
| || || || ||25\43|| || ||58.1935||
|-
| || || || || || ||57\98||58.1633||
|-
| || || || || ||32\55|| ||58.{{Overline|18}}||
|-
| || || || || || ||39\67||58.2090||The generator closest to a just [[7/5]] for EDOs less than 2400
|-
| ||7\12|| || || || || ||58.{{Overline|3}}||
|-
| || || || || || ||38\65||58.4615||
|-
| || || || || ||31\53|| ||58.4906||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||58,5106||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||58.5366||
|-
| || || || || || ||65\111||58.{{Overline|558}}||
|-
| || || || || ||41\70|| ||58.5714||
|-
| || || || || || ||58\99||58.{{Overline|58}}||
|-
| || || ||17\29|| || || ||58.6207||
|-
| || || || || || ||61\104||58.65385||
|-
| || || || || ||44\75|| ||58.{{Overline|6}}||
|-
| || || || || || ||71\121||58.6777||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||58.69565||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||58.7155||
|-
| || || || || ||37\63|| ||58.7302||
|-
| || || || || || ||47\80||58.7500||
|-
| || ||10\17|| || || || ||58.8235||
|-
| || || || || || ||43\73||58.9041||
|-
| || || || || ||33\56|| ||58.9286||
|-
| || || || || || ||56\95||58.9474||
|-
| || || || ||23\39|| || ||58.9744||
|-
| || || || || || ||59\100||59.0000||
|-
| || || || || ||36\61|| ||59.0164||
|-
| || || || || || ||49\83||59.0361||
|-
| || || ||13\22|| || || ||59.{{Overline|09}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||59.1549||
|-
| || || || || ||29\49|| ||59.1837||
|-
| || || || || || ||45\76||59.2105||
|-
| || || || ||16\27|| || ||59.{{Overline|259}}||
|-
| || || || || || ||35\59||59.3220||
|-
| || || || || ||19\32|| ||59.3750||
|-
| || || || || || ||22\37||59.{{Overline|459}}||
|-
|3\5|| || || || || || ||60.0000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>ebk3w20erqad9um9wp3mknnpx7gp3yi</sha1>
    </revision>
    <revision>
      <id>118499</id>
      <parentid>118498</parentid>
      <timestamp>2023-07-05T08:28:24Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>118499</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6068" sha1="7izcq7a4f3efi9w3oui0lz57zqoy6yk" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| |13/9
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
==Scale tree==
If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 57.14286 cents (4\7/12 = 1\21) to 60 cents (3\5/12 = 1\20)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7 || || || || || || ||57.1429||
|-
| || ||  || || || ||27\47||57.4468 ||
|-
|  || ||  || || ||23\40|| ||57.5000||
|-
| || ||  || || ||  ||42\73||57.53425||
|-
| ||  || || || 19\33|| || || 57.{{Overline|57}}||
|-
| || || || || || ||53\92||57.6087||
|-
| || || || || ||34\59|| ||57.6271||
|-
| || || || || || ||49\85||57.6471||
|-
| || || ||15\26|| || || ||57.6923||
|-
| || || || || || ||56\97||57.7320||
|-
| || || || || ||41\71|| ||57.7465||
|-
| || || || || || ||67\116||57.7586||
|-
| || || || ||26\45|| || ||57.{{Overline|7}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||57.7982||
|-
| || || || || ||37\64|| ||57.8125||
|-
| || || || || || ||48\83||57.8313||
|-
| || ||11\19|| || || || ||57.8947||
|-
| || || || || || ||51\88||57.9{{Overline|54}}||
|-
| || || || || ||40\69|| ||57.9710||
|-
| || || || || || ||69\119||57.9832||
|-
| || || || ||29\50|| || ||58.000||
|-
| || || || || || ||76\131||58.0153||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||58.0247||
|-
| || || || || || ||65\112||58.0357||
|-
| || || ||18\31|| || || ||58.0645||[[Meantone]] is in this region
|-
| || || || || || ||61\105||58.0952||
|-
| || || || || ||43\74|| ||58.{{Overline|108}}||
|-
| || || || || || ||68\117||58.1967||
|-
| || || || ||25\43|| || ||58.1935||
|-
| || || || || || ||57\98||58.1633||
|-
| || || || || ||32\55|| ||58.{{Overline|18}}||
|-
| || || || || || ||39\67||58.2090||The generator closest to a just [[7/5]] for EDOs less than 2400
|-
| ||7\12|| || || || || ||58.{{Overline|3}}||
|-
| || || || || || ||38\65||58.4615||
|-
| || || || || ||31\53|| ||58.4906||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||58,5106||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||58.5366||
|-
| || || || || || ||65\111||58.{{Overline|558}}||
|-
| || || || || ||41\70|| ||58.5714||
|-
| || || || || || ||58\99||58.{{Overline|58}}||
|-
| || || ||17\29|| || || ||58.6207||
|-
| || || || || || ||61\104||58.65385||
|-
| || || || || ||44\75|| ||58.{{Overline|6}}||
|-
| || || || || || ||71\121||58.6777||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||58.69565||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||58.7155||
|-
| || || || || ||37\63|| ||58.7302||
|-
| || || || || || ||47\80||58.7500||
|-
| || ||10\17|| || || || ||58.8235||
|-
| || || || || || ||43\73||58.9041||
|-
| || || || || ||33\56|| ||58.9286||
|-
| || || || || || ||56\95||58.9474||
|-
| || || || ||23\39|| || ||58.9744||
|-
| || || || || || ||59\100||59.0000||
|-
| || || || || ||36\61|| ||59.0164||
|-
| || || || || || ||49\83||59.0361||
|-
| || || ||13\22|| || || ||59.{{Overline|09}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||59.1549||
|-
| || || || || ||29\49|| ||59.1837||
|-
| || || || || || ||45\76||59.2105||
|-
| || || || ||16\27|| || ||59.{{Overline|259}}||
|-
| || || || || || ||35\59||59.3220||
|-
| || || || || ||19\32|| ||59.3750||
|-
| || || || || || ||22\37||59.{{Overline|459}}||
|-
|3\5|| || || || || || ||60.0000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>7izcq7a4f3efi9w3oui0lz57zqoy6yk</sha1>
    </revision>
    <revision>
      <id>123121</id>
      <parentid>118499</parentid>
      <timestamp>2023-08-31T18:50:41Z</timestamp>
      <contributor>
        <username>CellularAutomaton</username>
        <id>369</id>
      </contributor>
      <comment>/* Scale tree */ MMTM seems to have MMTMized this page, I deleted the offending portion</comment>
      <origin>123121</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1584" sha1="bt98ngcgiff5nxuukkvgct7xjowb033" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| |13/9
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>bt98ngcgiff5nxuukkvgct7xjowb033</sha1>
    </revision>
    <revision>
      <id>123126</id>
      <parentid>123121</parentid>
      <timestamp>2023-09-01T06:27:03Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <comment>Scale tree can be added back if there is an actual explanation of it</comment>
      <origin>123126</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6147" sha1="0mvz80vi1we8d0p8wv11fqe082b3qr7" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| |13/9
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
==Scale tree==

EDFs can be approximated by subdividing diatonic fifths between 4\7 and 3\5.

If 4\7 (four degrees of 7EDO) is at one extreme and 3\5 (three degrees of 5EDO) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking [[Mediant|"freshman sums"]] of the two edges - adding together the numerators, then adding together the denominators (i.e. adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12EDO.

If we carry this freshman-summing out a little further, new, larger [[EDO]]s pop up in our continuum.

Generator range: 57.14286 cents (4\7/12 = 1\21) to 60 cents (3\5/12 = 1\20)
{| class="wikitable center-all"
! colspan="7" |Fifth
!Cents
!Comments
|-
|4\7 || || || || || || ||57.1429||
|-
| || ||  || || || ||27\47||57.4468 ||
|-
|  || ||  || || ||23\40|| ||57.5000||
|-
| || ||  || || ||  ||42\73||57.53425||
|-
| ||  || || || 19\33|| || || 57.{{Overline|57}}||
|-
| || || || || || ||53\92||57.6087||
|-
| || || || || ||34\59|| ||57.6271||
|-
| || || || || || ||49\85||57.6471||
|-
| || || ||15\26|| || || ||57.6923||
|-
| || || || || || ||56\97||57.7320||
|-
| || || || || ||41\71|| ||57.7465||
|-
| || || || || || ||67\116||57.7586||
|-
| || || || ||26\45|| || ||57.{{Overline|7}}||[[Flattone]] is in this region
|-
| || || || || || ||63\109||57.7982||
|-
| || || || || ||37\64|| ||57.8125||
|-
| || || || || || ||48\83||57.8313||
|-
| || ||11\19|| || || || ||57.8947||
|-
| || || || || || ||51\88||57.9{{Overline|54}}||
|-
| || || || || ||40\69|| ||57.9710||
|-
| || || || || || ||69\119||57.9832||
|-
| || || || ||29\50|| || ||58.000||
|-
| || || || || || ||76\131||58.0153||[[Golden meantone]] (696.2145¢)
|-
| || || || || ||47\81|| ||58.0247||
|-
| || || || || || ||65\112||58.0357||
|-
| || || ||18\31|| || || ||58.0645||[[Meantone]] is in this region
|-
| || || || || || ||61\105||58.0952||
|-
| || || || || ||43\74|| ||58.{{Overline|108}}||
|-
| || || || || || ||68\117||58.1967||
|-
| || || || ||25\43|| || ||58.1935||
|-
| || || || || || ||57\98||58.1633||
|-
| || || || || ||32\55|| ||58.{{Overline|18}}||
|-
| || || || || || ||39\67||58.2090||The generator closest to a just [[7/5]] for EDOs less than 2400
|-
| ||7\12|| || || || || ||58.{{Overline|3}}||
|-
| || || || || || ||38\65||58.4615||
|-
| || || || || ||31\53|| ||58.4906||The fifth closest to a just [[3/2]] for EDOs less than 200
|-
| || || || || || ||55\94||58,5106||[[Garibaldi]] / [[Cassandra]]
|-
| || || || ||24\41|| || ||58.5366||
|-
| || || || || || ||65\111||58.{{Overline|558}}||
|-
| || || || || ||41\70|| ||58.5714||
|-
| || || || || || ||58\99||58.{{Overline|58}}||
|-
| || || ||17\29|| || || ||58.6207||
|-
| || || || || || ||61\104||58.65385||
|-
| || || || || ||44\75|| ||58.{{Overline|6}}||
|-
| || || || || || ||71\121||58.6777||Golden neogothic (704.0956¢)
|-
| || || || ||27\46|| || ||58.69565||[[Neogothic]] is in this region
|-
| || || || || || ||64\109||58.7155||
|-
| || || || || ||37\63|| ||58.7302||
|-
| || || || || || ||47\80||58.7500||
|-
| || ||10\17|| || || || ||58.8235||
|-
| || || || || || ||43\73||58.9041||
|-
| || || || || ||33\56|| ||58.9286||
|-
| || || || || || ||56\95||58.9474||
|-
| || || || ||23\39|| || ||58.9744||
|-
| || || || || || ||59\100||59.0000||
|-
| || || || || ||36\61|| ||59.0164||
|-
| || || || || || ||49\83||59.0361||
|-
| || || ||13\22|| || || ||59.{{Overline|09}}||[[Archy]] is in this region
|-
| || || || || || ||42\71||59.1549||
|-
| || || || || ||29\49|| ||59.1837||
|-
| || || || || || ||45\76||59.2105||
|-
| || || || ||16\27|| || ||59.{{Overline|259}}||
|-
| || || || || || ||35\59||59.3220||
|-
| || || || || ||19\32|| ||59.3750||
|-
| || || || || || ||22\37||59.{{Overline|459}}||
|-
|3\5|| || || || || || ||60.0000||
|}Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>0mvz80vi1we8d0p8wv11fqe082b3qr7</sha1>
    </revision>
    <revision>
      <id>123145</id>
      <parentid>123126</parentid>
      <timestamp>2023-09-01T15:20:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>This scale tree is totally irrelevant to this page and the temperaments are all wrong</comment>
      <origin>123145</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1584" sha1="bt98ngcgiff5nxuukkvgct7xjowb033" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| |13/9
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>bt98ngcgiff5nxuukkvgct7xjowb033</sha1>
    </revision>
    <revision>
      <id>129958</id>
      <parentid>123145</parentid>
      <timestamp>2023-12-29T04:12:15Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>129958</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1621" sha1="et962tkd7zx9ua4pxl6mmh8dbv3524u" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | [[28/27]], 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[10/9]], [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]], [[63/50]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| | [[13/9]]
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>et962tkd7zx9ua4pxl6mmh8dbv3524u</sha1>
    </revision>
    <revision>
      <id>140929</id>
      <parentid>129958</parentid>
      <timestamp>2024-04-11T18:50:20Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>140929</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1631" sha1="q1ge7klu5huepdidq58nio9xmpm4rp1" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | [[28/27]], 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[10/9]], [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]], [[63/50]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| | [[13/9]]
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>q1ge7klu5huepdidq58nio9xmpm4rp1</sha1>
    </revision>
    <revision>
      <id>143553</id>
      <parentid>140929</parentid>
      <timestamp>2024-05-19T07:36:26Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>143553</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1941" sha1="64raegtum4js8yxlud3fb9vqjatekeg" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

It is a strong [[half-prime subgroup|3/2.5/2.7/2 subgroup]] system, a fact first noted by [[User:CompactStar|CompactStar]], tempering out the commas [[10976/10935]] and [[3125/3087]], although the representation of [[11/2]] is more questionable. [[24edf]] (effectively 41edo) provides a correction for 11/2.


==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | [[28/27]], 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[10/9]], [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]], [[63/50]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| | [[13/9]]
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>64raegtum4js8yxlud3fb9vqjatekeg</sha1>
    </revision>
    <revision>
      <id>143554</id>
      <parentid>143553</parentid>
      <timestamp>2024-05-19T07:39:13Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>143554</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2103" sha1="sx6jsh5y8h3w6dks5ox5f54fwsth635" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

It is a strong [[half-prime subgroup|3/2.5/2.7/2 subgroup]] system, a fact first noted by [[User:CompactStar|CompactStar]], tempering out the commas [[10976/10935]] and [[3125/3087]], although the representation of [[11/2]] is more questionable. [[24edf]] (effectively 41edo) provides a correction for 11/2. It contains the [[macrodiatonic and microdiatonic scales|macrodiatonic]] scale that corresponds to 12edo's diatonic scale where [[2/1]] is compressed to [[3/2]].


==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | [[28/27]], 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[10/9]], [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]], [[63/50]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| | [[13/9]]
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>sx6jsh5y8h3w6dks5ox5f54fwsth635</sha1>
    </revision>
    <revision>
      <id>143555</id>
      <parentid>143554</parentid>
      <timestamp>2024-05-19T07:39:28Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>143555</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2109" sha1="g6z9uj1j8zhzs2xvkqsh8zvleqisv34" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

It is a strong [[half-prime subgroup|3/2.5/2.7/2 subgroup]] system, a fact first noted by [[User:CompactStar|CompactStar]], tempering out the commas [[10976/10935]] and [[3125/3087]], although the representation of [[11/2]] is more questionable. [[24edf]] (effectively 41edo) provides a correction for 11/2. It contains the [[macrodiatonic and microdiatonic scales|mairodiatonic]] scale that corresponds to 12edo's [[5L 2s|diatonic scale]] with [[2/1]] compressed to [[3/2]].


==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | [[28/27]], 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[10/9]], [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]], [[63/50]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| | [[13/9]]
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>g6z9uj1j8zhzs2xvkqsh8zvleqisv34</sha1>
    </revision>
    <revision>
      <id>143556</id>
      <parentid>143555</parentid>
      <timestamp>2024-05-19T07:39:37Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>143556</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2109" sha1="b5lfcpmf0p4br14nsbr931g4udemxqd" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

It is a strong [[half-prime subgroup|3/2.5/2.7/2 subgroup]] system, a fact first noted by [[User:CompactStar|CompactStar]], tempering out the commas [[10976/10935]] and [[3125/3087]], although the representation of [[11/2]] is more questionable. [[24edf]] (effectively 41edo) provides a correction for 11/2. It contains the [[macrodiatonic and microdiatonic scales|microdiatonic]] scale that corresponds to 12edo's [[5L 2s|diatonic scale]] with [[2/1]] compressed to [[3/2]].


==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | [[28/27]], 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[10/9]], [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]], [[63/50]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| | [[13/9]]
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>b5lfcpmf0p4br14nsbr931g4udemxqd</sha1>
    </revision>
    <revision>
      <id>172018</id>
      <parentid>143556</parentid>
      <timestamp>2024-12-21T03:34:22Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>add todo:expand, add harmonics</comment>
      <origin>172018</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2221" sha1="1w2m2s4iflbpaoax575gz79rqntmc76" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

It is a strong [[half-prime subgroup|3/2.5/2.7/2 subgroup]] system, a fact first noted by [[User:CompactStar|CompactStar]], tempering out the commas [[10976/10935]] and [[3125/3087]], although the representation of [[11/2]] is more questionable. [[24edf]] (effectively 41edo) provides a correction for 11/2. It contains the [[macrodiatonic and microdiatonic scales|microdiatonic]] scale that corresponds to 12edo's [[5L 2s|diatonic scale]] with [[2/1]] compressed to [[3/2]].

==Harmonics==
{{Harmonics in equal|12|3|2}}
{{Harmonics in equal|12|3|2|start=12|collapsed=1}}

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | [[28/27]], 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[10/9]], [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]], [[63/50]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| | [[13/9]]
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}

{{todo|expand}}
[[Category:Edf]]
[[Category:Edonoi]]</text>
      <sha1>1w2m2s4iflbpaoax575gz79rqntmc76</sha1>
    </revision>
    <revision>
      <id>205890</id>
      <parentid>172018</parentid>
      <timestamp>2025-08-01T19:00:08Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Edf]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>205890</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2204" sha1="4qq4n1g2acbqg0jxc66i8cor6b1sqvl" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

It is a strong [[half-prime subgroup|3/2.5/2.7/2 subgroup]] system, a fact first noted by [[User:CompactStar|CompactStar]], tempering out the commas [[10976/10935]] and [[3125/3087]], although the representation of [[11/2]] is more questionable. [[24edf]] (effectively 41edo) provides a correction for 11/2. It contains the [[macrodiatonic and microdiatonic scales|microdiatonic]] scale that corresponds to 12edo's [[5L 2s|diatonic scale]] with [[2/1]] compressed to [[3/2]].

==Harmonics==
{{Harmonics in equal|12|3|2}}
{{Harmonics in equal|12|3|2|start=12|collapsed=1}}

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | [[28/27]], 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[10/9]], [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]], [[63/50]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| | [[13/9]]
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}

{{todo|expand}}
[[Category:Edonoi]]</text>
      <sha1>4qq4n1g2acbqg0jxc66i8cor6b1sqvl</sha1>
    </revision>
    <revision>
      <id>205939</id>
      <parentid>205890</parentid>
      <timestamp>2025-08-01T19:20:37Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Edonoi]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>205939</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2184" sha1="fc07sg3qu35u0d5dgsqoxyb46gjws1z" xml:space="preserve">{{Infobox ET}}
'''12EDF''' is the [[EDF|equal division of the just perfect fifth]] into 12 parts of 58.49625 [[cent|cents]] each, corresponding to 20.5141 [[edo]] (similar to every second step of [[41edo]]). It is an intersection of [[3edf]]~[[5edo]] and [[4edf]]~[[7edo]] relations, and could pass as both [[20edo]] and [[21edo]], with both relations nearly breaking down by this point. It is related to the [[Tetracot family#Dodecacot|dodecacot temperament]], which tempers out 3087/3125 and 10976/10935 in the 7-limit.

It is a strong [[half-prime subgroup|3/2.5/2.7/2 subgroup]] system, a fact first noted by [[User:CompactStar|CompactStar]], tempering out the commas [[10976/10935]] and [[3125/3087]], although the representation of [[11/2]] is more questionable. [[24edf]] (effectively 41edo) provides a correction for 11/2. It contains the [[macrodiatonic and microdiatonic scales|microdiatonic]] scale that corresponds to 12edo's [[5L 2s|diatonic scale]] with [[2/1]] compressed to [[3/2]].

==Harmonics==
{{Harmonics in equal|12|3|2}}
{{Harmonics in equal|12|3|2|start=12|collapsed=1}}

==Intervals==
{| class="wikitable"
|-
! | degree
! | cents value
! | corresponding &lt;br&gt;JI intervals
! | comments
|-
! colspan="2" | 0
| | '''exact [[1/1]]'''
| | 
|-
| | 1
| | 58.49625
| | [[28/27]], 91/88, 88/85
| | 
|-
| | 2
| | 116.9925
| | [[15/14]]
| | 
|-
| | 3
| | 175.48875
| | [[10/9]], [[21/19]]
| | 
|-
| | 4
| | 233.9850
| | [[8/7]]
| | 
|-
| | 5
| | 292.48125
| | 45/38
| | 
|-
| | 6
| | 350.9775
| | [[11/9]], [[27/22]]
| | 
|-
| | 7
| | 409.47375
| | [[19/15]], [[63/50]]
| | 
|-
| | 8
| | 467.9700
| | [[21/16]]
| | 
|-
| | 9
| | 526.46625
| | [[19/14]]
| | 
|-
| | 10
| | 584.9625
| | [[7/5]]
| | 
|-
| | 11
| | 643.4588
| | [[13/9]]
| | 
|-
| | 12
| | 701.9550
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|13
|760.45125
|273/176, 132/85
|
|-
|14
|818.9475
|8/5
|
|-
|15
|877.44375
|63/38
|
|-
|16
|935.94
|12/7
|
|-
|17
|994.43625
|135/76
|
|-
|18
|1052.9325
|11/6, 81/44
|
|-
|19
|1111.42875
|19/10
|
|-
|20
|1169.925
|63/32
|
|-
|21
|1228.42125
|57/28
|
|-
|22
|1286.9175
|21/10
|
|-
|23
|1345.41375
|13/6
|
|-
|24
|1403.91
|'''exact''' 9/4
|
|}

{{todo|expand}}</text>
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  </page>
  <page>
    <title>12edo comma pump examples</title>
    <ns>0</ns>
    <id>134</id>
    <revision>
      <id>134</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
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        <id>0</id>
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      <text bytes="1856" sha1="8txm2513jsuezz2a7ixy0sjen7pvt2w" xml:space="preserve">The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: [[Meantone_comma_pump_examples|Meantone comma pump examples]]. These work in any meantone temperament, including [[19edo|19edo]], [[26edo|26edo]], [[31edo|31edo]], etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by [[12edo|12edo]] are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

=Augmented (128/125) comma pumps=
&lt;ul&gt;&lt;li&gt;''Looney Tunes'' theme song, c. 1940. ''[http://www.youtube.com/watch?v=F9S7yhD5M9A http://www.youtube.com/watch?v=F9S7yhD5M9A]''&lt;/li&gt;&lt;li&gt;''I Wish'', Skee-Lo, 1995. ''[http://www.youtube.com/watch?v=Zyf0YwUJcqk http://www.youtube.com/watch?v=Zyf0YwUJcqk]''&lt;/li&gt;&lt;li&gt;''Giant Steps'', John Coltrane 1959 [http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29 http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29] [https://www.youtube.com/watch?v=2kotK9FNEYU https://www.youtube.com/watch?v=2kotK9FNEYU]&lt;ul&gt;&lt;li&gt;'''Mike Battaglia note:''' you should be aware that, strictly speaking, Giant Steps doesn't actually involve a comma pump unless you deliberately want the "B" in the beginning at the top of the progression to be the same as the "B" at the end at the bottom. If you don't, it doesn't matter; then you can treat it starting at B and going down to Cb, and then ending on top at D#, and then it'll work in temperaments that don't temper out 128/125, such as 19-EDO. The important thing is that nothing drifts with each successive iteration of the head.&lt;/li&gt;&lt;/ul&gt;&lt;/li&gt;&lt;/ul&gt;

=Diminished (648/625) comma pumps=

=Srutal (2048/2025) comma pumps=</text>
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      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 408414242 - Original comment: **</comment>
      <origin>4623</origin>
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      <text bytes="6273" sha1="ehx7rnajt1cedpi465rwiosgdy3640g" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2013-02-20 05:29:48 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;408414242&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: [[Meantone comma pump examples]]. These work in any meantone temperament, including [[19edo]], [[26edo]], [[31edo]], etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by [[12edo]] are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

=Augmented (128/125) comma pumps= 
* //Looney Tunes// theme song, c. 1940. //[[http://www.youtube.com/watch?v=F9S7yhD5M9A]]//
* //I Wish//, Skee-Lo, 1995. //[[http://www.youtube.com/watch?v=Zyf0YwUJcqk]]//
* //Giant Steps//, John Coltrane 1959 http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29 https://www.youtube.com/watch?v=2kotK9FNEYU
** **Mike Battaglia note:** you should be aware that, strictly speaking, Giant Steps doesn't actually involve a comma pump unless you deliberately want the "B" in the beginning at the top of the progression to be the same as the "B" at the end at the bottom. If you don't, it doesn't matter; then you can treat it starting at B and going down to Cb, and then ending on top at D#, and then it'll work in temperaments that don't temper out 128/125, such as 19-EDO. The important thing is that nothing drifts with each successive iteration of the head.

=Diminished (648/625) comma pumps= 

=Srutal (2048/2025) comma pumps= &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo comma pump examples&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The most common kind of comma pump encountered in &amp;amp;quot;non-xenharmonic&amp;amp;quot; Western music is the meantone comma pump, for which there is a separate page: &amp;lt;a class="wiki_link" href="/Meantone%20comma%20pump%20examples"&amp;gt;Meantone comma pump examples&amp;lt;/a&amp;gt;. These work in any meantone temperament, including &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/26edo"&amp;gt;26edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, etc. Since standard notation is based on meantone, these don't appear unusual in any way.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
However, comma pumps of other commas tempered out by &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt; are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Augmented (128/125) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Augmented (128/125) comma pumps&amp;lt;/h1&amp;gt;
 &amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;Looney Tunes&amp;lt;/em&amp;gt; theme song, c. 1940. &amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=F9S7yhD5M9A" rel="nofollow"&amp;gt;http://www.youtube.com/watch?v=F9S7yhD5M9A&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;I Wish&amp;lt;/em&amp;gt;, Skee-Lo, 1995. &amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=Zyf0YwUJcqk" rel="nofollow"&amp;gt;http://www.youtube.com/watch?v=Zyf0YwUJcqk&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;Giant Steps&amp;lt;/em&amp;gt;, John Coltrane 1959 &amp;lt;!-- ws:start:WikiTextUrlRule:31:http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29 --&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:31 --&amp;gt; &amp;lt;!-- ws:start:WikiTextUrlRule:32:https://www.youtube.com/watch?v=2kotK9FNEYU --&amp;gt;&amp;lt;a class="wiki_link_ext" href="https://www.youtube.com/watch?v=2kotK9FNEYU" rel="nofollow"&amp;gt;https://www.youtube.com/watch?v=2kotK9FNEYU&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:32 --&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;strong&amp;gt;Mike Battaglia note:&amp;lt;/strong&amp;gt; you should be aware that, strictly speaking, Giant Steps doesn't actually involve a comma pump unless you deliberately want the &amp;amp;quot;B&amp;amp;quot; in the beginning at the top of the progression to be the same as the &amp;amp;quot;B&amp;amp;quot; at the end at the bottom. If you don't, it doesn't matter; then you can treat it starting at B and going down to Cb, and then ending on top at D#, and then it'll work in temperaments that don't temper out 128/125, such as 19-EDO. The important thing is that nothing drifts with each successive iteration of the head.&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Diminished (648/625) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Diminished (648/625) comma pumps&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
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        <id>0</id>
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      <comment>**Imported revision 408391916 - Original comment: **</comment>
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      <text bytes="5122" sha1="eyjrn0z7quxcapgwl6embpqadh8tlrz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:All_Your_Bass|All_Your_Bass]] and made on &lt;tt&gt;2013-02-20 03:03:29 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;408391916&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: [[Meantone comma pump examples]]. These work in any meantone temperament, including [[19edo]], [[26edo]], [[31edo]], etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by [[12edo]] are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

=Augmented (128/125) comma pumps= 
* //Looney Tunes// theme song, c. 1940. //[[http://www.youtube.com/watch?v=F9S7yhD5M9A]]//
* //I Wish//, Skee-Lo, 1995. //[[http://www.youtube.com/watch?v=Zyf0YwUJcqk]]//
* //Giant Steps//, John Coltrane 1959 http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29 https://www.youtube.com/watch?v=2kotK9FNEYU
&amp;gt; 

=Diminished (648/625) comma pumps= 

=Srutal (2048/2025) comma pumps= &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo comma pump examples&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The most common kind of comma pump encountered in &amp;amp;quot;non-xenharmonic&amp;amp;quot; Western music is the meantone comma pump, for which there is a separate page: &amp;lt;a class="wiki_link" href="/Meantone%20comma%20pump%20examples"&amp;gt;Meantone comma pump examples&amp;lt;/a&amp;gt;. These work in any meantone temperament, including &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/26edo"&amp;gt;26edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, etc. Since standard notation is based on meantone, these don't appear unusual in any way.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
However, comma pumps of other commas tempered out by &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt; are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Augmented (128/125) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Augmented (128/125) comma pumps&amp;lt;/h1&amp;gt;
 &amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;Looney Tunes&amp;lt;/em&amp;gt; theme song, c. 1940. &amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=F9S7yhD5M9A" rel="nofollow"&amp;gt;http://www.youtube.com/watch?v=F9S7yhD5M9A&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;I Wish&amp;lt;/em&amp;gt;, Skee-Lo, 1995. &amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=Zyf0YwUJcqk" rel="nofollow"&amp;gt;http://www.youtube.com/watch?v=Zyf0YwUJcqk&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;Giant Steps&amp;lt;/em&amp;gt;, John Coltrane 1959 &amp;lt;!-- ws:start:WikiTextUrlRule:29:http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29 --&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:29 --&amp;gt; &amp;lt;!-- ws:start:WikiTextUrlRule:30:https://www.youtube.com/watch?v=2kotK9FNEYU --&amp;gt;&amp;lt;a class="wiki_link_ext" href="https://www.youtube.com/watch?v=2kotK9FNEYU" rel="nofollow"&amp;gt;https://www.youtube.com/watch?v=2kotK9FNEYU&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:30 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
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      <timestamp>2013-02-20T03:03:14Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;All_Your_Bass</username>
        <id>0</id>
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      <comment>**Imported revision 408391884 - Original comment: **</comment>
      <origin>4625</origin>
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      <text bytes="5088" sha1="1ei6siq8z3kzxkrjws2w2ucgifk6mhl" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:All_Your_Bass|All_Your_Bass]] and made on &lt;tt&gt;2013-02-20 03:03:14 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;408391884&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: [[Meantone comma pump examples]]. These work in any meantone temperament, including [[19edo]], [[26edo]], [[31edo]], etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by [[12edo]] are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

=Augmented (128/125) comma pumps= 
* //Looney Tunes// theme song, c. 1940. ////[[http://www.youtube.com/watch?v=F9S7yhD5M9A]]////
* //I Wish//, Skee-Lo, 1995. ////[[http://www.youtube.com/watch?v=Zyf0YwUJcqk]]////
* //Giant Steps//, Johm Coltrane 1959 http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29 https://www.youtube.com/watch?v=2kotK9FNEYU
&amp;gt; 

=Diminished (648/625) comma pumps= 

=Srutal (2048/2025) comma pumps= &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo comma pump examples&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The most common kind of comma pump encountered in &amp;amp;quot;non-xenharmonic&amp;amp;quot; Western music is the meantone comma pump, for which there is a separate page: &amp;lt;a class="wiki_link" href="/Meantone%20comma%20pump%20examples"&amp;gt;Meantone comma pump examples&amp;lt;/a&amp;gt;. These work in any meantone temperament, including &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/26edo"&amp;gt;26edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, etc. Since standard notation is based on meantone, these don't appear unusual in any way.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
However, comma pumps of other commas tempered out by &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt; are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Augmented (128/125) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Augmented (128/125) comma pumps&amp;lt;/h1&amp;gt;
 &amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;Looney Tunes&amp;lt;/em&amp;gt; theme song, c. 1940. &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=F9S7yhD5M9A" rel="nofollow"&amp;gt;http://www.youtube.com/watch?v=F9S7yhD5M9A&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;I Wish&amp;lt;/em&amp;gt;, Skee-Lo, 1995. &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=Zyf0YwUJcqk" rel="nofollow"&amp;gt;http://www.youtube.com/watch?v=Zyf0YwUJcqk&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;Giant Steps&amp;lt;/em&amp;gt;, Johm Coltrane 1959 &amp;lt;!-- ws:start:WikiTextUrlRule:29:http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29 --&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:29 --&amp;gt; &amp;lt;!-- ws:start:WikiTextUrlRule:30:https://www.youtube.com/watch?v=2kotK9FNEYU --&amp;gt;&amp;lt;a class="wiki_link_ext" href="https://www.youtube.com/watch?v=2kotK9FNEYU" rel="nofollow"&amp;gt;https://www.youtube.com/watch?v=2kotK9FNEYU&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:30 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Diminished (648/625) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Diminished (648/625) comma pumps&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Srutal (2048/2025) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Srutal (2048/2025) comma pumps&amp;lt;/h1&amp;gt;
&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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    <revision>
      <id>4626</id>
      <timestamp>2013-02-20T03:01:26Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;All_Your_Bass</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 408391636 - Original comment: **</comment>
      <origin>4626</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5113" sha1="t4v15jbt4q78wyp5769le31rs5q8a88" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:All_Your_Bass|All_Your_Bass]] and made on &lt;tt&gt;2013-02-20 03:01:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;408391636&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: [[Meantone comma pump examples]]. These work in any meantone temperament, including [[19edo]], [[26edo]], [[31edo]], etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by [[12edo]] are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

=Augmented (128/125) comma pumps= 
* //Looney Tunes// theme song, c. 1940. //[[http://www.youtube.com/watch?v=F9S7yhD5M9A]]//
* //I Wish//, Skee-Lo, 1995. //[[http://www.youtube.com/watch?v=Zyf0YwUJcqk]]//
* ////Giant Steps, J////ohm Coltrane ////1959//// http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29 https://www.youtube.com/watch?v=2kotK9FNEYU
&amp;gt; 

=Diminished (648/625) comma pumps= 

=Srutal (2048/2025) comma pumps= &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo comma pump examples&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The most common kind of comma pump encountered in &amp;amp;quot;non-xenharmonic&amp;amp;quot; Western music is the meantone comma pump, for which there is a separate page: &amp;lt;a class="wiki_link" href="/Meantone%20comma%20pump%20examples"&amp;gt;Meantone comma pump examples&amp;lt;/a&amp;gt;. These work in any meantone temperament, including &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/26edo"&amp;gt;26edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, etc. Since standard notation is based on meantone, these don't appear unusual in any way.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
However, comma pumps of other commas tempered out by &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt; are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Augmented (128/125) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Augmented (128/125) comma pumps&amp;lt;/h1&amp;gt;
 &amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;Looney Tunes&amp;lt;/em&amp;gt; theme song, c. 1940. &amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=F9S7yhD5M9A" rel="nofollow"&amp;gt;http://www.youtube.com/watch?v=F9S7yhD5M9A&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;I Wish&amp;lt;/em&amp;gt;, Skee-Lo, 1995. &amp;lt;em&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=Zyf0YwUJcqk" rel="nofollow"&amp;gt;http://www.youtube.com/watch?v=Zyf0YwUJcqk&amp;lt;/a&amp;gt;&amp;lt;/em&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;Giant Steps, Johm Coltrane 1959 &amp;lt;!-- ws:start:WikiTextUrlRule:29:http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29 --&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29" rel="nofollow"&amp;gt;http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:29 --&amp;gt; &amp;lt;!-- ws:start:WikiTextUrlRule:30:https://www.youtube.com/watch?v=2kotK9FNEYU --&amp;gt;&amp;lt;a class="wiki_link_ext" href="https://www.youtube.com/watch?v=2kotK9FNEYU" rel="nofollow"&amp;gt;https://www.youtube.com/watch?v=2kotK9FNEYU&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:30 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Diminished (648/625) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Diminished (648/625) comma pumps&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Srutal (2048/2025) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Srutal (2048/2025) comma pumps&amp;lt;/h1&amp;gt;
&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>t4v15jbt4q78wyp5769le31rs5q8a88</sha1>
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    <revision>
      <id>4627</id>
      <timestamp>2012-06-06T11:05:34Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;phylingual</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 343195792 - Original comment: **</comment>
      <origin>4627</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4221" sha1="bzkz08epkn5qg3njmq06958oz8msqh7" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:phylingual|phylingual]] and made on &lt;tt&gt;2012-06-06 11:05:34 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;343195792&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: [[Meantone comma pump examples]]. These work in any meantone temperament, including [[19edo]], [[26edo]], [[31edo]], etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by [[12edo]] are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

=Augmented (128/125) comma pumps= 
* //Looney Tunes// theme song, c. 1940. [[http://www.youtube.com/watch?v=F9S7yhD5M9A]]
* //I Wish//, Skee-Lo, 1995. [[http://www.youtube.com/watch?v=Zyf0YwUJcqk]]

=Diminished (648/625) comma pumps= 

=Srutal (2048/2025) comma pumps= &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo comma pump examples&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The most common kind of comma pump encountered in &amp;amp;quot;non-xenharmonic&amp;amp;quot; Western music is the meantone comma pump, for which there is a separate page: &amp;lt;a class="wiki_link" href="/Meantone%20comma%20pump%20examples"&amp;gt;Meantone comma pump examples&amp;lt;/a&amp;gt;. These work in any meantone temperament, including &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/26edo"&amp;gt;26edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, etc. Since standard notation is based on meantone, these don't appear unusual in any way.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
However, comma pumps of other commas tempered out by &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt; are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Augmented (128/125) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Augmented (128/125) comma pumps&amp;lt;/h1&amp;gt;
 &amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;Looney Tunes&amp;lt;/em&amp;gt; theme song, c. 1940. &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=F9S7yhD5M9A" rel="nofollow"&amp;gt;http://www.youtube.com/watch?v=F9S7yhD5M9A&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;I Wish&amp;lt;/em&amp;gt;, Skee-Lo, 1995. &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=Zyf0YwUJcqk" rel="nofollow"&amp;gt;http://www.youtube.com/watch?v=Zyf0YwUJcqk&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Diminished (648/625) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Diminished (648/625) comma pumps&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Srutal (2048/2025) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Srutal (2048/2025) comma pumps&amp;lt;/h1&amp;gt;
&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>bzkz08epkn5qg3njmq06958oz8msqh7</sha1>
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    <revision>
      <id>4628</id>
      <timestamp>2012-03-26T19:07:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;keenanpepper</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 314832514 - Original comment: **</comment>
      <origin>4628</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3950" sha1="jr9l7xx7gfj3tdp21cwluizaw44hs1r" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on &lt;tt&gt;2012-03-26 19:07:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;314832514&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: [[Meantone comma pump examples]]. These work in any meantone temperament, including [[19edo]], [[26edo]], [[31edo]], etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by [[12edo]] are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

=Augmented (128/125) comma pumps= 
* //Looney Tunes// theme song, c. 1940. [[http://www.youtube.com/watch?v=F9S7yhD5M9A]]
* //I Wish//, Skee-Lo, 1995. [[http://www.youtube.com/watch?v=Zyf0YwUJcqk]]

=Diminished (648/625) comma pumps= &lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo comma pump examples&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;The most common kind of comma pump encountered in &amp;amp;quot;non-xenharmonic&amp;amp;quot; Western music is the meantone comma pump, for which there is a separate page: &amp;lt;a class="wiki_link" href="/Meantone%20comma%20pump%20examples"&amp;gt;Meantone comma pump examples&amp;lt;/a&amp;gt;. These work in any meantone temperament, including &amp;lt;a class="wiki_link" href="/19edo"&amp;gt;19edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/26edo"&amp;gt;26edo&amp;lt;/a&amp;gt;, &amp;lt;a class="wiki_link" href="/31edo"&amp;gt;31edo&amp;lt;/a&amp;gt;, etc. Since standard notation is based on meantone, these don't appear unusual in any way.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
However, comma pumps of other commas tempered out by &amp;lt;a class="wiki_link" href="/12edo"&amp;gt;12edo&amp;lt;/a&amp;gt; are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Augmented (128/125) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Augmented (128/125) comma pumps&amp;lt;/h1&amp;gt;
 &amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;Looney Tunes&amp;lt;/em&amp;gt; theme song, c. 1940. &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=F9S7yhD5M9A" rel="nofollow"&amp;gt;http://www.youtube.com/watch?v=F9S7yhD5M9A&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;em&amp;gt;I Wish&amp;lt;/em&amp;gt;, Skee-Lo, 1995. &amp;lt;a class="wiki_link_ext" href="http://www.youtube.com/watch?v=Zyf0YwUJcqk" rel="nofollow"&amp;gt;http://www.youtube.com/watch?v=Zyf0YwUJcqk&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="Diminished (648/625) comma pumps"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Diminished (648/625) comma pumps&amp;lt;/h1&amp;gt;
&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>jr9l7xx7gfj3tdp21cwluizaw44hs1r</sha1>
    </revision>
    <revision>
      <id>57906</id>
      <parentid>134</parentid>
      <timestamp>2020-12-30T09:40:38Z</timestamp>
      <contributor>
        <username>Lambda Fairy</username>
        <id>260</id>
      </contributor>
      <minor/>
      <comment>Fix Mediawiki syntax</comment>
      <origin>57906</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1806" sha1="4je4p7eb49z93ciel3nd9slsszhdc6x" xml:space="preserve">The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: [[Meantone_comma_pump_examples|Meantone comma pump examples]]. These work in any meantone temperament, including [[19edo|19edo]], [[26edo|26edo]], [[31edo|31edo]], etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by [[12edo|12edo]] are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

=Augmented (128/125) comma pumps=
* ''Looney Tunes'' theme song, c. 1940. [http://www.youtube.com/watch?v=F9S7yhD5M9A http://www.youtube.com/watch?v=F9S7yhD5M9A]
* ''I Wish'', Skee-Lo, 1995. [http://www.youtube.com/watch?v=Zyf0YwUJcqk http://www.youtube.com/watch?v=Zyf0YwUJcqk]
* ''Giant Steps'', John Coltrane 1959 [http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29 http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29] [https://www.youtube.com/watch?v=2kotK9FNEYU https://www.youtube.com/watch?v=2kotK9FNEYU]
** '''Mike Battaglia note:''' you should be aware that, strictly speaking, Giant Steps doesn't actually involve a comma pump unless you deliberately want the "B" in the beginning at the top of the progression to be the same as the "B" at the end at the bottom. If you don't, it doesn't matter; then you can treat it starting at B and going down to Cb, and then ending on top at D#, and then it'll work in temperaments that don't temper out 128/125, such as 19-EDO. The important thing is that nothing drifts with each successive iteration of the head.

=Diminished (648/625) comma pumps=

=Srutal (2048/2025) comma pumps=</text>
      <sha1>4je4p7eb49z93ciel3nd9slsszhdc6x</sha1>
    </revision>
    <revision>
      <id>57907</id>
      <parentid>57906</parentid>
      <timestamp>2020-12-30T09:50:59Z</timestamp>
      <contributor>
        <username>Lambda Fairy</username>
        <id>260</id>
      </contributor>
      <comment>Add Debussy reference</comment>
      <origin>57907</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2005" sha1="mpfjtt0l96182i9vbptcwwjg5mkkgdj" xml:space="preserve">The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: [[Meantone_comma_pump_examples|Meantone comma pump examples]]. These work in any meantone temperament, including [[19edo|19edo]], [[26edo|26edo]], [[31edo|31edo]], etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by [[12edo|12edo]] are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

=Augmented (128/125) comma pumps=
* ''Looney Tunes'' theme song, c. 1940. [http://www.youtube.com/watch?v=F9S7yhD5M9A http://www.youtube.com/watch?v=F9S7yhD5M9A]
* ''I Wish'', Skee-Lo, 1995. [http://www.youtube.com/watch?v=Zyf0YwUJcqk http://www.youtube.com/watch?v=Zyf0YwUJcqk]
* ''Giant Steps'', John Coltrane 1959 [http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29 http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29] [https://www.youtube.com/watch?v=2kotK9FNEYU https://www.youtube.com/watch?v=2kotK9FNEYU]
** '''Mike Battaglia note:''' you should be aware that, strictly speaking, Giant Steps doesn't actually involve a comma pump unless you deliberately want the "B" in the beginning at the top of the progression to be the same as the "B" at the end at the bottom. If you don't, it doesn't matter; then you can treat it starting at B and going down to Cb, and then ending on top at D#, and then it'll work in temperaments that don't temper out 128/125, such as 19-EDO. The important thing is that nothing drifts with each successive iteration of the head.
* ''Doctor Gradus ad Parnassum'', Claude Debussy, 1908. [https://www.youtube.com/watch?v=1k_OLrK4ZuE]
** This piece plays with the enharmonic equivalence between A♭ ([[8/5]]) and G♯ ([[25/16]]).

=Diminished (648/625) comma pumps=

=Srutal (2048/2025) comma pumps=</text>
      <sha1>mpfjtt0l96182i9vbptcwwjg5mkkgdj</sha1>
    </revision>
    <revision>
      <id>57917</id>
      <parentid>57907</parentid>
      <timestamp>2020-12-30T12:44:28Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>shift headings (h1 -&gt; h2), add links</comment>
      <origin>57917</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2125" sha1="fjf76r9es8a15kzf7qum13r8yc8njc3" xml:space="preserve">The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: [[Meantone_comma_pump_examples|Meantone comma pump examples]]. These work in any meantone temperament, including [[19edo|19edo]], [[26edo|26edo]], [[31edo|31edo]], etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by [[12edo|12edo]] are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

== Augmented (128/125) comma pumps ==

Comma pumps based on [[128/125]], the diesis.

* ''Looney Tunes'' theme song, c. 1940. [http://www.youtube.com/watch?v=F9S7yhD5M9A http://www.youtube.com/watch?v=F9S7yhD5M9A]
* ''I Wish'', Skee-Lo, 1995. [http://www.youtube.com/watch?v=Zyf0YwUJcqk http://www.youtube.com/watch?v=Zyf0YwUJcqk]
* ''Giant Steps'', John Coltrane 1959 [http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29 http://en.wikipedia.org/wiki/Giant_Steps_%28composition%29] [https://www.youtube.com/watch?v=2kotK9FNEYU https://www.youtube.com/watch?v=2kotK9FNEYU]
** '''Mike Battaglia note:''' you should be aware that, strictly speaking, Giant Steps doesn't actually involve a comma pump unless you deliberately want the "B" in the beginning at the top of the progression to be the same as the "B" at the end at the bottom. If you don't, it doesn't matter; then you can treat it starting at B and going down to Cb, and then ending on top at D#, and then it'll work in temperaments that don't temper out 128/125, such as 19-EDO. The important thing is that nothing drifts with each successive iteration of the head.
* ''Doctor Gradus ad Parnassum'', Claude Debussy, 1908. [https://www.youtube.com/watch?v=1k_OLrK4ZuE]
** This piece plays with the enharmonic equivalence between A♭ ([[8/5]]) and G♯ ([[25/16]]).

== Diminished (648/625) comma pumps ==

based on [[648/625]], todo.

== Srutal (2048/2025) comma pumps ==

based on [[2048/2025]], todo.</text>
      <sha1>fjf76r9es8a15kzf7qum13r8yc8njc3</sha1>
    </revision>
    <revision>
      <id>75291</id>
      <parentid>57917</parentid>
      <timestamp>2021-07-31T15:25:16Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>/* Augmented (128/125) comma pumps */ Updated two dead links, misc. edits</comment>
      <origin>75291</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1921" sha1="2f7nxwwx1rjtnajx5sbya0m2xqor7l6" xml:space="preserve">The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: [[Meantone_comma_pump_examples|Meantone comma pump examples]]. These work in any meantone temperament, including [[19edo|19edo]], [[26edo|26edo]], [[31edo|31edo]], etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by [[12edo|12edo]] are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

== Augmented (128/125) comma pumps ==

Comma pumps based on [[128/125]], the diesis.

* ''Looney Tunes'' theme song, Carl W. Stalling, c. 1940. https://youtu.be/0jTHNBKjMBU?t=19
* ''I Wish'', Skee-Lo, 1995. https://www.youtube.com/watch?v=cmXZOI7cM0M
* ''Giant Steps'', John Coltrane, 1959. [[Wikipedia: Giant Steps (composition)]], https://www.youtube.com/watch?v=2kotK9FNEYU
** '''Mike Battaglia note:''' you should be aware that, strictly speaking, Giant Steps doesn't actually involve a comma pump unless you deliberately want the "B" in the beginning at the top of the progression to be the same as the "B" at the end at the bottom. If you don't, it doesn't matter; then you can treat it starting at B and going down to Cb, and then ending on top at D#, and then it'll work in temperaments that don't temper out 128/125, such as 19-EDO. The important thing is that nothing drifts with each successive iteration of the head.
* ''Doctor Gradus ad Parnassum'', Claude Debussy, 1908. https://www.youtube.com/watch?v=1k_OLrK4ZuE
** This piece plays with the enharmonic equivalence between A♭ ([[8/5]]) and G♯ ([[25/16]]).

== Diminished (648/625) comma pumps ==

based on [[648/625]], todo.

== Srutal (2048/2025) comma pumps ==

based on [[2048/2025]], todo.</text>
      <sha1>2f7nxwwx1rjtnajx5sbya0m2xqor7l6</sha1>
    </revision>
    <revision>
      <id>93922</id>
      <parentid>75291</parentid>
      <timestamp>2022-08-04T19:27:30Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Categories</comment>
      <origin>93922</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1946" sha1="f4zbilsbmbv8n80zowpsaw0a9hrbd8h" xml:space="preserve">The most common kind of comma pump encountered in "non-xenharmonic" Western music is the meantone comma pump, for which there is a separate page: [[Meantone_comma_pump_examples|Meantone comma pump examples]]. These work in any meantone temperament, including [[19edo|19edo]], [[26edo|26edo]], [[31edo|31edo]], etc. Since standard notation is based on meantone, these don't appear unusual in any way.

However, comma pumps of other commas tempered out by [[12edo|12edo]] are also found. These necessarily involve spelling an interval unusually (for example spelling a major third as a diminished fourth), because they involve an enharmonic pun (e.g. G#~Ab).

== Augmented (128/125) comma pumps ==

Comma pumps based on [[128/125]], the diesis.

* ''Looney Tunes'' theme song, Carl W. Stalling, c. 1940. https://youtu.be/0jTHNBKjMBU?t=19
* ''I Wish'', Skee-Lo, 1995. https://www.youtube.com/watch?v=cmXZOI7cM0M
* ''Giant Steps'', John Coltrane, 1959. [[Wikipedia: Giant Steps (composition)]], https://www.youtube.com/watch?v=2kotK9FNEYU
** '''Mike Battaglia note:''' you should be aware that, strictly speaking, Giant Steps doesn't actually involve a comma pump unless you deliberately want the "B" in the beginning at the top of the progression to be the same as the "B" at the end at the bottom. If you don't, it doesn't matter; then you can treat it starting at B and going down to Cb, and then ending on top at D#, and then it'll work in temperaments that don't temper out 128/125, such as 19-EDO. The important thing is that nothing drifts with each successive iteration of the head.
* ''Doctor Gradus ad Parnassum'', Claude Debussy, 1908. https://www.youtube.com/watch?v=1k_OLrK4ZuE
** This piece plays with the enharmonic equivalence between A♭ ([[8/5]]) and G♯ ([[25/16]]).

== Diminished (648/625) comma pumps ==

based on [[648/625]], todo.

== Srutal (2048/2025) comma pumps ==

based on [[2048/2025]], todo.

[[Category:Comma pump]]</text>
      <sha1>f4zbilsbmbv8n80zowpsaw0a9hrbd8h</sha1>
    </revision>
  </page>
  <page>
    <title>12edo</title>
    <ns>0</ns>
    <id>135</id>
    <revision>
      <id>135</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>135</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9841" sha1="4vhefim5oz5ja9yh1rqmt9mxif47hk7" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[:de:12edo Deutsch]] - [[12平均律|日本語]]&lt;/span&gt;

12[[EDO|EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit|5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals_and_Tuning_Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
[[List_of_12et_rank_two_temperaments_by_badness|List of 12et rank two temperaments by badness]]

[[List_of_12et_rank_two_temperaments_by_complexity|List of 12et rank two temperaments by complexity]]

[[List_of_edo-distinct_12f_rank_two_temperaments|List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperaments
|-
| | 1
| | 1\12
| | [[Ripple|Ripple]]
|-
| | 1
| | 5\12
| | [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| | 2
| | 1\12
| | [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| | 3
| | 1\12
| | [[augmented|Augmented]]
|-
| | 4
| | 1\12
| | [[Diminished|Diminished]]
|-
| | 6
| | 1\12
| | [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 531441/524288
| | | -19 12 &amp;gt;
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 648/625
| | | 3 4 -4 &amp;gt;
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 128/125
| | | 7 0 -3 &amp;gt;
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 81/80
| | | -4 4 -1 &amp;gt;
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| | | 26 -12 -3 &amp;gt;
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 32805/32768
| | | -15 8 1 &amp;gt;
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| | | 161 -84 -12 &amp;gt;
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 36/35
| | | 2 2 -1 -1 &amp;gt;
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| | | 0 -2 5 -3 &amp;gt;
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 126/125
| | | 1 2 -3 1 &amp;gt;
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| | | 5 -4 3 -2 &amp;gt;
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| | | -9 8 -4 2 &amp;gt;
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| | | 6 0 -5 2 &amp;gt;
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| | | 10 -6 1 -1 &amp;gt;
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| | | 25 -14 0 -1 &amp;gt;
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| | | -11 2 7 -3 &amp;gt;
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| | | -4 6 -6 3 &amp;gt;
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 99/98
| | | -1 2 0 -2 1 &amp;gt;
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 100/99
| | | 2 -2 2 0 -1 &amp;gt;
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| | | 4 0 -2 -1 1 &amp;gt;
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| | | 7 -4 0 1 -1 &amp;gt;
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| | | -3 4 -2 -2 2 &amp;gt;
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
&lt;ul&gt;&lt;li&gt;[[MOS_in_12edo|MOS in 12edo]]&lt;/li&gt;&lt;/ul&gt;

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]      [[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>4vhefim5oz5ja9yh1rqmt9mxif47hk7</sha1>
    </revision>
    <revision>
      <id>4595</id>
      <timestamp>2017-07-22T19:00:18Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 615831645 - Original comment: added link (deorphaning)**</comment>
      <origin>4595</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="31052" sha1="qwcgxdanrin5nzzyx890np1bbure1il" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2017-07-22 19:00:18 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;615831645&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;added link (deorphaning)&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[xenharmonie/12edo|Deutsch]] - [[12平均律|日本語]]
&amp;lt;/span&amp;gt;
12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7_4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments== 
[[List of 12et rank two temperaments by badness]]
[[List of 12et rank two temperaments by complexity]]
[[List of edo-distinct 12f rank two temperaments]]
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||

==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Rational ||~ Monzo ||~ Size (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 33554432/33480783 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||

=Scales=
* [[MOS in 12edo]]

=Intervals= 

[[media type="custom" key="24802762"]]

[[file:12ed2-11-001.svg]]


An expanded version of the above, including some higher-limit intervals:
[[media type="custom" key="24802710"]]

[[file:12ed2-19-001e.svg]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/12edo"&amp;gt;Deutsch&amp;lt;/a&amp;gt; - &amp;lt;a class="wiki_link" href="/12%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The seventh partial (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;) is &amp;amp;quot;represented&amp;amp;quot; by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Rank two temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Rank two temperaments&amp;lt;/h2&amp;gt;
 &amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20badness"&amp;gt;List of 12et rank two temperaments by badness&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20complexity"&amp;gt;List of 12et rank two temperaments by complexity&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%20edo-distinct%2012f%20rank%20two%20temperaments"&amp;gt;List of edo-distinct 12f rank two temperaments&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Rational&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;33554432/33480783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Scales"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Scales&amp;lt;/h1&amp;gt;
&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a class="wiki_link" href="/MOS%20in%2012edo"&amp;gt;MOS in 12edo&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;br /&amp;gt;
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 &amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
An expanded version of the above, including some higher-limit intervals:&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802710?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802710&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/12ed2-19-001e.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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      <sha1>qwcgxdanrin5nzzyx890np1bbure1il</sha1>
    </revision>
    <revision>
      <id>4596</id>
      <timestamp>2016-12-29T17:12:11Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 602900782 - Original comment: **</comment>
      <origin>4596</origin>
      <model>wikitext</model>
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      <text bytes="30700" sha1="buqizivcpl82ihll5mqcdjuupti4paa" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2016-12-29 17:12:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;602900782&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[xenharmonie/12edo|Deutsch]] - [[12平均律|日本語]]
&amp;lt;/span&amp;gt;
12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7_4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments== 
[[List of 12et rank two temperaments by badness]]
[[List of 12et rank two temperaments by complexity]]
[[List of edo-distinct 12f rank two temperaments]]
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||

==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Rational ||~ Monzo ||~ Size (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 33554432/33480783 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||

=Intervals= 

[[media type="custom" key="24802762"]]

[[file:12ed2-11-001.svg]]


An expanded version of the above, including some higher-limit intervals:
[[media type="custom" key="24802710"]]

[[file:12ed2-19-001e.svg]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/12edo"&amp;gt;Deutsch&amp;lt;/a&amp;gt; - &amp;lt;a class="wiki_link" href="/12%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The seventh partial (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;) is &amp;amp;quot;represented&amp;amp;quot; by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Rank two temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Rank two temperaments&amp;lt;/h2&amp;gt;
 &amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20badness"&amp;gt;List of 12et rank two temperaments by badness&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20complexity"&amp;gt;List of 12et rank two temperaments by complexity&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%20edo-distinct%2012f%20rank%20two%20temperaments"&amp;gt;List of edo-distinct 12f rank two temperaments&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Rational&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Size (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;33554432/33480783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextFileRule:474:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/12ed2-11-001.svg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@12ed2-11-001.svg&amp;amp;quot; title=&amp;amp;quot;File: 12ed2-11-001.svg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/12ed2-11-001.svg/480709174/12ed2-11-001.svg" onclick="ws.common.trackFileLink('/file/view/12ed2-11-001.svg/480709174/12ed2-11-001.svg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="12ed2-11-001.svg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/12ed2-11-001.svg/480709174/12ed2-11-001.svg" onclick="ws.common.trackFileLink('/file/view/12ed2-11-001.svg/480709174/12ed2-11-001.svg');" class="filename" title="12ed2-11-001.svg"&amp;gt;12ed2-11-001.svg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/12ed2-11-001.svg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/12ed2-11-001.svg/480709174/12ed2-11-001.svg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;13 KB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:474 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
An expanded version of the above, including some higher-limit intervals:&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802710?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802710&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/12ed2-19-001e.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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      <sha1>buqizivcpl82ihll5mqcdjuupti4paa</sha1>
    </revision>
    <revision>
      <id>4597</id>
      <timestamp>2016-12-04T07:07:16Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 601321954 - Original comment: **</comment>
      <origin>4597</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="30544" sha1="imq2osboq43g0x7xnqk93f2n85bo0wz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-12-04 07:07:16 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;601321954&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[xenharmonie/12edo|Deutsch]] - [[12平均律|日本語]]
&amp;lt;/span&amp;gt;
12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7_4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments== 
[[List of 12et rank two temperaments by badness]]
[[List of 12et rank two temperaments by complexity]]
[[List of edo-distinct 12f rank two temperaments]]
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||

==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44/1 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||=   || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||=   || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||=   || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||=   || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||=   || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||=   || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||

=Intervals= 

[[media type="custom" key="24802762"]]

[[file:12ed2-11-001.svg]]


An expanded version of the above, including some higher-limit intervals:
[[media type="custom" key="24802710"]]

[[file:12ed2-19-001e.svg]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/12edo"&amp;gt;Deutsch&amp;lt;/a&amp;gt; - &amp;lt;a class="wiki_link" href="/12%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The seventh partial (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;) is &amp;amp;quot;represented&amp;amp;quot; by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Rank two temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Rank two temperaments&amp;lt;/h2&amp;gt;
 &amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20badness"&amp;gt;List of 12et rank two temperaments by badness&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20complexity"&amp;gt;List of 12et rank two temperaments by complexity&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%20edo-distinct%2012f%20rank%20two%20temperaments"&amp;gt;List of edo-distinct 12f rank two temperaments&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44/1 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
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&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802762?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802762&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" style="width:50%, height:50%" data="http://xenharmonic.wikispaces.com/file/view/12ed2-11-001.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;!-- ws:start:WikiTextFileRule:474:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/file/12ed2-11-001.svg?h=52&amp;amp;amp;w=320&amp;amp;quot; class=&amp;amp;quot;WikiFile&amp;amp;quot; id=&amp;amp;quot;wikitext@@file@@12ed2-11-001.svg&amp;amp;quot; title=&amp;amp;quot;File: 12ed2-11-001.svg&amp;amp;quot; width=&amp;amp;quot;320&amp;amp;quot; height=&amp;amp;quot;52&amp;amp;quot; /&amp;amp;gt; --&amp;gt;&amp;lt;div class="objectEmbed"&amp;gt;&amp;lt;a href="/file/view/12ed2-11-001.svg/480709174/12ed2-11-001.svg" onclick="ws.common.trackFileLink('/file/view/12ed2-11-001.svg/480709174/12ed2-11-001.svg');"&amp;gt;&amp;lt;img src="http://www.wikispaces.com/i/mime/32/empty.png" height="32" width="32" alt="12ed2-11-001.svg" /&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;div&amp;gt;&amp;lt;a href="/file/view/12ed2-11-001.svg/480709174/12ed2-11-001.svg" onclick="ws.common.trackFileLink('/file/view/12ed2-11-001.svg/480709174/12ed2-11-001.svg');" class="filename" title="12ed2-11-001.svg"&amp;gt;12ed2-11-001.svg&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;&amp;lt;ul&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/detail/12ed2-11-001.svg"&amp;gt;Details&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li&amp;gt;&amp;lt;a href="/file/view/12ed2-11-001.svg/480709174/12ed2-11-001.svg"&amp;gt;Download&amp;lt;/a&amp;gt;&amp;lt;/li&amp;gt;&amp;lt;li style="color: #666"&amp;gt;13 KB&amp;lt;/li&amp;gt;&amp;lt;/ul&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;/div&amp;gt;&amp;lt;!-- ws:end:WikiTextFileRule:474 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
An expanded version of the above, including some higher-limit intervals:&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802710?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802710&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/12ed2-19-001e.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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      <sha1>imq2osboq43g0x7xnqk93f2n85bo0wz</sha1>
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    <revision>
      <id>4598</id>
      <timestamp>2016-09-06T15:10:47Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;PiotrGrochowski</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 591168660 - Original comment: **</comment>
      <origin>4598</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="30712" sha1="ezyhdw8zqb8luikrn2qrca48exi4c35" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:PiotrGrochowski|PiotrGrochowski]] and made on &lt;tt&gt;2016-09-06 15:10:47 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;591168660&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[xenharmonie/12edo|Deutsch]] - [[12平均律|日本語]]
&amp;lt;/span&amp;gt;
12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7_4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments== 
[[List of 12et rank two temperaments by badness]]
[[List of 12et rank two temperaments by complexity]]
[[List of edo-distinct 12f rank two temperaments]]
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||

==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 33554432/33480783 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||

=Intervals= 

[[media type="custom" key="24802762"]]

[[file:12ed2-11-001.svg]]


An expanded version of the above, including some higher-limit intervals:
[[media type="custom" key="24802710"]]

[[file:12ed2-19-001e.svg]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/12edo"&amp;gt;Deutsch&amp;lt;/a&amp;gt; - &amp;lt;a class="wiki_link" href="/12%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The seventh partial (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;) is &amp;amp;quot;represented&amp;amp;quot; by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Rank two temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Rank two temperaments&amp;lt;/h2&amp;gt;
 &amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20badness"&amp;gt;List of 12et rank two temperaments by badness&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20complexity"&amp;gt;List of 12et rank two temperaments by complexity&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%20edo-distinct%2012f%20rank%20two%20temperaments"&amp;gt;List of edo-distinct 12f rank two temperaments&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;33554432/33480783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
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An expanded version of the above, including some higher-limit intervals:&amp;lt;br /&amp;gt;
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    <revision>
      <id>4599</id>
      <timestamp>2016-07-27T14:44:36Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;robertinventor</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 588211331 - Original comment: **</comment>
      <origin>4599</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="30678" sha1="mcmx3xvsd1lobl1vzh344nyj60vi0ko" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:robertinventor|robertinventor]] and made on &lt;tt&gt;2016-07-27 14:44:36 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;588211331&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[xenharmonie/12edo|Deutsch]] - [[12平均律|日本語]]
&amp;lt;/span&amp;gt;
12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7_4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments== 
[[List of 12et rank two temperaments by badness]]
[[List of 12et rank two temperaments by complexity]]
[[List of edo-distinct 12f rank two temperaments]]
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||

==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 33554432/33480783 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||

=Intervals= 

[[media type="custom" key="24802762"]]

[[file:12ed2-11-001.svg]]


An expanded version of the above, including some higher-limit intervals:
[[media type="custom" key="24802710"]]

[[file:12ed2-19-001e.svg]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/12edo"&amp;gt;Deutsch&amp;lt;/a&amp;gt; - &amp;lt;a class="wiki_link" href="/12%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The seventh partial (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;) is &amp;amp;quot;represented&amp;amp;quot; by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Rank two temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Rank two temperaments&amp;lt;/h2&amp;gt;
 &amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20badness"&amp;gt;List of 12et rank two temperaments by badness&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20complexity"&amp;gt;List of 12et rank two temperaments by complexity&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%20edo-distinct%2012f%20rank%20two%20temperaments"&amp;gt;List of edo-distinct 12f rank two temperaments&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;33554432/33480783&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
An expanded version of the above, including some higher-limit intervals:&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802710?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802710&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/12ed2-19-001e.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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      <sha1>mcmx3xvsd1lobl1vzh344nyj60vi0ko</sha1>
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    <revision>
      <id>4600</id>
      <timestamp>2015-08-04T09:29:19Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;hstraub</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 556160049 - Original comment: **</comment>
      <origin>4600</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="30660" sha1="gziwv95ziibilxkb4uqfu1mtoe05zaw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2015-08-04 09:29:19 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;556160049&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;[[xenharmonie/12edo|Deutsch]] - [[12平均律|日本語]]
&amp;lt;/span&amp;gt;
12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7_4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments== 
[[List of 12et rank two temperaments by badness]]
[[List of 12et rank two temperaments by complexity]]
[[List of edo-distinct 12f rank two temperaments]]
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||

==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||

=Intervals= 

[[media type="custom" key="24802762"]]

[[file:12ed2-11-001.svg]]


An expanded version of the above, including some higher-limit intervals:
[[media type="custom" key="24802710"]]

[[file:12ed2-19-001e.svg]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;span style="display: block; text-align: right;"&amp;gt;&amp;lt;a class="wiki_link" href="http://xenharmonie.wikispaces.com/12edo"&amp;gt;Deutsch&amp;lt;/a&amp;gt; - &amp;lt;a class="wiki_link" href="/12%E5%B9%B3%E5%9D%87%E5%BE%8B"&amp;gt;日本語&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The seventh partial (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;) is &amp;amp;quot;represented&amp;amp;quot; by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Rank two temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Rank two temperaments&amp;lt;/h2&amp;gt;
 &amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20badness"&amp;gt;List of 12et rank two temperaments by badness&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20complexity"&amp;gt;List of 12et rank two temperaments by complexity&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%20edo-distinct%2012f%20rank%20two%20temperaments"&amp;gt;List of edo-distinct 12f rank two temperaments&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Intervals"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Intervals&amp;lt;/h1&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
An expanded version of the above, including some higher-limit intervals:&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:1:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/24802710?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;24802710&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;object id="example" type="image/svg+xml" data="http://xenharmonic.wikispaces.com/file/view/12ed2-19-001e.svg"&amp;gt;alt : Your browser has no SVG support.&amp;lt;/object&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:1 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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      <sha1>gziwv95ziibilxkb4uqfu1mtoe05zaw</sha1>
    </revision>
    <revision>
      <id>4601</id>
      <timestamp>2014-01-05T13:50:38Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;spt3125</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 480711482 - Original comment: **</comment>
      <origin>4601</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="30261" sha1="hc7moowrgn273vcwj6ckkfe6c3jk9zk" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:spt3125|spt3125]] and made on &lt;tt&gt;2014-01-05 13:50:38 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;480711482&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7_4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments== 
[[List of 12et rank two temperaments by badness]]
[[List of 12et rank two temperaments by complexity]]
[[List of edo-distinct 12f rank two temperaments]]
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||

==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||

=Intervals= 

[[media type="custom" key="24802762"]]

[[file:12ed2-11-001.svg]]


An expanded version of the above, including some higher-limit intervals:
[[media type="custom" key="24802710"]]

[[file:12ed2-19-001e.svg]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The seventh partial (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;) is &amp;amp;quot;represented&amp;amp;quot; by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Rank two temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Rank two temperaments&amp;lt;/h2&amp;gt;
 &amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20badness"&amp;gt;List of 12et rank two temperaments by badness&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20complexity"&amp;gt;List of 12et rank two temperaments by complexity&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%20edo-distinct%2012f%20rank%20two%20temperaments"&amp;gt;List of edo-distinct 12f rank two temperaments&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
An expanded version of the above, including some higher-limit intervals:&amp;lt;br /&amp;gt;
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      <sha1>hc7moowrgn273vcwj6ckkfe6c3jk9zk</sha1>
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    <revision>
      <id>4602</id>
      <timestamp>2012-07-12T14:48:09Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;phylingual</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 352851850 - Original comment: **</comment>
      <origin>4602</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25853" sha1="stk8ped8ri4xenqtheb1v525rwqkuky" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:phylingual|phylingual]] and made on &lt;tt&gt;2012-07-12 14:48:09 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;352851850&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7_4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments== 
[[List of 12et rank two temperaments by badness]]
[[List of 12et rank two temperaments by complexity]]
[[List of edo-distinct 12f rank two temperaments]]
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||
==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The seventh partial (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;) is &amp;amp;quot;represented&amp;amp;quot; by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Rank two temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Rank two temperaments&amp;lt;/h2&amp;gt;
 &amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20badness"&amp;gt;List of 12et rank two temperaments by badness&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20complexity"&amp;gt;List of 12et rank two temperaments by complexity&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%20edo-distinct%2012f%20rank%20two%20temperaments"&amp;gt;List of edo-distinct 12f rank two temperaments&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>stk8ped8ri4xenqtheb1v525rwqkuky</sha1>
    </revision>
    <revision>
      <id>4603</id>
      <timestamp>2012-05-17T17:14:04Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 336798258 - Original comment: **</comment>
      <origin>4603</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25859" sha1="rbul6oc07wb90kzem55oheodqnvzx9z" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-05-17 17:14:04 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;336798258&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7_4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments== 
[[List of 12et rank two temperaments by badness]]
[[List of 12et rank two temperaments by complexity]]
[[List of edo-distinct 12f rank two temperaments]]
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||
==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The seventh partial (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;) is &amp;amp;quot;represented&amp;amp;quot; by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Rank two temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Rank two temperaments&amp;lt;/h2&amp;gt;
 &amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20badness"&amp;gt;List of 12et rank two temperaments by badness&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20complexity"&amp;gt;List of 12et rank two temperaments by complexity&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%20edo-distinct%2012f%20rank%20two%20temperaments"&amp;gt;List of edo-distinct 12f rank two temperaments&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>rbul6oc07wb90kzem55oheodqnvzx9z</sha1>
    </revision>
    <revision>
      <id>4604</id>
      <timestamp>2012-04-23T22:31:26Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;phylingual</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 324439100 - Original comment: **</comment>
      <origin>4604</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25639" sha1="0w2j2ghscjsbnkf6nnwebewpb08a13l" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:phylingual|phylingual]] and made on &lt;tt&gt;2012-04-23 22:31:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;324439100&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7_4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments== 
[[List of 12et rank two temperaments by badness]]
[[List of 12et rank two temperaments by complexity]]
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||
==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The seventh partial (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;) is &amp;amp;quot;represented&amp;amp;quot; by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Rank two temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Rank two temperaments&amp;lt;/h2&amp;gt;
 &amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20badness"&amp;gt;List of 12et rank two temperaments by badness&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20complexity"&amp;gt;List of 12et rank two temperaments by complexity&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>0w2j2ghscjsbnkf6nnwebewpb08a13l</sha1>
    </revision>
    <revision>
      <id>4605</id>
      <timestamp>2012-04-07T08:45:14Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 318361096 - Original comment: **</comment>
      <origin>4605</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25437" sha1="76gtnko7wgc9a4lsgbq318xmgxpxtix" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-04-07 08:45:14 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;318361096&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7_4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments== 
[[List of 12et rank two temperaments by badness]]

||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||
==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The seventh partial (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;) is &amp;amp;quot;represented&amp;amp;quot; by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Rank two temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Rank two temperaments&amp;lt;/h2&amp;gt;
 &amp;lt;a class="wiki_link" href="/List%20of%2012et%20rank%20two%20temperaments%20by%20badness"&amp;gt;List of 12et rank two temperaments by badness&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>76gtnko7wgc9a4lsgbq318xmgxpxtix</sha1>
    </revision>
    <revision>
      <id>4606</id>
      <timestamp>2012-04-05T13:40:46Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 317995458 - Original comment: "Its seventh partial" seemed misunderstandable to me**</comment>
      <origin>4606</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25240" sha1="4sf1tr0o1a3sjtlzgst8rdbq8hzmtiq" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2012-04-05 13:40:46 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;317995458&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;"Its seventh partial" seemed misunderstandable to me&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7_4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Linear temperaments== 
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||
==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
The seventh partial (&amp;lt;a class="wiki_link" href="/7_4"&amp;gt;7/4&amp;lt;/a&amp;gt;) is &amp;amp;quot;represented&amp;amp;quot; by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Linear temperaments&amp;lt;/h2&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>4sf1tr0o1a3sjtlzgst8rdbq8hzmtiq</sha1>
    </revision>
    <revision>
      <id>4607</id>
      <timestamp>2012-04-04T13:08:21Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;keenanpepper</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 317655934 - Original comment: **</comment>
      <origin>4607</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25039" sha1="dtkgezu3ac8smyfyh0r6eobpbrzun7k" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on &lt;tt&gt;2012-04-04 13:08:21 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;317655934&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Linear temperaments== 
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]]/[[injera]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||
==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Linear temperaments&amp;lt;/h2&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/injera"&amp;gt;injera&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>dtkgezu3ac8smyfyh0r6eobpbrzun7k</sha1>
    </revision>
    <revision>
      <id>4608</id>
      <timestamp>2012-02-15T15:57:24Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 302139320 - Original comment: **</comment>
      <origin>4608</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="24971" sha1="3dfl05fp2oflph9wkj0hrqcf43tl6p3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-02-15 15:57:24 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;302139320&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Linear temperaments== 
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||
==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Linear temperaments&amp;lt;/h2&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>3dfl05fp2oflph9wkj0hrqcf43tl6p3</sha1>
    </revision>
    <revision>
      <id>4609</id>
      <timestamp>2012-02-15T07:42:22Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 301936130 - Original comment: **</comment>
      <origin>4609</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="24883" sha1="1ni17jxyopwvx20n3qenwgd2m7o28ma" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2012-02-15 07:42:22 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;301936130&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Linear temperaments== 
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||
==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/12 Pythagorean comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Linear temperaments&amp;lt;/h2&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>1ni17jxyopwvx20n3qenwgd2m7o28ma</sha1>
    </revision>
    <revision>
      <id>4610</id>
      <timestamp>2011-08-14T12:32:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;keenanpepper</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 245874467 - Original comment: **</comment>
      <origin>4610</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="24865" sha1="mq4i2hlic440byisfbkk9gtug0j9ldz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on &lt;tt&gt;2011-08-14 12:32:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;245874467&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Linear temperaments== 
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||
==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Linear temperaments&amp;lt;/h2&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>mq4i2hlic440byisfbkk9gtug0j9ldz</sha1>
    </revision>
    <revision>
      <id>4611</id>
      <timestamp>2011-08-14T03:25:44Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;keenanpepper</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 245840171 - Original comment: **</comment>
      <origin>4611</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="24865" sha1="m5hvkz5kqbrgov2a7jtfe9p5ubo6oon" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on &lt;tt&gt;2011-08-14 03:25:44 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;245840171&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||
==Linear temperaments== 
||~ Periods
per octave ||~ Generator ||~ Temperaments ||
|| 1 || 1\12 || [[Ripple]] ||
|| 1 || 5\12 || [[Meantone]]/[[dominant]] ||
|| 2 || 1\12 || [[Srutal]]/[[pajara]] ||
|| 3 || 1\12 || [[Augmented]] ||
|| 4 || 1\12 || [[Diminished]] ||
|| 6 || 1\12 || [[Hexe]] ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc1"&amp;gt;&amp;lt;a name="x-Linear temperaments"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;Linear temperaments&amp;lt;/h2&amp;gt;
 

&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Periods&amp;lt;br /&amp;gt;
per octave&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Temperaments&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Ripple"&amp;gt;Ripple&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;Meantone&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/dominant"&amp;gt;dominant&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Srutal"&amp;gt;Srutal&amp;lt;/a&amp;gt;/&amp;lt;a class="wiki_link" href="/pajara"&amp;gt;pajara&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Augmented"&amp;gt;Augmented&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Diminished"&amp;gt;Diminished&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;6&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1\12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;a class="wiki_link" href="/Hexe"&amp;gt;Hexe&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>m5hvkz5kqbrgov2a7jtfe9p5ubo6oon</sha1>
    </revision>
    <revision>
      <id>4612</id>
      <timestamp>2011-06-29T09:59:03Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 239314707 - Original comment: **</comment>
      <origin>4612</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22619" sha1="ttflihcrk8zyqrlzh05owibj5f6qn7o" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 09:59:03 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239314707&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Commas== 
12 EDO [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO &amp;lt;a class="wiki_link" href="/tempering%20out"&amp;gt;tempers out&amp;lt;/a&amp;gt; the following &amp;lt;a class="wiki_link" href="/comma"&amp;gt;comma&amp;lt;/a&amp;gt;s. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>ttflihcrk8zyqrlzh05owibj5f6qn7o</sha1>
    </revision>
    <revision>
      <id>4613</id>
      <timestamp>2011-06-29T09:58:18Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 239314595 - Original comment: **</comment>
      <origin>4613</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22485" sha1="km7ttymfibb0w0t9brtpxc40cuq4fu4" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-29 09:58:18 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239314595&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Commas== 
12 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12&amp;lt;a class="wiki_link" href="/EDO"&amp;gt;EDO&amp;lt;/a&amp;gt;, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>km7ttymfibb0w0t9brtpxc40cuq4fu4</sha1>
    </revision>
    <revision>
      <id>4614</id>
      <timestamp>2011-06-26T15:15:02Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 238825979 - Original comment: **</comment>
      <origin>4614</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22444" sha1="s61p7rusua9u43yn8gq7uovq9xj7kbc" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-26 15:15:02 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;238825979&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Commas== 
12 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth &amp;lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&amp;gt;zeta integral edo&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>s61p7rusua9u43yn8gq7uovq9xj7kbc</sha1>
    </revision>
    <revision>
      <id>4615</id>
      <timestamp>2011-06-08T04:09:27Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 235089310 - Original comment: **</comment>
      <origin>4615</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22658" sha1="cj9qvatwlq3fw9r2ur7de31o1k1u7cx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-06-08 04:09:27 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;235089310&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, http://www.research.att.com/~njas/sequences/A117538.

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Commas== 
12 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, &amp;lt;!-- ws:start:WikiTextUrlRule:597:http://www.research.att.com/~njas/sequences/A117538 --&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow"&amp;gt;http://www.research.att.com/~njas/sequences/A117538&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:597 --&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>cj9qvatwlq3fw9r2ur7de31o1k1u7cx</sha1>
    </revision>
    <revision>
      <id>4616</id>
      <timestamp>2011-06-01T23:13:17Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jdfreivald</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 233665780 - Original comment: **</comment>
      <origin>4616</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22642" sha1="akwc7kj7w1d9vghbj3ckrk6n9sv73jh" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2011-06-01 23:13:17 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;233665780&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, http://www.research.att.com/~njas/sequences/A117538.

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Commas== 
12 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||=   || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, &amp;lt;!-- ws:start:WikiTextUrlRule:597:http://www.research.att.com/~njas/sequences/A117538 --&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow"&amp;gt;http://www.research.att.com/~njas/sequences/A117538&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:597 --&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>akwc7kj7w1d9vghbj3ckrk6n9sv73jh</sha1>
    </revision>
    <revision>
      <id>4617</id>
      <timestamp>2011-05-31T21:12:23Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jdfreivald</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 233334188 - Original comment: **</comment>
      <origin>4617</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22671" sha1="15ax2avd8jdsb33neylpwxru5lq3xy9" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2011-05-31 21:12:23 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;233334188&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, http://www.research.att.com/~njas/sequences/A117538.

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Commas== 
12 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||= 1465155/1465142 || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 0 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | 0 -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 0 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 0 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 0 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 0 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 0 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 0 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, &amp;lt;!-- ws:start:WikiTextUrlRule:597:http://www.research.att.com/~njas/sequences/A117538 --&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow"&amp;gt;http://www.research.att.com/~njas/sequences/A117538&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:597 --&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1465155/1465142&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 0 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 0 -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 0 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 0 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 0 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 0 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>15ax2avd8jdsb33neylpwxru5lq3xy9</sha1>
    </revision>
    <revision>
      <id>4618</id>
      <timestamp>2011-05-31T19:35:36Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jdfreivald</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 233313532 - Original comment: **</comment>
      <origin>4618</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="22639" sha1="dc8qd3bo3ell0gp2ih5l4jsg3v0uss3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2011-05-31 19:35:36 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;233313532&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, http://www.research.att.com/~njas/sequences/A117538.

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Commas== 
12 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
||= 531441/524288 || | -19 12 &amp;gt; ||&amp;gt; 23.46 ||= Pythagorean Comma ||=   ||=   ||
||= 648/625 || | 3 4 -4 &amp;gt; ||&amp;gt; 62.57 ||= Major Diesis ||= Diminished Comma ||=   ||
||= 128/125 || | 7 0 -3 &amp;gt; ||&amp;gt; 41.06 ||= Diesis ||= Augmented Comma ||=   ||
||= 81/80 || | -4 4 -1 &amp;gt; ||&amp;gt; 21.51 ||= Syntonic Comma ||= Didymos Comma ||= Meantone Comma ||
||= 2048/2025 || | 11 -4 -2 &amp;gt; ||&amp;gt; 19.55 ||= Diaschisma ||=   ||=   ||
||= 5201701/5149091 || | 26 -12 -3 &amp;gt; ||&amp;gt; 17.60 ||= Misty Comma ||=   ||=   ||
||= 32805/32768 || | -15 8 1 &amp;gt; ||&amp;gt; 1.95 ||= Schisma ||=   ||=   ||
||= 1465155/1465142 || | 161 -84 -12 &amp;gt; ||&amp;gt; 0.02 ||= Atom ||=   ||=   ||
||= 36/35 || | 2 2 -1 -1 &amp;gt; ||&amp;gt; 48.77 ||= Septimal Quarter Tone ||=   ||=   ||
||= 50/49 || | 1 2 -2 &amp;gt; ||&amp;gt; 34.98 ||= Tritonic Diesis ||= Jubilisma ||=   ||
||= 64/63 || | 6 -2 0 -1 &amp;gt; ||&amp;gt; 27.26 ||= Septimal Comma ||= Archytas' Comma ||= Leipziger Komma ||
||= 3125/3087 || | -2 5 -3 &amp;gt; ||&amp;gt; 21.18 ||= Gariboh ||=   ||=   ||
||= 126/125 || | 1 2 -3 1 &amp;gt; ||&amp;gt; 13.79 ||= Septimal Semicomma ||= Starling Comma ||=   ||
||= 4000/3969 || | 5 -4 3 -2 &amp;gt; ||&amp;gt; 13.47 ||= Octagar ||=   ||=   ||
||= 321489/320000 || | -9 8 -4 2 &amp;gt; ||&amp;gt; 8.04 ||= Varunisma ||=   ||=   ||
||= 225/224 || | -5 2 2 -1 &amp;gt; ||&amp;gt; 7.71 ||= Septimal Kleisma ||= Marvel Comma ||=   ||
||= 3136/3125 || | 6 -5 2 &amp;gt; ||&amp;gt; 6.08 ||= Hemimean ||=   ||=   ||
||= 5120/5103 || | 10 -6 1 -1 &amp;gt; ||&amp;gt; 5.76 ||= Hemifamity ||=   ||=   ||
||= 4994735/4983772 || | 25 -14 -1 &amp;gt; ||&amp;gt; 3.80 ||= Garischisma ||=   ||=   ||
||= 703125/702464 || | -11 2 7 -3 &amp;gt; ||&amp;gt; 1.63 ||= Meter ||=   ||=   ||
||= 250047/250000 || | -4 6 -6 3 &amp;gt; ||&amp;gt; 0.33 ||= Landscape Comma ||=   ||=   ||
||= 99/98 || | -1 2 -2 1 &amp;gt; ||&amp;gt; 17.58 ||= Mothwellsma ||=   ||=   ||
||= 100/99 || | 2 -2 2 -1 &amp;gt; ||&amp;gt; 17.40 ||= Ptolemisma ||=   ||=   ||
||= 176/175 || | 4 -2 -1 1 &amp;gt; ||&amp;gt; 9.86 ||= Valinorsma ||=   ||=   ||
||= 896/891 || | 7 -4 0 1 -1 &amp;gt; ||&amp;gt; 9.69 ||= Pentacircle ||=   ||=   ||
||= 441/440 || | -3 2 -1 2 -1 &amp;gt; ||&amp;gt; 3.93 ||= Werckisma ||=   ||=   ||
||= 9801/9800 || | -3 4 -2 -2 2 &amp;gt; ||&amp;gt; 0.18 ||= Kalisma ||= Gauss' Comma ||=   ||
||= 91/90 || | -1 -2 -1 1 1 &amp;gt; ||&amp;gt; 19.13 ||= Superleap ||=   ||=   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, &amp;lt;!-- ws:start:WikiTextUrlRule:597:http://www.research.att.com/~njas/sequences/A117538 --&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow"&amp;gt;http://www.research.att.com/~njas/sequences/A117538&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:597 --&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;1465155/1465142&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: right;"&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td style="text-align: center;"&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>dc8qd3bo3ell0gp2ih5l4jsg3v0uss3</sha1>
    </revision>
    <revision>
      <id>4619</id>
      <timestamp>2011-05-31T17:48:58Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;jdfreivald</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 233291310 - Original comment: Added comma table.**</comment>
      <origin>4619</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18541" sha1="qjlbam450i7kuk7a641i8wgjngada82" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:jdfreivald|jdfreivald]] and made on &lt;tt&gt;2011-05-31 17:48:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;233291310&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;Added comma table.&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, http://www.research.att.com/~njas/sequences/A117538.

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Commas== 
12 EDO tempers out the following commas. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

||~ Comma ||~ Monzo ||~ Value (Cents) ||~ Name 1 ||~ Name 2 ||~ Name 3 ||
|| 531441/524288 || | -19 12 &amp;gt; || 23.46 || Pythagorean Comma ||   ||   ||
|| 648/625 || | 3 4 -4 &amp;gt; || 62.57 || Major Diesis || Diminished Comma ||   ||
|| 128/125 || | 7 0 -3 &amp;gt; || 41.06 || Diesis || Augmented Comma ||   ||
|| 81/80 || | -4 4 -1 &amp;gt; || 21.51 || Syntonic Comma || Didymos Comma || Meantone Comma ||
|| 2048/2025 || | 11 -4 -2 &amp;gt; || 19.55 || Diaschisma ||   ||   ||
|| 5201701/5149091 || | 26 -12 -3 &amp;gt; || 17.60 || Misty Comma ||   ||   ||
|| 32805/32768 || | -15 8 1 &amp;gt; || 1.95 || Schisma ||   ||   ||
|| 1465155/1465142 || | 161 -84 -12 &amp;gt; || 0.02 || Atom ||   ||   ||
|| 36/35 || | 2 2 -1 -1 &amp;gt; || 48.77 || Septimal Quarter Tone ||   ||   ||
|| 50/49 || | 1 2 -2 &amp;gt; || 34.98 || Tritonic Diesis || Jubilisma ||   ||
|| 64/63 || | 6 -2 0 -1 &amp;gt; || 27.26 || Septimal Comma || Archytas' Comma || Leipziger Komma ||
|| 3125/3087 || | -2 5 -3 &amp;gt; || 21.18 || Gariboh ||   ||   ||
|| 126/125 || | 1 2 -3 1 &amp;gt; || 13.79 || Septimal Semicomma || Starling Comma ||   ||
|| 4000/3969 || | 5 -4 3 -2 &amp;gt; || 13.47 || Octagar ||   ||   ||
|| 321489/320000 || | -9 8 -4 2 &amp;gt; || 8.04 || Varunisma ||   ||   ||
|| 225/224 || | -5 2 2 -1 &amp;gt; || 7.71 || Septimal Kleisma || Marvel Comma ||   ||
|| 3136/3125 || | 6 -5 2 &amp;gt; || 6.08 || Hemimean ||   ||   ||
|| 5120/5103 || | 10 -6 1 -1 &amp;gt; || 5.76 || Hemifamity ||   ||   ||
|| 4994735/4983772 || | 25 -14 -1 &amp;gt; || 3.80 || Garischisma ||   ||   ||
|| 703125/702464 || | -11 2 7 -3 &amp;gt; || 1.63 || Meter ||   ||   ||
|| 250047/250000 || | -4 6 -6 3 &amp;gt; || 0.33 || Landscape Comma ||   ||   ||
|| 99/98 || | -1 2 -2 1 &amp;gt; || 17.58 || Mothwellsma ||   ||   ||
|| 100/99 || | 2 -2 2 -1 &amp;gt; || 17.40 || Ptolemisma ||   ||   ||
|| 176/175 || | 4 -2 -1 1 &amp;gt; || 9.86 || Valinorsma ||   ||   ||
|| 896/891 || | 7 -4 0 1 -1 &amp;gt; || 9.69 || Pentacircle ||   ||   ||
|| 441/440 || | -3 2 -1 2 -1 &amp;gt; || 3.93 || Werckisma ||   ||   ||
|| 9801/9800 || | -3 4 -2 -2 2 &amp;gt; || 0.18 || Kalisma || Gauss' Comma ||   ||
|| 91/90 || | -1 -2 -1 1 1 &amp;gt; || 19.13 || Superleap ||   ||   ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, &amp;lt;!-- ws:start:WikiTextUrlRule:597:http://www.research.att.com/~njas/sequences/A117538 --&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow"&amp;gt;http://www.research.att.com/~njas/sequences/A117538&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:597 --&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h2&amp;amp;gt; --&amp;gt;&amp;lt;h2 id="toc0"&amp;gt;&amp;lt;a name="x-Commas"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Commas&amp;lt;/h2&amp;gt;
 12 EDO tempers out the following commas. (Note: This assumes val &amp;amp;lt; 12 19 28 34 42 44 |.)&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th&amp;gt;Comma&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Monzo&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Value (Cents)&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 1&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 2&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;Name 3&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;531441/524288&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -19 12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23.46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Pythagorean Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;648/625&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 3 4 -4 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;62.57&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Major Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diminished Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;128/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 0 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;41.06&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Augmented Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;81/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 4 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21.51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Syntonic Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Didymos Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Meantone Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2048/2025&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 11 -4 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19.55&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Diaschisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5201701/5149091&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 26 -12 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.60&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Misty Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;32805/32768&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -15 8 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1.95&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Schisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1465155/1465142&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 161 -84 -12 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.02&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Atom&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;36/35&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 2 -1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;48.77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Septimal Quarter Tone&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;50/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;34.98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Tritonic Diesis&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Jubilisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;64/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -2 0 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27.26&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Septimal Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Archytas' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Leipziger Komma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3125/3087&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -2 5 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;21.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Gariboh&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;126/125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 1 2 -3 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13.79&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Septimal Semicomma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Starling Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4000/3969&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 5 -4 3 -2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13.47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Octagar&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;321489/320000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -9 8 -4 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8.04&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Varunisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;225/224&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -5 2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7.71&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Septimal Kleisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Marvel Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3136/3125&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 6 -5 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6.08&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemimean&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5120/5103&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 10 -6 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5.76&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Hemifamity&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4994735/4983772&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 25 -14 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3.80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Garischisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;703125/702464&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -11 2 7 -3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1.63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Meter&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;250047/250000&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -4 6 -6 3 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Landscape Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;99/98&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 2 -2 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Mothwellsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;100/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 2 -2 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17.40&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Ptolemisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;176/175&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 4 -2 -1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9.86&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Valinorsma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;896/891&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| 7 -4 0 1 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9.69&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Pentacircle&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;441/440&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 2 -1 2 -1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3.93&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Werckisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;9801/9800&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -3 4 -2 -2 2 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;0.18&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Kalisma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Gauss' Comma&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;91/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;| -1 -2 -1 1 1 &amp;amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19.13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;Superleap&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>qjlbam450i7kuk7a641i8wgjngada82</sha1>
    </revision>
    <revision>
      <id>4620</id>
      <timestamp>2011-03-31T03:12:16Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;xenwolf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 215735736 - Original comment: **</comment>
      <origin>4620</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6112" sha1="qt8ymb1kecb13go870qazvwt6jus3ha" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-03-31 03:12:16 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;215735736&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, http://www.research.att.com/~njas/sequences/A117538.

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/5-limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, &amp;lt;!-- ws:start:WikiTextUrlRule:9:http://www.research.att.com/~njas/sequences/A117538 --&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow"&amp;gt;http://www.research.att.com/~njas/sequences/A117538&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:9 --&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 144742107 - Original comment: **</comment>
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      <text bytes="6148" sha1="1qxmnpghuwhfbeoo1b0suq34f4t8zbk" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2010-05-25 22:48:37 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;144742107&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[Harmonic Limit|5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, http://www.research.att.com/~njas/sequences/A117538.

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12et is the fourth Zeta function integral scale, &amp;lt;!-- ws:start:WikiTextUrlRule:9:http://www.research.att.com/~njas/sequences/A117538 --&amp;gt;&amp;lt;a class="wiki_link_ext" href="http://www.research.att.com/~njas/sequences/A117538" rel="nofollow"&amp;gt;http://www.research.att.com/~njas/sequences/A117538&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextUrlRule:9 --&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 144146183 - Original comment: **</comment>
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      <text bytes="5575" sha1="i7ydznb1hpng534j100ufj8c9r5yutz" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2010-05-24 01:27:56 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;144146183&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[Harmonic Limit|5-limit]] harmony, and because as 1/11 comma meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19.

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edo&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;12EDO, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent &amp;lt;a class="wiki_link" href="/Harmonic%20Limit"&amp;gt;5-limit&amp;lt;/a&amp;gt; harmony, and because as 1/11 comma meantone, it represents &amp;lt;a class="wiki_link" href="/Meantone"&amp;gt;meantone&amp;lt;/a&amp;gt;.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Its seventh partial is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1-5/4-3/2-16/9, and while 12et officially supports septimal meantone via the &amp;lt;a class="wiki_link" href="/Vals%20and%20Tuning%20Space"&amp;gt;val&amp;lt;/a&amp;gt; &amp;amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymas comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal diesis, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>Text replacement - "(\[\[\:(dev|es|purdal|de|en):[^[:space:]\|]*) (.*])" to "$1|$3"</comment>
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      <text bytes="9841" sha1="hzt7newdcx4slzxufn97q8d6fp3baxj" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[:de:12edo|Deutsch]] - [[12平均律|日本語]]&lt;/span&gt;

12[[EDO|EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit|5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals_and_Tuning_Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
[[List_of_12et_rank_two_temperaments_by_badness|List of 12et rank two temperaments by badness]]

[[List_of_12et_rank_two_temperaments_by_complexity|List of 12et rank two temperaments by complexity]]

[[List_of_edo-distinct_12f_rank_two_temperaments|List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperaments
|-
| | 1
| | 1\12
| | [[Ripple|Ripple]]
|-
| | 1
| | 5\12
| | [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| | 2
| | 1\12
| | [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| | 3
| | 1\12
| | [[augmented|Augmented]]
|-
| | 4
| | 1\12
| | [[Diminished|Diminished]]
|-
| | 6
| | 1\12
| | [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 531441/524288
| | | -19 12 &amp;gt;
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 648/625
| | | 3 4 -4 &amp;gt;
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 128/125
| | | 7 0 -3 &amp;gt;
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 81/80
| | | -4 4 -1 &amp;gt;
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| | | 26 -12 -3 &amp;gt;
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 32805/32768
| | | -15 8 1 &amp;gt;
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| | | 161 -84 -12 &amp;gt;
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 36/35
| | | 2 2 -1 -1 &amp;gt;
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| | | 0 -2 5 -3 &amp;gt;
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 126/125
| | | 1 2 -3 1 &amp;gt;
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| | | 5 -4 3 -2 &amp;gt;
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| | | -9 8 -4 2 &amp;gt;
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| | | 6 0 -5 2 &amp;gt;
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| | | 10 -6 1 -1 &amp;gt;
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| | | 25 -14 0 -1 &amp;gt;
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| | | -11 2 7 -3 &amp;gt;
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| | | -4 6 -6 3 &amp;gt;
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 99/98
| | | -1 2 0 -2 1 &amp;gt;
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 100/99
| | | 2 -2 2 0 -1 &amp;gt;
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| | | 4 0 -2 -1 1 &amp;gt;
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| | | 7 -4 0 1 -1 &amp;gt;
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| | | -3 4 -2 -2 2 &amp;gt;
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
&lt;ul&gt;&lt;li&gt;[[MOS_in_12edo|MOS in 12edo]]&lt;/li&gt;&lt;/ul&gt;

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]      [[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>hzt7newdcx4slzxufn97q8d6fp3baxj</sha1>
    </revision>
    <revision>
      <id>33429</id>
      <parentid>32374</parentid>
      <timestamp>2018-09-22T19:23:16Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <comment>the table is 15edo, work in progress</comment>
      <origin>33429</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12549" sha1="3hhxtwqwcjrokn1dch8awzd9on9z5ru" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[:de:12edo|Deutsch]] - [[12平均律|日本語]]&lt;/span&gt;

12[[EDO|EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit|5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals_and_Tuning_Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
[[List_of_12et_rank_two_temperaments_by_badness|List of 12et rank two temperaments by badness]]

[[List_of_12et_rank_two_temperaments_by_complexity|List of 12et rank two temperaments by complexity]]

[[List_of_edo-distinct_12f_rank_two_temperaments|List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperaments
|-
| | 1
| | 1\12
| | [[Ripple|Ripple]]
|-
| | 1
| | 5\12
| | [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| | 2
| | 1\12
| | [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| | 3
| | 1\12
| | [[augmented|Augmented]]
|-
| | 4
| | 1\12
| | [[Diminished|Diminished]]
|-
| | 6
| | 1\12
| | [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 531441/524288
| | | -19 12 &amp;gt;
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 648/625
| | | 3 4 -4 &amp;gt;
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 128/125
| | | 7 0 -3 &amp;gt;
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 81/80
| | | -4 4 -1 &amp;gt;
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| | | 26 -12 -3 &amp;gt;
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 32805/32768
| | | -15 8 1 &amp;gt;
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| | | 161 -84 -12 &amp;gt;
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 36/35
| | | 2 2 -1 -1 &amp;gt;
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| | | 0 -2 5 -3 &amp;gt;
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 126/125
| | | 1 2 -3 1 &amp;gt;
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| | | 5 -4 3 -2 &amp;gt;
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| | | -9 8 -4 2 &amp;gt;
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| | | 6 0 -5 2 &amp;gt;
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| | | 10 -6 1 -1 &amp;gt;
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| | | 25 -14 0 -1 &amp;gt;
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| | | -11 2 7 -3 &amp;gt;
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| | | -4 6 -6 3 &amp;gt;
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 99/98
| | | -1 2 0 -2 1 &amp;gt;
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 100/99
| | | 2 -2 2 0 -1 &amp;gt;
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| | | 4 0 -2 -1 1 &amp;gt;
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| | | 7 -4 0 1 -1 &amp;gt;
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| | | -3 4 -2 -2 2 &amp;gt;
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
&lt;ul&gt;&lt;li&gt;[[MOS_in_12edo|MOS in 12edo]]&lt;/li&gt;&lt;/ul&gt;

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]      [[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).
&lt;!--
{| class="wikitable"
|-
| | '''Interval, complement'''
| | '''Error (abs., in [[cent|cents]])'''
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 3.382
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 4.359
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 5.004
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 7.741
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" | 8.682
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | 8.826
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 9.363
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | 13.686
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 17.508
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | 18.045
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 21.427
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 22.404
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 22.512
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 23.049
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 25.786
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 26.871
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | 27.408
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 30.790
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 31.731
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 36.090
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | 39.472
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 40.557
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 44.916
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 48.298
|}
--&gt;</text>
      <sha1>3hhxtwqwcjrokn1dch8awzd9on9z5ru</sha1>
    </revision>
    <revision>
      <id>33430</id>
      <parentid>33429</parentid>
      <timestamp>2018-09-22T19:32:10Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <comment>sleeping! table partially complete</comment>
      <origin>33430</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12546" sha1="6h39e3mnkevethflv9533l9pv0vlvo2" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[:de:12edo|Deutsch]] - [[12平均律|日本語]]&lt;/span&gt;

12[[EDO|EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit|5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals_and_Tuning_Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
[[List_of_12et_rank_two_temperaments_by_badness|List of 12et rank two temperaments by badness]]

[[List_of_12et_rank_two_temperaments_by_complexity|List of 12et rank two temperaments by complexity]]

[[List_of_edo-distinct_12f_rank_two_temperaments|List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperaments
|-
| | 1
| | 1\12
| | [[Ripple|Ripple]]
|-
| | 1
| | 5\12
| | [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| | 2
| | 1\12
| | [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| | 3
| | 1\12
| | [[augmented|Augmented]]
|-
| | 4
| | 1\12
| | [[Diminished|Diminished]]
|-
| | 6
| | 1\12
| | [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 531441/524288
| | | -19 12 &amp;gt;
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 648/625
| | | 3 4 -4 &amp;gt;
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 128/125
| | | 7 0 -3 &amp;gt;
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 81/80
| | | -4 4 -1 &amp;gt;
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| | | 26 -12 -3 &amp;gt;
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 32805/32768
| | | -15 8 1 &amp;gt;
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| | | 161 -84 -12 &amp;gt;
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 36/35
| | | 2 2 -1 -1 &amp;gt;
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| | | 0 -2 5 -3 &amp;gt;
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 126/125
| | | 1 2 -3 1 &amp;gt;
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| | | 5 -4 3 -2 &amp;gt;
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| | | -9 8 -4 2 &amp;gt;
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| | | 6 0 -5 2 &amp;gt;
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| | | 10 -6 1 -1 &amp;gt;
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| | | 25 -14 0 -1 &amp;gt;
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| | | -11 2 7 -3 &amp;gt;
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| | | -4 6 -6 3 &amp;gt;
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 99/98
| | | -1 2 0 -2 1 &amp;gt;
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 100/99
| | | 2 -2 2 0 -1 &amp;gt;
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| | | 4 0 -2 -1 1 &amp;gt;
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| | | 7 -4 0 1 -1 &amp;gt;
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| | | -3 4 -2 -2 2 &amp;gt;
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
&lt;ul&gt;&lt;li&gt;[[MOS_in_12edo|MOS in 12edo]]&lt;/li&gt;&lt;/ul&gt;

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]      [[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable"
|-
| | '''Interval, complement'''
| | '''Error (abs., in [[cent|cents]])'''
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 4.359
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | 8.826
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | 13.686
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 17.508
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | 18.045
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 22.404
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 22.512
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 26.871
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 31.731
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 36.090
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 40.557
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 34.996
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 35.084
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 36.618
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 36.951
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 38.573
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | 40.528
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" | 48.682
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 50.637
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 52.259
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | 52.592
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 54.214
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 71.702
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 89.210
|}</text>
      <sha1>6h39e3mnkevethflv9533l9pv0vlvo2</sha1>
    </revision>
    <revision>
      <id>33447</id>
      <parentid>33430</parentid>
      <timestamp>2018-09-23T05:10:08Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <comment>/* Selected just intervals by error */ completed with the help of https://scratch.mit.edu/projects/247892491/</comment>
      <origin>33447</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12546" sha1="0mbh16cm9tqg3gjzjj1l2c1fiqh136r" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[:de:12edo|Deutsch]] - [[12平均律|日本語]]&lt;/span&gt;

12[[EDO|EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit|5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals_and_Tuning_Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
[[List_of_12et_rank_two_temperaments_by_badness|List of 12et rank two temperaments by badness]]

[[List_of_12et_rank_two_temperaments_by_complexity|List of 12et rank two temperaments by complexity]]

[[List_of_edo-distinct_12f_rank_two_temperaments|List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperaments
|-
| | 1
| | 1\12
| | [[Ripple|Ripple]]
|-
| | 1
| | 5\12
| | [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| | 2
| | 1\12
| | [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| | 3
| | 1\12
| | [[augmented|Augmented]]
|-
| | 4
| | 1\12
| | [[Diminished|Diminished]]
|-
| | 6
| | 1\12
| | [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 531441/524288
| | | -19 12 &amp;gt;
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 648/625
| | | 3 4 -4 &amp;gt;
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 128/125
| | | 7 0 -3 &amp;gt;
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 81/80
| | | -4 4 -1 &amp;gt;
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| | | 26 -12 -3 &amp;gt;
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 32805/32768
| | | -15 8 1 &amp;gt;
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| | | 161 -84 -12 &amp;gt;
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 36/35
| | | 2 2 -1 -1 &amp;gt;
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| | | 0 -2 5 -3 &amp;gt;
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 126/125
| | | 1 2 -3 1 &amp;gt;
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| | | 5 -4 3 -2 &amp;gt;
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| | | -9 8 -4 2 &amp;gt;
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| | | 6 0 -5 2 &amp;gt;
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| | | 10 -6 1 -1 &amp;gt;
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| | | 25 -14 0 -1 &amp;gt;
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| | | -11 2 7 -3 &amp;gt;
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| | | -4 6 -6 3 &amp;gt;
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 99/98
| | | -1 2 0 -2 1 &amp;gt;
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 100/99
| | | 2 -2 2 0 -1 &amp;gt;
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| | | 4 0 -2 -1 1 &amp;gt;
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| | | 7 -4 0 1 -1 &amp;gt;
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| | | -3 4 -2 -2 2 &amp;gt;
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
&lt;ul&gt;&lt;li&gt;[[MOS_in_12edo|MOS in 12edo]]&lt;/li&gt;&lt;/ul&gt;

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]      [[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable"
|-
| | '''Interval, complement'''
| | '''Error (abs., in [[cent|cents]])'''
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | 1.955
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 3.910
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 11.731
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | 13.686
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 15.641
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 17.488
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 17.508
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 17.596
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 19.443
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | 31.174
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 33.129
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 34.996
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 35.084
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 36.618
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 36.951
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 38.573
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | 40.528
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" | 48.682
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 50.637
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 52.259
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | 52.592
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 54.214
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 71.702
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 89.210
|}</text>
      <sha1>0mbh16cm9tqg3gjzjj1l2c1fiqh136r</sha1>
    </revision>
    <revision>
      <id>33735</id>
      <parentid>33447</parentid>
      <timestamp>2018-09-26T14:22:58Z</timestamp>
      <contributor>
        <username>Hstraub</username>
        <id>11</id>
      </contributor>
      <comment>Language link</comment>
      <origin>33735</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12535" sha1="3v2vi574ualyafduy29njm2ik7tc8ne" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[de:12edo]] [[12平均律|日本語]]&lt;/span&gt;

12[[EDO|EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit|5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals_and_Tuning_Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
[[List_of_12et_rank_two_temperaments_by_badness|List of 12et rank two temperaments by badness]]

[[List_of_12et_rank_two_temperaments_by_complexity|List of 12et rank two temperaments by complexity]]

[[List_of_edo-distinct_12f_rank_two_temperaments|List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperaments
|-
| | 1
| | 1\12
| | [[Ripple|Ripple]]
|-
| | 1
| | 5\12
| | [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| | 2
| | 1\12
| | [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| | 3
| | 1\12
| | [[augmented|Augmented]]
|-
| | 4
| | 1\12
| | [[Diminished|Diminished]]
|-
| | 6
| | 1\12
| | [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 531441/524288
| | | -19 12 &amp;gt;
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 648/625
| | | 3 4 -4 &amp;gt;
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 128/125
| | | 7 0 -3 &amp;gt;
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 81/80
| | | -4 4 -1 &amp;gt;
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| | | 26 -12 -3 &amp;gt;
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 32805/32768
| | | -15 8 1 &amp;gt;
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| | | 161 -84 -12 &amp;gt;
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 36/35
| | | 2 2 -1 -1 &amp;gt;
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| | | 0 -2 5 -3 &amp;gt;
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 126/125
| | | 1 2 -3 1 &amp;gt;
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| | | 5 -4 3 -2 &amp;gt;
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| | | -9 8 -4 2 &amp;gt;
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| | | 6 0 -5 2 &amp;gt;
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| | | 10 -6 1 -1 &amp;gt;
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| | | 25 -14 0 -1 &amp;gt;
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| | | -11 2 7 -3 &amp;gt;
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| | | -4 6 -6 3 &amp;gt;
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 99/98
| | | -1 2 0 -2 1 &amp;gt;
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 100/99
| | | 2 -2 2 0 -1 &amp;gt;
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| | | 4 0 -2 -1 1 &amp;gt;
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| | | 7 -4 0 1 -1 &amp;gt;
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| | | -3 4 -2 -2 2 &amp;gt;
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
&lt;ul&gt;&lt;li&gt;[[MOS_in_12edo|MOS in 12edo]]&lt;/li&gt;&lt;/ul&gt;

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]      [[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable"
|-
| | '''Interval, complement'''
| | '''Error (abs., in [[cent|cents]])'''
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | 1.955
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 3.910
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 11.731
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | 13.686
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 15.641
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 17.488
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 17.508
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 17.596
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 19.443
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | 31.174
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 33.129
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 34.996
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 35.084
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 36.618
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 36.951
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 38.573
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | 40.528
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" | 48.682
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 50.637
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 52.259
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | 52.592
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 54.214
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 71.702
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 89.210
|}</text>
      <sha1>3v2vi574ualyafduy29njm2ik7tc8ne</sha1>
    </revision>
    <revision>
      <id>33939</id>
      <parentid>33735</parentid>
      <timestamp>2018-09-29T18:06:17Z</timestamp>
      <contributor>
        <username>PiotrGrochowski</username>
        <id>5</id>
      </contributor>
      <origin>33939</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12554" sha1="61oiqtd3hvv0egloc9rychexha85bq4" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[de:12edo]][[es:purdal:12-EDD]][[12平均律|日本語]]&lt;/span&gt;

12[[EDO|EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit|5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals_and_Tuning_Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
[[List_of_12et_rank_two_temperaments_by_badness|List of 12et rank two temperaments by badness]]

[[List_of_12et_rank_two_temperaments_by_complexity|List of 12et rank two temperaments by complexity]]

[[List_of_edo-distinct_12f_rank_two_temperaments|List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperaments
|-
| | 1
| | 1\12
| | [[Ripple|Ripple]]
|-
| | 1
| | 5\12
| | [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| | 2
| | 1\12
| | [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| | 3
| | 1\12
| | [[augmented|Augmented]]
|-
| | 4
| | 1\12
| | [[Diminished|Diminished]]
|-
| | 6
| | 1\12
| | [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 531441/524288
| | | -19 12 &amp;gt;
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 648/625
| | | 3 4 -4 &amp;gt;
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 128/125
| | | 7 0 -3 &amp;gt;
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 81/80
| | | -4 4 -1 &amp;gt;
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| | | 26 -12 -3 &amp;gt;
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 32805/32768
| | | -15 8 1 &amp;gt;
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| | | 161 -84 -12 &amp;gt;
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 36/35
| | | 2 2 -1 -1 &amp;gt;
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| | | 0 -2 5 -3 &amp;gt;
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 126/125
| | | 1 2 -3 1 &amp;gt;
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| | | 5 -4 3 -2 &amp;gt;
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| | | -9 8 -4 2 &amp;gt;
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| | | 6 0 -5 2 &amp;gt;
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| | | 10 -6 1 -1 &amp;gt;
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| | | 25 -14 0 -1 &amp;gt;
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| | | -11 2 7 -3 &amp;gt;
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| | | -4 6 -6 3 &amp;gt;
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 99/98
| | | -1 2 0 -2 1 &amp;gt;
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 100/99
| | | 2 -2 2 0 -1 &amp;gt;
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| | | 4 0 -2 -1 1 &amp;gt;
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| | | 7 -4 0 1 -1 &amp;gt;
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| | | -3 4 -2 -2 2 &amp;gt;
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
&lt;ul&gt;&lt;li&gt;[[MOS_in_12edo|MOS in 12edo]]&lt;/li&gt;&lt;/ul&gt;

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]      [[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable"
|-
| | '''Interval, complement'''
| | '''Error (abs., in [[cent|cents]])'''
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | 1.955
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 3.910
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 11.731
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | 13.686
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 15.641
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 17.488
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 17.508
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 17.596
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 19.443
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | 31.174
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 33.129
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 34.996
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 35.084
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 36.618
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 36.951
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 38.573
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | 40.528
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" | 48.682
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 50.637
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 52.259
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | 52.592
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 54.214
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 71.702
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 89.210
|}</text>
      <sha1>61oiqtd3hvv0egloc9rychexha85bq4</sha1>
    </revision>
    <revision>
      <id>33940</id>
      <parentid>33939</parentid>
      <timestamp>2018-09-29T18:34:47Z</timestamp>
      <contributor>
        <username>Mike Battaglia</username>
        <id>3</id>
      </contributor>
      <origin>33940</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12564" sha1="f6k76rgz7zmjqwo41e5fdtfwm52osup" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[de:12edo]][[es:12 EDO]][[purdal:12-EDD]][[12平均律|日本語]]&lt;/span&gt;

12[[EDO|EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit|5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals_and_Tuning_Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
[[List_of_12et_rank_two_temperaments_by_badness|List of 12et rank two temperaments by badness]]

[[List_of_12et_rank_two_temperaments_by_complexity|List of 12et rank two temperaments by complexity]]

[[List_of_edo-distinct_12f_rank_two_temperaments|List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperaments
|-
| | 1
| | 1\12
| | [[Ripple|Ripple]]
|-
| | 1
| | 5\12
| | [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| | 2
| | 1\12
| | [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| | 3
| | 1\12
| | [[augmented|Augmented]]
|-
| | 4
| | 1\12
| | [[Diminished|Diminished]]
|-
| | 6
| | 1\12
| | [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 531441/524288
| | | -19 12 &amp;gt;
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 648/625
| | | 3 4 -4 &amp;gt;
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 128/125
| | | 7 0 -3 &amp;gt;
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 81/80
| | | -4 4 -1 &amp;gt;
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| | | 26 -12 -3 &amp;gt;
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 32805/32768
| | | -15 8 1 &amp;gt;
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| | | 161 -84 -12 &amp;gt;
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 36/35
| | | 2 2 -1 -1 &amp;gt;
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| | | 0 -2 5 -3 &amp;gt;
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 126/125
| | | 1 2 -3 1 &amp;gt;
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| | | 5 -4 3 -2 &amp;gt;
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| | | -9 8 -4 2 &amp;gt;
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| | | 6 0 -5 2 &amp;gt;
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| | | 10 -6 1 -1 &amp;gt;
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| | | 25 -14 0 -1 &amp;gt;
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| | | -11 2 7 -3 &amp;gt;
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| | | -4 6 -6 3 &amp;gt;
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 99/98
| | | -1 2 0 -2 1 &amp;gt;
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 100/99
| | | 2 -2 2 0 -1 &amp;gt;
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| | | 4 0 -2 -1 1 &amp;gt;
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| | | 7 -4 0 1 -1 &amp;gt;
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| | | -3 4 -2 -2 2 &amp;gt;
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
&lt;ul&gt;&lt;li&gt;[[MOS_in_12edo|MOS in 12edo]]&lt;/li&gt;&lt;/ul&gt;

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]      [[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable"
|-
| | '''Interval, complement'''
| | '''Error (abs., in [[cent|cents]])'''
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | 1.955
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 3.910
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 11.731
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | 13.686
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 15.641
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 17.488
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 17.508
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 17.596
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 19.443
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | 31.174
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 33.129
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 34.996
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 35.084
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 36.618
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 36.951
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 38.573
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | 40.528
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" | 48.682
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 50.637
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 52.259
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | 52.592
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 54.214
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 71.702
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 89.210
|}</text>
      <sha1>f6k76rgz7zmjqwo41e5fdtfwm52osup</sha1>
    </revision>
    <revision>
      <id>33941</id>
      <parentid>33940</parentid>
      <timestamp>2018-09-29T18:35:11Z</timestamp>
      <contributor>
        <username>Mike Battaglia</username>
        <id>3</id>
      </contributor>
      <comment>Change to es, we're going to merge purdal into es at some point</comment>
      <origin>33941</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12547" sha1="kkbd7arp5hkv0le4rckdmkdylhqwhpo" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[de:12edo]][[es:12 EDO]][[12平均律|日本語]]&lt;/span&gt;

12[[EDO|EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit|5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[Meantone|meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4|7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals_and_Tuning_Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
[[List_of_12et_rank_two_temperaments_by_badness|List of 12et rank two temperaments by badness]]

[[List_of_12et_rank_two_temperaments_by_complexity|List of 12et rank two temperaments by complexity]]

[[List_of_edo-distinct_12f_rank_two_temperaments|List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! | Periods

per octave
! | Generator
! | Temperaments
|-
| | 1
| | 1\12
| | [[Ripple|Ripple]]
|-
| | 1
| | 5\12
| | [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| | 2
| | 1\12
| | [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| | 3
| | 1\12
| | [[augmented|Augmented]]
|-
| | 4
| | 1\12
| | [[Diminished|Diminished]]
|-
| | 6
| | 1\12
| | [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! | Rational
! | Monzo
! | Size (Cents)
! | Name 1
! | Name 2
! | Name 3
|-
| style="text-align:center;" | 531441/524288
| | | -19 12 &amp;gt;
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 648/625
| | | 3 4 -4 &amp;gt;
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 128/125
| | | 7 0 -3 &amp;gt;
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 81/80
| | | -4 4 -1 &amp;gt;
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| | | 11 -4 -2 &amp;gt;
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| | | 26 -12 -3 &amp;gt;
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 32805/32768
| | | -15 8 1 &amp;gt;
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| | | 161 -84 -12 &amp;gt;
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 36/35
| | | 2 2 -1 -1 &amp;gt;
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 50/49
| | | 1 0 2 -2 &amp;gt;
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 64/63
| | | 6 -2 0 -1 &amp;gt;
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| | | 0 -2 5 -3 &amp;gt;
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 126/125
| | | 1 2 -3 1 &amp;gt;
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| | | 5 -4 3 -2 &amp;gt;
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| | | -9 8 -4 2 &amp;gt;
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 225/224
| | | -5 2 2 -1 &amp;gt;
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| | | 6 0 -5 2 &amp;gt;
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| | | 10 -6 1 -1 &amp;gt;
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| | | 25 -14 0 -1 &amp;gt;
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| | | -11 2 7 -3 &amp;gt;
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| | | -4 6 -6 3 &amp;gt;
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 99/98
| | | -1 2 0 -2 1 &amp;gt;
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 100/99
| | | 2 -2 2 0 -1 &amp;gt;
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| | | 4 0 -2 -1 1 &amp;gt;
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| | | 7 -4 0 1 -1 &amp;gt;
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| | | -3 2 -1 2 -1 &amp;gt;
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| | | -3 4 -2 -2 2 &amp;gt;
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| | | -1 -2 -1 1 0 1 &amp;gt;
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
&lt;ul&gt;&lt;li&gt;[[MOS_in_12edo|MOS in 12edo]]&lt;/li&gt;&lt;/ul&gt;

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]      [[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable"
|-
| | '''Interval, complement'''
| | '''Error (abs., in [[cent|cents]])'''
|-
| style="text-align:center;" | [[4/3|4/3]], [[3/2|3/2]]
| style="text-align:center;" | 1.955
|-
| style="text-align:center;" | [[9/8|9/8]], [[16/9|16/9]]
| style="text-align:center;" | 3.910
|-
| style="text-align:center;" | [[16/15|16/15]], [[15/8|15/8]]
| style="text-align:center;" | 11.731
|-
| style="text-align:center;" | [[5/4|5/4]], [[8/5|8/5]]
| style="text-align:center;" | 13.686
|-
| style="text-align:center;" | [[6/5|6/5]], [[5/3|5/3]]
| style="text-align:center;" | 15.641
|-
| style="text-align:center;" | [[7/5|7/5]], [[10/7|10/7]]
| style="text-align:center;" | 17.488
|-
| style="text-align:center;" | [[14/11|14/11]], [[11/7|11/7]]
| style="text-align:center;" | 17.508
|-
| style="text-align:center;" | [[10/9|10/9]], [[9/5|9/5]]
| style="text-align:center;" | 17.596
|-
| style="text-align:center;" | [[15/14|15/14]], [[28/15|28/15]]
| style="text-align:center;" | 19.443
|-
| style="text-align:center;" | [[8/7|8/7]], [[7/4|7/4]]
| style="text-align:center;" | 31.174
|-
| style="text-align:center;" | [[7/6|7/6]], [[12/7|12/7]]
| style="text-align:center;" | 33.129
|-
| style="text-align:center;" | [[11/10|11/10]], [[20/11|20/11]]
| style="text-align:center;" | 34.996
|-
| style="text-align:center;" | [[9/7|9/7]], [[14/9|14/9]]
| style="text-align:center;" | 35.084
|-
| style="text-align:center;" | [[18/13|18/13]], [[13/9|13/9]]
| style="text-align:center;" | 36.618
|-
| style="text-align:center;" | [[15/11|15/11]], [[22/15|22/15]]
| style="text-align:center;" | 36.951
|-
| style="text-align:center;" | [[13/12|13/12]], [[24/13|24/13]]
| style="text-align:center;" | 38.573
|-
| style="text-align:center;" | [[16/13|16/13]], [[13/8|13/8]]
| style="text-align:center;" | 40.528
|-
| style="text-align:center;" | [[11/8|11/8]], [[16/11|16/11]]
| style="text-align:center;" | 48.682
|-
| style="text-align:center;" | [[12/11|12/11]], [[11/6|11/6]]
| style="text-align:center;" | 50.637
|-
| style="text-align:center;" | [[15/13|15/13]], [[26/15|26/15]]
| style="text-align:center;" | 52.259
|-
| style="text-align:center;" | [[11/9|11/9]], [[18/11|18/11]]
| style="text-align:center;" | 52.592
|-
| style="text-align:center;" | [[13/10|13/10]], [[20/13|20/13]]
| style="text-align:center;" | 54.214
|-
| style="text-align:center;" | [[14/13|14/13]], [[13/7|13/7]]
| style="text-align:center;" | 71.702
|-
| style="text-align:center;" | [[13/11|13/11]], [[22/13|22/13]]
| style="text-align:center;" | 89.210
|}</text>
      <sha1>kkbd7arp5hkv0le4rckdmkdylhqwhpo</sha1>
    </revision>
    <revision>
      <id>34294</id>
      <parentid>33941</parentid>
      <timestamp>2018-10-01T13:39:15Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>reworked: simplified tables, interval links, introduced [[Template:Monzo]], moved interwiki down</comment>
      <origin>34294</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10828" sha1="cqrmxzgvj3ioiyxbj4kpsmo821w6290" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[12平均律|日本語]]&lt;/span&gt;

12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br/&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Size (Cents)
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | 531441/524288
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 648/625
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 128/125
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 81/80
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 32805/32768
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 36/35
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 50/49
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 64/63
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 126/125
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 225/224
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 99/98
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 100/99
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| [[12/11]], [[11/6]]
| 50.637
|-
| [[15/13]], [[26/15]]
| 52.259
|-
| [[11/9]], [[18/11]]
| 52.592
|-
| [[13/10]], [[20/13]]
| 54.214
|-
| [[14/13]], [[13/7]]
| 71.702
|-
| [[13/11]], [[22/13]]
| 89.210
|}

[[Category:edo]]

&lt;!-- interwiki --&gt;
[[de:12edo]]
[[es:12 EDO]]</text>
      <sha1>cqrmxzgvj3ioiyxbj4kpsmo821w6290</sha1>
    </revision>
    <revision>
      <id>34295</id>
      <parentid>34294</parentid>
      <timestamp>2018-10-01T13:48:42Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>moved categories down</comment>
      <origin>34295</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10810" sha1="9ygtaelh16y3oxcle8t1xtgnrgbeqj0" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[12平均律|日本語]]&lt;/span&gt;

12[[EDO]], perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br/&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Size (Cents)
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | 531441/524288
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 648/625
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 128/125
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 81/80
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 32805/32768
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 36/35
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 50/49
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 64/63
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 126/125
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 225/224
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 99/98
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 100/99
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| [[12/11]], [[11/6]]
| 50.637
|-
| [[15/13]], [[26/15]]
| 52.259
|-
| [[11/9]], [[18/11]]
| 52.592
|-
| [[13/10]], [[20/13]]
| 54.214
|-
| [[14/13]], [[13/7]]
| 71.702
|-
| [[13/11]], [[22/13]]
| 89.210
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]

&lt;!-- interwiki --&gt;
[[de:12edo]]
[[es:12 EDO]]</text>
      <sha1>9ygtaelh16y3oxcle8t1xtgnrgbeqj0</sha1>
    </revision>
    <revision>
      <id>36472</id>
      <parentid>34295</parentid>
      <timestamp>2018-10-09T18:17:06Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>typographical changes, separated the [[edo]] link from lemma, added links to existing (comma) interval pages</comment>
      <origin>36472</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10890" sha1="7sa3tgwjnq2c3hl5cxq31ch6jn3bxwm" xml:space="preserve">&lt;span style="display: block; text-align: right;"&gt;[[12平均律|日本語]]&lt;/span&gt;

'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br/&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Size (Cents)
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| [[12/11]], [[11/6]]
| 50.637
|-
| [[15/13]], [[26/15]]
| 52.259
|-
| [[11/9]], [[18/11]]
| 52.592
|-
| [[13/10]], [[20/13]]
| 54.214
|-
| [[14/13]], [[13/7]]
| 71.702
|-
| [[13/11]], [[22/13]]
| 89.210
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]

&lt;!-- interwiki --&gt;
[[de:12edo]]
[[es:12 EDO]]</text>
      <sha1>7sa3tgwjnq2c3hl5cxq31ch6jn3bxwm</sha1>
    </revision>
    <revision>
      <id>36640</id>
      <parentid>36472</parentid>
      <timestamp>2018-10-12T12:00:50Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>introduced [[Template:Japanese version page]]</comment>
      <origin>36640</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10862" sha1="m2fb96oiql9wcgy5vxev8ypemlw5dc8" xml:space="preserve">{{Japanese version page| [[12平均律|日本語]] }}

'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br/&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Size (Cents)
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| [[12/11]], [[11/6]]
| 50.637
|-
| [[15/13]], [[26/15]]
| 52.259
|-
| [[11/9]], [[18/11]]
| 52.592
|-
| [[13/10]], [[20/13]]
| 54.214
|-
| [[14/13]], [[13/7]]
| 71.702
|-
| [[13/11]], [[22/13]]
| 89.210
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]

&lt;!-- interwiki --&gt;
[[de:12edo]]
[[es:12 EDO]]</text>
      <sha1>m2fb96oiql9wcgy5vxev8ypemlw5dc8</sha1>
    </revision>
    <revision>
      <id>36719</id>
      <parentid>36640</parentid>
      <timestamp>2018-10-12T21:56:17Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>replaced interwiki template</comment>
      <origin>36719</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10822" sha1="8krzyu0fxp5pxwcb5bjg89yf139crl7" xml:space="preserve">{{interwiki
| de = 12edo
| es = 12 EDO
| ja = 12平均律
}}

'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br/&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Size (Cents)
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| [[12/11]], [[11/6]]
| 50.637
|-
| [[15/13]], [[26/15]]
| 52.259
|-
| [[11/9]], [[18/11]]
| 52.592
|-
| [[13/10]], [[20/13]]
| 54.214
|-
| [[14/13]], [[13/7]]
| 71.702
|-
| [[13/11]], [[22/13]]
| 89.210
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>8krzyu0fxp5pxwcb5bjg89yf139crl7</sha1>
    </revision>
    <revision>
      <id>36744</id>
      <parentid>36719</parentid>
      <timestamp>2018-10-12T22:38:36Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>unified interwiki section for all related pages possible</comment>
      <origin>36744</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10835" sha1="62bad8kn73s5z77x7ijnp389fo679li" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}

'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br/&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Size (Cents)
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| [[12/11]], [[11/6]]
| 50.637
|-
| [[15/13]], [[26/15]]
| 52.259
|-
| [[11/9]], [[18/11]]
| 52.592
|-
| [[13/10]], [[20/13]]
| 54.214
|-
| [[14/13]], [[13/7]]
| 71.702
|-
| [[13/11]], [[22/13]]
| 89.210
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>62bad8kn73s5z77x7ijnp389fo679li</sha1>
    </revision>
    <revision>
      <id>37021</id>
      <parentid>36744</parentid>
      <timestamp>2018-10-14T22:38:49Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>removed newline at article start</comment>
      <origin>37021</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10834" sha1="ow2zatdfi43h5wyl8z7omg4jwsg7jg9" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br/&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Size (Cents)
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| [[12/11]], [[11/6]]
| 50.637
|-
| [[15/13]], [[26/15]]
| 52.259
|-
| [[11/9]], [[18/11]]
| 52.592
|-
| [[13/10]], [[20/13]]
| 54.214
|-
| [[14/13]], [[13/7]]
| 71.702
|-
| [[13/11]], [[22/13]]
| 89.210
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>ow2zatdfi43h5wyl8z7omg4jwsg7jg9</sha1>
    </revision>
    <revision>
      <id>39844</id>
      <parentid>37021</parentid>
      <timestamp>2019-01-29T22:25:08Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Selected just intervals by error */</comment>
      <origin>39844</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10874" sha1="ns9qiqh4wlhnm72fh11y9lvns0bfijw" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Size (Cents)
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>ns9qiqh4wlhnm72fh11y9lvns0bfijw</sha1>
    </revision>
    <revision>
      <id>41936</id>
      <parentid>39844</parentid>
      <timestamp>2019-06-14T19:56:12Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Commas */</comment>
      <origin>41936</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10966" sha1="5kyc527hdt0w7xnmzlgqzlmv00oahnv" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Size (Cents)
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|''32805/32768'']]
| ''{{Monzo| -15 8 1 }}''
| style="text-align:right;" | ''1.95''
| style="text-align:center;" | ''Schisma''
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | ''703125/702464''
| ''{{Monzo| -11 2 7 -3 }}''
| style="text-align:right;" | ''1.63''
| style="text-align:center;" | ''Meter''
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | ''250047/250000''
| ''{{Monzo| -4 6 -6 3 }}''
| style="text-align:right;" | ''0.33''
| style="text-align:center;" | ''Landscape Comma''
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | ''9801/9800''
| ''{{Monzo| -3 4 -2 -2 2 }}''
| style="text-align:right;" | ''0.18''
| style="text-align:center;" | ''Kalisma''
| style="text-align:center;" | ''Gauss' Comma''
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| ''{{Monzo| 161 -84 -12 }}''
| style="text-align:right;" | ''0.02''
| style="text-align:center;" | ''Atom''
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>5kyc527hdt0w7xnmzlgqzlmv00oahnv</sha1>
    </revision>
    <revision>
      <id>42087</id>
      <parentid>41936</parentid>
      <timestamp>2019-07-05T11:29:29Z</timestamp>
      <contributor>
        <username>Jkarimak</username>
        <id>136</id>
      </contributor>
      <minor/>
      <comment>/* Selected just intervals by error */</comment>
      <origin>42087</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10926" sha1="mhexoz288419zmn1d9xjmcp3exp1lvd" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Size (Cents)
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|''32805/32768'']]
| ''{{Monzo| -15 8 1 }}''
| style="text-align:right;" | ''1.95''
| style="text-align:center;" | ''Schisma''
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | ''703125/702464''
| ''{{Monzo| -11 2 7 -3 }}''
| style="text-align:right;" | ''1.63''
| style="text-align:center;" | ''Meter''
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | ''250047/250000''
| ''{{Monzo| -4 6 -6 3 }}''
| style="text-align:right;" | ''0.33''
| style="text-align:center;" | ''Landscape Comma''
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | ''9801/9800''
| ''{{Monzo| -3 4 -2 -2 2 }}''
| style="text-align:right;" | ''0.18''
| style="text-align:center;" | ''Kalisma''
| style="text-align:center;" | ''Gauss' Comma''
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| ''{{Monzo| 161 -84 -12 }}''
| style="text-align:right;" | ''0.02''
| style="text-align:center;" | ''Atom''
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>mhexoz288419zmn1d9xjmcp3exp1lvd</sha1>
    </revision>
    <revision>
      <id>42594</id>
      <parentid>42087</parentid>
      <timestamp>2019-12-12T12:20:03Z</timestamp>
      <contributor>
        <username>Jkarimak</username>
        <id>136</id>
      </contributor>
      <minor/>
      <comment>/* Commas */</comment>
      <origin>42594</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10866" sha1="jaayveoc0tqwu4euj1nxu0qrrrst9q0" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Size (Cents)
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| ''{{Monzo| -15 8 1 }}''
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| ''{{Monzo| -11 2 7 -3 }}''
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| ''{{Monzo| -4 6 -6 3 }}''
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| ''{{Monzo| -3 4 -2 -2 2 }}''
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| ''{{Monzo| 161 -84 -12 }}''
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>jaayveoc0tqwu4euj1nxu0qrrrst9q0</sha1>
    </revision>
    <revision>
      <id>42595</id>
      <parentid>42594</parentid>
      <timestamp>2019-12-12T12:20:52Z</timestamp>
      <contributor>
        <username>Jkarimak</username>
        <id>136</id>
      </contributor>
      <minor/>
      <comment>/* Commas */</comment>
      <origin>42595</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10859" sha1="k3h2ny5bwbkj2swcf898yttgz57dmsv" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| ''{{Monzo| -15 8 1 }}''
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| ''{{Monzo| -11 2 7 -3 }}''
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| ''{{Monzo| -4 6 -6 3 }}''
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| ''{{Monzo| -3 4 -2 -2 2 }}''
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| ''{{Monzo| 161 -84 -12 }}''
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>k3h2ny5bwbkj2swcf898yttgz57dmsv</sha1>
    </revision>
    <revision>
      <id>42596</id>
      <parentid>42595</parentid>
      <timestamp>2019-12-12T12:21:35Z</timestamp>
      <contributor>
        <username>Jkarimak</username>
        <id>136</id>
      </contributor>
      <minor/>
      <comment>/* Commas */</comment>
      <origin>42596</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10839" sha1="irxsc8w0jbv88lm5zx9cido62ggc1ga" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>irxsc8w0jbv88lm5zx9cido62ggc1ga</sha1>
    </revision>
    <revision>
      <id>42763</id>
      <parentid>42596</parentid>
      <timestamp>2019-12-13T07:50:49Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>/* Commas */  added color names</comment>
      <origin>42763</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11151" sha1="nzxii7rz0wvqecnm5ipbcxazniy54es" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
|Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
|Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
|Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
|Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
|Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
|Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
|Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
|Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
|Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
|Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
|Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
|Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
|Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
|Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
|Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
|Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
|Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
|Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
|Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
|Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
|Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
|Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
|Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
|Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
|Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
|Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
|Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
|Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| [[4/3]], [[3/2]]
| 1.955
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| [[5/4]], [[8/5]]
| 13.686
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| [[8/7]], [[7/4]]
| 31.174
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| [[16/13]], [[13/8]]
| 40.528
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| [[11/8]], [[16/11]]
| 48.682
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>nzxii7rz0wvqecnm5ipbcxazniy54es</sha1>
    </revision>
    <revision>
      <id>42819</id>
      <parentid>42763</parentid>
      <timestamp>2019-12-15T07:17:33Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>added an interval table</comment>
      <origin>42819</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12013" sha1="fhck44bhomousv5f23wvxa4gu4i8pl5" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!7-limit Ratios*
|-
|0
|0
|unison
|P1
|D
|1/1
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 21/20, 25/24, 28/27
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 28/15, 40/21, 48/25, 27/14
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7 subgroup temperament; other approaches are possible.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
|Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
|Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
|Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
|Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
|Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
|Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
|Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
|Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
|Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
|Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
|Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
|Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
|Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
|Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
|Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
|Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
|Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
|Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
|Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
|Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
|Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
|Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
|Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
|Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
|Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
|Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
|Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
|Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>fhck44bhomousv5f23wvxa4gu4i8pl5</sha1>
    </revision>
    <revision>
      <id>42857</id>
      <parentid>42819</parentid>
      <timestamp>2019-12-16T18:30:10Z</timestamp>
      <contributor>
        <username>Jkarimak</username>
        <id>136</id>
      </contributor>
      <comment>/* Intervals */ Added higher prime limit intervals to a separate column</comment>
      <origin>42857</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12077" sha1="525ovlp0sk84tab4pv3t67uxen4ie32" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!7-limit Ratios*
!Other
|-
|0
|0
|unison
|P1
|D
|1/1
|
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 21/20, 25/24, 28/27
|18/17
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9
|
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5
|19/16
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7
|17/12, 12/17
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7
|32/19
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 28/15, 40/21, 48/25, 27/14
|17/9
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7 subgroup temperament; other approaches are possible.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
|Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
|Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
|Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
|Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
|Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
|Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
|Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
|Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
|Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
|Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
|Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
|Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
|Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
|Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
|Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
|Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
|Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
|Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
|Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
|Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
|Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
|Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
|Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
|Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
|Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
|Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
|Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
|Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>525ovlp0sk84tab4pv3t67uxen4ie32</sha1>
    </revision>
    <revision>
      <id>42881</id>
      <parentid>42857</parentid>
      <timestamp>2019-12-22T02:24:47Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>/* Intervals */  merged the two columns of JI ratios into one</comment>
      <origin>42881</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12879" sha1="rhf2v5329utnlqk2a861xei2jmh7ha4" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
|0
|0
|unison
|P1
|D
|1/1
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 18/17, 21/20, 25/24, 28/27
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5, 19/16
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7, 17/12, 24/17
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7, 32/19
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" |Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" |Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" |Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" |Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" |Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" |Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" |Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" |Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" |Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" |Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" |Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" |Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" |Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" |Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" |Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" |Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" |Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" |Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" |Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" |Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).

{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>rhf2v5329utnlqk2a861xei2jmh7ha4</sha1>
    </revision>
    <revision>
      <id>43932</id>
      <parentid>42881</parentid>
      <timestamp>2020-04-17T08:42:12Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>/* Selected just intervals by error */ added a table that shows each prime's error and fifthspan</comment>
      <origin>43932</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13098" sha1="jssqfb06hffa495cont4lfvkg5tb44z" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
|0
|0
|unison
|P1
|D
|1/1
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 18/17, 21/20, 25/24, 28/27
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5, 19/16
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7, 17/12, 24/17
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7, 32/19
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

==Rank two temperaments==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

==Commas==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" |Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" |Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" |Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" |Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" |Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" |Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" |Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" |Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" |Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" |Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" |Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" |Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" |Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" |Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" |Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" |Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" |Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" |Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" |Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" |Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

=Scales=
* [[MOS in 12edo]]

=Intervals=

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

==Selected just intervals by error==
{| class="wikitable" style="text-align:center;"
!
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
|-
!error
|0¢
| -1.96¢
| +13.7¢
| +31.2¢
| +48.7¢
| -40.5¢
|-
![[fifthspan]]
|0
| +1
| +4
| -2
| +6
| -4
|}
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).
{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>jssqfb06hffa495cont4lfvkg5tb44z</sha1>
    </revision>
    <revision>
      <id>45244</id>
      <parentid>43932</parentid>
      <timestamp>2020-05-29T09:58:38Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>fixed heading structure</comment>
      <origin>45244</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13114" sha1="l05okl95hfl5fvgut9223w3t3nxcozb" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
|0
|0
|unison
|P1
|D
|1/1
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 18/17, 21/20, 25/24, 28/27
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5, 19/16
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7, 17/12, 24/17
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7, 32/19
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" |Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" |Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" |Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" |Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" |Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" |Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" |Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" |Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" |Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" |Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" |Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" |Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" |Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" |Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" |Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" |Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" |Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" |Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" |Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" |Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

== Scales ==
* [[MOS in 12edo]]

== Intervals ==

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
|-
!error
|0¢
| -1.96¢
| +13.7¢
| +31.2¢
| +48.7¢
| -40.5¢
|-
![[fifthspan]]
|0
| +1
| +4
| -2
| +6
| -4
|}
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).
{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>l05okl95hfl5fvgut9223w3t3nxcozb</sha1>
    </revision>
    <revision>
      <id>45245</id>
      <parentid>45244</parentid>
      <timestamp>2020-05-29T10:05:12Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>45245</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13128" sha1="cjqlg463yy40i1zneiu6b5yxmd1lgnj" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
|0
|0
|unison
|P1
|D
|1/1
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 18/17, 21/20, 25/24, 28/27
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9, 17/15, 19/17
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5, 19/16
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7, 17/12, 24/17
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7, 32/19
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" |Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" |Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" |Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" |Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" |Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" |Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" |Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" |Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" |Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" |Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" |Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" |Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" |Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" |Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" |Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" |Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" |Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" |Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" |Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" |Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

== Scales ==
* [[MOS in 12edo]]

== Intervals ==

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
|-
!error
|0¢
| -1.96¢
| +13.7¢
| +31.2¢
| +48.7¢
| -40.5¢
|-
![[fifthspan]]
|0
| +1
| +4
| -2
| +6
| -4
|}
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).
{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>cjqlg463yy40i1zneiu6b5yxmd1lgnj</sha1>
    </revision>
    <revision>
      <id>45246</id>
      <parentid>45245</parentid>
      <timestamp>2020-05-29T10:07:12Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>45246</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13135" sha1="9979rb6czru739rnqpro2sk77fsy6z5" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
|0
|0
|unison
|P1
|D
|1/1
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9, 17/15, 19/17
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5, 19/16
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7, 17/12, 24/17
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7, 32/19
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" |Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" |Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" |Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" |Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" |Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" |Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" |Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" |Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" |Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" |Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" |Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" |Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" |Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" |Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" |Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" |Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" |Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" |Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" |Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" |Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

== Scales ==
* [[MOS in 12edo]]

== Intervals ==

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
|-
!error
|0¢
| -1.96¢
| +13.7¢
| +31.2¢
| +48.7¢
| -40.5¢
|-
![[fifthspan]]
|0
| +1
| +4
| -2
| +6
| -4
|}
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).
{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>9979rb6czru739rnqpro2sk77fsy6z5</sha1>
    </revision>
    <revision>
      <id>45248</id>
      <parentid>45246</parentid>
      <timestamp>2020-05-29T10:11:31Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>45248</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13134" sha1="110jmjryg99sdv87v1qjk2infl51g73" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[Vals and Tuning Space|val]] &amp;lt;12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
|0
|0
|unison
|P1
|D
|1/1
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9, 17/15, 19/17
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5, 19/16
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7, 17/12, 24/17
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7, 32/19
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes val &amp;lt; 12 19 28 34 42 44 |.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" |Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" |Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" |Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" |Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" |Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" |Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" |Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" |Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" |Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" |Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" |Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" |Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" |Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" |Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" |Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" |Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" |Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" |Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" |Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" |Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

== Scales ==
* [[MOS in 12edo]]

== Intervals ==

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
|-
!error
|0¢
| -1.96¢
| +13.7¢
| +31.2¢
| +48.7¢
| -40.5¢
|-
![[fifthspan]]
|0
| +1
| +4
| -2
| +6
| -4
|}
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).
{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>110jmjryg99sdv87v1qjk2infl51g73</sha1>
    </revision>
    <revision>
      <id>45271</id>
      <parentid>45248</parentid>
      <timestamp>2020-05-29T17:23:39Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>use of [[template:val]] + link [[val]]</comment>
      <origin>45271</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13122" sha1="igfjo4bl24m60ce8umvtvv0w2gqbeds" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
|0
|0
|unison
|P1
|D
|1/1
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9, 17/15, 19/17
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5, 19/16
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7, 17/12, 24/17
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7, 32/19
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" |Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" |Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" |Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" |Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" |Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" |Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" |Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" |Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" |Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" |Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" |Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" |Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" |Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" |Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" |Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" |Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" |Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" |Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" |Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" |Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

== Scales ==
* [[MOS in 12edo]]

== Intervals ==

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
|-
!error
|0¢
| -1.96¢
| +13.7¢
| +31.2¢
| +48.7¢
| -40.5¢
|-
![[fifthspan]]
|0
| +1
| +4
| -2
| +6
| -4
|}
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).
{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>igfjo4bl24m60ce8umvtvv0w2gqbeds</sha1>
    </revision>
    <revision>
      <id>46183</id>
      <parentid>45271</parentid>
      <timestamp>2020-06-09T16:54:31Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>46183</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13116" sha1="phc79azbv7ef8wh97lsk2278yj2unid" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
|0
|0
|unison
|P1
|D
|1/1
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9, 17/15, 19/17
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5, 19/16
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7, 17/12, 24/17
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7, 32/19
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" |Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" |Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" |Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" |Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" |Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" |Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" |Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" |Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" |Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" |Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" |Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" |Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" |Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" |Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" |Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" |Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" |Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" |Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" |Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" |Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

== Scales ==
* [[MOS in 12edo]]

== Intervals ==

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
|-
!error
|0¢
| -1.96¢
| +13.7¢
| +31.2¢
| +48.7¢
| -40.5¢
|-
![[fifthspan]]
|0
| +1
| +4
| -2
| +6
| -4
|}
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).
{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[15/13]], [[26/15]]
| 2.259
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>phc79azbv7ef8wh97lsk2278yj2unid</sha1>
    </revision>
    <revision>
      <id>47783</id>
      <parentid>46183</parentid>
      <timestamp>2020-07-19T21:08:32Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <comment>what is 15/</comment>
      <origin>47783</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13117" sha1="rj5e0xjnkn5974uu02ihfpv9onkitbg" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
|0
|0
|unison
|P1
|D
|1/1
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9, 17/15, 19/17
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5, 19/16
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7, 17/12, 24/17
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7, 32/19
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" |Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" |Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" |Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" |Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" |Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" |Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" |Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" |Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" |Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" |Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" |Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" |Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" |Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" |Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" |Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" |Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" |Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" |Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" |Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" |Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

== Scales ==
* [[MOS in 12edo]]

== Intervals ==

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
|-
!error
|0¢
| -1.96¢
| +13.7¢
| +31.2¢
| +48.7¢
| -40.5¢
|-
![[fifthspan]]
|0
| +1
| +4
| -2
| +6
| -4
|}
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).
{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| [[15/13]], [[26/15]]
| 47.741
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>rj5e0xjnkn5974uu02ihfpv9onkitbg</sha1>
    </revision>
    <revision>
      <id>47784</id>
      <parentid>47783</parentid>
      <timestamp>2020-07-19T21:10:32Z</timestamp>
      <contributor>
        <username>Yourmusic Productions</username>
        <id>199</id>
      </contributor>
      <comment>What is 15/13 doing in the wrong place on the error table?</comment>
      <origin>47784</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="13118" sha1="kictdzgu5yg5opbjavd4zk2xyg6ieh4" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
|0
|0
|unison
|P1
|D
|1/1
|-
|1
|100
|aug 1sn, minor 2nd
|A1, m2
|D#, Eb
|15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
|2
|200
|major 2nd
|M2
|E
|8/7, 9/8, 10/9, 17/15, 19/17
|-
|3
|300
|minor 3rd
|m3
|F
|7/6, 6/5, 19/16
|-
|4
|400
|major 3rd
|M3
|F#
|5/4, 9/7
|-
|5
|500
|perfect 4th
|P4
|G
|4/3
|-
|6
|600
|aug 4th, dim 5th
|A4, d5
|G#, Ab
|7/5, 10/7, 17/12, 24/17
|-
|7
|700
|perfect 5th
|P5
|A
|3/2
|-
|8
|800
|minor 6th
|m6
|Bb
|8/5, 14/9
|-
|9
|900
|major 6th
|M6
|B
|5/3, 12/7, 32/19
|-
|10
|1000
|minor 7th
|m7
|C
|7/4, 9/5, 16/9
|-
|11
|1100
|major 7th
|M7
|C#
|15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
|19
|1200
|perfect 8ve
|P8
|D
|2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple|Ripple]]
|-
| 1
| 5\12
| [[Meantone|Meantone]]/[[dominant|dominant]]
|-
| 2
| 1\12
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]]
|-
| 3
| 1\12
| [[augmented|Augmented]]
|-
| 4
| 1\12
| [[Diminished|Diminished]]
|-
| 6
| 1\12
| [[Hexe|Hexe]]
|}

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|comma]]s. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| style="text-align:center;" | [[648/625]]
| {{Monzo| 3 4 -4 }}
| style="text-align:right;" | 62.57
| style="text-align:center;" |Quadgu
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| style="text-align:right;" | 48.77
| style="text-align:center;" |Rugu
| style="text-align:center;" | Septimal Quarter Tone
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[128/125]]
| {{Monzo| 7 0 -3 }}
| style="text-align:right;" | 41.06
| style="text-align:center;" |Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| style="text-align:right;" | 34.98
| style="text-align:center;" |Biruyo
| style="text-align:center;" | Tritonic Diesis
| style="text-align:center;" | Jubilisma
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| style="text-align:right;" | 27.26
| style="text-align:center;" |Ru
| style="text-align:center;" | Septimal Comma
| style="text-align:center;" | Archytas' Comma
| style="text-align:center;" | Leipziger Komma
|-
| style="text-align:center;" | [[531441/524288]]
| {{Monzo| -19 12 }}
| style="text-align:right;" | 23.46
| style="text-align:center;" |Lalawa
| style="text-align:center;" | Pythagorean Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
| style="text-align:right;" | 21.51
| style="text-align:center;" |Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| style="text-align:right;" | 21.18
| style="text-align:center;" |Triru-aquinyo
| style="text-align:center;" | Gariboh
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 2048/2025
| {{Monzo| 11 -4 -2 }}
| style="text-align:right;" | 19.55
| style="text-align:center;" |Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| style="text-align:right;" | 19.13
| style="text-align:center;" |Thozogu
| style="text-align:center;" | Superleap
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| style="text-align:right;" | 17.60
| style="text-align:center;" |Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| style="text-align:right;" | 17.58
| style="text-align:center;" |Loruru
| style="text-align:center;" | Mothwellsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| style="text-align:right;" | 17.40
| style="text-align:center;" |Luyoyo
| style="text-align:center;" | Ptolemisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| style="text-align:right;" | 13.79
| style="text-align:center;" |Zotrigu
| style="text-align:center;" | Septimal Semicomma
| style="text-align:center;" | Starling Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| style="text-align:right;" | 13.47
| style="text-align:center;" |Rurutriyo
| style="text-align:center;" | Octagar
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| style="text-align:right;" | 9.86
| style="text-align:center;" |Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| style="text-align:right;" | 9.69
| style="text-align:center;" |Saluzo
| style="text-align:center;" | Pentacircle
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 321489/320000
| {{Monzo| -9 8 -4 2 }}
| style="text-align:right;" | 8.04
| style="text-align:center;" |Labizogugu
| style="text-align:center;" | Varunisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| style="text-align:right;" | 7.71
| style="text-align:center;" |Ruyoyo
| style="text-align:center;" | Septimal Kleisma
| style="text-align:center;" | Marvel Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 3136/3125
| {{Monzo| 6 0 -5 2 }}
| style="text-align:right;" | 6.08
| style="text-align:center;" |Zozoquingu
| style="text-align:center;" | Hemimean
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| style="text-align:right;" | 5.76
| style="text-align:center;" |Saruyo
| style="text-align:center;" | Hemifamity
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| style="text-align:right;" | 3.93
| style="text-align:center;" |Luzozogu
| style="text-align:center;" | Werckisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| style="text-align:right;" | 3.80
| style="text-align:center;" |Sasaru
| style="text-align:center;" | Garischisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| style="text-align:right;" | 1.95
| style="text-align:center;" |Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 703125/702464
| {{Monzo| -11 2 7 -3 }}
| style="text-align:right;" | 1.63
| style="text-align:center;" |Latriru-asepyo
| style="text-align:center;" | Meter
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 250047/250000
| {{Monzo| -4 6 -6 3 }}
| style="text-align:right;" | 0.33
| style="text-align:center;" |Trizogugu
| style="text-align:center;" | Landscape Comma
| style="text-align:center;" | 
| style="text-align:center;" | 
|-
| style="text-align:center;" | 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| style="text-align:right;" | 0.18
| style="text-align:center;" |Bilorugu
| style="text-align:center;" | Kalisma
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" | 
|-
| style="text-align:center;" | 
| {{Monzo| 161 -84 -12 }}
| style="text-align:right;" | 0.02
| style="text-align:center;" |Sepbisa-quadtrigu
| style="text-align:center;" | Atom
| style="text-align:center;" | 
| style="text-align:center;" | 
|}

== Scales ==
* [[MOS in 12edo]]

== Intervals ==

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-11-001.svg|12ed2-11-001.svg]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]]

=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
!prime 2
!prime 3
!prime 5
!prime 7
!prime 11
!prime 13
|-
!error
|0¢
| -1.96¢
| +13.7¢
| +31.2¢
| +48.7¢
| -40.5¢
|-
![[fifthspan]]
|0
| +1
| +4
| -2
| +6
| -4
|}
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error).
{| class="wikitable" style="text-align:center;"
|-
! Interval, complement
! Error (abs., in [[cent|cents]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| [[13/11]], [[22/13]]
| 10.790
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| [[14/13]], [[13/7]]
| 28.298
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| [[13/10]], [[20/13]]
| 45.786
|-
| [[11/9]], [[18/11]]
| 47.408
|-
| [[15/13]], [[26/15]]
| 47.741 
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| [[12/11]], [[11/6]]
| 49.323
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>kictdzgu5yg5opbjavd4zk2xyg6ieh4</sha1>
    </revision>
    <revision>
      <id>47786</id>
      <parentid>47784</parentid>
      <timestamp>2020-07-20T07:20:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>General cleanup. Really confusing when there were two sections of intervals.</comment>
      <origin>47786</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8406" sha1="0weiq52zn20fxq8iqsvyzrhvma15gnv" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-2 right-3"
|-
! Rational
! Monzo
! Cents
![[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| 2048/2025
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|-
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>0weiq52zn20fxq8iqsvyzrhvma15gnv</sha1>
    </revision>
    <revision>
      <id>47787</id>
      <parentid>47786</parentid>
      <timestamp>2020-07-20T07:37:44Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Comma table cleanup</comment>
      <origin>47787</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8557" sha1="fkog53dloygnizlxy14lycryakex8ol" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]].

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| 2048/2025
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>fkog53dloygnizlxy14lycryakex8ol</sha1>
    </revision>
    <revision>
      <id>47909</id>
      <parentid>47787</parentid>
      <timestamp>2020-07-25T01:57:22Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Let</comment>
      <origin>47909</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9297" sha1="dp9ixo22rywz487h1zhxtjx4f286l4x" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| 0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| 2048/2025
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>dp9ixo22rywz487h1zhxtjx4f286l4x</sha1>
    </revision>
    <revision>
      <id>47911</id>
      <parentid>47909</parentid>
      <timestamp>2020-07-25T02:01:51Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Let's try adding TE temperament measures</comment>
      <origin>47911</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9309" sha1="acch3ed0igmbznlqgmzkkidzxqsd323" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable"
!Degree
!Cents
! colspan="3" |Interval
!Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| 2048/2025
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>acch3ed0igmbznlqgmzkkidzxqsd323</sha1>
    </revision>
    <revision>
      <id>48956</id>
      <parentid>47911</parentid>
      <timestamp>2020-09-08T20:49:50Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ two columns right aligned</comment>
      <origin>48956</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9329" sha1="qd9vel7irhrnu98kl1qy9oohvvmg1pp" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| 2048/2025
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:edo]]
[[Category:meantone]]
[[Category:todo:unify_precision]]
[[Category:zeta]]</text>
      <sha1>qd9vel7irhrnu98kl1qy9oohvvmg1pp</sha1>
    </revision>
    <revision>
      <id>48957</id>
      <parentid>48956</parentid>
      <timestamp>2020-09-08T20:50:37Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>precision seems to be unified</comment>
      <origin>48957</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9295" sha1="ju0z3l8u58v4wmq4l8t4zmocgcx3scl" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| 2048/2025
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768|32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| 176/175
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| 896/891
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Edo]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>ju0z3l8u58v4wmq4l8t4zmocgcx3scl</sha1>
    </revision>
    <revision>
      <id>52725</id>
      <parentid>48957</parentid>
      <timestamp>2020-11-18T16:48:26Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>some links added</comment>
      <origin>52725</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9299" sha1="1wokki5drrjhiur8bitm9my81cttw4m" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Edo]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>1wokki5drrjhiur8bitm9my81cttw4m</sha1>
    </revision>
    <revision>
      <id>52789</id>
      <parentid>52725</parentid>
      <timestamp>2020-11-20T20:35:22Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* 15-odd-limit interval mappings */ linked [[consistent]]</comment>
      <origin>52789</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9303" sha1="ilbwyblsgwipv48su5mlnn0chh8pj5n" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. Prime harmonics are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Edo]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>ilbwyblsgwipv48su5mlnn0chh8pj5n</sha1>
    </revision>
    <revision>
      <id>52792</id>
      <parentid>52789</parentid>
      <timestamp>2020-11-20T20:46:44Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* 15-odd-limit interval mappings */ linked [[prime harmonics]]</comment>
      <origin>52792</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9307" sha1="idlpy65a8bfu3h9i31isb6gxyko84cv" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Edo]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>idlpy65a8bfu3h9i31isb6gxyko84cv</sha1>
    </revision>
    <revision>
      <id>54146</id>
      <parentid>52792</parentid>
      <timestamp>2020-12-03T18:54:15Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>+infobox</comment>
      <origin>54146</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9648" sha1="khmyd2tftacp1dfub4o2xld9576e65l" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup * 3
| Subgroup = 2.3.5.7.17.19
| Step size = 100¢
| Fifth type = Meantone 7\12 700¢ (-1.955¢)
| Important MOSes = diatonic (meantone) 5L2s 2221221 (generator = 7\12)&lt;br/&gt;pentatonic (meantone) 2L3s 22323 (generator = 7\12)&lt;br/&gt;diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Edo]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>khmyd2tftacp1dfub4o2xld9576e65l</sha1>
    </revision>
    <revision>
      <id>54147</id>
      <parentid>54146</parentid>
      <timestamp>2020-12-03T18:54:27Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>54147</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9649" sha1="gybpw8j9uf6no4qt5uqmkhy47guetw7" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; * 3
| Subgroup = 2.3.5.7.17.19
| Step size = 100¢
| Fifth type = Meantone 7\12 700¢ (-1.955¢)
| Important MOSes = diatonic (meantone) 5L2s 2221221 (generator = 7\12)&lt;br/&gt;pentatonic (meantone) 2L3s 22323 (generator = 7\12)&lt;br/&gt;diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Edo]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>gybpw8j9uf6no4qt5uqmkhy47guetw7</sha1>
    </revision>
    <revision>
      <id>54188</id>
      <parentid>54147</parentid>
      <timestamp>2020-12-03T21:53:02Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>normalize ET template parameters</comment>
      <origin>54188</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9649" sha1="br9k1l8r81tusm4ey8gojzv1g4ixlgz" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; * 3
| Subgroup = 2.3.5.7.17.19
| Step size = 100.000
| Fifth type = Meantone 7\12 700¢ (-1.955¢)
| Important MOS = diatonic (meantone) 5L2s 2221221 (generator = 7\12)&lt;br/&gt;pentatonic (meantone) 2L3s 22323 (generator = 7\12)&lt;br/&gt;diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Edo]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>br9k1l8r81tusm4ey8gojzv1g4ixlgz</sha1>
    </revision>
    <revision>
      <id>54982</id>
      <parentid>54188</parentid>
      <timestamp>2020-12-04T23:13:07Z</timestamp>
      <contributor>
        <username>Keenan Pepper category edits</username>
        <id>257</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Edo]] to [[Category:Equal divisions of the octave]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54982</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9675" sha1="qg0599zygap7q6iep7jgy5al7t0jl66" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; * 3
| Subgroup = 2.3.5.7.17.19
| Step size = 100.000
| Fifth type = Meantone 7\12 700¢ (-1.955¢)
| Important MOS = diatonic (meantone) 5L2s 2221221 (generator = 7\12)&lt;br/&gt;pentatonic (meantone) 2L3s 22323 (generator = 7\12)&lt;br/&gt;diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
|-
! Error (¢)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
|}

==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>qg0599zygap7q6iep7jgy5al7t0jl66</sha1>
    </revision>
    <revision>
      <id>56229</id>
      <parentid>54982</parentid>
      <timestamp>2020-12-09T03:00:13Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>moved the prime-errors table up to the top</comment>
      <origin>56229</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9850" sha1="cd8v1zfigydw4i1zi53v3wogi5xke5n" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; * 3
| Subgroup = 2.3.5.7.17.19
| Step size = 100.000
| Fifth type = Meantone 7\12 700¢
| Important MOS = diatonic (meantone) 5L2s 2221221 (generator = 7\12)&lt;br/&gt;pentatonic (meantone) 2L3s 22323 (generator = 7\12)&lt;br/&gt;diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
!prime 17
!prime 19
|-
! Error (¢)
| 0
|  -1.96
|  +13.7
|  +31.2
|  +48.7
|  -40.5
| +5.0
| -2.5
|-
!Error (%)
|0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| +5.0
| -2.5
|-
![[Patent val|nearest edomapping]]
|12
|7
|4
|10
|6
|8
|1
|3
|-
![[Fifthspan]]
| 0
|  +1
|  +4
|  -2
|  +6
|  -4
| -5
| -3
|}
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by erro ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>cd8v1zfigydw4i1zi53v3wogi5xke5n</sha1>
    </revision>
    <revision>
      <id>56230</id>
      <parentid>56229</parentid>
      <timestamp>2020-12-09T03:05:59Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <origin>56230</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9936" sha1="ew930cemqnos0p2q2q1k1esb42vpwfv" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; * 3
| Subgroup = 2.3.5.7.17.19
| Step size = 100.000
| Fifth type = Meantone 7\12 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
| Important MOS = diatonic (meantone) 5L2s 2221221 (generator = 7\12)&lt;br/&gt;pentatonic (meantone) 2L3s 22323 (generator = 7\12)&lt;br/&gt;diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
!prime 17
!prime 19
|-
! Error (¢)
| 0
|  -1.96
|  +13.7
|  +31.2
|  +48.7
|  -40.5
| +5.0
| -2.5
|-
!Error (%)
|0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| +5.0
| -2.5
|-
![[Patent val|nearest edomapping]]
|12
|7
|4
|10
|6
|8
|1
|3
|-
![[Fifthspan]]
| 0
|  +1
|  +4
|  -2
|  +6
|  -4
| -5
| -3
|}
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>ew930cemqnos0p2q2q1k1esb42vpwfv</sha1>
    </revision>
    <revision>
      <id>56235</id>
      <parentid>56230</parentid>
      <timestamp>2020-12-09T03:22:10Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
      <origin>56235</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9940" sha1="jxa9jzoue3ktr6ac83n7kc19e9wzj48" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; * 3
| Subgroup = 2.3.5.7.17.19
| Step size = 100.000
| Fifth type = Meantone 7\12 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
| Important MOS = diatonic (meantone) 5L2s 2221221 (generator = 7\12)&lt;br/&gt;pentatonic (meantone) 2L3s 22323 (generator = 7\12)&lt;br/&gt;diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
!prime 17
!prime 19
|-
! Error (¢)
| 0
|  -1.96
|  +13.7
|  +31.2
|  +48.7
|  -40.5
|  -5.0
|  +2.5
|-
!Error (%)
|0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|  -5.0
|  +2.5
|-
![[Patent val|nearest edomapping]]
|12
|7
|4
|10
|6
|8
|1
|3
|-
![[Fifthspan]]
| 0
|  +1
|  +4
|  -2
|  +6
|  -4
| -5
| -3
|}
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>jxa9jzoue3ktr6ac83n7kc19e9wzj48</sha1>
    </revision>
    <revision>
      <id>56901</id>
      <parentid>56235</parentid>
      <timestamp>2020-12-18T06:02:38Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <origin>56901</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9975" sha1="8xj91irhjqn28t6lxypv7dw044l9nks" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; * 3
| Step size = 100.000¢
| Fifth type = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable" style="text-align:center;"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
!prime 17
!prime 19
|-
! Error (¢)
| 0
|  -1.96
|  +13.7
|  +31.2
|  +48.7
|  -40.5
|  -5.0
|  +2.5
|-
!Error (%)
|0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
|  -5.0
|  +2.5
|-
![[Patent val|nearest edomapping]]
|12
|7
|4
|10
|6
|8
|1
|3
|-
![[Fifthspan]]
| 0
|  +1
|  +4
|  -2
|  +6
|  -4
| -5
| -3
|}
12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Degree
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
The two most common 12-edo MOS scales are meantone [5] and meantone [7].

'''diatonic (meantone) 5L2s 2221221 (generator = 7\12)'''

'''pentatonic (meantone) 2L3s 22323 (generator = 7\12)'''

'''diminished 4L4s 12121212 (generator = 1\12, period = 3\12)'''
{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>8xj91irhjqn28t6lxypv7dw044l9nks</sha1>
    </revision>
    <revision>
      <id>56989</id>
      <parentid>56901</parentid>
      <timestamp>2020-12-18T15:06:09Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>ET parameter name, cleanup</comment>
      <origin>56989</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9954" sha1="d3vkq3y7dqid2fw0fhbumc3w801nym9" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
The two most common 12-edo MOS scales are meantone [5] and meantone [7].

'''diatonic (meantone) 5L2s 2221221 (generator = 7\12)'''

'''pentatonic (meantone) 2L3s 22323 (generator = 7\12)'''

'''diminished 4L4s 12121212 (generator = 1\12, period = 3\12)'''
{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Rational
! Monzo
! Cents
! [[Color notation/Temperament Names|Color Name]]
! Name 1
! Name 2
! Name 3
|-
| 3
| [[531441/524288]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| Pythagorean comma
| 
| 
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis
| Diminished comma
| 
|-
| "
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis
| Augmented comma
| 
|-
| "
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
| Didymos comma
| Meantone comma
|-
| "
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
| 
| 
|-
| "
| 5201701/5149091
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| Misty comma
| 
| 
|-
| "
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
| 
| 
|-
| "
| 
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| Atom
| 
| 
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
| 
| 
|-
| "
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis
| Jubilisma
| 
|-
| "
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
| Archytas' comma
| Leipziger Komma
|-
| "
| 3125/3087
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
| 
| 
|-
| "
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma
| Starling comma
| 
|-
| "
| 4000/3969
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
| 
| 
|-
| "
| 321489/320000
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| Varunisma
| 
| 
|-
| "
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma
| Marvel comma
| 
|-
| "
| 3136/3125
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
| 
| 
|-
| "
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
| 
| 
|-
| "
| 33554432/33480783
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| Garischisma
| 
| 
|-
| "
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| Meter
| 
| 
|-
| "
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| Landscape comma
| 
| 
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
| 
| 
|-
| "
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
| 
| 
|-
| "
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
| 
| 
|-
| "
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
| 
| 
|-
| "
| 441/440
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
| 
| 
|-
| "
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
| Gauss' comma
| 
|-
| 13
| 91/90
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
| 
| 
|}

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>d3vkq3y7dqid2fw0fhbumc3w801nym9</sha1>
    </revision>
    <revision>
      <id>57142</id>
      <parentid>56989</parentid>
      <timestamp>2020-12-20T14:10:27Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Commas */ reworked ratio column</comment>
      <origin>57142</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10197" sha1="0ivdcbp7wlcg4ngx5ibu0nejc8lxrx0" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
The two most common 12-edo MOS scales are meantone [5] and meantone [7].

'''diatonic (meantone) 5L2s 2221221 (generator = 7\12)'''

'''pentatonic (meantone) 2L3s 22323 (generator = 7\12)'''

'''diminished 4L4s 12121212 (generator = 1\12, period = 3\12)'''
{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! Ratio&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! Monzo
! Cents
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, Diminished comma
|-
| 5
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, Augmented comma
|-
| 5
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymos comma, Meantone comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]] 
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, Jubilisma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, Starling comma
|-
| 7
| [[4000/3969]]
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, Marvel comma
|-
| 7
| [[3136/3125]]
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| &lt;abbr title="703125/702464"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>0ivdcbp7wlcg4ngx5ibu0nejc8lxrx0</sha1>
    </revision>
    <revision>
      <id>57144</id>
      <parentid>57142</parentid>
      <timestamp>2020-12-20T14:14:43Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* Commas */ link some column headers</comment>
      <origin>57144</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10205" sha1="3moz57skufz639l6d1eil3uw2eoqk2e" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
The two most common 12-edo MOS scales are meantone [5] and meantone [7].

'''diatonic (meantone) 5L2s 2221221 (generator = 7\12)'''

'''pentatonic (meantone) 2L3s 22323 (generator = 7\12)'''

'''diminished 4L4s 12121212 (generator = 1\12, period = 3\12)'''
{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempering_out|tempers out]] the following [[Comma|commas]]. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! Cents
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, Diminished comma
|-
| 5
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, Augmented comma
|-
| 5
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymos comma, Meantone comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]] 
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, Jubilisma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, Starling comma
|-
| 7
| [[4000/3969]]
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, Marvel comma
|-
| 7
| [[3136/3125]]
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| &lt;abbr title="703125/702464"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>3moz57skufz639l6d1eil3uw2eoqk2e</sha1>
    </revision>
    <revision>
      <id>57146</id>
      <parentid>57144</parentid>
      <timestamp>2020-12-20T14:37:27Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>/* Commas */ add class "commatable" to ease finding</comment>
      <origin>57146</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10200" sha1="spfmvroi2lb5m99k2hsts3oihhcxfnz" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
The two most common 12-edo MOS scales are meantone [5] and meantone [7].

'''diatonic (meantone) 5L2s 2221221 (generator = 7\12)'''

'''pentatonic (meantone) 2L3s 22323 (generator = 7\12)'''

'''diminished 4L4s 12121212 (generator = 1\12, period = 3\12)'''
{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, Diminished comma
|-
| 5
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, Augmented comma
|-
| 5
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymos comma, Meantone comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]] 
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, Jubilisma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, Starling comma
|-
| 7
| [[4000/3969]]
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, Marvel comma
|-
| 7
| [[3136/3125]]
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| 5120/5103
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| &lt;abbr title="703125/702464"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>spfmvroi2lb5m99k2hsts3oihhcxfnz</sha1>
    </revision>
    <revision>
      <id>57269</id>
      <parentid>57146</parentid>
      <timestamp>2020-12-21T06:57:23Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Commas */ more cleanup</comment>
      <origin>57269</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10186" sha1="m7lseobg217sbuj5dwqopfa5mumd2n5" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
The two most common 12-edo MOS scales are meantone [5] and meantone [7].

'''diatonic (meantone) 5L2s 2221221 (generator = 7\12)'''

'''pentatonic (meantone) 2L3s 22323 (generator = 7\12)'''

'''diminished 4L4s 12121212 (generator = 1\12, period = 3\12)'''
{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 12 19 28 34 42 44 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, Jubilisma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, Starling comma
|-
| 7
| [[4000/3969]]
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>m7lseobg217sbuj5dwqopfa5mumd2n5</sha1>
    </revision>
    <revision>
      <id>57270</id>
      <parentid>57269</parentid>
      <timestamp>2020-12-21T06:59:06Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Commas */ +1</comment>
      <origin>57270</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10178" sha1="j3jj16r61ogyhtfmd1bxqvglv4ki4zm" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 19
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
The two most common 12-edo MOS scales are meantone [5] and meantone [7].

'''diatonic (meantone) 5L2s 2221221 (generator = 7\12)'''

'''pentatonic (meantone) 2L3s 22323 (generator = 7\12)'''

'''diminished 4L4s 12121212 (generator = 1\12, period = 3\12)'''
{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>j3jj16r61ogyhtfmd1bxqvglv4ki4zm</sha1>
    </revision>
    <revision>
      <id>57908</id>
      <parentid>57270</parentid>
      <timestamp>2020-12-30T12:03:18Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <minor/>
      <origin>57908</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10178" sha1="8v5o74agbajw6x08fg7aqr0wr7w36fa" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
The two most common 12-edo MOS scales are meantone [5] and meantone [7].

'''diatonic (meantone) 5L2s 2221221 (generator = 7\12)'''

'''pentatonic (meantone) 2L3s 22323 (generator = 7\12)'''

'''diminished 4L4s 12121212 (generator = 1\12, period = 3\12)'''
{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
''Main article: [[MOS in 12edo]]''

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>8v5o74agbajw6x08fg7aqr0wr7w36fa</sha1>
    </revision>
    <revision>
      <id>58019</id>
      <parentid>57908</parentid>
      <timestamp>2021-01-01T08:39:28Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Fix what's in the wrong section</comment>
      <origin>58019</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10151" sha1="67zw47swzpmuwwg7eak4w41zjgexbib" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>67zw47swzpmuwwg7eak4w41zjgexbib</sha1>
    </revision>
    <revision>
      <id>58182</id>
      <parentid>58019</parentid>
      <timestamp>2021-01-03T19:44:15Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <comment>link to purdal wiki</comment>
      <origin>58182</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10190" sha1="ao26c6qwfqpye2y5cwbgimxwp3maujx" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{Monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{Monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{Monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{Monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{Monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{Monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{Monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{Monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{Monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{Monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{Monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{Monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{Monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{Monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{Monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{Monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{Monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{Monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{Monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{Monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{Monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{Monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{Monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{Monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{Monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{Monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{Monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>ao26c6qwfqpye2y5cwbgimxwp3maujx</sha1>
    </revision>
    <revision>
      <id>59167</id>
      <parentid>58182</parentid>
      <timestamp>2021-01-17T11:45:12Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
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      <text bytes="10190" sha1="fbtxju7b0188ukkxhgr3hmu8rc8n9c0" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable"
|-
! Periods &lt;br /&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>fbtxju7b0188ukkxhgr3hmu8rc8n9c0</sha1>
    </revision>
    <revision>
      <id>63055</id>
      <parentid>59167</parentid>
      <timestamp>2021-03-14T11:10:44Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <comment>/* Rank two temperaments */</comment>
      <origin>63055</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10237" sha1="pqrlcf0rcdfxosk5ca6abgug6fb2itv" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>pqrlcf0rcdfxosk5ca6abgug6fb2itv</sha1>
    </revision>
    <revision>
      <id>64846</id>
      <parentid>63055</parentid>
      <timestamp>2021-03-28T04:04:07Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>64846</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10275" sha1="o1az0chu5mzv3zvk15xo3elxrg80d4q" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>o1az0chu5mzv3zvk15xo3elxrg80d4q</sha1>
    </revision>
    <revision>
      <id>66000</id>
      <parentid>64846</parentid>
      <timestamp>2021-04-08T23:41:08Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
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      <origin>66000</origin>
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      <format>text/x-wiki</format>
      <text bytes="10282" sha1="395cah2n2r6y1lxlkt3iyn16um927eb" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[patent val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>395cah2n2r6y1lxlkt3iyn16um927eb</sha1>
    </revision>
    <revision>
      <id>66001</id>
      <parentid>66000</parentid>
      <timestamp>2021-04-08T23:41:40Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
      <origin>66001</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8995" sha1="te7teeyix3zt94smbndzq60f7spu29d" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>te7teeyix3zt94smbndzq60f7spu29d</sha1>
    </revision>
    <revision>
      <id>66002</id>
      <parentid>66001</parentid>
      <timestamp>2021-04-08T23:42:21Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>66002</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10315" sha1="5swfmu78z24mq38on2jgnwde9k510nj" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== As a regular temperament ==
The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[patent val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

=== Rank two temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>5swfmu78z24mq38on2jgnwde9k510nj</sha1>
    </revision>
    <revision>
      <id>66003</id>
      <parentid>66002</parentid>
      <timestamp>2021-04-08T23:46:23Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>66003</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9585" sha1="a7ltlrh9ypcamiojqgmyr0i9ntopw76" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== As a regular temperament ==
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

=== Rank two temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>a7ltlrh9ypcamiojqgmyr0i9ntopw76</sha1>
    </revision>
    <revision>
      <id>66004</id>
      <parentid>66003</parentid>
      <timestamp>2021-04-08T23:47:25Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
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      <origin>66004</origin>
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      <format>text/x-wiki</format>
      <text bytes="10319" sha1="4jqcu2l0zjoc95tuq24bml1xmg8idc6" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports [[septimal meantone]] via the [[patent val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== As a regular temperament ==
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

=== Rank two temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>4jqcu2l0zjoc95tuq24bml1xmg8idc6</sha1>
    </revision>
    <revision>
      <id>66005</id>
      <parentid>66004</parentid>
      <timestamp>2021-04-08T23:48:04Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
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      <text bytes="10317" sha1="ce4rgz1e4a8f9mn34sspcm3j8jqv742" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it arose as a temperament, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports [[septimal meantone]] via the [[patent val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== As a regular temperament ==
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

=== Rank two temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>ce4rgz1e4a8f9mn34sspcm3j8jqv742</sha1>
    </revision>
    <revision>
      <id>66006</id>
      <parentid>66005</parentid>
      <timestamp>2021-04-08T23:50:16Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>66006</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7306" sha1="gcix3tieeu2138n5e2ywju5i30kzj64" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it arose as a temperament, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports [[septimal meantone]] via the [[patent val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== As a regular temperament ==
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

=== Rank two temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>gcix3tieeu2138n5e2ywju5i30kzj64</sha1>
    </revision>
    <revision>
      <id>66007</id>
      <parentid>66006</parentid>
      <timestamp>2021-04-08T23:50:34Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <origin>66007</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10319" sha1="kud8nmae5kbfi4yk9buffvl84jsaa2n" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it arose as a temperament, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports [[septimal meantone]] via the [[patent val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== As a regular temperament ==
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

=== Rank two temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

=== Commas ===
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>kud8nmae5kbfi4yk9buffvl84jsaa2n</sha1>
    </revision>
    <revision>
      <id>66008</id>
      <parentid>66007</parentid>
      <timestamp>2021-04-08T23:57:03Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <minor/>
      <comment>/* Theory */Clearer word choice; explain what septimal meantone is without referring to patent vals; give an example of a better septimal meantone edo</comment>
      <origin>66008</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10464" sha1="p9abiio7h8x5bnmmw5fgaygx8d8z01u" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it arose as a temperament, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "approximated" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports [[septimal meantone]] (meaning that the [[syntonic comma]] tempered out, and going 10 fifths up, i.e. C-A#, give 12edo's best approximation to 7/4), its credentials in the 7-limit department are distinctly cheesy. ([[31edo]] is a much better tuning for septimal meantone.) It cannot be said to approximate 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== As a regular temperament ==
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

=== Rank two temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

=== Commas ===
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
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    <revision>
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        <id>76</id>
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      <text bytes="10467" sha1="502gsalofit1xqikoa0f6ghajrwkmvo" xml:space="preserve">{{User:IlL/Template:RTT restriction}}
{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it arose as a temperament, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "approximated" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports [[septimal meantone]] (meaning that the [[syntonic comma]] is tempered out, and going 10 fifths up, i.e. C-A#, give 12edo's best approximation to 7/4), its credentials in the 7-limit department are distinctly cheesy. ([[31edo]] is a much better tuning for septimal meantone.) It cannot be said to approximate 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== As a regular temperament ==
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

=== Rank two temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

=== Commas ===
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
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    </revision>
    <revision>
      <id>66010</id>
      <parentid>66009</parentid>
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      <contributor>
        <username>Inthar</username>
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{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it arose as a temperament, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "approximated" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports [[septimal meantone]] (meaning that 12edo tempers out the [[syntonic comma]], and going 10 fifths up, i.e. C-A#, gives 12edo's best approximation to 7/4), its credentials in the 7-limit department are distinctly cheesy. ([[31edo]] is a much better tuning for septimal meantone.) It cannot be said to approximate 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== As a regular temperament ==
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

=== Rank two temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

=== Commas ===
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>ppzb7vwvbrgx73su7jk6yivzhazqd9e</sha1>
    </revision>
    <revision>
      <id>66593</id>
      <parentid>66010</parentid>
      <timestamp>2021-04-13T06:54:28Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>66593</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10432" sha1="ahegw49rcgeadwd4rbqvykkbzssv2zg" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it arose as a temperament, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "approximated" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports [[septimal meantone]] (meaning that 12edo tempers out the [[syntonic comma]], and going 10 fifths up, i.e. C-A#, gives 12edo's best approximation to 7/4), its credentials in the 7-limit department are distinctly cheesy. ([[31edo]] is a much better tuning for septimal meantone.) It cannot be said to approximate 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== As a regular temperament ==
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

=== Rank two temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

=== Commas ===
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>ahegw49rcgeadwd4rbqvykkbzssv2zg</sha1>
    </revision>
    <revision>
      <id>66958</id>
      <parentid>66593</parentid>
      <timestamp>2021-04-14T01:32:43Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Scales */</comment>
      <origin>66958</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10562" sha1="lcb3878azczenddis9h69hdrktnlnht" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it arose as a temperament, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "approximated" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports [[septimal meantone]] (meaning that 12edo tempers out the [[syntonic comma]], and going 10 fifths up, i.e. C-A#, gives 12edo's best approximation to 7/4), its credentials in the 7-limit department are distinctly cheesy. ([[31edo]] is a much better tuning for septimal meantone.) It cannot be said to approximate 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== As a regular temperament ==
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

=== Rank two temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

=== Commas ===
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
==Pathological Modes==
3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>lcb3878azczenddis9h69hdrktnlnht</sha1>
    </revision>
    <revision>
      <id>66973</id>
      <parentid>66958</parentid>
      <timestamp>2021-04-14T01:51:16Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Pathological Modes */</comment>
      <origin>66973</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10562" sha1="iij9ofkcqorzjfnhdpmibowubjbcmqf" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it arose as a temperament, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "approximated" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports [[septimal meantone]] (meaning that 12edo tempers out the [[syntonic comma]], and going 10 fifths up, i.e. C-A#, gives 12edo's best approximation to 7/4), its credentials in the 7-limit department are distinctly cheesy. ([[31edo]] is a much better tuning for septimal meantone.) It cannot be said to approximate 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== As a regular temperament ==
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

=== Rank two temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

=== Commas ===
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
==Pathological Modes==
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>iij9ofkcqorzjfnhdpmibowubjbcmqf</sha1>
    </revision>
    <revision>
      <id>67003</id>
      <parentid>66973</parentid>
      <timestamp>2021-04-14T05:33:32Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <comment>revert</comment>
      <origin>67003</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10237" sha1="pqrlcf0rcdfxosk5ca6abgug6fb2itv" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>pqrlcf0rcdfxosk5ca6abgug6fb2itv</sha1>
    </revision>
    <revision>
      <id>67033</id>
      <parentid>67003</parentid>
      <timestamp>2021-04-14T06:02:39Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
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      <text bytes="10366" sha1="0cstygzi3ov20dmmpwt4gx44rj2n5jg" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized by the rank) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}

* 12et has a lower relative error than any previous edos in the 3-, 5-, 7-, and 11-limit. The next ET that does better in these subgroups is 41, 19, 19, and 22, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72. 

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
==Pathological Modes==
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS
== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>0cstygzi3ov20dmmpwt4gx44rj2n5jg</sha1>
    </revision>
    <revision>
      <id>68897</id>
      <parentid>67033</parentid>
      <timestamp>2021-04-27T08:14:20Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Temperament measures */ update</comment>
      <origin>68897</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10378" sha1="awupghydwfsxfc0z2vi4s5ztkxxohmz" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! Error (%)
| 0
| -1.96
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}
* 12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
==Pathological Modes==
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS
== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>awupghydwfsxfc0z2vi4s5ztkxxohmz</sha1>
    </revision>
    <revision>
      <id>71570</id>
      <parentid>68897</parentid>
      <timestamp>2021-06-02T04:10:35Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>71570</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10441" sha1="oabz77t7ngebyivqu7f06aidhai968v" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}
* 12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
==Pathological Modes==
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS
== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>oabz77t7ngebyivqu7f06aidhai968v</sha1>
    </revision>
    <revision>
      <id>71679</id>
      <parentid>71570</parentid>
      <timestamp>2021-06-02T22:20:14Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <comment>/* Intervals */</comment>
      <origin>71679</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10630" sha1="hniuw5uvrt01iaz6ebrnu3gkvghopkc" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
P0
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
aug 1sn, minor 1st
| A1, m2
A0, m1
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
major 1st
| M2
M1
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
minor 2nd
| m3
m2
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
major 2nd
| M3
M2
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
perfect 3rd
| P4
P3
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
aug 3rd, dim 4th
| A4, d5
A3, d4
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
perfect 4th
| P5
P4
| A
| 3/2
|-
| 8
| 800
| minor 6th
minor 5th
| m6
m5
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
major 5th
| M6
M5
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
minor 6th
| m7
m6
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
major 6th
| M7
M6
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
perfect 7th
| P8
P7
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}
* 12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
==Pathological Modes==
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS
== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>hniuw5uvrt01iaz6ebrnu3gkvghopkc</sha1>
    </revision>
    <revision>
      <id>71710</id>
      <parentid>71679</parentid>
      <timestamp>2021-06-03T04:04:59Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Undo revision 71679 by [[Special:Contributions/Moremajorthanmajor|Moremajorthanmajor]] ([[User talk:Moremajorthanmajor|talk]])</comment>
      <origin>71710</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10441" sha1="oabz77t7ngebyivqu7f06aidhai968v" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12edo''', perhaps better known as '''12et''' since it really is a temperament, is the predominating tuning system in the world today. 

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}
* 12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
==Pathological Modes==
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS
== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>oabz77t7ngebyivqu7f06aidhai968v</sha1>
    </revision>
    <revision>
      <id>71922</id>
      <parentid>71710</parentid>
      <timestamp>2021-06-05T13:24:22Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>uppercase EDO</comment>
      <origin>71922</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10454" sha1="55dxq2y90bcfnm6pg1oakv4sxofqjcb" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12 EDO''', also known as '''12-ET''' or '''12-TET''' since it really is a temperament, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12 EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12 EDO is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12 EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}
* 12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
==Pathological Modes==
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS
== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>55dxq2y90bcfnm6pg1oakv4sxofqjcb</sha1>
    </revision>
    <revision>
      <id>72129</id>
      <parentid>71922</parentid>
      <timestamp>2021-06-07T21:26:45Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>step size as special case</comment>
      <origin>72129</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10474" sha1="gmee2rkbodnvuy4d0ndyallc7chs0kb" xml:space="preserve">{{interwiki
| de = 12edo
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12 EDO''', also known as '''12-ET''' or '''12-TET''' since it really is a temperament, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12 EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12 EDO is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12 EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}
* 12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
==Pathological Modes==
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS
== Commas ==
12 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>gmee2rkbodnvuy4d0ndyallc7chs0kb</sha1>
    </revision>
    <revision>
      <id>73485</id>
      <parentid>72129</parentid>
      <timestamp>2021-06-30T23:22:18Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>prepare move</comment>
      <origin>73485</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10481" sha1="3c0nje0hbc0viplrv04xgcnjlkjfnfs" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12-EDO''', also known as '''12-ET''' or '''12-TET''' since it really is a [[temperament]], is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12-EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12-EDO is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}
* 12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
==Pathological Modes==
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Commas ==
12-EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>3c0nje0hbc0viplrv04xgcnjlkjfnfs</sha1>
    </revision>
    <revision>
      <id>73486</id>
      <parentid>73485</parentid>
      <timestamp>2021-06-30T23:26:41Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>Xenwolf moved page [[12edo]] to [[12-EDO]] over redirect: is more common and consistent: EDO is actually an abbreviation and the number in front of it can be grasped at a glance</comment>
      <origin>73485</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10481" sha1="3c0nje0hbc0viplrv04xgcnjlkjfnfs" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12-EDO''', also known as '''12-ET''' or '''12-TET''' since it really is a [[temperament]], is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12-EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12-EDO is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}
* 12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
==Pathological Modes==
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Commas ==
12-EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>3c0nje0hbc0viplrv04xgcnjlkjfnfs</sha1>
    </revision>
    <revision>
      <id>73663</id>
      <parentid>73486</parentid>
      <timestamp>2021-07-04T13:17:01Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Improve intro</comment>
      <origin>73663</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10563" sha1="6vdzmqoyhi9rqmwl2xdqxjgxoei4ate" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12 equal divisions of the octave''' ('''12-EDO'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12-ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
!
! prime 2
! prime 3
! prime 5
! prime 7
! prime 11
! prime 13
! prime 17
! prime 19
|-
! Error (¢)
| 0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12-EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12-EDO is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Just approximation ==
=== Selected just intervals by error ===
==== 15-odd-limit interval mappings ====

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

==== Selected 19-limit intervals ====

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

=== Temperament measures ===
Shown below are [[TE temperament measures]] (RMS normalized) of 12et. 
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}
* 12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 
* 12et is most prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

== Rank two temperaments ==
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}
The two most common 12-EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)
==Pathological Modes==
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Commas ==
12-EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>6vdzmqoyhi9rqmwl2xdqxjgxoei4ate</sha1>
    </revision>
    <revision>
      <id>73664</id>
      <parentid>73663</parentid>
      <timestamp>2021-07-04T13:24:16Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Some cleanup</comment>
      <origin>73664</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10498" sha1="i1f7cyxms49jx9kzg5g9pjfr4kva01x" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12 equal divisions of the octave''' ('''12-EDO'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12-ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
|+ Approximation of prime intervals in 12-EDO
! Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|-
! Error (¢)
| 0.0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|Nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12-EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12-EDO is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-all"
! colspan="2" |
! 3-limit
! 5-limit
! 7-limit
! 2.3.5.7.17.19
|-
! colspan="2" |Octave stretch (¢)
| +0.617
| -1.56
| -3.95
| -2.53
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.617
| 3.11
| 4.92
| 4.52
|-
! [[TE simple badness|relative]] (%)
| 0.617
| 3.11
| 4.94
| 4.53
|}
12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12-EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}

The two most common 12-EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>i1f7cyxms49jx9kzg5g9pjfr4kva01x</sha1>
    </revision>
    <revision>
      <id>73665</id>
      <parentid>73664</parentid>
      <timestamp>2021-07-04T13:32:42Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ improve the RTT property table</comment>
      <origin>73665</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10893" sha1="nqt8ptzbbgh8immeqcgwtqllp21y61i" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12 equal divisions of the octave''' ('''12-EDO'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12-ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
|+ Approximation of prime intervals in 12-EDO
! Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|-
! Error (¢)
| 0.0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|Nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12-EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12-EDO is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12-EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}

The two most common 12-EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>nqt8ptzbbgh8immeqcgwtqllp21y61i</sha1>
    </revision>
    <revision>
      <id>74188</id>
      <parentid>73665</parentid>
      <timestamp>2021-07-12T15:17:14Z</timestamp>
      <contributor>
        <username>SupahstarSaga</username>
        <id>281</id>
      </contributor>
      <minor/>
      <comment>SupahstarSaga moved page [[12-EDO]] to [[12edo]] over redirect: Reversing the recent move for consistency</comment>
      <origin>73665</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10893" sha1="nqt8ptzbbgh8immeqcgwtqllp21y61i" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12 equal divisions of the octave''' ('''12-EDO'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12-ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
|+ Approximation of prime intervals in 12-EDO
! Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|-
! Error (¢)
| 0.0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|Nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12-EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12-EDO is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12-EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}

The two most common 12-EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>nqt8ptzbbgh8immeqcgwtqllp21y61i</sha1>
    </revision>
    <revision>
      <id>75456</id>
      <parentid>74188</parentid>
      <timestamp>2021-08-04T18:30:04Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
      <origin>75456</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11158" sha1="ku9eiyou9mh9dtbgb1hqtalc1y3tr5u" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12 equal divisions of the octave''' ('''12-EDO'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12-ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
|+ Approximation of prime intervals in 12-EDO
! Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|-
! Error (¢)
| 0.0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|Nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12-EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12-EDO is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

===Differences between distributionally-even scales and smaller edos===
{| class="wikitable"
|+
!N
!L-Nedo
!s-Nedo
|-
|5
|60¢
| -40¢
|-
| 7
|28.571¢
|  -71.429
|-
|8
|50¢
|  -50¢
|-
|9
| 66.667¢
| -33.333¢
|-
|10
|80¢
| -20¢
|-
|11
|90.909¢
| -9.091¢
|}
== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12-EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main| MOS in 12edo }}

The two most common 12-EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>ku9eiyou9mh9dtbgb1hqtalc1y3tr5u</sha1>
    </revision>
    <revision>
      <id>75928</id>
      <parentid>75456</parentid>
      <timestamp>2021-08-09T15:20:20Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>/* Scales */ Updated internal link</comment>
      <origin>75928</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11171" sha1="832vb930mtnio8t5fwsidcr1clgjxbx" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 = 700¢
| Major 2nd = 2\12 = 200¢
| Minor 2nd = 1\12 = 100¢
| Augmented 1sn = 1\12 = 100¢
}}
'''12 equal divisions of the octave''' ('''12-EDO'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12-ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
|+ Approximation of prime intervals in 12-EDO
! Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|-
! Error (¢)
| 0.0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|Nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12-EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12-EDO is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

===Differences between distributionally-even scales and smaller edos===
{| class="wikitable"
|+
!N
!L-Nedo
!s-Nedo
|-
|5
|60¢
| -40¢
|-
| 7
|28.571¢
|  -71.429
|-
|8
|50¢
|  -50¢
|-
|9
| 66.667¢
| -33.333¢
|-
|10
|80¢
| -20¢
|-
|11
|90.909¢
| -9.091¢
|}
== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12-EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12-EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>832vb930mtnio8t5fwsidcr1clgjxbx</sha1>
    </revision>
    <revision>
      <id>79236</id>
      <parentid>75928</parentid>
      <timestamp>2021-10-08T23:50:59Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Update infobox</comment>
      <origin>79236</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11183" sha1="deopmro3cszxc6ovpkjoj53yehzzjhj" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
'''12 equal divisions of the octave''' ('''12-EDO'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12-ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
|+ Approximation of prime intervals in 12-EDO
! Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|-
! Error (¢)
| 0.0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|Nearest edomapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12-EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12-EDO is the fourth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12et is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

===Differences between distributionally-even scales and smaller edos===
{| class="wikitable"
|+
!N
!L-Nedo
!s-Nedo
|-
|5
|60¢
| -40¢
|-
| 7
|28.571¢
|  -71.429
|-
|8
|50¢
|  -50¢
|-
|9
| 66.667¢
| -33.333¢
|-
|10
|80¢
| -20¢
|-
|11
|90.909¢
| -9.091¢
|}
== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12-EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===

The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===

[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12-EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12-EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== See also ==

* [[:purdal:12-EDD]]


[[Category:12edo]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>deopmro3cszxc6ovpkjoj53yehzzjhj</sha1>
    </revision>
    <revision>
      <id>79500</id>
      <parentid>79236</parentid>
      <timestamp>2021-10-13T23:57:47Z</timestamp>
      <contributor>
        <username>Xenllium</username>
        <id>73</id>
      </contributor>
      <minor/>
      <origin>79500</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11187" sha1="5x29p4n40ofagly7kh8pk43qlh34cn5" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
'''12 equal divisions of the octave''' ('''12EDO'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
|+ Approximation of prime intervals in 12 EDO
! Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|-
! Error (¢)
| 0.0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|Nearest EDO-mapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12ET officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12EDO is the fourth [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral EDO]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12ET in specific ways, and tuning systems which share the comma in question will be similar to 12ET in precisely those ways.

12EDO is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
!N
!L-Nedo
!s-Nedo
|-
| 5
| 60¢
| -40¢
|-
| 7
| 28.571¢
| -71.429
|-
| 8
| 50¢
| -50¢
|-
| 9
| 66.667¢
| -33.333¢
|-
| 10
| 80¢
| -20¢
|-
| 11
| 90.909¢
| -9.091¢
|}
== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12EDO. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12ET (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12ET is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>5x29p4n40ofagly7kh8pk43qlh34cn5</sha1>
    </revision>
    <revision>
      <id>80012</id>
      <parentid>79500</parentid>
      <timestamp>2021-10-27T16:07:00Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Added music section (for pieces that are partially in 12edo)</comment>
      <origin>80012</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11236" sha1="sv9e6n6tryvnpyjg6e6origy2sis7o1" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
'''12 equal divisions of the octave''' ('''12EDO'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
|+ Approximation of prime intervals in 12 EDO
! Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|-
! Error (¢)
| 0.0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|Nearest EDO-mapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12ET officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12EDO is the fourth [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral EDO]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12ET in specific ways, and tuning systems which share the comma in question will be similar to 12ET in precisely those ways.

12EDO is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

=== Differences between distributionally-even scales and smaller EDOs ===
{| class="wikitable"
|+
!N
!L-Nedo
!s-Nedo
|-
| 5
| 60¢
| -40¢
|-
| 7
| 28.571¢
| -71.429
|-
| 8
| 50¢
| -50¢
|-
| 9
| 66.667¢
| -33.333¢
|-
| 10
| 80¢
| -20¢
|-
| 11
| 90.909¢
| -9.091¢
|}
== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12EDO. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12ET (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12ET is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>sv9e6n6tryvnpyjg6e6origy2sis7o1</sha1>
    </revision>
    <revision>
      <id>83756</id>
      <parentid>80012</parentid>
      <timestamp>2022-01-13T00:03:38Z</timestamp>
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        <username>Sintel</username>
        <id>330</id>
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      <text bytes="10961" sha1="rayjir5rpm77pwp6ier8o376wqmgmyf" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
'''12 equal divisions of the octave''' ('''12EDO'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
|+ Approximation of prime intervals in 12 EDO
! Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|-
! Error (¢)
| 0.0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|Nearest EDO-mapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12ET officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12EDO is the fourth [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral EDO]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12ET in specific ways, and tuning systems which share the comma in question will be similar to 12ET in precisely those ways.

12EDO is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12EDO. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12ET (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12ET is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>rayjir5rpm77pwp6ier8o376wqmgmyf</sha1>
    </revision>
    <revision>
      <id>84017</id>
      <parentid>83756</parentid>
      <timestamp>2022-01-15T01:21:48Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <origin>84017</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10961" sha1="f0mka7zcoce4mwosn0tckjf4h3xsx0v" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
|+ Approximation of prime intervals in 12 EDO
! Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|-
! Error (¢)
| 0.0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|Nearest EDO-mapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12EDO achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12ET officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12EDO is the fourth [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral EDO]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12ET in specific ways, and tuning systems which share the comma in question will be similar to 12ET in precisely those ways.

12EDO is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]].

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12EDO. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12ET (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12ET is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12EDO MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>f0mka7zcoce4mwosn0tckjf4h3xsx0v</sha1>
    </revision>
    <revision>
      <id>84019</id>
      <parentid>84017</parentid>
      <timestamp>2022-01-15T01:26:48Z</timestamp>
      <contributor>
        <username>Aura</username>
        <id>220</id>
      </contributor>
      <comment>Additional links and the like</comment>
      <origin>84019</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11036" sha1="iz079yzjzgcbz82ngk7fvpt4j6yhbbd" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{| class="wikitable center-all"
|+ Approximation of prime intervals in 12 EDO
! Prime number
! 2
! 3
! 5
! 7
! 11
! 13
! 17
! 19
|-
! Error (¢)
| 0.0
| -2.0
| +13.7
| +31.2
| +48.7
| -40.5
| -5.0
| +2.5
|-
! [[Patent val|Nearest EDO-mapping]]
| 12
| 7
| 4
| 10
| 6
| 8
| 1
| 3
|-
! [[Fifthspan]]
| 0
| +1
| +4
| -2
| +6
| -4
| -5
| -3
|}

12edo achieved its position because it is the smallest equal division of the octave ([[EDO]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12ET officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral EDO]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12ET in specific ways, and tuning systems which share the comma in question will be similar to 12ET in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]], and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12EDO as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12EDO. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12ET (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12ET is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>iz079yzjzgcbz82ngk7fvpt4j6yhbbd</sha1>
    </revision>
    <revision>
      <id>84130</id>
      <parentid>84019</parentid>
      <timestamp>2022-01-15T20:27:54Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>New harmonics template, lowercase edo</comment>
      <origin>84130</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10720" sha1="fy6j3efdtaoijq6k1e35729u5j41ynu" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{{Harmonics in equal}}

12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12ET officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12ET in specific ways, and tuning systems which share the comma in question will be similar to 12ET in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]], and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12ET (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12ET is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>fy6j3efdtaoijq6k1e35729u5j41ynu</sha1>
    </revision>
    <revision>
      <id>84160</id>
      <parentid>84130</parentid>
      <timestamp>2022-01-16T23:06:33Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Added Wikipedia box</comment>
      <origin>84160</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10755" sha1="oua9acoi9kextouzclz3bzuj8o4iu0a" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{{Harmonics in equal}}

12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12ET officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12ET in specific ways, and tuning systems which share the comma in question will be similar to 12ET in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]], and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12ET (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12ET is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>oua9acoi9kextouzclz3bzuj8o4iu0a</sha1>
    </revision>
    <revision>
      <id>84221</id>
      <parentid>84160</parentid>
      <timestamp>2022-01-18T22:26:46Z</timestamp>
      <contributor>
        <username>Moremajorthanmajor</username>
        <id>28</id>
      </contributor>
      <origin>84221</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10784" sha1="sx7vy8hy9y79lfnk4cirfrnjy2q6axi" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
Relative Radian = 15.91549¢
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{{Harmonics in equal}}

12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12ET officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12ET in specific ways, and tuning systems which share the comma in question will be similar to 12ET in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]], and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12ET (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12ET is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>sx7vy8hy9y79lfnk4cirfrnjy2q6axi</sha1>
    </revision>
    <revision>
      <id>84234</id>
      <parentid>84221</parentid>
      <timestamp>2022-01-19T00:28:48Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Undo relative radian: this type of addition should be discussed in Template:Infobox ET</comment>
      <origin>84234</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10755" sha1="oua9acoi9kextouzclz3bzuj8o4iu0a" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{{Harmonics in equal}}

12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12ET officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12ET in specific ways, and tuning systems which share the comma in question will be similar to 12ET in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]], and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12ET (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12ET is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]</text>
      <sha1>oua9acoi9kextouzclz3bzuj8o4iu0a</sha1>
    </revision>
    <revision>
      <id>84317</id>
      <parentid>84234</parentid>
      <timestamp>2022-01-20T12:59:07Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <origin>84317</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10884" sha1="6f6ht4aan88jp1fb2ta8c1q0c43lc17" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{{Harmonics in equal}}

12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12ET officially supports septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12ET in specific ways, and tuning systems which share the comma in question will be similar to 12ET in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]], and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12ET (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12ET is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>6f6ht4aan88jp1fb2ta8c1q0c43lc17</sha1>
    </revision>
    <revision>
      <id>84854</id>
      <parentid>84317</parentid>
      <timestamp>2022-01-25T19:05:00Z</timestamp>
      <contributor>
        <username>Cmloegcmluin</username>
        <id>125</id>
      </contributor>
      <comment>link to new page for Supports</comment>
      <origin>84854</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10888" sha1="ncv28imil6qqt96va0678ujiugtxjl1" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12-TET''', '''12ET''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{{Harmonics in equal}}

12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12ET, the style of the music changed so that the defects of 12ET appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12ET officially [[support]]s septimal meantone via the [[val]] {{val|12 19 28 34}}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12ET in specific ways, and tuning systems which share the comma in question will be similar to 12ET in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]], and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all"
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12ET (12f val) is lower in relative error than any previous edos in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroup are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12ET is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]]/[[dominant]]
|-
| 2
| 1\12
| [[Srutal]]/[[pajara]]/[[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{main|List of MOS scales in 12edo}}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Pathological Modes ===
2 1 1 1 1 2 1 1 1 1 [[2L 8s]] MOS

3 1 1 1 1 1 1 1 1 1 [[1L 9s]] MOS

2 1 1 1 1 1 1 1 1 1 1 [[1L 10s]] MOS

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>ncv28imil6qqt96va0678ujiugtxjl1</sha1>
    </revision>
    <revision>
      <id>87043</id>
      <parentid>84854</parentid>
      <timestamp>2022-02-25T12:12:16Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Clarify, styling and misc</comment>
      <origin>87043</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10918" sha1="4mgox0oejqrrmyffc60ozn0vq9p90m2" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{{Harmonics in equal}}

12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
|-
| [[9/8]], [[16/9]]
| 3.910
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
|-
| [[16/15]], [[15/8]]
| 11.731
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
|-
| [[6/5]], [[5/3]]
| 15.641
|-
| [[7/5]], [[10/7]]
| 17.488
|-
| [[14/11]], [[11/7]]
| 17.508
|-
| [[10/9]], [[9/5]]
| 17.596
|-
| [[15/14]], [[28/15]]
| 19.443
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
|-
| [[7/6]], [[12/7]]
| 33.129
|-
| [[11/10]], [[20/11]]
| 34.996
|-
| [[9/7]], [[14/9]]
| 35.084
|-
| [[18/13]], [[13/9]]
| 36.618
|-
| [[15/11]], [[22/15]]
| 36.951
|-
| [[13/12]], [[24/13]]
| 38.573
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741 ''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
|}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>4mgox0oejqrrmyffc60ozn0vq9p90m2</sha1>
    </revision>
    <revision>
      <id>87047</id>
      <parentid>87043</parentid>
      <timestamp>2022-02-25T13:26:38Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+relative error for odd-limit error table, +automated odd-limit error table by patent val</comment>
      <origin>87047</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11188" sha1="if9xdpgjo5wb2xoj0hvo4so6fpvtn78" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
{{Harmonics in equal}}

12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number.

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]]
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>if9xdpgjo5wb2xoj0hvo4so6fpvtn78</sha1>
    </revision>
    <revision>
      <id>87048</id>
      <parentid>87047</parentid>
      <timestamp>2022-02-25T13:29:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Prime error table sectioning</comment>
      <origin>87048</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11213" sha1="3mj3n0w9yny91z3dl0qglsdhgiqdyk0" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>3mj3n0w9yny91z3dl0qglsdhgiqdyk0</sha1>
    </revision>
    <revision>
      <id>87621</id>
      <parentid>87048</parentid>
      <timestamp>2022-02-26T20:20:16Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>/* See also */ +1</comment>
      <origin>87621</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11246" sha1="755n2qrzdlmk00mn80q1fhw4evjx9io" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
| Monotonicity = 11
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>755n2qrzdlmk00mn80q1fhw4evjx9io</sha1>
    </revision>
    <revision>
      <id>90403</id>
      <parentid>87621</parentid>
      <timestamp>2022-03-26T12:57:40Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>-monotonicity</comment>
      <origin>90403</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11226" sha1="3w5vxycxs3durruokhrr4nkmaat4wd2" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>3w5vxycxs3durruokhrr4nkmaat4wd2</sha1>
    </revision>
    <revision>
      <id>90472</id>
      <parentid>90403</parentid>
      <timestamp>2022-03-28T15:02:02Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>90472</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11300" sha1="77muox8e0iy94v02wf2maz94w5q2rzh" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only EDO that is both highly melodic and zeta.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:12edo| ]]
[[Category:Equal divisions of the octave]]
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>77muox8e0iy94v02wf2maz94w5q2rzh</sha1>
    </revision>
    <revision>
      <id>92575</id>
      <parentid>90472</parentid>
      <timestamp>2022-06-14T13:29:30Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>92575</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11349" sha1="7md0jm25vsndsc86k4sfzre2e8l67mi" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Major 2nd = 2\12 (200¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only EDO that is both highly melodic and zeta.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:12edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>7md0jm25vsndsc86k4sfzre2e8l67mi</sha1>
    </revision>
    <revision>
      <id>94748</id>
      <parentid>92575</parentid>
      <timestamp>2022-08-19T18:53:24Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>-major 2nd in infobox</comment>
      <origin>94748</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11326" sha1="bf9hj6qx8zqlioc2a4kcbk4yfzvjm95" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only EDO that is both highly melodic and zeta.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! colspan="3" | Interval
! Approximate JI Ratios*
|-
| 0
| 0
| unison
| P1
| D
| 1/1
|-
| 1
| 100
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
|-
| 2
| 200
| major 2nd
| M2
| E
| 8/7, 9/8, 10/9, 17/15, 19/17
|-
| 3
| 300
| minor 3rd
| m3
| F
| 7/6, 6/5, 19/16
|-
| 4
| 400
| major 3rd
| M3
| F#
| 5/4, 9/7
|-
| 5
| 500
| perfect 4th
| P4
| G
| 4/3
|-
| 6
| 600
| aug 4th, dim 5th
| A4, d5
| G#, Ab
| 7/5, 10/7, 17/12, 24/17
|-
| 7
| 700
| perfect 5th
| P5
| A
| 3/2
|-
| 8
| 800
| minor 6th
| m6
| Bb
| 8/5, 14/9
|-
| 9
| 900
| major 6th
| M6
| B
| 5/3, 12/7, 32/19
|-
| 10
| 1000
| minor 7th
| m7
| C
| 7/4, 9/5, 16/9
|-
| 11
| 1100
| major 7th
| M7
| C#
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
|-
| 12
| 1200
| perfect 8ve
| P8
| D
| 2/1
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
! Temperaments
|-
| 1
| 1\12
| [[Ripple]]
|-
| 1
| 5\12
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| [[Augmented]]
|-
| 4
| 1\12
| [[Diminished]]
|-
| 6
| 1\12
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:12edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>bf9hj6qx8zqlioc2a4kcbk4yfzvjm95</sha1>
    </revision>
    <revision>
      <id>95179</id>
      <parentid>94748</parentid>
      <timestamp>2022-08-31T12:01:15Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>added trad. solfege, added pergens</comment>
      <origin>95179</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11490" sha1="k82uhe7jmqgecfj6q21i541eik2c9xz" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only EDO that is both highly melodic and zeta.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
!Pergen
! Temperaments
|-
| 1
| 1\12
|(P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
|(P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
|(P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
|(P8/3, P5)
| [[Augmented]]
|-
| 4
| 1\12
|(P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
|(P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:12edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>k82uhe7jmqgecfj6q21i541eik2c9xz</sha1>
    </revision>
    <revision>
      <id>95297</id>
      <parentid>95179</parentid>
      <timestamp>2022-09-07T17:16:33Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>95297</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11587" sha1="fm0q7d76ajv736kz9v0l59fsek7w1ie" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET
| Prime factorization = 2&lt;sup&gt;2&lt;/sup&gt; × 3
| Step size = 100.000¢&lt;sup&gt;by definition&lt;/sup&gt;
| Fifth = 7\12 (700¢)
| Semitones = 1:1 (100¢:100¢)
| Consistency = 9
}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
!Pergen
! Temperaments
|-
| 1
| 1\12
|(P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
|(P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
|(P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
|(P8/3, P5)
| [[Augmented]]
|-
| 4
| 1\12
|(P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
|(P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:12edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>fm0q7d76ajv736kz9v0l59fsek7w1ie</sha1>
    </revision>
    <revision>
      <id>96088</id>
      <parentid>95297</parentid>
      <timestamp>2022-10-04T16:04:37Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET now computes most parameters automatically</comment>
      <origin>96088</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11422" sha1="nvvs81kcz5ial0hue8hjie0jsi8cum4" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
!Pergen
! Temperaments
|-
| 1
| 1\12
|(P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
|(P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
|(P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
|(P8/3, P5)
| [[Augmented]]
|-
| 4
| 1\12
|(P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
|(P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:12edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>nvvs81kcz5ial0hue8hjie0jsi8cum4</sha1>
    </revision>
    <revision>
      <id>96872</id>
      <parentid>96088</parentid>
      <timestamp>2022-10-06T14:35:33Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Scales */</comment>
      <origin>96872</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11723" sha1="2aucbpg0644sdfncujoclx4kiyt9ip0" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
!Pergen
! Temperaments
|-
| 1
| 1\12
|(P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
|(P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
|(P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
|(P8/3, P5)
| [[Augmented]]
|-
| 4
| 1\12
|(P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
|(P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-MOS scales ===
Due to 12edo's dominance, some non-MOS scales are also widely used in many musical practices around the world.

* Harmonic major - 2212132
* Melodic major - 2212122
* Hungarian minor - 2131131
* Maqam Hijaz / Double Harmonic major = 1312131
* 5-limit tonality diamond: 3112113

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:12edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly melodic]]</text>
      <sha1>2aucbpg0644sdfncujoclx4kiyt9ip0</sha1>
    </revision>
    <revision>
      <id>97111</id>
      <parentid>96872</parentid>
      <timestamp>2022-10-09T18:44:44Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:Highly melodic]] to [[Category:Highly composite]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>97111</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11725" sha1="1rh7qkwjw9ddxk24rmyindc3j0r6e76" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
'''12 equal divisions of the octave''' ('''12edo'''), or '''12(-tone) equal temperament''' ('''12tet''', '''12et''') when viewed from a [[regular temperament]] perspective, is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
!Pergen
! Temperaments
|-
| 1
| 1\12
|(P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
|(P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
|(P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
|(P8/3, P5)
| [[Augmented]]
|-
| 4
| 1\12
|(P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
|(P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-MOS scales ===
Due to 12edo's dominance, some non-MOS scales are also widely used in many musical practices around the world.

* Harmonic major - 2212132
* Melodic major - 2212122
* Hungarian minor - 2131131
* Maqam Hijaz / Double Harmonic major = 1312131
* 5-limit tonality diamond: 3112113

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:12edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly composite]]</text>
      <sha1>1rh7qkwjw9ddxk24rmyindc3j0r6e76</sha1>
    </revision>
    <revision>
      <id>100019</id>
      <parentid>97111</parentid>
      <timestamp>2022-12-15T00:04:51Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>a bit of semantic - it is predominant tuning system no matter what, not just because from a "regular temperament perspective".</comment>
      <origin>100019</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11572" sha1="d6odqs7tk03gb18bcdkxwxrsoactndn" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
!Pergen
! Temperaments
|-
| 1
| 1\12
|(P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
|(P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
|(P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
|(P8/3, P5)
| [[Augmented]]
|-
| 4
| 1\12
|(P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
|(P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-MOS scales ===
Due to 12edo's dominance, some non-MOS scales are also widely used in many musical practices around the world.

* Harmonic major - 2212132
* Melodic major - 2212122
* Hungarian minor - 2131131
* Maqam Hijaz / Double Harmonic major = 1312131
* 5-limit tonality diamond: 3112113

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:12edo| ]] &lt;!-- main article --&gt;
[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Meantone]]
[[Category:Zeta]]
[[Category:Highly composite]]</text>
      <sha1>d6odqs7tk03gb18bcdkxwxrsoactndn</sha1>
    </revision>
    <revision>
      <id>100653</id>
      <parentid>100019</parentid>
      <timestamp>2022-12-29T18:27:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>-redundant categories addressed by the infobox</comment>
      <origin>100653</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11411" sha1="1e6c2wtq56hr622ocped0e8btn3h9nz" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Major diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Tritonic diesis, jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Septimal semicomma, starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Septimal kleisma, marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma, Gauss' comma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per octave
! Generator
!Pergen
! Temperaments
|-
| 1
| 1\12
|(P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
|(P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
|(P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
|(P8/3, P5)
| [[Augmented]]
|-
| 4
| 1\12
|(P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
|(P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-MOS scales ===
Due to 12edo's dominance, some non-MOS scales are also widely used in many musical practices around the world.

* Harmonic major - 2212132
* Melodic major - 2212122
* Hungarian minor - 2131131
* Maqam Hijaz / Double Harmonic major = 1312131
* 5-limit tonality diamond: 3112113

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Meantone]]</text>
      <sha1>1e6c2wtq56hr622ocped0e8btn3h9nz</sha1>
    </revision>
    <revision>
      <id>101890</id>
      <parentid>100653</parentid>
      <timestamp>2023-01-25T08:14:18Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>Style and cleanup</comment>
      <origin>101890</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11297" sha1="ieurca1iq9uq3k8vej0uc36n7ht76ih" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Meantone]]</text>
      <sha1>ieurca1iq9uq3k8vej0uc36n7ht76ih</sha1>
    </revision>
    <revision>
      <id>101891</id>
      <parentid>101890</parentid>
      <timestamp>2023-01-25T08:15:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>+well temperaments</comment>
      <origin>101891</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11500" sha1="ei0y3jdc92z9a50o2ewl5vifwnqd3sm" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Meantone]]</text>
      <sha1>ei0y3jdc92z9a50o2ewl5vifwnqd3sm</sha1>
    </revision>
    <revision>
      <id>102039</id>
      <parentid>101891</parentid>
      <timestamp>2023-01-28T19:50:11Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Intervals */ same thing as on 5L 2s page.</comment>
      <origin>102039</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11565" sha1="ao8vunw8isdvtt0d9g76kyriala7nju" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Meantone]]</text>
      <sha1>ao8vunw8isdvtt0d9g76kyriala7nju</sha1>
    </revision>
    <revision>
      <id>102632</id>
      <parentid>102039</parentid>
      <timestamp>2023-02-11T16:40:33Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Rank-2 temperaments */ lithium is also</comment>
      <origin>102632</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11579" sha1="taav2d3l8rcgbe7fghkbsemveahib0k" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Meantone]]</text>
      <sha1>taav2d3l8rcgbe7fghkbsemveahib0k</sha1>
    </revision>
    <revision>
      <id>106422</id>
      <parentid>102632</parentid>
      <timestamp>2023-03-30T22:04:20Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>106422</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12003" sha1="tvdq3kfn5phnq1881d2g8cfpr1ydb9f" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
| 
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:Audacity pluck 1 12.wav]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:Audacity pluck 2 12.wav]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:Audacity pluck 3 12.wav]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:Audacity pluck  4 12.wav]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:Audacity pluck 5 12.wav]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:Audacity pluck 6 12.wav]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:Audacity pluck 7 12.wav]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:Audacity pluck 8 12.wav]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:Audacity pluck 9 12.wav]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:Audacity pluck 10 12.wav]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:Audacity pluck 11 12.wav]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:Audacity pluck octave.wav]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Meantone]]</text>
      <sha1>tvdq3kfn5phnq1881d2g8cfpr1ydb9f</sha1>
    </revision>
    <revision>
      <id>109732</id>
      <parentid>106422</parentid>
      <timestamp>2023-04-27T01:16:07Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Added to categories "equal divisions of the octave" and "historical"</comment>
      <origin>109732</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12069" sha1="rp0pt39yzdv5trldsbsgja91fo1oxpm" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
| 
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:Audacity pluck 1 12.wav]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:Audacity pluck 2 12.wav]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:Audacity pluck 3 12.wav]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:Audacity pluck  4 12.wav]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:Audacity pluck 5 12.wav]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:Audacity pluck 6 12.wav]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:Audacity pluck 7 12.wav]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:Audacity pluck 8 12.wav]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:Audacity pluck 9 12.wav]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:Audacity pluck 10 12.wav]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:Audacity pluck 11 12.wav]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:Audacity pluck octave.wav]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{See also|:Category:12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]
[[Category:Equal divisions of the octave]]
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>rp0pt39yzdv5trldsbsgja91fo1oxpm</sha1>
    </revision>
    <revision>
      <id>115634</id>
      <parentid>109732</parentid>
      <timestamp>2023-06-03T14:30:48Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>/* Music */ Catrel</comment>
      <origin>115634</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12057" sha1="flwujvjmjeeiwihv22y08fszzi8xnjg" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
| 
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:Audacity pluck 1 12.wav]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:Audacity pluck 2 12.wav]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:Audacity pluck 3 12.wav]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:Audacity pluck  4 12.wav]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:Audacity pluck 5 12.wav]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:Audacity pluck 6 12.wav]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:Audacity pluck 7 12.wav]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:Audacity pluck 8 12.wav]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:Audacity pluck 9 12.wav]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:Audacity pluck 10 12.wav]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:Audacity pluck 11 12.wav]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:Audacity pluck octave.wav]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]
[[Category:Equal divisions of the octave]]
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>flwujvjmjeeiwihv22y08fszzi8xnjg</sha1>
    </revision>
    <revision>
      <id>116540</id>
      <parentid>115634</parentid>
      <timestamp>2023-06-13T01:33:45Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>116540</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11996" sha1="aapm74y5nz2kxo73l9uqqd36q4xlxqw" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]
[[Category:Equal divisions of the octave]]
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>aapm74y5nz2kxo73l9uqqd36q4xlxqw</sha1>
    </revision>
    <revision>
      <id>116541</id>
      <parentid>116540</parentid>
      <timestamp>2023-06-13T01:34:10Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>116541</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="11997" sha1="7d0x64uy85ncgi6j9ldkfny3jcoz0u1" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]
[[Category:Equal divisions of the octave]]
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>7d0x64uy85ncgi6j9ldkfny3jcoz0u1</sha1>
    </revision>
    <revision>
      <id>116711</id>
      <parentid>116541</parentid>
      <timestamp>2023-06-13T21:06:03Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Sort key</comment>
      <origin>116711</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12025" sha1="qcf11d761zqcwvr3zu0tydyctb7qf4g" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>qcf11d761zqcwvr3zu0tydyctb7qf4g</sha1>
    </revision>
    <revision>
      <id>117262</id>
      <parentid>116711</parentid>
      <timestamp>2023-06-21T04:22:14Z</timestamp>
      <contributor>
        <username>Gud2228</username>
        <id>176</id>
      </contributor>
      <comment>/* Rank-2 temperaments */</comment>
      <origin>117262</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12039" sha1="kxrf9pkkkdp3aj0yjab0ljg3iercizp" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
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    <revision>
      <id>120432</id>
      <parentid>117262</parentid>
      <timestamp>2023-07-15T18:50:54Z</timestamp>
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      <text bytes="12149" sha1="gdo9cgfz1xfzymyhy4h0tkrl5cwfh67" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}
[https://www.youtube.com/watch?v=dQw4w9WgXcQ Never Gonna Give You Up] - [[wikipedia:Rick Astley|Rick Astley]]

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>gdo9cgfz1xfzymyhy4h0tkrl5cwfh67</sha1>
    </revision>
    <revision>
      <id>120436</id>
      <parentid>120432</parentid>
      <timestamp>2023-07-15T18:59:20Z</timestamp>
      <contributor>
        <username>Secretcoffee</username>
        <id>399</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>120436</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12372" sha1="6d8i8c9hx0at7344fjlhiy0x8tjt6gf" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony - for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}
[https://www.youtube.com/watch?v=dQw4w9WgXcQ Never Gonna Give You Up] - [[wikipedia:Rick Astley|Rick Astley]]

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>6d8i8c9hx0at7344fjlhiy0x8tjt6gf</sha1>
    </revision>
    <revision>
      <id>120437</id>
      <parentid>120436</parentid>
      <timestamp>2023-07-15T18:59:51Z</timestamp>
      <contributor>
        <username>Secretcoffee</username>
        <id>399</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
      <origin>120437</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12371" sha1="ehj33unztkbalxf7idmym9f0q2ofs9u" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}
[https://www.youtube.com/watch?v=dQw4w9WgXcQ Never Gonna Give You Up] - [[wikipedia:Rick Astley|Rick Astley]]

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>ehj33unztkbalxf7idmym9f0q2ofs9u</sha1>
    </revision>
    <revision>
      <id>120889</id>
      <parentid>120437</parentid>
      <timestamp>2023-07-23T04:55:20Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Add 13-limit uniform maps</comment>
      <origin>120889</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12423" sha1="9bsee0t635u2bi5j4vsbus9xb92v98s" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12-EDO
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}
[https://www.youtube.com/watch?v=dQw4w9WgXcQ Never Gonna Give You Up] - [[wikipedia:Rick Astley|Rick Astley]]

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>9bsee0t635u2bi5j4vsbus9xb92v98s</sha1>
    </revision>
    <revision>
      <id>124039</id>
      <parentid>120889</parentid>
      <timestamp>2023-09-13T02:49:09Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Match English wiki page title</comment>
      <origin>124039</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12422" sha1="1nr4jtbmgx91hlolywbpe568v88spfk" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}
[https://www.youtube.com/watch?v=dQw4w9WgXcQ Never Gonna Give You Up] - [[wikipedia:Rick Astley|Rick Astley]]

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Equal divisions of the octave|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>1nr4jtbmgx91hlolywbpe568v88spfk</sha1>
    </revision>
    <revision>
      <id>125624</id>
      <parentid>124039</parentid>
      <timestamp>2023-10-06T14:20:16Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Sup with this song?</comment>
      <origin>125624</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12356" sha1="42znof7azdpkz9f4l59ani4ox0hbps7" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

; {{W|Rick Astley}}
* [https://www.youtube.com/watch?v=dQw4w9WgXcQ ''Never Gonna Give You Up'']&lt;sup&gt;[why?]&lt;/sup&gt;

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>42znof7azdpkz9f4l59ani4ox0hbps7</sha1>
    </revision>
    <revision>
      <id>127766</id>
      <parentid>125624</parentid>
      <timestamp>2023-11-22T08:26:44Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Music */ unexplained for extended period</comment>
      <origin>127766</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12242" sha1="co3a8cl472qb5mung6tmkjloxx2fqf1" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| 7/6, 6/5, 19/16
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| 5/4, 9/7
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| 8/5, 14/9
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>co3a8cl472qb5mung6tmkjloxx2fqf1</sha1>
    </revision>
    <revision>
      <id>129265</id>
      <parentid>127766</parentid>
      <timestamp>2023-12-18T01:27:06Z</timestamp>
      <contributor>
        <username>YoVariable</username>
        <id>434</id>
      </contributor>
      <minor/>
      <comment>Added new JI ratios to 12edo /* Intervals */</comment>
      <origin>129265</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12305" sha1="84n8zkqvb3ewr33wdmpkm4qzozhm00r" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| 1/1
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| 15/14, 16/15, 17/16, 18/17, 21/20, 25/24, 28/27
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| 8/7, 9/8, 10/9, 17/15, 19/17, 55/49, 64/57
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| 7/6, 6/5, 19/16, 25/21
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| 5/4, 9/7, 81/64, 63/50, 34/27
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| 4/3
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| 7/5, 10/7, 17/12, 24/17, 99/70
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| 3/2
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| 8/5, 14/9, 27/17
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| 5/3, 12/7, 32/19
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| 7/4, 9/5, 16/9, 25/14
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| 15/8, 17/9, 28/15, 40/21, 48/25, 27/14
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| 2/1
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>84n8zkqvb3ewr33wdmpkm4qzozhm00r</sha1>
    </revision>
    <revision>
      <id>129267</id>
      <parentid>129265</parentid>
      <timestamp>2023-12-18T01:37:29Z</timestamp>
      <contributor>
        <username>YoVariable</username>
        <id>434</id>
      </contributor>
      <comment>Added hyperlinks to 12edo approximate JI ratios in /* Intervals */</comment>
      <origin>129267</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="12508" sha1="ijqhetxdhpgspv3nvo3v7relvrt6nxz" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
12edo intervals have standard names from classical music theory.
{| class="wikitable right-1 right-2"
! Steps
! Cents
! Approximate JI Ratios*
! colspan="3" | Interval
!solfege
! Audio
|-
| 0
| 0
| [[1/1]]
| unison
| P1
| D
|Do
|[[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| [[15/14]], [[16/15]], [[17/16]], [[18/17]], [[21/20]], [[25/24]], [[28/27]]
| aug 1sn, minor 2nd
| A1, m2
| D#, Eb
|Di / Ra
|[[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| [[8/7]], [[9/8]], [[10/9]], [[17/15]], [[19/17]], [[55/49]], [[64/57]]
| major 2nd
| M2
| E
|Re
|[[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| [[7/6]], [[6/5]], [[32/27]], [[19/16]], [[25/21]]
| minor 3rd
| m3
| F
|Ri / Me
|[[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| [[5/4]], [[9/7]], [[81/64]], [[63/50]], [[34/27]]
| major 3rd
| M3
| F#
|Mi
|[[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| [[4/3]]
| perfect 4th
| P4
| G
|Fa
|[[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[7/5]], [[10/7]], [[17/12]], [[24/17]], [[99/70]]
| aug 4th, dim 5th
| A4, d5
| G#, Ab
|Fi / Se
|[[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| [[3/2]]
| perfect 5th
| P5
| A
|So
|[[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| [[8/5]], [[14/9]], [[27/17]]
| minor 6th
| m6
| Bb
|Si / Le
|[[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| [[5/3]], [[12/7]], [[32/19]]
| major 6th
| M6
| B
|La
|[[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| [[7/4]], [[9/5]], [[16/9]], [[25/14]]
| minor 7th
| m7
| C
|Li / Te
|[[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| [[15/8]], [[17/9]], [[28/15]], [[40/21]], [[48/25]], [[27/14]]
| major 7th
| M7
| C#
|Ti
|[[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| [[2/1]]
| perfect 8ve
| P8
| D
|Do
|[[File:piano_1_1edo.mp3]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>ijqhetxdhpgspv3nvo3v7relvrt6nxz</sha1>
    </revision>
    <revision>
      <id>130088</id>
      <parentid>129267</parentid>
      <timestamp>2023-12-30T21:44:44Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>Split intervals table into 3 (intervals, notation, solfege)</comment>
      <origin>130088</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15661" sha1="cfk0i9fcft71etaktivsjtixxhy7ny2" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| 1/1 (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| Tritone
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| '''Do'''
| '''Da'''
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| '''Re'''
| '''Ra'''
|-
| 3
| 300
| Ri (A2)&lt;br&gt;'''Me''' (m3)
| Ru (A2)&lt;br&gt;'''Na''' (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| '''So'''
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| '''La'''
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;'''Te''' (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| '''Do'''
| '''Da'''
|}

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>cfk0i9fcft71etaktivsjtixxhy7ny2</sha1>
    </revision>
    <revision>
      <id>130091</id>
      <parentid>130088</parentid>
      <timestamp>2023-12-30T22:29:53Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>/* Notation */ Mention edos that use 12edo subset notation</comment>
      <origin>130091</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15755" sha1="mn7ew2zl9hjzph6om1fnlvyjshv2dh8" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| 1/1 (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| Tritone
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| '''Do'''
| '''Da'''
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| '''Re'''
| '''Ra'''
|-
| 3
| 300
| Ri (A2)&lt;br&gt;'''Me''' (m3)
| Ru (A2)&lt;br&gt;'''Na''' (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| '''So'''
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| '''La'''
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;'''Te''' (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| '''Do'''
| '''Da'''
|}

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>mn7ew2zl9hjzph6om1fnlvyjshv2dh8</sha1>
    </revision>
    <revision>
      <id>130114</id>
      <parentid>130091</parentid>
      <timestamp>2023-12-31T02:20:22Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>/* Solfege */ Remove bold (not relevant without the nominals)</comment>
      <origin>130114</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15689" sha1="2yo2m6he9lhiyjwczt4zz8ezqckgj31" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| 1/1 (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| Tritone
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>2yo2m6he9lhiyjwczt4zz8ezqckgj31</sha1>
    </revision>
    <revision>
      <id>130129</id>
      <parentid>130114</parentid>
      <timestamp>2023-12-31T04:15:53Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>/* Intervals */ Add 44/37 and 37/22 as closer approximations to m3 and M6</comment>
      <origin>130129</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15733" sha1="plc86p6yex9ud4ppm4ozco0n3e03niq" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| 1/1 (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| Tritone
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>plc86p6yex9ud4ppm4ozco0n3e03niq</sha1>
    </revision>
    <revision>
      <id>130334</id>
      <parentid>130129</parentid>
      <timestamp>2024-01-02T08:34:09Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>/* Intervals */ Add 19/12 and 24/19</comment>
      <origin>130334</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15777" sha1="rqkqc28q8097vjtqukmv5e3ra3qcwfn" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| 1/1 (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| Tritone
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>rqkqc28q8097vjtqukmv5e3ra3qcwfn</sha1>
    </revision>
    <revision>
      <id>130335</id>
      <parentid>130334</parentid>
      <timestamp>2024-01-02T08:39:23Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ Link to 1/1</comment>
      <origin>130335</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15781" sha1="hb4bvqfn0km78rsk39c8kbe0q11v6sp" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| Tritone
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>hb4bvqfn0km78rsk39c8kbe0q11v6sp</sha1>
    </revision>
    <revision>
      <id>130830</id>
      <parentid>130335</parentid>
      <timestamp>2024-01-05T04:48:02Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ Link to tritone</comment>
      <origin>130830</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15785" sha1="moz3zm2sq1tmr0c67v8pf5mh6eeda0x" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest equal division of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13 + 2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15 + 2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via the [[val]] {{val| 12 19 28 34 }}, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[Well tempered nonet|wtn]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>moz3zm2sq1tmr0c67v8pf5mh6eeda0x</sha1>
    </revision>
    <revision>
      <id>138626</id>
      <parentid>130830</parentid>
      <timestamp>2024-03-09T08:53:29Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
      <origin>138626</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15364" sha1="7qcdg28qechus7ua0s9v0ucwx13zots" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by some 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

===Prime harmonics===
{{Harmonics in equal}}

==Intervals==
{| class="wikitable center-all"
|+Intervals of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" |[[Interval region]]
! colspan="4" |Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" |Audio
|-
![[3-limit]]
![[5-limit]]
![[7-limit]]
!Other
|-
|0
|0
|Unison (prime)
|[[1/1]] (just)
|
|
|
|[[File:piano_0_1edo.mp3]]
|-
|1
| 100
|Minor second
|
|[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
|[[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
|[[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
|[[File:piano_1_12edo.mp3]]
|-
|2
| 200
|Major second
|[[9/8]] (-3.910)
|[[10/9]] (+17.596)
|[[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
|[[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
|[[File:piano_1_6edo.mp3]]
|-
|3
| 300
|Minor third
|[[32/27]] (+5.865)
|[[6/5]] (-15.641)
|[[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
|[[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
|[[File:piano_1_4edo.mp3]]
|-
|4
| 400
|Major third
|[[81/64]] (-7.820)
|[[5/4]] (+13.686)
|[[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
|[[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
|[[File:piano_1_3edo.mp3]]
|-
|5
|500
|Fourth
|[[4/3]] (+1.955)
|
|
|
|[[File:piano_5_12edo.mp3]]
|-
|6
|600
|[[Tritone]]
|
|
|[[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
|[[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
|[[File:piano_1_2edo.mp3]]
|-
|7
|700
|Fifth
|[[3/2]] (-1.955)
|
|
|
|[[File:piano_7_12edo.mp3]]
|-
|8
| 800
|Minor sixth
|[[128/81]] (+7.820)
|[[8/5]] (-13.686)
|[[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
|[[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
|[[File:piano_2_3edo.mp3]]
|-
|9
| 900
|Major sixth
|[[27/16]] (-5.865)
|[[5/3]] (+15.641)
|[[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
|[[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
|[[File:piano_3_4edo.mp3]]
|-
| 10
|1000
|Minor seventh
|[[16/9]] (+3.910)
|[[9/5]] (-17.596)
|[[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
|[[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
|[[File:piano_5_6edo.mp3]]
|-
| 11
|1100
|Major seventh
| 
|[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
|[[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
|[[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
|[[File:piano_11_12edo.mp3]]
|-
| 12
|1200
|Octave
|[[2/1]] (just)
|
|
|
|[[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

==Notation==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
![[5L 2s|Diatonic]] interval names
!Note names (on D)
|-
|0
|0
|'''Perfect unison (P1)'''
|'''D'''
|-
|1
|100
|Augmented unison (A1)&lt;br&gt;Minor second (m2)
|D#&lt;br&gt;Eb
|-
|2
|200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
|3
|300
|Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
|E#&lt;br&gt;'''F'''
|-
|4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
|F#&lt;br&gt;Gb
|-
|5
|500
|'''Perfect fourth (P4)'''
|'''G'''
|-
|6
|600
|Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
|G#&lt;br&gt;Ab
|-
|7
|700
|'''Perfect fifth (P5)'''
|A
|-
|8
|800
|Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
|A#&lt;br&gt;Bb
|-
|9
|900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
|1000
|Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
|B#&lt;br&gt;'''C'''
|-
| 11
|1100
|Major seventh (M7)&lt;br&gt;Diminished octave (d8)
|C#&lt;br&gt;Db
|-
| 12
|1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
*[[ups and downs notation]] is identical to standard notation;
*mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

==Solfege==
{| class="wikitable center-all"
|+Solfege of 12edo
![[Degree]]
![[Cents]]
!Standard [[solfege]]&lt;br&gt;(movable do)
![[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
|0
|0
|Do
| Da
|-
|1
|100
|Di (A1)&lt;br&gt;Ra (m2)
|Du (A1)&lt;br&gt;Fra (m2)
|-
|2
|200
|Re
| Ra
|-
|3
|300
|Ri (A2)&lt;br&gt;Me (m3)
|Ru (A2)&lt;br&gt;Na (m3)
|-
|4
|400
|Mi
|Ma (M3)&lt;br&gt;Fo (d4)
|-
|5
|500
|Fa
|Mu (A3)&lt;br&gt;Fa (P4)
|-
|6
|600
|Fi (A4)&lt;br&gt;Se (d5)
|Pa (A4)&lt;br&gt;Sha (d5)
|-
|7
|700
|So
| Sa
|-
|8
|800
|Si (A5)&lt;br&gt;Le (m6)
|Su (A5)&lt;br&gt;Fla (m6)
|-
|9
|900
|La
|La (M6)&lt;br&gt;Tho (d7)
|-
| 10
|1000
|Li (A6)&lt;br&gt;Te (m7)
|Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
|Ti
|Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
|Do
|Da
|}

==JI approximation==
===15-odd-limit interval mappings===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space:nowrap" |15-odd-limit intervals by direct approximation (even if inconsistent)
|-
!Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
|'''[[4/3]], [[3/2]]'''
|'''1.955'''
|'''2.0'''
|-
|[[9/8]], [[16/9]]
|3.910
|3.9
|-
|''[[13/11]], [[22/13]]''
|''10.790''
|''10.8''
|-
|[[16/15]], [[15/8]]
|11.731
|11.7
|-
|'''[[5/4]], [[8/5]]'''
|'''13.686'''
|'''13.7'''
|-
|[[6/5]], [[5/3]]
|15.641
|15.6
|-
|[[7/5]], [[10/7]]
|17.488
|17.5
|-
|[[14/11]], [[11/7]]
|17.508
|17.5
|-
|[[10/9]], [[9/5]]
|17.596
|17.6
|-
|[[15/14]], [[28/15]]
|19.443
|19.4
|-
|''[[14/13]], [[13/7]]''
|''28.298''
|''28.3''
|-
|'''[[8/7]], [[7/4]]'''
|'''31.174'''
|'''31.2'''
|-
|[[7/6]], [[12/7]]
|33.129
|33.1
|-
|[[11/10]], [[20/11]]
|34.996
|35.0
|-
|[[9/7]], [[14/9]]
|35.084
|35.1
|-
|[[18/13]], [[13/9]]
|36.618
|36.7
|-
|[[15/11]], [[22/15]]
|36.951
|37.0
|-
|[[13/12]], [[24/13]]
|38.573
|38.6
|-
|'''[[16/13]], [[13/8]]'''
|'''40.528'''
|'''40.5'''
|-
|''[[13/10]], [[20/13]]''
|''45.786''
|''45.8''
|-
|''[[11/9]], [[18/11]]''
|''47.408''
|''47.4''
|-
|''[[15/13]], [[26/15]]''
|''47.741''
|''47.7''
|-
|'''[[11/8]], [[16/11]]'''
|'''48.682'''
|'''48.7'''
|-
|''[[12/11]], [[11/6]]''
|''49.323''
|''49.3''
|}
{{15-odd-limit}}

===Selected 19-limit intervals===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

==Regular temperament properties==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo| -19 12 }}
|[{{val| 12 19 }}]
| +0.617
|0.617
|0.617
|-
|2.3.5
|81/80, 128/125
|[{{val| 12 19 28 }}]
| -1.56
|3.11
|3.11
|-
|2.3.5.7
|36/35, 50/49, 64/63
|[{{val| 12 19 28 34 }}]
| -3.95
|4.92
|4.94
|-
|2.3.5.7.17
|36/35, 50/49, 51/49, 64/63
|[{{val| 12 19 28 34 49 }}]
| -2.92
|4.86
|4.87
|-
|2.3.5.7.17.19
|36/35, 50/49, 51/49, 57/56, 64/63
|[{{val| 12 19 28 34 49 51 }}]
| -2.53
|4.52
|4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

===Uniform maps===
{{Uniform map|13|11.5|12.5}} 

===Commas===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
![[Harmonic limit|Prime&lt;br&gt;Limit]]
![[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
![[Monzo]]
![[Cent]]s
![[Color name|Color Name]]
!Name
|-
|3
|[[531441/524288|(12 digits)]]
|{{monzo| -19 12 }}
|23.46
|Lalawa
|[[Pythagorean comma]]
|-
|5
|[[648/625]]
|{{monzo| 3 4 -4 }}
|62.57
|Quadgu
|Diminished comma
|-
|5
|[[128/125]]
|{{monzo| 7 0 -3 }}
|41.06
|Trigu
|Augmented comma
|-
|5
|[[81/80]]
|{{monzo| -4 4 -1 }}
|21.51
|Gu
|Syntonic comma
|-
|5
|[[2048/2025]]
|{{monzo| 11 -4 -2 }}
|19.55
|Sagugu
|Diaschisma
|-
|5
|[[67108864/66430125|(16 digits)]]
|{{monzo| 26 -12 -3 }}
|17.60
|Sasa-trigu
|[[Misty comma]]
|-
|5
|[[32805/32768]]
|{{monzo| -15 8 1 }}
|1.95
|Layo
|Schisma
|-
|5
|&lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
|{{monzo| 161 -84 -12 }}
|0.02
|Sepbisa-quadtrigu
|[[Kirnberger's atom]]
|-
|7
|[[36/35]]
|{{monzo| 2 2 -1 -1 }}
|48.77
|Rugu
|Septimal quartertone
|-
|7
|[[50/49]]
|{{monzo| 1 0 2 -2 }}
|34.98
|Biruyo
|Jubilisma
|-
|7
|[[64/63]]
|{{monzo| 6 -2 0 -1 }}
|27.26
|Ru
|Septimal comma
|-
|7
|[[3125/3087]]
|{{monzo| 0 -2 5 -3 }}
|21.18
|Triru-aquinyo
|Gariboh
|-
|7
|[[126/125]]
|{{monzo| 1 2 -3 1 }}
|13.79
|Zotrigu
|Starling comma
|-
|7
|[[4000/3969]]
|{{monzo| 5 -4 3 -2 }}
|13.47
| Rurutriyo
|Octagar comma
|-
|7
|&lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
|{{monzo| -9 8 -4 2 }}
|8.04
|Labizogugu
|[[Varunisma]]
|-
|7
|[[225/224]]
|{{monzo| -5 2 2 -1 }}
|7.71
|Ruyoyo
|Marvel comma
|-
|7
|[[3136/3125]]
|{{monzo| 6 0 -5 2 }}
|6.08
|Zozoquingu
|Hemimean
|-
|7
|[[5120/5103]]
|{{monzo| 10 -6 1 -1 }}
|5.76
|Saruyo
|Hemifamity
|-
|7
|[[33554432/33480783|(16 digits)]]
|{{monzo| 25 -14 0 -1 }}
|3.80
|Sasaru
|[[Garischisma]]
|-
|7
|[[703125/702464|(12 digits)]]
|{{monzo| -11 2 7 -3 }}
|1.63
|Latriru-asepyo
|[[Meter]]
|-
|7
|&lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
|{{monzo| -4 6 -6 3 }}
|0.33
| Trizogugu
|[[Landscape comma]]
|-
|11
|[[128/121]]
|{{monzo| 7 0 0 0 -2 }}
|97.36
|1uu2
|Axirabian limma
|-
|11
|[[99/98]]
|{{monzo| -1 2 0 -2 1 }}
|17.58
|Loruru
|Mothwellsma
|-
|11
|[[100/99]]
|{{monzo| 2 -2 2 0 -1 }}
|17.40
|Luyoyo
|Ptolemisma
|-
|11
|[[176/175]]
|{{monzo| 4 0 -2 -1 1 }}
|9.86
|Lorugugu
|Valinorsma
|-
|11
|[[896/891]]
|{{monzo| 7 -4 0 1 -1 }}
|9.69
|Saluzo
|Pentacircle
|-
|11
|[[441/440]]
|{{monzo| -3 2 -1 2 -1 }}
|3.93
|Luzozogu
|Werckisma
|-
|11
|[[9801/9800]]
|{{monzo| -3 4 -2 -2 2 }}
|0.18
|Bilorugu
|Kalisma
|-
|13
|[[91/90]]
|{{monzo| -1 -2 -1 1 0 1 }}
|19.13
|Thozogu
|Superleap
|}
&lt;references /&gt;

===Rank-2 temperaments===
*[[List of 12et rank two temperaments by badness]]
*[[List of 12et rank two temperaments by complexity]]
*[[List of edo-distinct 12f rank two temperaments]]
*[[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
!Periods &lt;br&gt; per 8ve
! Generator
!Pergen
!Temperaments
|-
|1
| 1\12
|(P8, P4/5)
|[[Ripple]] / [[passion]]
|-
|1
| 5\12
|(P8, P5)
|[[Meantone]] / [[dominant]]
|-
|2
|1\12
|(P8/2, P5)
|[[Srutal]] / [[pajara]] / [[injera]]
|-
|3
|1\12
|(P8/3, P5)
|[[Augmented]] / [[lithium]]
|-
|4
|1\12
|(P8/4, P5)
|[[Diminished]]
|-
|6
|1\12
|(P8/6, P5)
|[[Hexe]]
|}

==Scales==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
*Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
*Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
*Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

===Non-mos scales===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

*Harmonic major – 2212132
*Melodic major – 2212122
*Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

==Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

*[[Cauldron]]
*[[Bifrost]]
*[[Grail]]
*[[Secor5 23TX]]
*[[Secor wt10]]
*[[Sabat1]]
*[[Sabat2]]

==Music==
{{Catrel|12edo tracks}}

==See also==
*[[Lumatone mapping for 12edo]]
*[[purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>7qcdg28qechus7ua0s9v0ucwx13zots</sha1>
    </revision>
    <revision>
      <id>138631</id>
      <parentid>138626</parentid>
      <timestamp>2024-03-09T09:10:13Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
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      <origin>138631</origin>
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      <text bytes="15925" sha1="05vhf9w8xdcgoqadj3ymxe76s9i8zgt" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by some 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
|{{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>05vhf9w8xdcgoqadj3ymxe76s9i8zgt</sha1>
    </revision>
    <revision>
      <id>138669</id>
      <parentid>138631</parentid>
      <timestamp>2024-03-09T10:48:39Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>138669</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16331" sha1="kzoiqj2so5ydhajmwweni4y42v0ltt8" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by some 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| [[67108864/66430125|(16 digits)]]
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
|{{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| 117/110
| {{monzo| -1 2 -1 0 -1 1 }}
| 106.81
| 
| 
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 2 }}
| 6.00
| Soso
| Semitonisma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>kzoiqj2so5ydhajmwweni4y42v0ltt8</sha1>
    </revision>
    <revision>
      <id>138735</id>
      <parentid>138669</parentid>
      <timestamp>2024-03-09T20:57:38Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>138735</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16349" sha1="1fkidomsd5qkqxefgm7nbaq91nj3s4s" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by some 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| [[531441/524288|(12 digits)]]
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| 117/110
| {{monzo| -1 2 -1 0 -1 1 }}
| 106.81
| 
| 
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 2 }}
| 6.00
| Soso
| Semitonisma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>1fkidomsd5qkqxefgm7nbaq91nj3s4s</sha1>
    </revision>
    <revision>
      <id>138749</id>
      <parentid>138735</parentid>
      <timestamp>2024-03-10T00:55:09Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>138749</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16434" sha1="gi9oc7g7mf61qo6t090yog88sfdm2tc" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by some 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== JI approximation ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| 117/110
| {{monzo| -1 2 -1 0 -1 1 }}
| 106.81
| 
| 
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 2 }}
| 6.00
| Soso
| Semitonisma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>gi9oc7g7mf61qo6t090yog88sfdm2tc</sha1>
    </revision>
    <revision>
      <id>138867</id>
      <parentid>138749</parentid>
      <timestamp>2024-03-11T04:24:57Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* JI approximation */ -&gt; approximation to JI</comment>
      <origin>138867</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16437" sha1="kpvd3kr5m8ib48gzpmqrmbylhq381w2" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by some 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

=== Selected 19-limit intervals ===
[[File:12ed2-11-001.svg|alt=alt : Your browser has no SVG support.]]

An expanded version of the above, including some higher-limit intervals:

[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]]

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| 117/110
| {{monzo| -1 2 -1 0 -1 1 }}
| 106.81
| 
| 
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 2 }}
| 6.00
| Soso
| Semitonisma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>kpvd3kr5m8ib48gzpmqrmbylhq381w2</sha1>
    </revision>
    <revision>
      <id>139353</id>
      <parentid>138867</parentid>
      <timestamp>2024-03-16T21:07:42Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Approximation to JI */</comment>
      <origin>139353</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16322" sha1="ln4glka4zdaqk6kfqop4sofwu1yw44v" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by some 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| 117/110
| {{monzo| -1 2 -1 0 -1 1 }}
| 106.81
| 
| 
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 2 }}
| 6.00
| Soso
| Semitonisma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>ln4glka4zdaqk6kfqop4sofwu1yw44v</sha1>
    </revision>
    <revision>
      <id>139378</id>
      <parentid>139353</parentid>
      <timestamp>2024-03-17T00:41:52Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <comment>/* Scales */</comment>
      <origin>139378</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16353" sha1="kowh9zi31718au89yz6ski35oi7lqjy" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by some 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| 117/110
| {{monzo| -1 2 -1 0 -1 1 }}
| 106.81
| 
| 
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 2 }}
| 6.00
| Soso
| Semitonisma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>kowh9zi31718au89yz6ski35oi7lqjy</sha1>
    </revision>
    <revision>
      <id>140196</id>
      <parentid>139378</parentid>
      <timestamp>2024-03-26T12:35:14Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
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      <text bytes="16380" sha1="bgxyogpoxq7b77m4roa2l4vkqgkw64o" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>bgxyogpoxq7b77m4roa2l4vkqgkw64o</sha1>
    </revision>
    <revision>
      <id>140352</id>
      <parentid>140196</parentid>
      <timestamp>2024-03-29T14:18:06Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Commas */</comment>
      <origin>140352</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16548" sha1="faa5l68qzttsdpogjlpt577jpdexwiw" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>faa5l68qzttsdpogjlpt577jpdexwiw</sha1>
    </revision>
    <revision>
      <id>140353</id>
      <parentid>140352</parentid>
      <timestamp>2024-03-29T14:21:25Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>140353</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16630" sha1="tvjzjqg0vnb0rbk3ing1iy7o3scb7i4" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>tvjzjqg0vnb0rbk3ing1iy7o3scb7i4</sha1>
    </revision>
    <revision>
      <id>140473</id>
      <parentid>140353</parentid>
      <timestamp>2024-03-31T18:24:09Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Commas */</comment>
      <origin>140473</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16706" sha1="twe5ldmaj8jvhukdorlkvzwbjfknelb" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>twe5ldmaj8jvhukdorlkvzwbjfknelb</sha1>
    </revision>
    <revision>
      <id>140488</id>
      <parentid>140473</parentid>
      <timestamp>2024-03-31T19:42:48Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Commas */</comment>
      <origin>140488</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17147" sha1="safqajauioz208r9p9bjvxfdr1y5vd8" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>safqajauioz208r9p9bjvxfdr1y5vd8</sha1>
    </revision>
    <revision>
      <id>140489</id>
      <parentid>140488</parentid>
      <timestamp>2024-03-31T19:43:23Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Commas */</comment>
      <origin>140489</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17147" sha1="cx9h3s4vxrng3pfd8xyjtbjffblkt85" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>cx9h3s4vxrng3pfd8xyjtbjffblkt85</sha1>
    </revision>
    <revision>
      <id>140699</id>
      <parentid>140489</parentid>
      <timestamp>2024-04-04T17:47:37Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>140699</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17289" sha1="iei5rx1doartfcp9w761s60w8ftw5n4" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave shrinking or stretching is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both highly melodic and zeta, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>iei5rx1doartfcp9w761s60w8ftw5n4</sha1>
    </revision>
    <revision>
      <id>141594</id>
      <parentid>140699</parentid>
      <timestamp>2024-04-22T12:39:16Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>141594</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17485" sha1="pf1w78gqm7c7bj4nzypuup7chz6ezi1" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>pf1w78gqm7c7bj4nzypuup7chz6ezi1</sha1>
    </revision>
    <revision>
      <id>141645</id>
      <parentid>141594</parentid>
      <timestamp>2024-04-22T15:36:06Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>141645</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17571" sha1="gal30dl4pmjt4n7tn7a1ary6sg352q3" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| 0 -1 2 -1 1 -1 }}
| 12.64
| Thuloruyoyo
| Gassorma
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>gal30dl4pmjt4n7tn7a1ary6sg352q3</sha1>
    </revision>
    <revision>
      <id>141651</id>
      <parentid>141645</parentid>
      <timestamp>2024-04-22T15:40:20Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
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| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>3h6271i5zs636yb5jxgetoa3d95rd0f</sha1>
    </revision>
    <revision>
      <id>141660</id>
      <parentid>141651</parentid>
      <timestamp>2024-04-22T15:51:24Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>141660</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17673" sha1="7zow3vvmt11vsfzmmvk0de4d2qf2fk0" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quartertone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>7zow3vvmt11vsfzmmvk0de4d2qf2fk0</sha1>
    </revision>
    <revision>
      <id>141676</id>
      <parentid>141660</parentid>
      <timestamp>2024-04-22T18:31:30Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Commas */</comment>
      <origin>141676</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17759" sha1="7m3exdov4pj3isl9cx0k7zxr0mqkpwg" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>7m3exdov4pj3isl9cx0k7zxr0mqkpwg</sha1>
    </revision>
    <revision>
      <id>141887</id>
      <parentid>141676</parentid>
      <timestamp>2024-04-25T13:56:53Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Commas */</comment>
      <origin>141887</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17951" sha1="h65niqqcwbzia0u0b7glf4evhxtjd4q" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>h65niqqcwbzia0u0b7glf4evhxtjd4q</sha1>
    </revision>
    <revision>
      <id>141898</id>
      <parentid>141887</parentid>
      <timestamp>2024-04-25T14:42:30Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Notation */</comment>
      <origin>141898</origin>
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      <format>text/x-wiki</format>
      <text bytes="17976" sha1="rzqanay7r56kxz5s273ohhjl9c0lb1s" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>rzqanay7r56kxz5s273ohhjl9c0lb1s</sha1>
    </revision>
    <revision>
      <id>141951</id>
      <parentid>141898</parentid>
      <timestamp>2024-04-25T19:40:12Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Commas */</comment>
      <origin>141951</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18058" sha1="l5je4ja706y59sxxxey1m90xnwtryyc" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>l5je4ja706y59sxxxey1m90xnwtryyc</sha1>
    </revision>
    <revision>
      <id>142460</id>
      <parentid>141951</parentid>
      <timestamp>2024-05-06T19:12:59Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>142460</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17974" sha1="owyxs22qtqzk2nx6m5gp8sgf189vqb7" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo, and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.17
| 51/50, 81/80, 128/125
| [{{val| 12 19 28 34 49 }}]
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| [{{val| 12 19 28 34 49 51 }}]
| −0.81
| 2.64
| 2.64
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>owyxs22qtqzk2nx6m5gp8sgf189vqb7</sha1>
    </revision>
    <revision>
      <id>142469</id>
      <parentid>142460</parentid>
      <timestamp>2024-05-06T20:01:24Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */</comment>
      <origin>142469</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17971" sha1="bv7fzbqx7rr5bhfuhtygw1q5woupzw5" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo, and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.17
| 51/50, 81/80, 128/125
| [{{val| 12 19 28 34 49 }}]
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| [{{val| 12 19 28 34 49 51 }}]
| −0.81
| 2.64
| 2.64
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>bv7fzbqx7rr5bhfuhtygw1q5woupzw5</sha1>
    </revision>
    <revision>
      <id>142470</id>
      <parentid>142469</parentid>
      <timestamp>2024-05-06T20:02:01Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
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      <text bytes="17967" sha1="gw1hmp5qme6c6wcek5t6y2efsjaszq4" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo, and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|5-Limit just intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.17
| 51/50, 81/80, 128/125
| [{{val| 12 19 28 34 49 }}]
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| [{{val| 12 19 28 34 49 51 }}]
| −0.81
| 2.64
| 2.64
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>gw1hmp5qme6c6wcek5t6y2efsjaszq4</sha1>
    </revision>
    <revision>
      <id>142471</id>
      <parentid>142470</parentid>
      <timestamp>2024-05-06T20:02:28Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */</comment>
      <origin>142471</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17961" sha1="12vmx00h2djsuwh4ijt4xb2jdpyzfet" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo, and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|5-Limit just intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.17
| 51/50, 81/80, 128/125
| [{{val| 12 19 28 49 }}]
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| [{{val| 12 19 28 49 51 }}]
| −0.81
| 2.64
| 2.64
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>12vmx00h2djsuwh4ijt4xb2jdpyzfet</sha1>
    </revision>
    <revision>
      <id>142494</id>
      <parentid>142471</parentid>
      <timestamp>2024-05-07T10:27:29Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <origin>142494</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18058" sha1="l5je4ja706y59sxxxey1m90xnwtryyc" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.617
| 0.617
| 0.617
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| -1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| -3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| -2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| -2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>l5je4ja706y59sxxxey1m90xnwtryyc</sha1>
    </revision>
    <revision>
      <id>142500</id>
      <parentid>142494</parentid>
      <timestamp>2024-05-07T10:42:53Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Regular temperament properties */</comment>
      <origin>142500</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18063" sha1="eiqlu757b1kcb7fmhlkaez0xzdh8ix7" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| −2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>eiqlu757b1kcb7fmhlkaez0xzdh8ix7</sha1>
    </revision>
    <revision>
      <id>142630</id>
      <parentid>142500</parentid>
      <timestamp>2024-05-08T18:24:56Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Regular temperament properties */</comment>
      <origin>142630</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18252" sha1="n4p68zyad4fxwh5245qwllinuwlik4v" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.17
| 51/50, 81/80, 128/125
| [{{val| 12 19 28 49 }}]
| −0.87
| 2.94
| 2.95
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| [{{val| 12 19 28 49 51 }}]
| −0.81
| 2.64
| 2.65
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| −2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>n4p68zyad4fxwh5245qwllinuwlik4v</sha1>
    </revision>
    <revision>
      <id>143023</id>
      <parentid>142630</parentid>
      <timestamp>2024-05-12T19:52:34Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Regular temperament properties */</comment>
      <origin>143023</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18252" sha1="hsk7gfuca7oiyhcjwr4cjw47kj1xwqh" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-7Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.17
| 51/50, 81/80, 128/125
| [{{val| 12 19 28 49 }}]
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| [{{val| 12 19 28 49 51 }}]
| −0.81
| 2.64
| 2.64
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| −2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>hsk7gfuca7oiyhcjwr4cjw47kj1xwqh</sha1>
    </revision>
    <revision>
      <id>143445</id>
      <parentid>143023</parentid>
      <timestamp>2024-05-17T23:58:39Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Approximation to JI */</comment>
      <origin>143445</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18252" sha1="l4br1cit63gzsppmyxit2efy2t1lcw9" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 7-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.17
| 51/50, 81/80, 128/125
| [{{val| 12 19 28 49 }}]
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| [{{val| 12 19 28 49 51 }}]
| −0.81
| 2.64
| 2.64
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| −2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>l4br1cit63gzsppmyxit2efy2t1lcw9</sha1>
    </revision>
    <revision>
      <id>143446</id>
      <parentid>143445</parentid>
      <timestamp>2024-05-17T23:58:57Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Approximation to JI */</comment>
      <origin>143446</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18252" sha1="tghu3gepfwybab13zlc8cdpmrofwgm7" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{15-odd-limit}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.17
| 51/50, 81/80, 128/125
| [{{val| 12 19 28 49 }}]
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| [{{val| 12 19 28 49 51 }}]
| −0.81
| 2.64
| 2.64
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| −2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>tghu3gepfwybab13zlc8cdpmrofwgm7</sha1>
    </revision>
    <revision>
      <id>144097</id>
      <parentid>143446</parentid>
      <timestamp>2024-05-20T19:55:26Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Approximation to JI */</comment>
      <origin>144097</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18278" sha1="85lnh639ijk3t1e6jwsrs53k0ege6m9" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]
{{clear}}

=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{Q-odd-limit intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.17
| 51/50, 81/80, 128/125
| [{{val| 12 19 28 49 }}]
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| [{{val| 12 19 28 49 51 }}]
| −0.81
| 2.64
| 2.64
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| −2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>85lnh639ijk3t1e6jwsrs53k0ege6m9</sha1>
    </revision>
    <revision>
      <id>144098</id>
      <parentid>144097</parentid>
      <timestamp>2024-05-20T19:55:48Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Approximation to JI */</comment>
      <origin>144098</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18278" sha1="ipamvit87e14x8erhoo85wq4fbp4c3v" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[ups and downs notation]] is identical to standard notation;
* mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{Q-odd-limit intervals}}
{{clear}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.17
| 51/50, 81/80, 128/125
| [{{val| 12 19 28 49 }}]
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| [{{val| 12 19 28 49 51 }}]
| −0.81
| 2.64
| 2.64
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| −2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>ipamvit87e14x8erhoo85wq4fbp4c3v</sha1>
    </revision>
    <revision>
      <id>144099</id>
      <parentid>144098</parentid>
      <timestamp>2024-05-20T19:59:36Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Notation */</comment>
      <origin>144099</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18292" sha1="la2vpjq3kqfyc9ezwvucuqaln4ujsqg" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{Q-odd-limit intervals}}
{{clear}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.17
| 51/50, 81/80, 128/125
| [{{val| 12 19 28 49 }}]
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| [{{val| 12 19 28 49 51 }}]
| −0.81
| 2.64
| 2.64
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| −2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>la2vpjq3kqfyc9ezwvucuqaln4ujsqg</sha1>
    </revision>
    <revision>
      <id>144146</id>
      <parentid>144099</parentid>
      <timestamp>2024-05-21T06:26:22Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Approximation to JI */ I can't recommend this huge whitespace</comment>
      <origin>144146</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18282" sha1="kc15zv0yrtxz14plx6druymdj1upu2i" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{Q-odd-limit intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| [{{val| 12 19 }}]
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| [{{val| 12 19 28 }}]
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| [{{val| 12 19 28 34 }}]
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.17
| 51/50, 81/80, 128/125
| [{{val| 12 19 28 49 }}]
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| [{{val| 12 19 28 34 49 }}]
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| [{{val| 12 19 28 49 51 }}]
| −0.81
| 2.64
| 2.64
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| [{{val| 12 19 28 34 49 51 }}]
| −2.53
| 4.52
| 4.53
|}

12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next ETs doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next ET that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>kc15zv0yrtxz14plx6druymdj1upu2i</sha1>
    </revision>
    <revision>
      <id>144147</id>
      <parentid>144146</parentid>
      <timestamp>2024-05-21T06:32:55Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ cleanup</comment>
      <origin>144147</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18355" sha1="8xyobcmjyelpth49a5zwim6zxi3mrva" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''. 

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{Q-odd-limit intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>8xyobcmjyelpth49a5zwim6zxi3mrva</sha1>
    </revision>
    <revision>
      <id>144762</id>
      <parentid>144147</parentid>
      <timestamp>2024-05-23T18:03:40Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* 15-odd-limit interval mappings */</comment>
      <origin>144762</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18492" sha1="mzpq4ytsgphnpdqxco456z7vdkoz5e2" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''.

Note that since the [[cent]] was defined specifically in terms of 12edo, the absolute and relative errors are the same.

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap; font-size: 110%;" | 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{Q-odd-limit intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>mzpq4ytsgphnpdqxco456z7vdkoz5e2</sha1>
    </revision>
    <revision>
      <id>144795</id>
      <parentid>144762</parentid>
      <timestamp>2024-05-23T21:47:11Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Approximation to JI */</comment>
      <origin>144795</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18475" sha1="3ju86h3bgvrt2yus1mc9xwftfaqmmuh" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''.

Note that since the [[cent]] was defined specifically in terms of 12edo, the absolute and relative errors are the same.

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | 15-odd-limit intervals by direct approximation (even if inconsistent)
|-
! Interval, complement
! Error (abs, [[cent|¢]])
! Error (rel, [[relative cent|%]])
|-
| '''[[4/3]], [[3/2]]'''
| '''1.955'''
| '''2.0'''
|-
| [[9/8]], [[16/9]]
| 3.910
| 3.9
|-
| ''[[13/11]], [[22/13]]''
| ''10.790''
| ''10.8''
|-
| [[16/15]], [[15/8]]
| 11.731
| 11.7
|-
| '''[[5/4]], [[8/5]]'''
| '''13.686'''
| '''13.7'''
|-
| [[6/5]], [[5/3]]
| 15.641
| 15.6
|-
| [[7/5]], [[10/7]]
| 17.488
| 17.5
|-
| [[14/11]], [[11/7]]
| 17.508
| 17.5
|-
| [[10/9]], [[9/5]]
| 17.596
| 17.6
|-
| [[15/14]], [[28/15]]
| 19.443
| 19.4
|-
| ''[[14/13]], [[13/7]]''
| ''28.298''
| ''28.3''
|-
| '''[[8/7]], [[7/4]]'''
| '''31.174'''
| '''31.2'''
|-
| [[7/6]], [[12/7]]
| 33.129
| 33.1
|-
| [[11/10]], [[20/11]]
| 34.996
| 35.0
|-
| [[9/7]], [[14/9]]
| 35.084
| 35.1
|-
| [[18/13]], [[13/9]]
| 36.618
| 36.7
|-
| [[15/11]], [[22/15]]
| 36.951
| 37.0
|-
| [[13/12]], [[24/13]]
| 38.573
| 38.6
|-
| '''[[16/13]], [[13/8]]'''
| '''40.528'''
| '''40.5'''
|-
| ''[[13/10]], [[20/13]]''
| ''45.786''
| ''45.8''
|-
| ''[[11/9]], [[18/11]]''
| ''47.408''
| ''47.4''
|-
| ''[[15/13]], [[26/15]]''
| ''47.741''
| ''47.7''
|-
| '''[[11/8]], [[16/11]]'''
| '''48.682'''
| '''48.7'''
|-
| ''[[12/11]], [[11/6]]''
| ''49.323''
| ''49.3''
|}
{{Q-odd-limit intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>3ju86h3bgvrt2yus1mc9xwftfaqmmuh</sha1>
    </revision>
    <revision>
      <id>144832</id>
      <parentid>144795</parentid>
      <timestamp>2024-05-24T04:22:55Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Approximation to JI */</comment>
      <origin>144832</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17093" sha1="d0u87jbrmmaxwx4s5uqib4a5307kcgu" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''.

Note that since the [[cent]] was defined specifically in terms of 12edo, the absolute and relative errors are the same.

{{Q-odd-limit intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>d0u87jbrmmaxwx4s5uqib4a5307kcgu</sha1>
    </revision>
    <revision>
      <id>144833</id>
      <parentid>144832</parentid>
      <timestamp>2024-05-24T04:23:24Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Approximation to JI */</comment>
      <origin>144833</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17092" sha1="2vcfg0ec7t7hvct77wt0caenbkpir7r" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 12edo. [[Prime harmonics]] are in '''bold'''; in[[consistent]] intervals are in ''italic''.

Note that since the [[cent]] was defined specifically in terms of 12edo, the absolute and relative errors are the same.
{{Q-odd-limit intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>2vcfg0ec7t7hvct77wt0caenbkpir7r</sha1>
    </revision>
    <revision>
      <id>144854</id>
      <parentid>144833</parentid>
      <timestamp>2024-05-24T04:55:16Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>144854</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16806" sha1="hmhfdc0yt9qdze921vm1wnk519059ci" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>hmhfdc0yt9qdze921vm1wnk519059ci</sha1>
    </revision>
    <revision>
      <id>145142</id>
      <parentid>144854</parentid>
      <timestamp>2024-05-28T13:00:14Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Approximation to JI */</comment>
      <origin>145142</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16803" sha1="31tl0sm6b7xpt0ihpqhfefcynalnlbf" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>31tl0sm6b7xpt0ihpqhfefcynalnlbf</sha1>
    </revision>
    <revision>
      <id>145144</id>
      <parentid>145142</parentid>
      <timestamp>2024-05-28T13:11:13Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
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      <text bytes="16816" sha1="mmoezfhb6nwdodwexjocvzrugn2z4db" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>mmoezfhb6nwdodwexjocvzrugn2z4db</sha1>
    </revision>
    <revision>
      <id>145153</id>
      <parentid>145144</parentid>
      <timestamp>2024-05-28T13:52:26Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <origin>145153</origin>
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      <format>text/x-wiki</format>
      <text bytes="16813" sha1="bl7zhvt0u1xu4ky5x97n53tbbh0zv67" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>bl7zhvt0u1xu4ky5x97n53tbbh0zv67</sha1>
    </revision>
    <revision>
      <id>145178</id>
      <parentid>145153</parentid>
      <timestamp>2024-05-28T14:43:38Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>Undid revision 145153 by [[Special:Contributions/ArrowHead294|ArrowHead294]] ([[User talk:ArrowHead294|talk]])</comment>
      <origin>145178</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16816" sha1="mmoezfhb6nwdodwexjocvzrugn2z4db" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless octave stretching or compression is employed. It has a fifth which is quite good at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic]] interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>mmoezfhb6nwdodwexjocvzrugn2z4db</sha1>
    </revision>
    <revision>
      <id>146448</id>
      <parentid>145178</parentid>
      <timestamp>2024-06-20T13:15:57Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
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      <origin>146448</origin>
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      <format>text/x-wiki</format>
      <text bytes="16891" sha1="0jcca0tfld8pc8k9qkao4ppdjigiwy0" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both full zeta and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>0jcca0tfld8pc8k9qkao4ppdjigiwy0</sha1>
    </revision>
    <revision>
      <id>146449</id>
      <parentid>146448</parentid>
      <timestamp>2024-06-20T13:16:21Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Theory */</comment>
      <origin>146449</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16932" sha1="heklrijkrjcme38z7texzm58u264x3j" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>heklrijkrjcme38z7texzm58u264x3j</sha1>
    </revision>
    <revision>
      <id>146831</id>
      <parentid>146449</parentid>
      <timestamp>2024-06-30T21:49:44Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>External link</comment>
      <origin>146831</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17057" sha1="4ogrvwh4b7f0rqiukifxp8ncy1dmvpc" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composte number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>4ogrvwh4b7f0rqiukifxp8ncy1dmvpc</sha1>
    </revision>
    <revision>
      <id>149002</id>
      <parentid>146831</parentid>
      <timestamp>2024-07-28T18:16:22Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
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      <text bytes="17058" sha1="bb9ucvnh9n0yagussc5lwo9net833vp" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>bb9ucvnh9n0yagussc5lwo9net833vp</sha1>
    </revision>
    <revision>
      <id>151261</id>
      <parentid>149002</parentid>
      <timestamp>2024-08-21T06:54:09Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Commas */ Test comma list header</comment>
      <origin>151261</origin>
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      <text bytes="16872" sha1="mjfk56wtmwvizwdsq9ns7x7m8pj06e6" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{{Comma list begin
| classes=center-all left-3 right-4 left-6
}}
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
{{Comma list end}}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>mjfk56wtmwvizwdsq9ns7x7m8pj06e6</sha1>
    </revision>
    <revision>
      <id>151265</id>
      <parentid>151261</parentid>
      <timestamp>2024-08-21T07:06:17Z</timestamp>
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        <username>ArrowHead294</username>
        <id>448</id>
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| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{{Comma list begin
| classes=center-all left-3 right-4 left-6
| rprec=yes
}}
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
{{Comma list end|cols=6|rprec=10}}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>t6uog5k90ukhn8afo0o2wp1uo7nfztr</sha1>
    </revision>
    <revision>
      <id>151270</id>
      <parentid>151265</parentid>
      <timestamp>2024-08-21T07:15:33Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
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      <text bytes="16893" sha1="j6jo7844iicvqadqwh943g9jqy8xonl" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{{Comma list begin
| classes=center-all left-3 right-4 left-6
| rprec=yes
}}
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
{{Comma list end|rprec=10}}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>j6jo7844iicvqadqwh943g9jqy8xonl</sha1>
    </revision>
    <revision>
      <id>151283</id>
      <parentid>151270</parentid>
      <timestamp>2024-08-21T10:07:46Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Substituted template shouldn't change the result</comment>
      <origin>151283</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17058" sha1="bb9ucvnh9n0yagussc5lwo9net833vp" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref&gt;Ratios longer than 10 digits are presented by placeholders with informative hints&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>bb9ucvnh9n0yagussc5lwo9net833vp</sha1>
    </revision>
    <revision>
      <id>151310</id>
      <parentid>151283</parentid>
      <timestamp>2024-08-21T13:49:35Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>151310</origin>
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      <format>text/x-wiki</format>
      <text bytes="17044" sha1="d7g5ogcsvdg5tvvolbkl5ta07hl4c9i" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals* ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

&lt;nowiki&gt;*&lt;/nowiki&gt; based on treating 12edo as a 2.3.5.7.17.19 subgroup temperament; other approaches are possible.

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd|10}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

=== Notes ===
&lt;references group="note" /&gt;

=== External links ===
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>d7g5ogcsvdg5tvvolbkl5ta07hl4c9i</sha1>
    </revision>
    <revision>
      <id>151311</id>
      <parentid>151310</parentid>
      <timestamp>2024-08-21T13:50:39Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
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      <text bytes="16980" sha1="hfbbv8fakn0qpgw8t6vgquepfaodlex" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|12edo|sg=2.3.5.7.17.19}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd|10}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

=== Notes ===
&lt;references group="note" /&gt;

=== External links ===
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>hfbbv8fakn0qpgw8t6vgquepfaodlex</sha1>
    </revision>
    <revision>
      <id>151314</id>
      <parentid>151311</parentid>
      <timestamp>2024-08-21T13:52:40Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
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      <text bytes="16992" sha1="0ljaez24t7u2rl7jnocv6xlottgm9w8" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|12edo|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd|10}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

=== Notes ===
&lt;references group="note" /&gt;

=== External links ===
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>0ljaez24t7u2rl7jnocv6xlottgm9w8</sha1>
    </revision>
    <revision>
      <id>151344</id>
      <parentid>151314</parentid>
      <timestamp>2024-08-21T14:24:13Z</timestamp>
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        <username>ArrowHead294</username>
        <id>448</id>
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      <text bytes="16944" sha1="8xlc3fplt8rx79hb6suk5ai05f38bje" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals{{sg|12edo|limit=2.3.5.7.17.19 subgroup}} ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]{{rd|10}}
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

=== Notes ===
&lt;references group="note" /&gt;

=== External links ===
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>8xlc3fplt8rx79hb6suk5ai05f38bje</sha1>
    </revision>
    <revision>
      <id>151359</id>
      <parentid>151344</parentid>
      <timestamp>2024-08-21T14:49:55Z</timestamp>
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        <id>448</id>
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      <text bytes="16986" sha1="q8lm2hhiyioy698w3rt6vqvyt0z6tgd" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd|10}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references/&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

=== Notes ===
&lt;references group="note" /&gt;

=== External links ===
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>q8lm2hhiyioy698w3rt6vqvyt0z6tgd</sha1>
    </revision>
    <revision>
      <id>151360</id>
      <parentid>151359</parentid>
      <timestamp>2024-08-21T14:50:23Z</timestamp>
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        <username>ArrowHead294</username>
        <id>448</id>
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      <text bytes="16972" sha1="7iybz7hucyg15r312iilhscf1n4pk05" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd|10}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

=== Notes ===
&lt;references group="note" /&gt;

=== External links ===
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>7iybz7hucyg15r312iilhscf1n4pk05</sha1>
    </revision>
    <revision>
      <id>151365</id>
      <parentid>151360</parentid>
      <timestamp>2024-08-21T14:54:22Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Commas */</comment>
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      <text bytes="16969" sha1="eq3jlekxya77rf5bqfoau8ay0xvh5iw" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

=== Notes ===
&lt;references group="note" /&gt;

=== External links ===
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>eq3jlekxya77rf5bqfoau8ay0xvh5iw</sha1>
    </revision>
    <revision>
      <id>151800</id>
      <parentid>151365</parentid>
      <timestamp>2024-08-23T14:40:34Z</timestamp>
      <contributor>
        <username>Bcmills</username>
        <id>450</id>
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      <text bytes="16992" sha1="98een8x9jvd4dp3hyb5dnvmm62x8nhs" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

=== Notes ===
&lt;references group="note" /&gt;

=== External links ===
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>98een8x9jvd4dp3hyb5dnvmm62x8nhs</sha1>
    </revision>
    <revision>
      <id>153949</id>
      <parentid>151800</parentid>
      <timestamp>2024-09-04T02:48:34Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
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      <text bytes="16988" sha1="hbaj3d2oix44ibudml9mj5iajgxf642" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. As of right now, it is also the only known EDO that is both [[The Riemann zeta function and tuning|full zeta]] and highly composite, and the only one with a step size larger than the just noticeable difference (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>hbaj3d2oix44ibudml9mj5iajgxf642</sha1>
    </revision>
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      <id>154573</id>
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      <timestamp>2024-09-10T14:52:15Z</timestamp>
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        <id>448</id>
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      <minor/>
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      <text bytes="16992" sha1="iv43e2jvw43kehblzz8g4r0q9pqj94w" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. 12edo is also the only known EDO that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite, and the only strict zeta EDO with a step size larger than the [[just-noticeable difference]] (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>iv43e2jvw43kehblzz8g4r0q9pqj94w</sha1>
    </revision>
    <revision>
      <id>154574</id>
      <parentid>154573</parentid>
      <timestamp>2024-09-10T14:52:31Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
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      <text bytes="17013" sha1="nbs6qaoiqp0impqiqtubx7px8h5fies" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. 12edo is also the only known EDO that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite, and the only strict zeta EDO that is composite and has a step size larger than the [[just-noticeable difference]] (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name|Color Name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Greater diesis, diminished comma
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Lesser diesis, augmented comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th Partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>nbs6qaoiqp0impqiqtubx7px8h5fies</sha1>
    </revision>
    <revision>
      <id>165072</id>
      <parentid>154574</parentid>
      <timestamp>2024-11-12T12:51:28Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ style and comma names</comment>
      <origin>165072</origin>
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      <format>text/x-wiki</format>
      <text bytes="17004" sha1="fyb0jy2qalbpm7ikfkyitz19e7xugpc" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[Stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be {{dash|{{frac|1|1}}, {{frac|5|4}}, {{frac|3|2}}, {{frac|16|9}}|s=hair|d=long}}, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[The Riemann Zeta Function and Tuning #Zeta edo lists|zeta integral edo]].

The commas it tempers out include the Pythagorean comma, 3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;, the Didymus comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly melodic EDO]], 12 being both a superabundant and a highly composite number. 12edo is also the only known EDO that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite, and the only strict zeta EDO that is composite and has a step size larger than the [[just-noticeable difference]] (~3-4 cents).

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>fyb0jy2qalbpm7ikfkyitz19e7xugpc</sha1>
    </revision>
    <revision>
      <id>165078</id>
      <parentid>165072</parentid>
      <timestamp>2024-11-12T12:59:50Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ revert presentation of chords for readability. -incorrect statements (even discarding 72edo, 270edo's step size is 4.44 cents, larger than the 4-cent criterion given here)</comment>
      <origin>165078</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16793" sha1="17k3nfg7ikpbelxp48x2i1bhcltajkt" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>17k3nfg7ikpbelxp48x2i1bhcltajkt</sha1>
    </revision>
    <revision>
      <id>165285</id>
      <parentid>165078</parentid>
      <timestamp>2024-11-14T06:38:03Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>/* See also */ Added Near12</comment>
      <origin>165285</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16890" sha1="i6q8aom1oq6odeciy2uhsyf76xxssiy" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>i6q8aom1oq6odeciy2uhsyf76xxssiy</sha1>
    </revision>
    <revision>
      <id>165286</id>
      <parentid>165285</parentid>
      <timestamp>2024-11-14T06:38:24Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* See also */ mark dead link</comment>
      <origin>165286</origin>
      <model>wikitext</model>
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      <text bytes="16903" sha1="5cg5imh92e8ijvqzk6g3uot2eav5oi4" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods &lt;br&gt; per 8ve
! Generator
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>5cg5imh92e8ijvqzk6g3uot2eav5oi4</sha1>
    </revision>
    <revision>
      <id>165803</id>
      <parentid>165286</parentid>
      <timestamp>2024-11-16T02:52:58Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */</comment>
      <origin>165803</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="16697" sha1="m2jr2w5ik1r9yi3cchaymq02u7aejgv" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{{comma basis begin}}
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
{{comma basis end}}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}
{{orf}}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
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        <id>448</id>
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      <text bytes="17084" sha1="jvudc8zd1avac51lvx7gp6jhls7qd7e" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br /&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;Limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br /&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 1\12
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 1\12
| (P8/3, P5)
| [[Augmented]] / [[lithium]]
|-
| 4
| 1\12
| (P8/4, P5)
| [[Diminished]]
|-
| 6
| 1\12
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki /&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>jvudc8zd1avac51lvx7gp6jhls7qd7e</sha1>
    </revision>
    <revision>
      <id>167154</id>
      <parentid>166343</parentid>
      <timestamp>2024-11-21T11:28:43Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ cleanup and review</comment>
      <origin>167154</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17150" sha1="pfwoo05phk4t4tz09pnnr1psrb8kbjd" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>pfwoo05phk4t4tz09pnnr1psrb8kbjd</sha1>
    </revision>
    <revision>
      <id>169662</id>
      <parentid>167154</parentid>
      <timestamp>2024-12-10T16:07:25Z</timestamp>
      <contributor>
        <username>Tristanbay</username>
        <id>418</id>
      </contributor>
      <comment>/* Prime harmonics */Added more prime harmonics (up to 89th)</comment>
      <origin>169662</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17274" sha1="5warzbowwq37mjp6rc91oc32mvmzm5j" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|columns=12|prec=2}}
{{Harmonics in equal|12|start=13|columns=12|prec=2|title=Approximation of prime harmonics in 12edo (continued)}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic-Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>5warzbowwq37mjp6rc91oc32mvmzm5j</sha1>
    </revision>
    <revision>
      <id>169666</id>
      <parentid>169662</parentid>
      <timestamp>2024-12-10T16:16:04Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
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      <text bytes="17276" sha1="rdd35a9d391dovhbounfmqvgw15imd7" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|columns=12|prec=2}}
{{Harmonics in equal|12|start=13|columns=12|prec=2|title=Approximation of prime harmonics in 12edo (continued)}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>rdd35a9d391dovhbounfmqvgw15imd7</sha1>
    </revision>
    <revision>
      <id>169919</id>
      <parentid>169666</parentid>
      <timestamp>2024-12-12T11:27:45Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Undo revision 169662 by [[Special:Contributions/Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]). Insane for 12edo</comment>
      <origin>169919</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17152" sha1="an3ntl836p6ri4y63eo3qffrmjaj78f" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

{| class="wikitable center-all"
|+ Notation of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>an3ntl836p6ri4y63eo3qffrmjaj78f</sha1>
    </revision>
    <revision>
      <id>172651</id>
      <parentid>169919</parentid>
      <timestamp>2024-12-23T12:20:15Z</timestamp>
      <contributor>
        <username>TallKite</username>
        <id>23</id>
      </contributor>
      <comment>/* Notation */ added a line about 12edo's EI</comment>
      <origin>172651</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="17275" sha1="fcuekx7naliwca4hjchgaq3fpi8k071" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic interval|respelled enharmonically]] by adding or subtracting a diminished 2nd to/from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>fcuekx7naliwca4hjchgaq3fpi8k071</sha1>
    </revision>
    <revision>
      <id>172881</id>
      <parentid>172651</parentid>
      <timestamp>2024-12-24T11:11:21Z</timestamp>
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        <id>23</id>
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      <text bytes="17280" sha1="5hpfybvva12h5k4od0jwgii5lguctnj" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>5hpfybvva12h5k4od0jwgii5lguctnj</sha1>
    </revision>
    <revision>
      <id>173620</id>
      <parentid>172881</parentid>
      <timestamp>2024-12-29T06:56:14Z</timestamp>
      <contributor>
        <username>Dave Keenan</username>
        <id>218</id>
      </contributor>
      <comment>/* Notation */ added Sagittal notation</comment>
      <origin>173620</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18254" sha1="4v15p0m3wbe9shpk6xylj4i47g4v4k9" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===
This notation is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]] and a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====

&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
====Revo flavor====

&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>4v15p0m3wbe9shpk6xylj4i47g4v4k9</sha1>
    </revision>
    <revision>
      <id>173923</id>
      <parentid>173620</parentid>
      <timestamp>2024-12-30T23:10:58Z</timestamp>
      <contributor>
        <username>Dave Keenan</username>
        <id>218</id>
      </contributor>
      <comment>/* Sagittal notation */ added same-sagittal-sequence EDOs</comment>
      <origin>173923</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18396" sha1="kocizn3x5g093u5nf9d91gvulwnt0pd" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], and is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]] and a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====

&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
====Revo flavor====

&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Meter comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>kocizn3x5g093u5nf9d91gvulwnt0pd</sha1>
    </revision>
    <revision>
      <id>174987</id>
      <parentid>173923</parentid>
      <timestamp>2025-01-07T09:08:24Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ meter, or metric comma</comment>
      <origin>174987</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18397" sha1="0n3re7y55wtmdm7y4vtxr8ptii1ezvu" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], and is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]] and a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====

&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
====Revo flavor====

&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>0n3re7y55wtmdm7y4vtxr8ptii1ezvu</sha1>
    </revision>
    <revision>
      <id>175645</id>
      <parentid>174987</parentid>
      <timestamp>2025-01-11T01:03:22Z</timestamp>
      <contributor>
        <username>Dave Keenan</username>
        <id>218</id>
      </contributor>
      <comment>/* Sagittal notation */ Removed extraneous non-final "and". Changed final "and" to ", and is".</comment>
      <origin>175645</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18397" sha1="0y05p3lricr6c4tyo4ap8yh1stqv5n5" xml:space="preserve">{{interwiki
| de = 12-EDO
| en = 12edo
| es = 12 EDO
| ja = 12平均律
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====

&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
====Revo flavor====

&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
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      <origin>176090</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18410" sha1="j1h5df2pkg991017df53x7ezmqplk14" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}, and it also contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. 12edo is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====

&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
====Revo flavor====

&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>j1h5df2pkg991017df53x7ezmqplk14</sha1>
    </revision>
    <revision>
      <id>176093</id>
      <parentid>176090</parentid>
      <timestamp>2025-01-15T13:28:22Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ +octave stretch, +subsets and supersets</comment>
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      <format>text/x-wiki</format>
      <text bytes="19323" sha1="0jxn2t4ks0zz4zn547aiwjlw3hdyfzm" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[19edt]] or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, provides a great correction for the approximate harmonics 11 and 13. [[36edo]], which triples it, provides a great correction for the approximate harmonic 7. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====

&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
====Revo flavor====

&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| &amp;minus;1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| &amp;minus;3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| &amp;minus;2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| &amp;minus;2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| &amp;minus;0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| &amp;minus;0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>0jxn2t4ks0zz4zn547aiwjlw3hdyfzm</sha1>
    </revision>
    <revision>
      <id>176661</id>
      <parentid>176093</parentid>
      <timestamp>2025-01-17T12:19:38Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ cleanup</comment>
      <origin>176661</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19313" sha1="fe478a05h7s6d73fq2022k428lua5zt" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[19edt]] or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, provides a great correction for the approximate harmonics 11 and 13. [[36edo]], which triples it, provides a great correction for the approximate harmonic 7. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====

&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
====Revo flavor====

&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>fe478a05h7s6d73fq2022k428lua5zt</sha1>
    </revision>
    <revision>
      <id>176921</id>
      <parentid>176661</parentid>
      <timestamp>2025-01-18T20:16:22Z</timestamp>
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        <id>191</id>
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      <text bytes="19719" sha1="q1do9lrmcsfxoit44d779cj9itr0rx4" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[19edt]] or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, provides a great correction for the approximate harmonics 11 and 13. [[36edo]], which triples it, provides a great correction for the approximate harmonic 7. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====

&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
====Revo flavor====

&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Zeta properties ==

=== Zeta Peak Index ===
{| class="wikitable"
! colspan="3" |Tuning
! colspan="3" |Strength
! colspan="2" |Closest EDO
! colspan="2" |Integer limit
|-
!ZPI
!Steps per octave
!Step size (cents)
!Height
!Integral
!Gap
!EDO
!Octave (cents)
!Consistent
!Distinct
|-
|[[34zpi]]
|12.0231830072926
|99.8071807833375
|5.193290
| 1.269599
|15.899282
|12edo
|1197.68616940005
|10
|6
|}
== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>q1do9lrmcsfxoit44d779cj9itr0rx4</sha1>
    </revision>
    <revision>
      <id>176922</id>
      <parentid>176921</parentid>
      <timestamp>2025-01-18T20:18:57Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
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      <format>text/x-wiki</format>
      <text bytes="19719" sha1="qim6kp94iv0idwjky10bifzok6mkffn" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[19edt]] or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, provides a great correction for the approximate harmonics 11 and 13. [[36edo]], which triples it, provides a great correction for the approximate harmonic 7. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====

&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
====Revo flavor====

&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Zeta properties ==

=== Zeta peak index ===
{| class="wikitable"
! colspan="3" |Tuning
! colspan="3" |Strength
! colspan="2" |Closest EDO
! colspan="2" |Integer limit
|-
!ZPI
!Steps per octave
!Step size (cents)
!Height
!Integral
!Gap
!EDO
!Octave (cents)
!Consistent
!Distinct
|-
|[[34zpi]]
|12.0231830072926
|99.8071807833375
|5.193290
| 1.269599
|15.899282
|12edo
|1197.68616940005
|10
|6
|}
== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
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| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
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{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[19edt]] or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, provides a great correction for the approximate harmonics 11 and 13. [[36edo]], which triples it, provides a great correction for the approximate harmonic 7. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====

&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
====Revo flavor====

&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Zeta properties ==

=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest EDO
! colspan="2" | Integer limit
|-
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! EDO
! Octave (cents)
! Consistent
! Distinct
|-
| [[34zpi]]
| 12.0231830072926
| 99.8071807833375
| 5.193290
| 1.269599
| 15.899282
| 12edo
| 1197.68616940005
| 10
| 6
|}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>h1c6gj58clu8iigkiar5jdlzuztqj1c</sha1>
    </revision>
    <revision>
      <id>177087</id>
      <parentid>176995</parentid>
      <timestamp>2025-01-19T08:29:19Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Consolidate sections</comment>
      <origin>177087</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19729" sha1="q5jj2ldlsdqycgj9zrgi7ezhsqz54ea" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[19edt]] or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]] and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, provides a great correction for the approximate harmonics 11 and 13. [[36edo]], which triples it, provides a great correction for the approximate harmonic 7. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ Intervals of 12edo
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (-11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (-19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (-4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (-3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (-31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (-0.532)&lt;br&gt;[[17/15]] (-16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (-15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (-1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (-7.820)
| [[5/4]] (+13.686)
| [[63/50]] (-0.108)&lt;br&gt;[[9/7]] (-35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (-4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (-17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (-0.088)&lt;br&gt;[[17/12]] (-3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (-1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (-13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (-0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (-5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (-33.129)
| [[37/22]] (-0.026)&lt;br&gt;[[32/19]] (-2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (-17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (-3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (-0.020)&lt;br&gt;[[34/19]] (-7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (-29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (-15.533)&lt;br&gt;[[27/14]] (-37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (-1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+Notation of 12edo
! rowspan="2" |[[Degree]]
! rowspan="2" |[[Cent]]s
! colspan="2" |[[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L 2s]]) interval names
! Note names (on D)
|-
| 0
| 0
|'''Perfect unison (P1)'''
|'''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
|'''Major second (M2)'''&lt;br&gt;Diminished third (d3)
|'''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
|'''Perfect fourth (P4)'''
|'''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
|'''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
|'''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
|'''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
|'''Perfect octave (P8)'''
|'''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

===Sagittal notation===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].
====Evo flavor====

&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.
====Revo flavor====

&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ Solfege of 12edo
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[34zpi]]
| 12.0231830072926
| 99.8071807833375
| 5.193290
| 1.269599
| 15.899282
| 12edo
| 1197.68616940005
| 10
| 6
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] - a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>q5jj2ldlsdqycgj9zrgi7ezhsqz54ea</sha1>
    </revision>
    <revision>
      <id>177534</id>
      <parentid>177087</parentid>
      <timestamp>2025-01-21T14:19:49Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
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      <format>text/x-wiki</format>
      <text bytes="20029" sha1="jwy7eecqrb9jn2umxvlpbxzgh4fw0gw" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| 
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[34zpi]]
| 12.0231830072926
| 99.8071807833375
| 5.193290
| 1.269599
| 15.899282
| 12edo
| 1197.68616940005
| 10
| 6
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>jwy7eecqrb9jn2umxvlpbxzgh4fw0gw</sha1>
    </revision>
    <revision>
      <id>178590</id>
      <parentid>177534</parentid>
      <timestamp>2025-01-30T15:30:21Z</timestamp>
      <contributor>
        <username>Dummy index</username>
        <id>318</id>
      </contributor>
      <comment>/* Intervals */ +5-limit tritones (45-odd-limit is not very large in this table)</comment>
      <origin>178590</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20069" sha1="4olzaftccjh1qw9joxztdi9ubqlv0t1" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (-9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[34zpi]]
| 12.0231830072926
| 99.8071807833375
| 5.193290
| 1.269599
| 15.899282
| 12edo
| 1197.68616940005
| 10
| 6
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>4olzaftccjh1qw9joxztdi9ubqlv0t1</sha1>
    </revision>
    <revision>
      <id>179042</id>
      <parentid>178590</parentid>
      <timestamp>2025-01-31T16:50:50Z</timestamp>
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      <text bytes="20071" sha1="q3fht0o51chghrsmh6dxwt6c0r3z52d" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[34zpi]]
| 12.0231830072926
| 99.8071807833375
| 5.193290
| 1.269599
| 15.899282
| 12edo
| 1197.68616940005
| 10
| 6
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (12f val) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>q3fht0o51chghrsmh6dxwt6c0r3z52d</sha1>
    </revision>
    <revision>
      <id>179307</id>
      <parentid>179042</parentid>
      <timestamp>2025-02-03T13:21:16Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ Brief explanation for musicians who don't know theory</comment>
      <origin>179307</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20152" sha1="njzxn4uribuj9gm58jllobvc0zqi6oj" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{EDO intro|12}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[34zpi]]
| 12.0231830072926
| 99.8071807833375
| 5.193290
| 1.269599
| 15.899282
| 12edo
| 1197.68616940005
| 10
| 6
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>njzxn4uribuj9gm58jllobvc0zqi6oj</sha1>
    </revision>
    <revision>
      <id>181339</id>
      <parentid>179307</parentid>
      <timestamp>2025-02-18T17:01:02Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
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      <text bytes="20148" sha1="fyl65go1zm0se3f1iu5b7b45y8e2ewq" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[34zpi]]
| 12.0231830072926
| 99.8071807833375
| 5.193290
| 1.269599
| 15.899282
| 12edo
| 1197.68616940005
| 10
| 6
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>fyl65go1zm0se3f1iu5b7b45y8e2ewq</sha1>
    </revision>
    <revision>
      <id>184921</id>
      <parentid>181339</parentid>
      <timestamp>2025-03-06T19:03:36Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* 15-odd-limit interval mappings */ Add 12F mapping (where 13/8&amp;nbsp;←&amp;nbsp;9\12 instead of 8\12)</comment>
      <origin>184921</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20256" sha1="27wkoq5534isvp50yb77nv6j07pq5hi" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12F val mapping}}

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[34zpi]]
| 12.0231830072926
| 99.8071807833375
| 5.193290
| 1.269599
| 15.899282
| 12edo
| 1197.68616940005
| 10
| 6
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>27wkoq5534isvp50yb77nv6j07pq5hi</sha1>
    </revision>
    <revision>
      <id>184942</id>
      <parentid>184921</parentid>
      <timestamp>2025-03-06T19:26:57Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* 15-odd-limit interval mappings */</comment>
      <origin>184942</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20256" sha1="sapg7e645g40rk8hf8ydf7u6ocbsutp" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{| class="wikitable center-all"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest edo
! colspan="2" | Integer limit
|-
|-
! ZPI
! Steps per octave
! Step size (cents)
! Height
! Integral
! Gap
! Edo
! Octave (cents)
! Consistent
! Distinct
|-
| [[34zpi]]
| 12.0231830072926
| 99.8071807833375
| 5.193290
| 1.269599
| 15.899282
| 12edo
| 1197.68616940005
| 10
| 6
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>sapg7e645g40rk8hf8ydf7u6ocbsutp</sha1>
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    <revision>
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      <minor/>
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      <text bytes="20090" sha1="op3ywoofot62goizx63vv5oyhgq3aom" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| TO height = 5.193290
| PO height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>op3ywoofot62goizx63vv5oyhgq3aom</sha1>
    </revision>
    <revision>
      <id>187037</id>
      <parentid>187035</parentid>
      <timestamp>2025-03-19T05:10:55Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <comment>/* Zeta peak index */</comment>
      <origin>187037</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20098" sha1="2e0byyethhv6h011qhoybgo3iov8t5n" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>2e0byyethhv6h011qhoybgo3iov8t5n</sha1>
    </revision>
    <revision>
      <id>187044</id>
      <parentid>187037</parentid>
      <timestamp>2025-03-19T05:25:09Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Zeta peak index */</comment>
      <origin>187044</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19873" sha1="akzzu68e3fsxnxp40wrvd2x0wtor3tu" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
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      <text bytes="20090" sha1="op3ywoofot62goizx63vv5oyhgq3aom" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| TO height = 5.193290
| PO height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>op3ywoofot62goizx63vv5oyhgq3aom</sha1>
    </revision>
    <revision>
      <id>187049</id>
      <parentid>187048</parentid>
      <timestamp>2025-03-19T05:28:09Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Zeta peak index */</comment>
      <origin>187049</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20098" sha1="2e0byyethhv6h011qhoybgo3iov8t5n" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|13|11.5|12.5}} 

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>2e0byyethhv6h011qhoybgo3iov8t5n</sha1>
    </revision>
    <revision>
      <id>188768</id>
      <parentid>187049</parentid>
      <timestamp>2025-03-29T17:04:40Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>/* Uniform maps */ use new args</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20091" sha1="m676fuvjipaeptxhvf16l5vmusbgpwo" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

=== Miscellany ===
12edo is the largest equal division of the octave which uniquely patently alternates with an *ed(9/8) in a [[well tempered nonet]]{{clarify}}.

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>m676fuvjipaeptxhvf16l5vmusbgpwo</sha1>
    </revision>
    <revision>
      <id>189164</id>
      <parentid>188768</parentid>
      <timestamp>2025-03-31T16:03:57Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>/* Theory */ Remove nonet section</comment>
      <origin>189164</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19928" sha1="fzyxair6x3cbw4r0g999i7qh89bvur5" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
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{{High priority}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>tbv330tm1mygnqflymz1swqhzwtp1ww</sha1>
    </revision>
    <revision>
      <id>190224</id>
      <parentid>190220</parentid>
      <timestamp>2025-04-08T12:04:16Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Protected "[[12edo]]": semi-protected this page as high priority ([Edit=Allow only autoconfirmed users] (indefinite) [Move=Allow only autoconfirmed users] (indefinite))</comment>
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      <text bytes="19946" sha1="tbv330tm1mygnqflymz1swqhzwtp1ww" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{High priority}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>tbv330tm1mygnqflymz1swqhzwtp1ww</sha1>
    </revision>
    <revision>
      <id>190618</id>
      <parentid>190224</parentid>
      <timestamp>2025-04-10T17:01:52Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Move high priority to talk page</comment>
      <origin>190618</origin>
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      <format>text/x-wiki</format>
      <text bytes="19928" sha1="fzyxair6x3cbw4r0g999i7qh89bvur5" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable rank-2 temperaments that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>fzyxair6x3cbw4r0g999i7qh89bvur5</sha1>
    </revision>
    <revision>
      <id>190933</id>
      <parentid>190618</parentid>
      <timestamp>2025-04-11T06:25:41Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Subsets and supersets */ add link</comment>
      <origin>190933</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19932" sha1="4mz209srduwfnav17jlm2gd76ifcqdy" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>4mz209srduwfnav17jlm2gd76ifcqdy</sha1>
    </revision>
    <revision>
      <id>194768</id>
      <parentid>190933</parentid>
      <timestamp>2025-04-30T07:07:45Z</timestamp>
      <contributor>
        <username>Tristanbay</username>
        <id>418</id>
      </contributor>
      <comment>/* Subsets and supersets */Corrected sentence about edos that are both strict zeta and highly composite</comment>
      <origin>194768</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="19948" sha1="3gi1x04bvrivu8aiocqpl6s8gckqnw3" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the Didymus' comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the Archytas' comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>3gi1x04bvrivu8aiocqpl6s8gckqnw3</sha1>
    </revision>
    <revision>
      <id>198234</id>
      <parentid>194768</parentid>
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      <text bytes="20980" sha1="6yxhpm5zzdzgh5rdp5owebcwwqcax6m" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

12edo is the basic example of an equidodecatonic scale, or more simply, a 12-[[well temperament]]. It is in the position after 7edo as convergent to a musically "natural" [[golden meantone]] fifth.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include:

- the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]] (splitting the octave into twelve parts, 7 of which represent a perfect fifth 3/2), 

- the Didymus' comma, [[81/80]] (resulting in the diatonic major third being equated to the just major third 5/4), 

- the lesser diesis, [[128/125]] (resulting in 5/4 stacking three times to reach the octave), 

- the diaschisma, [[2048/2025]] (resulting in a semitone 16/15 being exactly half of a whole tone 9/8, and in two 5-limit tritones, 45/32 and 64/45, being equated to the "true" tritone of [[sqrt(2/1)]] or 600c), 

- the Archytas' comma, [[64/63]] (equating the dominant seventh chord and harmonic seventh chord), 

- the septimal quartertone, [[36/35]] (resulting in the lack of a distinction in the 7-limit beyond "major" and "minor" intervals), 

- the jubilisma, [[50/49]] (which equates [[10/7]] and [[7/5]], two simple 7-limit tritones, to the semioctave), 

- the septimal semicomma, [[126/125]] (the difference between the previous two intervals), 

- and the septimal kleisma, [[225/224]] (the difference between simple 7-limit intervals and 5-limit augmented or diminished intervals). 

Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>6yxhpm5zzdzgh5rdp5owebcwwqcax6m</sha1>
    </revision>
    <revision>
      <id>198242</id>
      <parentid>198234</parentid>
      <timestamp>2025-05-31T13:17:55Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ don't make this section too long. There are plenty of room to explain these commas in their own pages.</comment>
      <origin>198242</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20151" sha1="3wkpjqxu3okw6gq20gcctofevcvkqbz" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.

12edo is the basic example of an equidodecatonic scale, or more simply, a 12-[[well temperament]]. It is in the position after 7edo as convergent to a musically "natural" [[golden meantone]] fifth.

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the syntonic comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the septimal comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>3wkpjqxu3okw6gq20gcctofevcvkqbz</sha1>
    </revision>
    <revision>
      <id>198540</id>
      <parentid>198242</parentid>
      <timestamp>2025-06-03T01:21:43Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>198540</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20493" sha1="ovtshpn3iruiq2xovmu3smljzcim10t" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which, while reasonable for its size, is unsatisfactory for many (especially among those who come to xenharmony), and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are sometimes reduced by the tuning adaptations of the performers. Another change that resulted from the adoption of 12edo is the general detachment from the actual meantone and 5-limit system which originally justified it, with theories such as serialism and much of jazz theory deriving from 12edo's nature as a system itself rather than its underlying temperament structure.

12edo is the basic example of an equidodecatonic scale, or more simply, a 12-[[well temperament]]. It is in the position after 7edo as convergent to a musically "natural" [[golden meantone]] fifth. 

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a tetrad. Such tetrads are often used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1–5/4–3/2–16/9, and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of 7-limit intervals are not very accurate. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The commas it tempers out include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the syntonic comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the septimal comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[The Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>ovtshpn3iruiq2xovmu3smljzcim10t</sha1>
    </revision>
    <revision>
      <id>198568</id>
      <parentid>198540</parentid>
      <timestamp>2025-06-03T11:32:39Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Theory */ review. Linking improvements</comment>
      <origin>198568</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20537" sha1="fjnh0zw3r1hgisfyk2l7qopnskua7ue" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which, while reasonable for its size, is unsatisfactory for some, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers. In particular, there is a general detachment from the actual meantone and 5-limit system which originally justified it, with theories such as {{w|serialism}} and much of {{w|jazz}} theory deriving from 12edo's nature as a system itself rather than its underlying temperament structure.

12edo is the basic example of an equidodecatonic scale, or more simply, a 12-tone [[well temperament]]. It is in the position after 7edo as convergent to a musically "natural" [[golden meantone]] fifth. 

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The [[comma]]s it [[tempering out|tempers out]] include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the syntonic comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the septimal comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>fjnh0zw3r1hgisfyk2l7qopnskua7ue</sha1>
    </revision>
    <revision>
      <id>198697</id>
      <parentid>198568</parentid>
      <timestamp>2025-06-05T04:07:20Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>+ category</comment>
      <origin>198697</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20597" sha1="464k8eb9hvy8k8d97xbmripp8uik1jo" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which, while reasonable for its size, is unsatisfactory for some, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers. In particular, there is a general detachment from the actual meantone and 5-limit system which originally justified it, with theories such as {{w|serialism}} and much of {{w|jazz}} theory deriving from 12edo's nature as a system itself rather than its underlying temperament structure.

12edo is the basic example of an equidodecatonic scale, or more simply, a 12-tone [[well temperament]]. It is in the position after 7edo as convergent to a musically "natural" [[golden meantone]] fifth. 

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

The [[comma]]s it [[tempering out|tempers out]] include the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], the syntonic comma, [[81/80]], the lesser diesis, [[128/125]], the diaschisma, [[2048/2025]], the septimal comma, [[64/63]], the septimal quartertone, [[36/35]], the jubilisma, [[50/49]], the septimal semicomma, [[126/125]], and the septimal kleisma, [[225/224]]. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>464k8eb9hvy8k8d97xbmripp8uik1jo</sha1>
    </revision>
    <revision>
      <id>198792</id>
      <parentid>198697</parentid>
      <timestamp>2025-06-06T04:46:14Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
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      <text bytes="21013" sha1="jgzjpr3ummp44gc85urrdlxt516micj" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a fifth which is quite accurate at two cents flat. It has a major third which is 13.7 cents sharp, which, while reasonable for its size, is unsatisfactory for some, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers. In particular, there is a general detachment from the actual meantone and 5-limit system which originally justified it, with theories such as {{w|serialism}} and much of {{w|jazz}} theory deriving from 12edo's nature as a system itself rather than its underlying temperament structure.

12edo maps [[3/2]] to 700c, meaning that it tempers out the Pythagorean comma [[531441/524288]] (equating Pythagorean enharmonic equivalents), and 5/4 to 400c, meaning that it tempers out the augmented comma [[128/125]] (thus equating classical enharmonic equivalents).

12edo is the basic example of an equidodecatonic scale, or more simply, a 12-tone [[well temperament]]. It is in the position after 7edo as convergent to a musically "natural" [[golden meantone]] fifth. 

The seventh partial ([[7/4]]) is "represented" by an interval which is sharp by 31 cents, which is why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. This interval is the same as the diatonic [[minor seventh]], and as such 12edo tempers out [[64/63]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone via its patent [[val]] of {{val| 12 19 28 34}} (tempering out [[225/224]] and [[126/125]]), its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Other [[comma]]s 12edo [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each comma tempered out by 12edo affects its structure in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways. For example, other diaschismic edos like 34edo separate 16/15 and 3/2 by the semioctave, and other augmented edos like 15edo conflate classical enharmonic equivalents (for example, the augmented third to the perfect fourth).

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>jgzjpr3ummp44gc85urrdlxt516micj</sha1>
    </revision>
    <revision>
      <id>198824</id>
      <parentid>198792</parentid>
      <timestamp>2025-06-06T12:47:47Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>There's no "Pythagorean enharmonic equivalents" or "classical enharmonic equivalents". This was so much prior arts ignored</comment>
      <origin>198824</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20912" sha1="k7o0uu5qkoh2rb3z0yqamojuoek2u8r" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a major third which is 13.7 cents sharp, which, while reasonable for its size, is unsatisfactory for some, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers. In particular, there is a general detachment from the actual meantone and 5-limit system which originally justified it, with theories such as {{w|serialism}} and much of {{w|jazz}} theory deriving from 12edo's nature as a system itself rather than its underlying temperament structure.

12edo is the basic example of an equidodecatonic scale, or more simply, a 12-tone [[well temperament]]. It is in the position after 7edo as convergent to a musically "natural" [[golden meantone]] fifth. 

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>k7o0uu5qkoh2rb3z0yqamojuoek2u8r</sha1>
    </revision>
    <revision>
      <id>206145</id>
      <parentid>198824</parentid>
      <timestamp>2025-08-02T08:29:27Z</timestamp>
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      <text bytes="21036" sha1="je13i19q2l3eq28t1kvj9bwopy0ee8x" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because as {{frac|1|12}} Pythagorean comma (approximately {{frac|1|11}} syntonic comma or full schisma) meantone, it represents [[meantone]]. It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a major third which is 13.7 cents sharp, which, while reasonable for its size, is unsatisfactory for some, and a minor third which is flat by even more, 15.6 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the "defects" of 12et appeared less evident. For example, Picardy thirds became less common, likely due to the stability of [[16:19:24]], 12edo's minor triad. However, it should be borne in mind that in actual performance these defects are often reduced by the tuning adaptations of the performers. In particular, there is a general detachment from the actual meantone and 5-limit system which originally justified it, with theories such as {{w|serialism}} and much of {{w|jazz}} theory deriving from 12edo's nature as a system itself rather than its underlying temperament structure.

12edo is the basic example of an equidodecatonic scale, or more simply, a 12-tone [[well temperament]]. It is in the position after 7edo as convergent to a musically "natural" [[golden meantone]] fifth. 

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>je13i19q2l3eq28t1kvj9bwopy0ee8x</sha1>
    </revision>
    <revision>
      <id>206246</id>
      <parentid>206145</parentid>
      <timestamp>2025-08-03T15:25:06Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
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      <comment>/* Theory */ Remove some questionable claims, improve flow</comment>
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      <text bytes="20915" sha1="krlsmhfc73qhe8ttvj34woqldems0kx" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position because it is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone). It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a major third which is 13.7 cents sharp, which, while reasonable for its size, is unsatisfactory for some, and a minor third which is flat by even more, 15.6 cents. 12edo became the standard Western tuning through a gradual historical process, primarily due to its practical advantages for keyboard instruments and its reasonable approximation of common intervals across all keys. However, it should be borne in mind that in actual performance these deviations from just intonation are often reduced by the tuning adaptations of the performers. In particular, there is a general detachment from the actual meantone and 5-limit system which originally justified it, with theories such as {{w|serialism}} and much of {{w|jazz}} theory deriving from 12edo's nature as a system itself rather than its underlying temperament structure.

12edo is the basic example of an equidodecatonic scale, or more simply, a 12-tone [[well temperament]].

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>krlsmhfc73qhe8ttvj34woqldems0kx</sha1>
    </revision>
    <revision>
      <id>206247</id>
      <parentid>206246</parentid>
      <timestamp>2025-08-03T15:26:31Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>/* Theory */ Improve flow</comment>
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      <text bytes="21084" sha1="61vvgieiet814m6nnd6rdnf42umopwd" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

=== Zeta peak index ===
{{ZPI
| zpi = 34
| steps = 12.0231830072926
| step size = 99.8071807833375
| tempered height = 5.193290
| pure height = 5.084467
| integral = 1.269599
| gap = 15.899282
| octave = 1197.68616940005
| consistent = 10
| distinct = 6
}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>61vvgieiet814m6nnd6rdnf42umopwd</sha1>
    </revision>
    <revision>
      <id>207037</id>
      <parentid>206247</parentid>
      <timestamp>2025-08-15T23:59:06Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>/* Approximation to JI */ -zeta peak index</comment>
      <origin>207037</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20826" sha1="gvetou0gvopjqk9ml8mkwvv5sg4r9ot" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>gvetou0gvopjqk9ml8mkwvv5sg4r9ot</sha1>
    </revision>
    <revision>
      <id>207224</id>
      <parentid>207037</parentid>
      <timestamp>2025-08-17T01:21:56Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <comment>clarify what the 12edo thirds are being compared to. We are a wiki, not a 31edo propaganda channel</comment>
      <origin>207224</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20840" sha1="8rejk9ddeutkqznkpt7b17zgwroxq9o" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of 5/4, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of 6/5 by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>8rejk9ddeutkqznkpt7b17zgwroxq9o</sha1>
    </revision>
    <revision>
      <id>207538</id>
      <parentid>207224</parentid>
      <timestamp>2025-08-19T14:59:26Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
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      <format>text/x-wiki</format>
      <text bytes="20842" sha1="rozqg8s5f45mkqntgf6cx4pg1edd8kr" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Octave stretch ===
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[The Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>rozqg8s5f45mkqntgf6cx4pg1edd8kr</sha1>
    </revision>
    <revision>
      <id>207988</id>
      <parentid>207538</parentid>
      <timestamp>2025-08-21T23:38:47Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Octave stretch */</comment>
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      <text bytes="20279" sha1="sqfb451ot93qtoomjwpbtu6qd41uu4l" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>sqfb451ot93qtoomjwpbtu6qd41uu4l</sha1>
    </revision>
    <revision>
      <id>207989</id>
      <parentid>207988</parentid>
      <timestamp>2025-08-21T23:39:15Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
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      <comment>/* Scales */ octave stretch/compression section</comment>
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      <text bytes="24937" sha1="4961vkk7f3jn0am79vr5x3siex7za3c" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
What follows is a comparison of stretched- and compressed-octave 12edo tunings.

; [[WE|12et, 7-limit WE tuning]] 
* Step size: 99.664{{c}}, octave size: 1196.0{{c}}
Compressing the octave of 12edo by 4{{c}} results in much improved primes 5, 7 and 11, but a much worse prime 3. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. [[40ed10]] does this as well. An argument could be made that such tunings [[7-limit|harmonies involving the 7th harmonic]] to regular old 12edo without even needing to add any new notes to the octave. This adds in brand new harmonic possibilities without breaking any common 12-tone music theory.
{{Harmonics in cet|99.664|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning}}
{{Harmonics in cet|99.664|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 7-limit WE tuning (continued)}}

; [[zpi|34zpi]] 
* Step size: 99.807{{c}}, octave size: 1197.7{{c}}
Compressing the octave of 12edo by around 2{{c}} results in improved primes 5 and 7, but a worse prime 3. The tuning 34zpi does this. It might be a good tuning for 5-limit [[meantone]], for composers seeking more pure thirds and sixths than regular 12edo. It would be well suited for playing classic pieces written for [[historical temperaments]], as well as being well suited to playing simultaneously with other instruments or voices that use [[just intonation]].
{{Harmonics in cet|99.807|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi}}
{{Harmonics in cet|99.807|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 34zpi (continued)}}

; [[WE|12et, 5-limit WE tuning]] 
* Step size: 99.868{{c}}, octave size: 1198.4{{c}}
Compressing the octave of 12edo by around 1{{c}} results in slightly improved primes 5 and 7, but a slightly prime 3. Its 5-limit WE tuning and 5-limit [[TE]] tuning both do this. This has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi.
{{Harmonics in cet|99.868|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning}}
{{Harmonics in cet|99.868|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12et, 5-limit WE tuning (continued)}}

; 12edo
* Step size: 100.000{{c}}, octave size: 1200.0{{c}} 
Pure-octaves 12edo performs well on harmonics 2, 3 and 5 but poorly on harmonics 7, 11 and 13 compared to other edos with a similar number of notes per octave.
{{Harmonics in equal|12|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 12edo}}
{{Harmonics in equal|12|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 12edo (continued)}}

; [[31ed6]] 
* Step size: 100.063{{c}}, octave size: 1200.8{{c}}
Stretching the octave of 12edo by a little less than 1{{c}} results in an improved prime 3, but worse prime 5. This loosely resembles the stretched-octave tunings commonly used on pianos. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 31ed6 does this.
{{Harmonics in equal|31|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6}}
{{Harmonics in equal|31|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 31ed6 (continued)}}

; [[19edt]] 
* Step size: 101.103{{c}}, octave size: 1201.2{{c}}
Stretching the octave of 12edo by a little more than 1{{c}} results in an improved prime 3, but worse prime 5. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 19edt does this.
{{Harmonics in equal|19|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt}}
{{Harmonics in equal|19|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 19edt (continued)}}

; [[7edf]] 
* Step size: 100.3{{c}}, octave size: 1203.35{{c}}
Stretching the octave of 12edo by around 3{{c}} results in improved primes 3 and 13, but much worse primes 5 and 7. This has similar benefits and drawbacks to [[Pythagorean]] tuning. Most modern music probably won't sound very good here because of the off 5th harmonic. The tuning 7edf does this.
{{Harmonics in equal|7|3|2|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edf}}
{{Harmonics in equal|7|3|2|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edf (continued)}}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>4961vkk7f3jn0am79vr5x3siex7za3c</sha1>
    </revision>
    <revision>
      <id>208023</id>
      <parentid>207989</parentid>
      <timestamp>2025-08-22T07:47:40Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>Restore the summary</comment>
      <origin>208023</origin>
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      <format>text/x-wiki</format>
      <text bytes="25396" sha1="8msa82rm32ku8y02oa1i4o4c48gnhzv" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[the Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics 5 and 7 at the cost of worse 2 and 3, while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

; [[WE|12et, 7-limit WE tuning]] 
* Step size: 99.664{{c}}, octave size: 1196.0{{c}}
Compressing the octave of 12edo by 4{{c}} results in much improved primes 5, 7 and 11, but a much worse prime 3. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. [[40ed10]] does this as well. An argument could be made that such tunings [[7-limit|harmonies involving the 7th harmonic]] to regular old 12edo without even needing to add any new notes to the octave. This adds in brand new harmonic possibilities without breaking any common 12-tone music theory.
{{Harmonics in cet|99.664|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 7-limit WE tuning}}
{{Harmonics in cet|99.664|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12et, 7-limit WE tuning (continued)}}

; [[zpi|34zpi]] 
* Step size: 99.807{{c}}, octave size: 1197.7{{c}}
Compressing the octave of 12edo by around 2{{c}} results in improved primes 5 and 7, but a worse prime 3. The tuning 34zpi does this. It might be a good tuning for 5-limit [[meantone]], for composers seeking more pure thirds and sixths than regular 12edo. It would be well suited for playing classic pieces written for [[historical temperaments]], as well as being well suited to playing simultaneously with other instruments or voices that use [[just intonation]].
{{Harmonics in cet|99.807|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 34zpi}}
{{Harmonics in cet|99.807|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 34zpi (continued)}}

; [[WE|12et, 5-limit WE tuning]] 
* Step size: 99.868{{c}}, octave size: 1198.4{{c}}
Compressing the octave of 12edo by around 1{{c}} results in slightly improved primes 5 and 7, but a slightly prime 3. Its 5-limit WE tuning and 5-limit [[TE]] tuning both do this. This has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi.
{{Harmonics in cet|99.868|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 5-limit WE tuning}}
{{Harmonics in cet|99.868|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12et, 5-limit WE tuning (continued)}}

; 12edo
* Step size: 100.000{{c}}, octave size: 1200.0{{c}} 
Pure-octaves 12edo performs well on harmonics 2, 3 and 5 but poorly on harmonics 7, 11 and 13 compared to other edos with a similar number of notes per octave.
{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12edo}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12edo (continued)}}

; [[31ed6]] 
* Step size: 100.063{{c}}, octave size: 1200.8{{c}}
Stretching the octave of 12edo by a little less than 1{{c}} results in an improved prime 3, but worse prime 5. This loosely resembles the stretched-octave tunings commonly used on pianos. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 31ed6 does this.
{{Harmonics in equal|31|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 31ed6}}
{{Harmonics in equal|31|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31ed6 (continued)}}

; [[19edt]] 
* Step size: 101.103{{c}}, octave size: 1201.2{{c}}
Stretching the octave of 12edo by a little more than 1{{c}} results in an improved prime 3, but worse prime 5. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 19edt does this.
{{Harmonics in equal|19|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 19edt}}
{{Harmonics in equal|19|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 19edt (continued)}}

; [[7edf]] 
* Step size: 100.3{{c}}, octave size: 1203.35{{c}}
Stretching the octave of 12edo by around 3{{c}} results in improved primes 3 and 13, but much worse primes 5 and 7. This has similar benefits and drawbacks to [[Pythagorean]] tuning. Most modern music probably won't sound very good here because of the off 5th harmonic. The tuning 7edf does this.
{{Harmonics in equal|7|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 7edf}}
{{Harmonics in equal|7|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 7edf (continued)}}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>8msa82rm32ku8y02oa1i4o4c48gnhzv</sha1>
    </revision>
    <revision>
      <id>208024</id>
      <parentid>208023</parentid>
      <timestamp>2025-08-22T08:34:50Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Octave stretch or compression */ unify precision. Note prime 2. Misc. fixes</comment>
      <origin>208024</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="25474" sha1="ede7q4t20uy4nvje4x9u94r4uu74xu0" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and the [[the Riemann zeta function and tuning|zeta-optimized]] 99.81{{c}} step size shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

; [[WE|12et, 7-limit WE tuning]] 
* Step size: 99.664{{c}}, octave size: 1195.971{{c}}
Compressing the octave of 12edo by 4{{c}} results in much improved primes 5, 7 and 11, but much worse primes 2 and 3. Both 7-limit [[WE]] and [[TE]] tuning do this. [[40ed10]] does this as well. An argument could be made that such tunings enable [[7-limit|harmonies involving the 7th harmonic]] to regular old 12edo without even needing to add any new notes to the octave. This adds in brand new harmonic possibilities without breaking any common 12-tone music theory.
{{Harmonics in cet|99.664256|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 7-limit WE tuning}}
{{Harmonics in cet|99.664256|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12et, 7-limit WE tuning (continued)}}

; [[ZPI|34zpi]] 
* Step size: 99.807{{c}}, octave size: 1197.686{{c}}
Compressing the octave of 12edo by around 2{{c}} results in improved primes 5 and 7, but worse primes 2 and 3. The tuning 34zpi does this. It might be a good tuning for 5-limit [[meantone]], for composers seeking more pure thirds and sixths than regular 12edo. It would be well suited for playing classic pieces written for [[historical temperaments]], as well as being well suited to playing simultaneously with other instruments or voices that use [[just intonation]].
{{Harmonics in cet|99.807|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 34zpi}}
{{Harmonics in cet|99.807|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 34zpi (continued)}}

; [[WE|12et, 5-limit WE tuning]] 
* Step size: 99.868{{c}}, octave size: 1198.416{{c}}
Compressing the octave of 12edo by around 1{{c}} results in slightly improved primes 5 and 7, but slightly worse primes 2 and 3. Both 5-limit WE and TE tuning do this. This has the same benefits and drawbacks as 34zpi, but both are less intense here compared to 34zpi.
{{Harmonics in cet|99.868021|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12et, 5-limit WE tuning}}
{{Harmonics in cet|99.868021|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12et, 5-limit WE tuning (continued)}}

; 12edo
* Step size: 100.000{{c}}, octave size: 1200.000{{c}} 
Pure-octaves 12edo performs well on harmonics 2, 3 and 5 but poorly on harmonics 7, 11 and 13 compared to other edos with a similar number of notes per octave.
{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 12edo}}
{{Harmonics in equal|12|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 12edo (continued)}}

; [[31ed6]] 
* Step size: 100.063{{c}}, octave size: 1200.757{{c}}
Stretching the octave of 12edo by a little less than 1{{c}} results in an improved prime 3, but worse primes 2, 5, and 7. This loosely resembles the stretched-octave tunings commonly used on pianos. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 31ed6 does this.
{{Harmonics in equal|31|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 31ed6}}
{{Harmonics in equal|31|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 31ed6 (continued)}}

; [[19edt]] 
* Step size: 101.103{{c}}, octave size: 1201.235{{c}}
Stretching the octave of 12edo by a little more than 1{{c}} results in an improved prime 3, but worse primes 2, 5, and 7. It may better match the [[timbre|slightly inharmonic partials]] of some string instruments. The tuning 19edt does this.
{{Harmonics in equal|19|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 19edt}}
{{Harmonics in equal|19|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 19edt (continued)}}

; [[7edf]] 
* Step size: 100.279{{c}}, octave size: 1203.351{{c}}
Stretching the octave of 12edo by around 3{{c}} results in improved primes 3 and 13, but much worse primes 2, 5, and 7. This has similar benefits and drawbacks to [[Pythagorean]] tuning. Most modern music probably will not sound very good here because of the off 5th harmonic. The tuning 7edf does this.
{{Harmonics in equal|7|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 7edf}}
{{Harmonics in equal|7|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 7edf (continued)}}

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>ede7q4t20uy4nvje4x9u94r4uu74xu0</sha1>
    </revision>
    <revision>
      <id>210490</id>
      <parentid>208024</parentid>
      <timestamp>2025-09-23T02:23:35Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <comment>/* Octave stretch or compression */</comment>
      <origin>210490</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20819" sha1="aohv0khvm9unj7xotzg0uwpx6tjbtz7" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
&lt;imagemap&gt;
File:12-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 384 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Evo_Sagittal.svg]]
&lt;/imagemap&gt;

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
&lt;imagemap&gt;
File:12-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 399 0 559 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
default [[File:12-EDO_Revo_Sagittal.svg]]
&lt;/imagemap&gt;

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>aohv0khvm9unj7xotzg0uwpx6tjbtz7</sha1>
    </revision>
    <revision>
      <id>214309</id>
      <parentid>210490</parentid>
      <timestamp>2025-10-24T02:07:40Z</timestamp>
      <contributor>
        <username>Tristanbay</username>
        <id>418</id>
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      <comment>/* Sagittal notation */  Switched to template</comment>
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      <text bytes="20371" sha1="p16v2avfio7iv8g1u5vlgmjlzcesc0v" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>p16v2avfio7iv8g1u5vlgmjlzcesc0v</sha1>
    </revision>
    <revision>
      <id>214958</id>
      <parentid>214309</parentid>
      <timestamp>2025-10-28T01:46:03Z</timestamp>
      <contributor>
        <username>Contribution</username>
        <id>191</id>
      </contributor>
      <comment>/* See also */</comment>
      <origin>214958</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20501" sha1="c7x0edlo4nfnzk6hvum2kjsn4z68rn1" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step
* [https://en.xen.wiki/w/Equal-step_tuning#Alpha-beta-gamma_family_of_equal_divisions Alpha-beta-gamma_family_of_equal_divisions]

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>c7x0edlo4nfnzk6hvum2kjsn4z68rn1</sha1>
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      <id>215075</id>
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      <timestamp>2025-10-28T12:12:39Z</timestamp>
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      <text bytes="21912" sha1="n0ta2lz7greorh3chpl0s1s45hf0xjx" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

==== Commas of the 2.3.5 subgroup tempered out in 12edo ====
Commas with numerator &lt; 1000000000:

[[81/80|81/80 [-4 4 -1⟩]], [[128/125|128/125 [7 0 -3⟩]], [[648/625|648/625 [3 4 -4⟩]], [[2048/2025|2048/2025 [11 -4 -2⟩]], [[6561/6250|6561/6250 [-1 8 -5⟩]], [[6561/6400|6561/6400 [-8 8 -2⟩]], [[16384/15625|16384/15625 [14 0 -6⟩]], [[32805/32768|32805/32768 [-15 8 1⟩]], [[82944/78125|82944/78125 [10 4 -7⟩]], [[262144/253125|262144/253125 [18 -4 -5⟩]], [[419904/390625|419904/390625 [6 8 -8⟩]], [[531441/500000|531441/500000 [-5 12 -6⟩]], [[531441/512000|531441/512000 [-12 12 -3⟩]], [[531441/524288|531441/524288 [-19 12 0⟩]], [[2097152/1953125|2097152/1953125 [21 0 -9⟩]], [[2125764/1953125|2125764/1953125 [2 12 -9⟩]], [[4194304/4100625|4194304/4100625 [22 -8 -4⟩]], [[10616832/9765625|10616832/9765625 [17 4 -10⟩]], [[33554432/31640625|33554432/31640625 [25 -4 -8⟩]], [[43046721/39062500|43046721/39062500 [-2 16 -10⟩]], [[43046721/40000000|43046721/40000000 [-9 16 -7⟩]], [[43046721/40960000|43046721/40960000 [-16 16 -4⟩]], [[43046721/41943040|43046721/41943040 [-23 16 -1⟩]], [[53747712/48828125|53747712/48828125 [13 8 -11⟩]], [[67108864/66430125|67108864/66430125 [26 -12 -3⟩]], [[268435456/244140625|268435456/244140625 [28 0 -12⟩]], [[272097792/244140625|272097792/244140625 [9 12 -12⟩]], [[536870912/512578125|536870912/512578125 [29 -8 -7⟩]]

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step
* [https://en.xen.wiki/w/Equal-step_tuning#Alpha-beta-gamma_family_of_equal_divisions Alpha-beta-gamma_family_of_equal_divisions]

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>n0ta2lz7greorh3chpl0s1s45hf0xjx</sha1>
    </revision>
    <revision>
      <id>215081</id>
      <parentid>215075</parentid>
      <timestamp>2025-10-28T14:20:13Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>We don't need this. Thanks!</comment>
      <origin>215081</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20371" sha1="p16v2avfio7iv8g1u5vlgmjlzcesc0v" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>p16v2avfio7iv8g1u5vlgmjlzcesc0v</sha1>
    </revision>
    <revision>
      <id>216353</id>
      <parentid>215081</parentid>
      <timestamp>2025-11-07T09:15:21Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ Add the kalismic counterpart of 99/70 = 140/99</comment>
      <origin>216353</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20394" sha1="ommdezh98xn0w0r8bps849c92n8ns8j" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament (specifically {{frac|1|12}} Pythagorean comma meantone, or approximately {{frac|1|11}} syntonic comma or full schisma meantone).

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s each unless [[stretched and compressed tuning|octave stretching or compression]] is employed. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is flat of just by even more, 15.6 cents.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[140/99]] (+0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>ommdezh98xn0w0r8bps849c92n8ns8j</sha1>
    </revision>
    <revision>
      <id>219583</id>
      <parentid>216353</parentid>
      <timestamp>2025-12-24T09:25:53Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Theory */ more beginner-friendly</comment>
      <origin>219583</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20185" sha1="qwx8bjvn35f4z6550fh1982v8wx9y9y" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality.
It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and because it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Historically, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest [[well temperament]], where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[140/99]] (+0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>qwx8bjvn35f4z6550fh1982v8wx9y9y</sha1>
    </revision>
    <revision>
      <id>219713</id>
      <parentid>219583</parentid>
      <timestamp>2025-12-25T09:00:31Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Theory */ note historical temperaments</comment>
      <origin>219713</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20328" sha1="pywkk64vnl73a59m5eqjtmz39g1skpf" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[140/99]] (+0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== Notes ==
&lt;references group="note" /&gt;

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>pywkk64vnl73a59m5eqjtmz39g1skpf</sha1>
    </revision>
    <revision>
      <id>222023</id>
      <parentid>219713</parentid>
      <timestamp>2026-01-20T10:50:05Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>Move notes to be under the corresponding tables</comment>
      <origin>222023</origin>
      <model>wikitext</model>
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      <text bytes="20343" sha1="pcuozyo10ycjx3csogg4mtegpnhzyob" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[140/99]] (+0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}
&lt;references group="note" /&gt;

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Schismina
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>pcuozyo10ycjx3csogg4mtegpnhzyob</sha1>
    </revision>
    <revision>
      <id>225095</id>
      <parentid>222023</parentid>
      <timestamp>2026-03-03T11:01:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <minor/>
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      <text bytes="20341" sha1="tefcsihp4j95tw8om6dhkh0pyr0jxch" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[140/99]] (+0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}
&lt;references group="note" /&gt;

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>tefcsihp4j95tw8om6dhkh0pyr0jxch</sha1>
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        <id>186</id>
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      <minor/>
      <comment>Text replacement - "[[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct" to "[[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct"</comment>
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      <text bytes="20396" sha1="afeb0rjcc4co02z8zu5a5cthucxo2jq" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[140/99]] (+0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}
&lt;references group="note" /&gt;

== Notation ==
12edo intervals and notes have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

{{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>afeb0rjcc4co02z8zu5a5cthucxo2jq</sha1>
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| en = 12edo
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| ja = 12平均律
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{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[140/99]] (+0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of 13/8 instead of 8 steps) is lower in relative error than any previous equal temperaments in the 3-, 5-, 7-, 11-, 13-, and 19-limit. The next equal temperaments doing better in those subgroups are 41, 19, 19, 22, 19/19e, and 19egh, respectively. 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is 72.

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>m4pi9p3muwjihzijql7bbg8e6tuz1wt</sha1>
    </revision>
    <revision>
      <id>226518</id>
      <parentid>226517</parentid>
      <timestamp>2026-03-20T18:39:56Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <minor/>
      <comment>/* Regular temperament properties */ split and links</comment>
      <origin>226518</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20527" sha1="qp69x5pf58knw4dmndcvrihsgvf2thd" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[140/99]] (+0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of [[13/8]] instead of 8 steps) is lower in relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>qp69x5pf58knw4dmndcvrihsgvf2thd</sha1>
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    <revision>
      <id>226519</id>
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      <timestamp>2026-03-20T18:44:27Z</timestamp>
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      <text bytes="20691" sha1="nlgfg5yepvgu5u1l4hjf85o04drmnvb" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="4" | Approximated [[JI]] intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt; ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[3-limit]]
! [[5-limit]]
! [[7-limit]]
! Other
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| 
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| 
| [[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039)&lt;br&gt;[[21/20]] (+15.533)&lt;br&gt;[[15/14]] (−19.443)
| [[18/17]] (+1.045)&lt;br&gt;[[17/16]] (−4.955)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)
| [[10/9]] (+17.596)
| [[28/25]] (+3.802)&lt;br&gt;[[8/7]] (−31.174)
| [[19/17]] (+7.442)&lt;br&gt;[[55/49]] (+0.020)&lt;br&gt;[[64/57]] (−0.532)&lt;br&gt;[[17/15]] (−16.687)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)
| [[6/5]] (−15.641)
| [[7/6]] (+33.129)&lt;br&gt;[[25/21]] (−1.847)
| [[19/16]] (+2.487)&lt;br&gt;[[44/37]] (+0.026)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)
| [[5/4]] (+13.686)
| [[63/50]] (−0.108)&lt;br&gt;[[9/7]] (−35.084)
| [[34/27]] (+0.910)&lt;br&gt;[[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)
| 
| 
| 
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| 
| [[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488)&lt;br&gt;[[10/7]] (−17.488)
| [[24/17]] (+3.000)&lt;br&gt;[[99/70]] (−0.088)&lt;br&gt;[[140/99]] (+0.088)&lt;br&gt;[[17/12]] (−3.000)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)
| 
| 
| 
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)
| [[8/5]] (−13.686)
| [[14/9]] (+35.084)&lt;br&gt;[[100/63]] (+0.108)
| [[19/12]] (+4.442)&lt;br&gt;[[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)
| [[5/3]] (+15.641)
| [[42/25]] (+1.847)&lt;br&gt;[[12/7]] (−33.129)
| [[37/22]] (−0.026)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)
| [[9/5]] (−17.596)
| [[7/4]] (+31.174)&lt;br&gt;[[25/14]] (−3.802)
| [[30/17]] (+16.687)&lt;br&gt;[[57/32]] (+0.532)&lt;br&gt;[[98/55]] (−0.020)&lt;br&gt;[[34/19]] (−7.442)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| 
| [[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443)&lt;br&gt;[[40/21]] (−15.533)&lt;br&gt;[[27/14]] (−37.039)
| [[32/17]] (+4.955)&lt;br&gt;[[17/9]] (−1.045)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| 
| 
| [[File:piano_1_1edo.mp3]]
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of [[13/8]] instead of 8 steps) is lower in relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>nlgfg5yepvgu5u1l4hjf85o04drmnvb</sha1>
    </revision>
    <revision>
      <id>226569</id>
      <parentid>226519</parentid>
      <timestamp>2026-03-20T23:21:48Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
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      <comment>/* Intervals */ four columns is overkill</comment>
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      <text bytes="20904" sha1="2zk0smufb3wb0kn1kqth07mumhb3w7r" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! rowspan="2" | [[Interval region]]
! colspan="2" | Approximated [[JI]] intervals ([[error]] in [[¢]])
! rowspan="2" | Audio
|-
! [[5-limit]]
! style="width: 330px;" | Higher limits&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| 
| [[File:piano_0_1edo.mp3]]
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
| [[File:piano_1_12edo.mp3]]
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442), [[64/57]] (−0.532)
| [[File:piano_1_6edo.mp3]]
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
| [[File:piano_1_4edo.mp3]]
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
| [[File:piano_1_3edo.mp3]]
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[21/16]] (-29.219)
| [[File:piano_5_12edo.mp3]]
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
| [[File:piano_1_2edo.mp3]]
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[32/21]] (+29.219)
| [[File:piano_7_12edo.mp3]]
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
| [[File:piano_2_3edo.mp3]]
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
| [[File:piano_3_4edo.mp3]]
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[57/32]] (+0.532)
| [[File:piano_5_6edo.mp3]]
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
| [[File:piano_11_12edo.mp3]]
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| 
| [[File:piano_1_1edo.mp3]]
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of [[13/8]] instead of 8 steps) is lower in relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>2zk0smufb3wb0kn1kqth07mumhb3w7r</sha1>
    </revision>
    <revision>
      <id>226571</id>
      <parentid>226569</parentid>
      <timestamp>2026-03-21T00:58:15Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
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      <comment>/* Intervals */ place audio in the middle for better comparison</comment>
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      <text bytes="20851" sha1="r5bvf771oe5d1x54oclnnhou3sfbbjy" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit intervals&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of [[13/8]] instead of 8 steps) is lower in relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>r5bvf771oe5d1x54oclnnhou3sfbbjy</sha1>
    </revision>
    <revision>
      <id>226682</id>
      <parentid>226571</parentid>
      <timestamp>2026-03-22T23:12:41Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Intervals */ they're interpretations of the tempered intervals</comment>
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| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;{{sg|limit=2.3.5.7.17.19 subgroup}}&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of [[13/8]] instead of 8 steps) is lower in relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>pp3z7nyc5p47c5anh5immrbycw4f823</sha1>
    </revision>
    <revision>
      <id>227440</id>
      <parentid>226682</parentid>
      <timestamp>2026-04-07T18:05:00Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Intervals */ add back 55/49 and 98/55</comment>
      <origin>227440</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="20943" sha1="8991e4rv41scppq5yzqzm6alyn2pwdn" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a diminished 2nd to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps is used as the approximation of [[13/8]] instead of 8 steps) is lower in relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 12 19 28 34 42 44 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawa
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquingu
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Trigu
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Sagugu
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-trigu
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layo
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtrigu
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Rurugu
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyo
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyo
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaru
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizogugu
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| 1uu2
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyo
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruru
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Bilorugu
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thulu
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathurugu
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Sogugu
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Sutho
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Sogu
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Sugu
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Soso
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothugugu
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nugu
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nuso
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosugu
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusu
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonogu
| Go comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki/&gt;* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]], while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense.

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>8991e4rv41scppq5yzqzm6alyn2pwdn</sha1>
    </revision>
    <revision>
      <id>231420</id>
      <parentid>227440</parentid>
      <timestamp>2026-06-02T00:10:25Z</timestamp>
      <contributor>
        <username>Eufalesio</username>
        <id>581</id>
      </contributor>
      <comment>Added many things: rewording / detemperaments / commas</comment>
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      <text bytes="21868" sha1="eei804fzd5g1vpc26r0yent1sp5at95" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, specially not 11, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo with an impecable 11-limit, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps - a major sixth - is used as the approximation of [[13/8]] instead of 8 steps) is lower in relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[Detempering|detempered]]:

* [[Schismic]] / [[Schismatic family#Schismic, schismatic, a.k.a. helmholtz|helmholtz]] (12 &amp; 53)
* [[Compton]] (12 &amp; 72)
* [[Atomic]] (12 &amp; 612)
* [[Garibaldi]] / [[cassandra]] (41 &amp; 53)*
* [[Heptacot]] (311 &amp; 323)*
* [[Misty]] / murky** (87 &amp; 99ef)
* [[Septimal diaschismic]] (46 &amp; 58)**

Rank-3 temperaments to which 12et can be detempered:

* [[Odin]] (12 &amp; 84 &amp; 270)
* [[Marvel]] (31 &amp; 41 &amp; 53)*
* [[Hemifamity|Aberschismic]] / [[Pele]] (41 &amp; 46 &amp; 53)*
* [[Cassaschismic]] (41 &amp; 53 &amp; 270)*

&lt;nowiki&gt;*&lt;/nowiki&gt;Through the 12e val

&lt;nowiki&gt;**&lt;/nowiki&gt;Through the 12f val

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>eei804fzd5g1vpc26r0yent1sp5at95</sha1>
    </revision>
    <revision>
      <id>231444</id>
      <parentid>231420</parentid>
      <timestamp>2026-06-02T04:06:31Z</timestamp>
      <contributor>
        <username>Lucius Chiaraviglio</username>
        <id>442</id>
      </contributor>
      <minor/>
      <comment>Fix a couple of misspelled words</comment>
      <origin>231444</origin>
      <model>wikitext</model>
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      <text bytes="21870" sha1="fonw6amzdaz0st3y66c6pnr1wwrgonh" xml:space="preserve">{{interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation.
In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. 
Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, especially not 11, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo with an impeccable 11-limit, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps - a major sixth - is used as the approximation of [[13/8]] instead of 8 steps) is lower in relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[Detempering|detempered]]:

* [[Schismic]] / [[Schismatic family#Schismic, schismatic, a.k.a. helmholtz|helmholtz]] (12 &amp; 53)
* [[Compton]] (12 &amp; 72)
* [[Atomic]] (12 &amp; 612)
* [[Garibaldi]] / [[cassandra]] (41 &amp; 53)*
* [[Heptacot]] (311 &amp; 323)*
* [[Misty]] / murky** (87 &amp; 99ef)
* [[Septimal diaschismic]] (46 &amp; 58)**

Rank-3 temperaments to which 12et can be detempered:

* [[Odin]] (12 &amp; 84 &amp; 270)
* [[Marvel]] (31 &amp; 41 &amp; 53)*
* [[Hemifamity|Aberschismic]] / [[Pele]] (41 &amp; 46 &amp; 53)*
* [[Cassaschismic]] (41 &amp; 53 &amp; 270)*

&lt;nowiki&gt;*&lt;/nowiki&gt;Through the 12e val

&lt;nowiki&gt;**&lt;/nowiki&gt;Through the 12f val

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>fonw6amzdaz0st3y66c6pnr1wwrgonh</sha1>
    </revision>
    <revision>
      <id>231454</id>
      <parentid>231444</parentid>
      <timestamp>2026-06-02T07:35:00Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>No exaggerated language bs.</comment>
      <origin>231454</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21822" sha1="770p9nzfq5suc9wf94mrmhslyamw3gm" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -19 12 }}
| {{mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et (using the 12f val, where 9 steps - a major sixth - is used as the approximation of [[13/8]] instead of 8 steps) is lower in relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], [[11-limit|11-]], [[13-limit|13-]], and [[19-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], 19/19e, and 19egh, respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–Pythagorean equivalence continuum]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[Detempering|detempered]]:

* [[Schismic]] / [[Schismatic family#Schismic, schismatic, a.k.a. helmholtz|helmholtz]] (12 &amp; 53)
* [[Compton]] (12 &amp; 72)
* [[Atomic]] (12 &amp; 612)
* [[Garibaldi]] / [[cassandra]] (41 &amp; 53)*
* [[Heptacot]] (311 &amp; 323)*
* [[Misty]] / murky** (87 &amp; 99ef)
* [[Septimal diaschismic]] (46 &amp; 58)**

Rank-3 temperaments to which 12et can be detempered:

* [[Odin]] (12 &amp; 84 &amp; 270)
* [[Marvel]] (31 &amp; 41 &amp; 53)*
* [[Hemifamity|Aberschismic]] / [[Pele]] (41 &amp; 46 &amp; 53)*
* [[Cassaschismic]] (41 &amp; 53 &amp; 270)*

&lt;nowiki&gt;*&lt;/nowiki&gt;Through the 12e val

&lt;nowiki&gt;**&lt;/nowiki&gt;Through the 12f val

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>770p9nzfq5suc9wf94mrmhslyamw3gm</sha1>
    </revision>
    <revision>
      <id>231455</id>
      <parentid>231454</parentid>
      <timestamp>2026-06-02T07:50:16Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ no more relative error note beyond its monotonicity capability. Reduce duplication. - rank-3 stuff</comment>
      <origin>231455</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21291" sha1="r8k3f502nrinunbfac7l4azulbh3eoq" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[schismic]] (12 &amp; 53), and [[diaschismic]] (46 &amp; 58). See [[Schismic–commatic equivalence continuum]] for a more comprehensive list.

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>r8k3f502nrinunbfac7l4azulbh3eoq</sha1>
    </revision>
    <revision>
      <id>231481</id>
      <parentid>231455</parentid>
      <timestamp>2026-06-02T17:23:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ schismic -&gt; garibaldi</comment>
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      <model>wikitext</model>
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      <text bytes="21292" sha1="dc9k5358hxs3it1skn51fflu72uw5qe" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]] / [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[Dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Srutal]] / [[pajara]] / [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[lithium]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[garibaldi]] (41 &amp; 53), and [[diaschismic]] (46 &amp; 58). See [[Schismic–commatic equivalence continuum]] for a more comprehensive list.

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>dc9k5358hxs3it1skn51fflu72uw5qe</sha1>
    </revision>
    <revision>
      <id>231482</id>
      <parentid>231481</parentid>
      <timestamp>2026-06-02T17:29:10Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ no more srutal. Replace lithium with august, which is simpler</comment>
      <origin>231482</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21276" sha1="cs13crul29eu1l68fbqds03ze5jj36p" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]

{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[garibaldi]] (41 &amp; 53), and [[diaschismic]] (46 &amp; 58). See [[Schismic–commatic equivalence continuum]] for a more comprehensive list.

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>cs13crul29eu1l68fbqds03ze5jj36p</sha1>
    </revision>
    <revision>
      <id>231489</id>
      <parentid>231482</parentid>
      <timestamp>2026-06-02T21:31:54Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
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      <comment>/* Regular temperament properties */ re-organize to make the connection more obvious</comment>
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      <text bytes="21278" sha1="4yn096b2xm9m9yt5mrzlf0iwznqz55p" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[garibaldi]] (41 &amp; 53), and [[diaschismic]] (46 &amp; 58). For more comprehensive lists, see: 
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
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{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 '''de'''temperaments to which 12et can be [[detempering|detempered]] include: 

* [[compton]] (12 &amp; 72)
* [[garibaldi]] (41 &amp; 53)
* [[diaschismic]] (46 &amp; 58)
* [[atomic]] (12 &amp; 612)

For more comprehensive lists, see:
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>0ixxwoleqikfpmo1gocgapztk1v3cet</sha1>
    </revision>
    <revision>
      <id>231501</id>
      <parentid>231499</parentid>
      <timestamp>2026-06-03T09:16:52Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>No one cares about atomic</comment>
      <origin>231501</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21278" sha1="4yn096b2xm9m9yt5mrzlf0iwznqz55p" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[garibaldi]] (41 &amp; 53), and [[diaschismic]] (46 &amp; 58). For more comprehensive lists, see: 
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>4yn096b2xm9m9yt5mrzlf0iwznqz55p</sha1>
    </revision>
    <revision>
      <id>231511</id>
      <parentid>231501</parentid>
      <timestamp>2026-06-03T13:22:12Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Regular temperament properties */ add rank-3 back, after discussion with Eufalesio</comment>
      <origin>231511</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21448" sha1="73yi371parumh0yc3hu9vddxgsmhg1a" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[25/24]] (+29.328)&lt;br&gt;[[16/15]] (−11.731)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[schismic]]/[[garibaldi]] (41 &amp; 53), and [[diaschismic]] (46 &amp; 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see: 
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>73yi371parumh0yc3hu9vddxgsmhg1a</sha1>
    </revision>
    <revision>
      <id>231631</id>
      <parentid>231511</parentid>
      <timestamp>2026-06-06T05:11:51Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <minor/>
      <comment>/* Intervals */ swap these</comment>
      <origin>231631</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21448" sha1="93b1wfywnwtcg141vjkgn0f5dtdfc3z" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament.

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of just. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[16/15]] (−11.731)&lt;br&gt;[[25/24]] (+29.328)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[schismic]]/[[garibaldi]] (41 &amp; 53), and [[diaschismic]] (46 &amp; 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see: 
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>93b1wfywnwtcg141vjkgn0f5dtdfc3z</sha1>
    </revision>
    <revision>
      <id>231633</id>
      <parentid>231631</parentid>
      <timestamp>2026-06-06T05:27:59Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Theory */ various additions</comment>
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      <text bytes="21817" sha1="e4dj4lnr4qcv056vcaez75nzpcv47u2" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament, tempering out [[81/80]], equating four [[3/2|perfect fifths]] with the [[5/1|5th harmonic]].

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of [[just intonation]]. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. The [[duodene]] is an unequal 12-note scale in 5-limit just intonation which observes these commas, and can be considered a [[detempering|detemper]] of 12edo. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], and the schisma, [[32805/32768]], and in the 7-limit, the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented, as well as the [[16:19:24]] otonal minor triad.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[16/15]] (−11.731)&lt;br&gt;[[25/24]] (+29.328)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[schismic]]/[[garibaldi]] (41 &amp; 53), and [[diaschismic]] (46 &amp; 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see: 
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{Main| List of MOS scales in 12edo }}

The two most common 12edo mos scales are meantone[5] and meantone[7].
* Diatonic (meantone) 5L2s 2221221 (generator = 7\12)
* Pentatonic (meantone) 2L3s 22323 (generator = 7\12)
* Diminished 4L4s 12121212 (generator = 1\12, period = 3\12)

=== Non-mos scales ===
Due to 12edo's dominance, some non-mos scales are also widely used in many musical practices around the world.

* Harmonic major – 2212132
* Melodic major – 2212122
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131
* 5-odd-limit tonality diamond – 3112113
[[File:12edo modes.pdf|thumb]]

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>e4dj4lnr4qcv056vcaez75nzpcv47u2</sha1>
    </revision>
    <revision>
      <id>231852</id>
      <parentid>231633</parentid>
      <timestamp>2026-06-06T20:37:01Z</timestamp>
      <contributor>
        <username>Sintel</username>
        <id>330</id>
      </contributor>
      <comment>/* Scales */ Wording, give more obvious examples, remove pdf link</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21811" sha1="akl4i2gz5tzkm3zmq3j2bx2vqa9tivn" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament, tempering out [[81/80]], equating four [[3/2|perfect fifths]] with the [[5/1|5th harmonic]].

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of [[just intonation]]. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. The [[duodene]] is an unequal 12-note scale in 5-limit just intonation which observes these commas, and can be considered a [[detempering|detemper]] of 12edo. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], and the schisma, [[32805/32768]], and in the 7-limit, the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented, as well as the [[16:19:24]] otonal minor triad.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[16/15]] (−11.731)&lt;br&gt;[[25/24]] (+29.328)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[schismic]]/[[garibaldi]] (41 &amp; 53), and [[diaschismic]] (46 &amp; 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see: 
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{See also| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic: [[5L 2s]] – 2221221 (generator = 7\12)
* Pentatonic: [[2L 3s]] – 22323 (generator = 7\12)

The diminished and augmented scales are also MOS scales.
* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12)
* Augmented: [[3L 3s]] – 13131313 (generator = 1\12, period = 4\12)

Other widely used scales include:
* Melodic minor – 2122221
* Harmonic minor – 2122131
* Harmonic major – 2212131
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>akl4i2gz5tzkm3zmq3j2bx2vqa9tivn</sha1>
    </revision>
    <revision>
      <id>231858</id>
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      <text bytes="21809" sha1="438q0zhgfzjuj88lb4euy4agbtysje0" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament, tempering out [[81/80]], equating four [[3/2|perfect fifths]] with the [[5/1|5th harmonic]].

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of [[just intonation]]. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. The [[duodene]] is an unequal 12-note scale in 5-limit just intonation which observes these commas, and can be considered a [[detempering|detemper]] of 12edo. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], and the schisma, [[32805/32768]], and in the 7-limit, the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented, as well as the [[16:19:24]] otonal minor triad.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Notable [[rank-2 temperament]]s that augment 12edo with extra [[generator]]s include [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[16/15]] (−11.731)&lt;br&gt;[[25/24]] (+29.328)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[schismic]]/[[garibaldi]] (41 &amp; 53), and [[diaschismic]] (46 &amp; 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see: 
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{See also| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic: [[5L 2s]] – 2221221 (generator = 7\12)
* Pentatonic: [[2L 3s]] – 22323 (generator = 7\12)

The diminished and augmented scales are also MOS scales.
* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12)
* Augmented: [[3L 3s]] – 131313 (generator = 1\12, period = 4\12)

Other widely used scales include:
* Melodic minor – 2122221
* Harmonic minor – 2122131
* Harmonic major – 2212131
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>438q0zhgfzjuj88lb4euy4agbtysje0</sha1>
    </revision>
    <revision>
      <id>232152</id>
      <parentid>231858</parentid>
      <timestamp>2026-06-11T08:25:29Z</timestamp>
      <contributor>
        <username>Eliora</username>
        <id>274</id>
      </contributor>
      <comment>/* Subsets and supersets */ fractional octave temperaments</comment>
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      <model>wikitext</model>
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      <text bytes="21817" sha1="hd2gwnvi395f5ucz1plhedyrsdh5ev0" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament, tempering out [[81/80]], equating four [[3/2|perfect fifths]] with the [[5/1|5th harmonic]].

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has a [[3/2|fifth]] which is quite accurate at 700 cents, two cents flat of [[just intonation]]. It has a [[5/4|major third]] which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5|minor third]] is even less accurate, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. The [[duodene]] is an unequal 12-note scale in 5-limit just intonation which observes these commas, and can be considered a [[detempering|detemper]] of 12edo. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], and the schisma, [[32805/32768]], and in the 7-limit, the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented, as well as the [[16:19:24]] otonal minor triad.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Various [[12th-octave temperaments]] that augment 12edo exist, most prominent examples being [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[16/15]] (−11.731)&lt;br&gt;[[25/24]] (+29.328)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[schismic]]/[[garibaldi]] (41 &amp; 53), and [[diaschismic]] (46 &amp; 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see: 
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{See also| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic: [[5L 2s]] – 2221221 (generator = 7\12)
* Pentatonic: [[2L 3s]] – 22323 (generator = 7\12)

The diminished and augmented scales are also MOS scales.
* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12)
* Augmented: [[3L 3s]] – 131313 (generator = 1\12, period = 4\12)

Other widely used scales include:
* Melodic minor – 2122221
* Harmonic minor – 2122131
* Harmonic major – 2212131
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>hd2gwnvi395f5ucz1plhedyrsdh5ev0</sha1>
    </revision>
    <revision>
      <id>232450</id>
      <parentid>232152</parentid>
      <timestamp>2026-06-20T00:21:42Z</timestamp>
      <contributor>
        <username>Overthink</username>
        <id>595</id>
      </contributor>
      <comment>/* Theory */ tone; separate interval names from ratios</comment>
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      <text bytes="21852" sha1="q1m9kfc9ktfujvxgmf625a2qu31tyxs" xml:space="preserve">{{Interwiki
| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
}}
{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament, tempering out [[81/80]], equating four [[3/2]] perfect fifths with the [[5/1|5th harmonic]].

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has an approximate [[3/2]] perfect fifth which is quite accurate at 700 cents, two cents flat of [[just intonation]]. It has a [[5/4]] major third which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5]] minor third is slightly less accurate than that, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. The [[duodene]] is an unequal 12-note scale in 5-limit just intonation which observes these commas, and can be considered a [[detempering|detemper]] of 12edo. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], and the schisma, [[32805/32768]], and in the 7-limit, the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented, as well as the [[16:19:24]] otonal minor triad.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[the Riemann zeta function and tuning|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Various [[12th-octave temperaments]] that augment 12edo exist, most prominent examples being [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[16/15]] (−11.731)&lt;br&gt;[[25/24]] (+29.328)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[schismic]]/[[garibaldi]] (41 &amp; 53), and [[diaschismic]] (46 &amp; 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see: 
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{See also| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic: [[5L 2s]] – 2221221 (generator = 7\12)
* Pentatonic: [[2L 3s]] – 22323 (generator = 7\12)

The diminished and augmented scales are also MOS scales.
* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12)
* Augmented: [[3L 3s]] – 131313 (generator = 1\12, period = 4\12)

Other widely used scales include:
* Melodic minor – 2122221
* Harmonic minor – 2122131
* Harmonic major – 2212131
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
      <sha1>q1m9kfc9ktfujvxgmf625a2qu31tyxs</sha1>
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| en = 12edo
| de = 12-EDO
| es = 12 EDO
| ja = 12平均律
| ro = 12DEO
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{{Infobox ET}}
{{Wikipedia|12 equal temperament}}
{{ED intro}} It is the predominating tuning system in the world today.

== Theory ==
12edo achieved its position as the standard Western tuning system through a combination of theoretical properties and practicality. It is the smallest number of equal divisions of the octave ([[edo]]) which can seriously claim to represent [[5-limit]] harmony, and it represents a [[meantone]] temperament, tempering out [[81/80]], equating four [[3/2]] perfect fifths with the [[5/1|5th harmonic]].

It divides the octave into twelve equal parts, each of exactly 100 [[cent]]s. It has an approximate [[3/2]] perfect fifth which is quite accurate at 700 cents, two cents flat of [[just intonation]]. It has a [[5/4]] major third which is 13.7 cents sharp of just, which, while reasonable for its size, is unsatisfactory for some. The [[6/5]] minor third is slightly less accurate than that, being 15.6 cents flat of just.

Before people used 12edo, people used a variety of [[historical temperaments]] such as [[quarter-comma meantone]], and later [[well temperament]]s. By the 20th century, 12edo became dominant primarily due to practical considerations for keyboard instruments and its ability to handle modulation across all keys with reasonable intonation. In actual performance, these deviations from just intonation are often reduced by the tuning adaptations of skilled performers. Modern music theory has increasingly treated 12edo as a system in its own right rather than as an approximation of just intonation or meantone, leading to theoretical approaches such as {{w|serialism}} and much of {{w|jazz harmony}} that derive from 12edo's structure as an equal division rather than its underlying temperament properties.{{cn}}

12edo is the basic example of a [[:Category:12-tone scales|dodecatonic]] scale and can be considered the simplest well temperament, where all twelve fifths are the same.

The 7th harmonic ([[7/4]]) is represented by the diatonic [[minor seventh]], which is sharp by 31 cents, and as such 12edo tempers out [[64/63]]. The deviation explains why minor sevenths tend to stand out distinctly from the rest of the chord in a [[tetrad]]. Such tetrads are often used as [[dominant seventh chord]]s in [[diatonic functional harmony|functional harmony]], for which the 5-limit JI version would be [[36:45:54:64|1–5/4–3/2–16/9]], and while 12et officially [[support]]s septimal meantone for tempering out [[126/125]] and [[225/224]] via its [[patent val]] of {{val| 12 19 28 34}}, its approximations of [[7-limit]] intervals are not very accurate. It cannot be said to represent [[11/1|11]] or [[13/1|13]] at all, though it does a quite credible [[17/1|17]] and an even better [[19/1|19]]. Nevertheless, its relative tuning accuracy is quite high, and 12edo is the fourth [[zeta integral edo]].

Stacking the fifth twelve times returns the pitch to the starting point, so that the Pythagorean comma, [[Pythagorean comma|3&lt;sup&gt;12&lt;/sup&gt;/2&lt;sup&gt;19&lt;/sup&gt;]], is tempered out. Three major thirds equal an octave, so the lesser diesis, [[128/125]], is tempered out. Four minor thirds also equal an octave, so the greater diesis, [[648/625]], is tempered out. These features have been widely utilized in contemporary music. The [[duodene]] is an unequal 12-note scale in 5-limit just intonation which observes these commas, and can be considered a [[detempering|detemper]] of 12edo. Other [[comma]]s 12et [[tempering out|tempers out]] include the diaschisma, [[2048/2025]], and the schisma, [[32805/32768]], and in the 7-limit, the septimal quartertone, [[36/35]], and the jubilisma, [[50/49]]. Each affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.

12edo also offers very good approximations to intervals in the 2.3.17.19 subgroup. This indicates one way to use 12edo that deviates from common-practice harmony; for instance the cluster chord 8:17:36:76 is well represented, as well as the [[16:19:24]] otonal minor triad.

=== Prime harmonics ===
{{Harmonics in equal|12|prec=2}}

=== Subsets and supersets ===
12edo contains [[2edo]], [[3edo]], [[4edo]], and [[6edo]] as subsets. It is the 5th [[highly composite edo]], 12 being both a superabundant and a highly composite number. 12edo is also the only known edo aside from 2edo that is both [[Riemann zeta function|strict zeta]] and highly composite. 

[[24edo]], which doubles it, improves significantly on approximations to 11 and 13, with 13 tuned sharp. [[36edo]], which triples it, improves on harmonics 7 and 13, but has the 13 tuned flat instead of sharp. [[72edo]] is a notable zeta-record edo, and [[60edo|60-]], [[84edo|84-]], and [[96edo]] all see utilities. Various [[12th-octave temperaments]] that augment 12edo exist, most prominent examples being [[compton]] and [[catler]].

== Intervals ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Intervals of 12edo
|-
! [[Degree]]
! [[Cent]]s
! [[Interval region]]
! style="width: 165px;" | Approximated 5-limit&lt;br&gt;JI intervals (error in [[¢]])
! Audio
! style="width: 330px;" | Higher limit interpretations&lt;ref group="note"&gt;Intervals of the 2.3.5.7.17.19 subgroup, with a few additional interpretations&lt;/ref&gt;
|-
| 0
| 0
| Unison (prime)
| [[1/1]] (just)
| [[File:piano_0_1edo.mp3]]
|
|-
| 1
| 100
| Minor second
| [[256/243]] (+9.775)&lt;br&gt;[[16/15]] (−11.731)&lt;br&gt;[[25/24]] (+29.328)
| [[File:piano_1_12edo.mp3]]
| [[28/27]] (+37.039), [[21/20]] (+15.533), [[15/14]] (−19.443)&lt;br&gt;[[17/16]] (−4.955), [[18/17]] (+1.045)&lt;br&gt;[[19/18]] (+6.397), [[20/19]] (+11.199)
|-
| 2
| 200
| Major second
| [[9/8]] (−3.910)&lt;br&gt;[[10/9]] (+17.596)
| [[File:piano_1_6edo.mp3]]
| [[8/7]] (−31.174), [[28/25]] (+3.802)&lt;br&gt;[[17/15]] (−16.687), [[19/17]] (+7.442),&lt;br&gt;[[55/49]] (+0.020), [[64/57]] (−0.532)
|-
| 3
| 300
| Minor third
| [[32/27]] (+5.865)&lt;br&gt;[[6/5]] (−15.641)&lt;br&gt;[[75/64]] (+25.418)
| [[File:piano_1_4edo.mp3]]
| [[7/6]] (+33.129), [[25/21]] (−1.847)&lt;br&gt;[[19/16]] (+2.487)
|-
| 4
| 400
| Major third
| [[81/64]] (−7.820)&lt;br&gt;[[5/4]] (+13.686)&lt;br&gt; [[32/25]] (-27.373)
| [[File:piano_1_3edo.mp3]]
| [[63/50]] (−0.108), [[9/7]] (−35.084)&lt;br&gt;[[34/27]] (+0.910), [[24/19]] (−4.442)
|-
| 5
| 500
| Fourth
| [[4/3]] (+1.955)&lt;br&gt; [[27/20]] (-19.551)
| [[File:piano_5_12edo.mp3]]
| [[21/16]] (-29.219)
|-
| 6
| 600
| [[Tritone]]
| [[25/18]] (+31.283)&lt;br&gt;[[36/25]] (-31.283)&lt;br&gt;[[45/32]] (+9.776)&lt;br&gt;[[64/45]] (−9.776)
| [[File:piano_1_2edo.mp3]]
| [[7/5]] (+17.488), [[10/7]] (−17.488)&lt;br&gt;[[24/17]] (+3.000), [[17/12]] (−3.000)&lt;br&gt;[[99/70]] (−0.088), [[140/99]] (+0.088)
|-
| 7
| 700
| Fifth
| [[3/2]] (−1.955)&lt;br&gt;[[40/27]] (+19.551)
| [[File:piano_7_12edo.mp3]]
| [[32/21]] (+29.219)
|-
| 8
| 800
| Minor sixth
| [[128/81]] (+7.820)&lt;br&gt;[[8/5]] (−13.686)&lt;br&gt;[[25/16]] (+27.373)
| [[File:piano_2_3edo.mp3]]
| [[14/9]] (+35.084), [[100/63]] (+0.108)&lt;br&gt;[[19/12]] (+4.442), [[27/17]] (−0.910)
|-
| 9
| 900
| Major sixth
| [[27/16]] (−5.865)&lt;br&gt;[[5/3]] (+15.641)&lt;br&gt;[[128/75]] (-25.418)
| [[File:piano_3_4edo.mp3]]
| [[12/7]] (−33.129), [[42/25]] (+1.847)&lt;br&gt;[[32/19]] (−2.487)
|-
| 10
| 1000
| Minor seventh
| [[16/9]] (+3.910)&lt;br&gt;[[9/5]] (−17.596)
| [[File:piano_5_6edo.mp3]]
| [[7/4]] (+31.174), [[25/14]] (−3.802)&lt;br&gt;[[30/17]] (+16.687), [[34/19]] (−7.442)&lt;br&gt;[[98/55]] (-0.020), [[57/32]] (+0.532)
|-
| 11
| 1100
| Major seventh
| [[243/128]] (-9.775)&lt;br&gt;[[15/8]] (+11.731)&lt;br&gt;[[48/25]] (−29.328)
| [[File:piano_11_12edo.mp3]]
| [[28/15]] (+19.443), [[40/21]] (−15.533), [[27/14]] (−37.039)&lt;br&gt;[[32/17]] (+4.955), [[17/9]] (−1.045)&lt;br&gt;[[36/19]] (-6.397), [[19/10]] (-11.199)
|-
| 12
| 1200
| Octave
| [[2/1]] (just)
| [[File:piano_1_1edo.mp3]]
|
|}
&lt;references group="note" /&gt;

== Notation ==
The intervals and notes of 12edo have standard names from classical music theory. This classical notation system, which was in use before 12edo with other tuning systems based on chains of fifths, is sometimes called the [[chain-of-fifths notation]] or extended Pythagorean notation.

{{Sharpness-sharp1|12}}

The subsets {{EDOs|1edo, 2edo, 3edo, 4edo and 6edo}} can all be written using 12edo [[subset notation]].

Any 12edo note or interval can be [[Enharmonic unison|respelled enharmonically]] by adding a [[pythagorean comma]] to it or subtracting one from it.

{| class="wikitable center-all"
|+ style="font-size: 105%;" | Notation of 12edo
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! Diatonic ([[5L&amp;nbsp;2s]]) interval names
! Note names (on D)
|-
| 0
| 0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 100
| Augmented unison (A1)&lt;br&gt;Minor second (m2)
| D#&lt;br&gt;Eb
|-
| 2
| 200
| '''Major second (M2)'''&lt;br&gt;Diminished third (d3)
| '''E'''&lt;br&gt;Fb
|-
| 3
| 300
| Augmented second (A2)&lt;br&gt;'''Minor third (m3)'''
| E#&lt;br&gt;'''F'''
|-
| 4
| 400
| Major third (M3)&lt;br&gt;Diminished fourth (d4)
| F#&lt;br&gt;Gb
|-
| 5
| 500
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 6
| 600
| Augmented fourth (A4)&lt;br&gt;Diminished fifth (d5)
| G#&lt;br&gt;Ab
|-
| 7
| 700
| '''Perfect fifth (P5)'''
| A
|-
| 8
| 800
| Augmented fifth (A5)&lt;br&gt;Minor sixth (m6)
| A#&lt;br&gt;Bb
|-
| 9
| 900
| '''Major sixth (M6)'''&lt;br&gt;Diminished seventh (d7)
| '''B'''&lt;br&gt;Cb
|-
| 10
| 1000
| Augmented sixth (A6)&lt;br&gt;'''Minor seventh (m7)'''
| B#&lt;br&gt;'''C'''
|-
| 11
| 1100
| Major seventh (M7)&lt;br&gt;Diminished octave (d8)
| C#&lt;br&gt;Db
|-
| 12
| 1200
| '''Perfect octave (P8)'''
| '''D'''
|}

In 12edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (&amp;#x266F;) and flats (&amp;#x266D;) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

=== Sagittal notation ===
This notation uses the same sagittal sequence as EDOs [[5edo#Sagittal notation|5]], [[19edo#Sagittal notation|19]], and [[26edo#Sagittal notation|26]], is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[36edo#Sagittal notation|36]], [[48edo#Sagittal notation|48]], [[60edo#Sagittal notation|60]], [[72edo#Sagittal notation|72]], and [[84edo#Sagittal notation|84]], and is a superset of the notation for [[6edo#Sagittal notation|6-EDO]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

== Solfege ==
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Solfege of 12edo
|-
! [[Degree]]
! [[Cents]]
! Standard [[solfege]]&lt;br&gt;(movable do)
! [[Uniform solfege]]&lt;br&gt;(2-3 vowels)
|-
| 0
| 0
| Do
| Da
|-
| 1
| 100
| Di (A1)&lt;br&gt;Ra (m2)
| Du (A1)&lt;br&gt;Fra (m2)
|-
| 2
| 200
| Re
| Ra
|-
| 3
| 300
| Ri (A2)&lt;br&gt;Me (m3)
| Ru (A2)&lt;br&gt;Na (m3)
|-
| 4
| 400
| Mi
| Ma (M3)&lt;br&gt;Fo (d4)
|-
| 5
| 500
| Fa
| Mu (A3)&lt;br&gt;Fa (P4)
|-
| 6
| 600
| Fi (A4)&lt;br&gt;Se (d5)
| Pa (A4)&lt;br&gt;Sha (d5)
|-
| 7
| 700
| So
| Sa
|-
| 8
| 800
| Si (A5)&lt;br&gt;Le (m6)
| Su (A5)&lt;br&gt;Fla (m6)
|-
| 9
| 900
| La
| La (M6)&lt;br&gt;Tho (d7)
|-
| 10
| 1000
| Li (A6)&lt;br&gt;Te (m7)
| Lu (A6)&lt;br&gt;Tha (m7)
|-
| 11
| 1100
| Ti
| Ta (M7)&lt;br&gt;Do (d8)
|-
| 12
| 1200
| Do
| Da
|}

== Approximation to JI ==
[[File:12ed2-5Limit.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 5-limit intervals approximated in 12edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|12}}
{{Q-odd-limit intervals|12.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 12f val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal&lt;br&gt;8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -19 12 }}
| {{Mapping| 12 19 }}
| +0.62
| 0.62
| 0.62
|-
| 2.3.5
| 81/80, 128/125
| {{Mapping| 12 19 28 }}
| −1.56
| 3.11
| 3.11
|-
| 2.3.5.7
| 36/35, 50/49, 64/63
| {{Mapping| 12 19 28 34 }}
| −3.95
| 4.92
| 4.94
|-
| 2.3.5.7.17
| 36/35, 50/49, 51/49, 64/63
| {{Mapping| 12 19 28 34 49 }}
| −2.92
| 4.86
| 4.87
|-
| 2.3.5.7.17.19
| 36/35, 50/49, 51/49, 57/56, 64/63
| {{Mapping| 12 19 28 34 49 51 }}
| −2.53
| 4.52
| 4.53
|- style="border-top: double;"
| 2.3.5.17
| 51/50, 81/80, 128/125
| {{Mapping| 12 19 28 49 }}
| −0.87
| 2.95
| 2.95
|-
| 2.3.5.17.19
| 51/50, 76/75, 81/80, 128/125
| {{Mapping| 12 19 28 49 51 }}
| −0.81
| 2.64
| 2.64
|}
* 12et is monotonic to the [[11-odd-limit]]. It is the first equal temperament to achieve this. 
* 12et has a lower relative error than any previous equal temperaments in the [[3-limit|3-]], [[5-limit|5-]], [[7-limit|7-]], and [[11-limit]]. The next equal temperaments doing better in those subgroups are [[41edo|41]], [[19edo|19]], 19, [[22edo|22]], respectively.
* 12et is even more prominent in the 2.3.5.7.17.19 subgroup, and the next equal temperament that does this better is [[72edo|72]].

=== Uniform maps ===
{{Uniform map|edo=12}}

=== Commas ===
12edo [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 12 19 28 34 42 44 49 51}}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime&lt;br&gt;limit]]
! [[Ratio]]&lt;ref group="note"&gt;{{rd}}&lt;/ref&gt;
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 3
| &lt;abbr title="531441/524288"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -19 12 }}
| 23.46
| Lalawama / Poma
| [[Pythagorean comma]]
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadguma
| Diminished comma, greater diesis
|-
| 5
| &lt;abbr title="262144/253125"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| 18 -4 -5 }}
| 60.61
| Saquinguma
| [[Passion comma]]
|-
| 5
| [[128/125]]
| {{monzo| 7 0 -3 }}
| 41.06
| Triguma
| Augmented comma, lesser diesis
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Guma
| Syntonic comma, Didymus' comma, meantone comma
|-
| 5
| [[2048/2025]]
| {{monzo| 11 -4 -2 }}
| 19.55
| Saguguma
| Diaschisma
|-
| 5
| &lt;abbr title="67108864/66430125"&gt;(16 digits)&lt;/abbr&gt;
| {{monzo| 26 -12 -3 }}
| 17.60
| Sasa-triguma
| [[Misty comma]]
|-
| 5
| [[32805/32768]]
| {{monzo| -15 8 1 }}
| 1.95
| Layoma
| Schisma
|-
| 5
| &lt;abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625"&gt;(98 digits)&lt;/abbr&gt;
| {{monzo| 161 -84 -12 }}
| 0.02
| Sepbisa-quadtriguma
| [[Kirnberger's atom]]
|-
| 7
| [[256/245]]
| {{monzo| 8 0 -1 -2 }}
| 76.03
| Ruruguma
| Bapbo comma
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laruma
| Harrison's comma
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Ruguma
| Mint comma, septimal quarter tone
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyoma
| Jubilisma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyoma
| Schismean comma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ruma
| Septimal comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyoma
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotriguma
| Starling comma
|-
| 7
| [[4000/3969]]
| {{monzo| 5 -4 3 -2 }}
| 13.47
| Rurutriyoma
| Octagar comma
|-
| 7
| &lt;abbr title="321489/320000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizoguguma
| [[Varunisma]]
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyoma
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquinguma
| Hemimean comma
|-
| 7
| [[5120/5103]]
| {{monzo| 10 -6 1 -1 }}
| 5.76
| Saruyoma
| Hemifamity comma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| {{monzo| 25 -14 0 -1 }}
| 3.80
| Sasaruma
| [[Garischisma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyoma
| [[Metric comma]]
|-
| 7
| &lt;abbr title="250047/250000"&gt;(12 digits)&lt;/abbr&gt;
| {{monzo| -4 6 -6 3 }}
| 0.33
| Trizoguguma
| [[Landscape comma]]
|-
| 11
| [[128/121]]
| {{monzo| 7 0 0 0 -2 }}
| 97.36
| Lulubima
| Axirabian limma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyoma
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzoguma
| Undecimal tritonic comma
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Luluzozoyoma
| Frostma
|-
| 11
| [[99/98]]
| {{monzo| -1 2 0 -2 1 }}
| 17.58
| Loruruma
| Mothwellsma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyoma
| Ptolemisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Loruguguma
| Valinorsma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzoma
| Pentacircle comma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozoguma
| Werckisma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Biloruguma
| Kalisma
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyoma
| Wilsorma
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozoguma
| Superleap comma, biome comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Thuluma
| Grossma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotriguma
| Fairytale comma, sinbadma
|-
| 13
| [[4096/4095]]
| {{monzo| 12 -2 -1 -1 0 -1 }}
| 0.42
| Sathuruguma
| Minisma
|-
| 17
| [[51/50]]
| {{monzo| -1 1 -2 0 0 0 1 }}
| 34.28
| Soguguma
| Large septendecimal sixth tone
|-
| 17
| [[52/51]]
| {{monzo| 2 -1 0 0 0 1 -1 }}
| 33.62
| Suthoma
| Small septendecimal sixth tone
|-
| 17
| [[136/135]]
| {{monzo| 3 -3 -1 0 0 0 1 }}
| 12.78
| Soguma
| Diatisma, fiventeen comma
|-
| 17
| [[256/255]]
| {{monzo| 8 -1 -1 0 0 0 -1 }}
| 6.78
| Suguma
| Charisma, septendecimal kleisma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Sosoma
| Semitonisma
|-
| 17
| [[2601/2600]]
| {{monzo| -3 2 -2 0 0 -1 2 }}
| 0.67
| Sosothuguguma
| Sextantonisma
|-
| 19
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nuthoma
| Undevicesimal two-ninth tone
|-
| 19
| [[96/95]]
| {{monzo| 5 1 -1 0 0 0 0 -1 }}
| 18.13
| Nuguma
| 19th-partial chroma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1}}
| 11.35
| Nusoma
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1 }}
| 10.15
| Nosuguma
| Malcolmisma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Nusuma
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Nonoguma
| Go comma
|-
|19
|[[513/512]]
|{{Monzo|9 3 0 0 0 0 0 -1}}
|3.37
|Lanoma
|Boethius' comma
|}
&lt;references group="note" /&gt;

=== Rank-2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods&lt;br&gt;per 8ve
! Generator*
! Pergen
! Temperaments
|-
| 1
| 1\12
| (P8, P4/5)
| [[Ripple]], [[passion]]
|-
| 1
| 5\12
| (P8, P5)
| [[Meantone]] / [[dominant (temperament)|dominant]]
|-
| 2
| 5\12 (1\12)
| (P8/2, P5)
| [[Pajara]], [[injera]]
|-
| 3
| 5\12 (1\12)
| (P8/3, P5)
| [[Augmented (temperament)|Augmented]] / [[august]]
|-
| 4
| 5\12 (1\12)
| (P8/4, P5)
| [[Diminished (temperament)|Diminished]]
|-
| 6
| 5\12 (1\12)
| (P8/6, P5)
| [[Hexe]]
|}
&lt;nowiki&gt;*&lt;/nowiki&gt; [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Rank-2 temperaments to which 12et can be [[detempering|detempered]] include [[compton]] (12 &amp; 72), [[schismic]]/[[garibaldi]] (41 &amp; 53), and [[diaschismic]] (46 &amp; 58). Rank-3 temperaments to which 12et can be detempered include [[marvel]], [[starling]]/[[thrush]], [[aberschismic]], [[orthoschismic]], and [[cassaschismic]]. For more comprehensive lists, see: 
* [[List of 12et rank two temperaments by badness]]
* [[List of 12et rank two temperaments by complexity]]
* [[List of edo-distinct 12f rank two temperaments]]
* [[Schismic–commatic equivalence continuum]]

== Octave stretch or compression ==
Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch and compression]] for 12edo varies by context. A slight compression such as what is given by [[40ed10]] and [[zpi|34zpi]] shows improved intonation of harmonics [[5/1|5]] and [[7/1|7]] at the cost of worse [[2/1|2]] and [[3/1|3]]; while stretching the octave for a purer 3 and for a better match of the inharmonicity on string instruments, like those in [[7edf]], [[19edt]], or [[31ed6]], also makes sense. 

== Scales ==
{{See also| List of MOS scales in 12edo }}

The two most common 12edo MOS scales are meantone[5] and meantone[7].
* Diatonic: [[5L 2s]] – 2221221 (generator = 7\12)
* Pentatonic: [[2L 3s]] – 22323 (generator = 7\12)

The diminished and augmented scales are also MOS scales.
* Diminished: [[4L 4s]] – 12121212 (generator = 1\12, period = 3\12)
* Augmented: [[3L 3s]] – 131313 (generator = 1\12, period = 4\12)

Other widely used scales include:
* Melodic minor – 2122221
* Harmonic minor – 2122131
* Harmonic major – 2212131
* Hungarian minor – 2131131
* Maqam hijaz / double harmonic major – 1312131

== Well temperaments ==
:''For a list of historical well temperaments, see [[Well temperament]].''

* [[Cauldron]]
* [[Bifrost]]
* [[Grail]]
* [[Secor5 23TX]]
* [[Secor wt10]]
* [[Sabat1]]
* [[Sabat2]]

== Music ==
{{Catrel|12edo tracks}}

The vast majority of music today is in 12edo or 12edo with slight modifications. As such, one can easily find 12edo music by going on any music streaming service.

== See also ==
* [[Lumatone mapping for 12edo]]
* [[:purdal:12-EDD]]{{dead link}}
* [[Near12]] – a just intonation scale where every interval is within 12.5 cents of a 12edo step

== External links ==
* [http://tonalsoft.com/enc/number/12edo.aspx 12-tone equal-temperament] on [[Tonalsoft Encyclopedia]]

[[Category:3-limit record edos|##]] &lt;!-- 2-digit number --&gt;
[[Category:Historical]]
[[Category:Meantone]]</text>
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  <page>
    <title>12edt</title>
    <ns>0</ns>
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    <revision>
      <id>136</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
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        <id>0</id>
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      <text bytes="1766" sha1="gob5ncraaw1ilzk8nrsg2ft3s9n5aqq" xml:space="preserve">=Division of the tritave (3/1) into 12 equal parts=
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.

=A scala formatted description of the tuning=

! C:\Cakewalk\scales\tritave-in-12.scl

!

3/1 in 12

12

!

158.49625

316.99250

475.48875

633.98500

792.48125

950.97750

1109.47375

1267.97000

1426.46625

1584.96250

1743.45875

3/1

=Exactly analogous to meantone=
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

=Compositions=
[http://www.seraph.it/XenoTunes3.html Instant Gamelan] [http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3 play] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      [[Category:edonoi]]
[[Category:edt]]
[[Category:equal]]
[[Category:listen]]</text>
      <sha1>gob5ncraaw1ilzk8nrsg2ft3s9n5aqq</sha1>
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    <revision>
      <id>4629</id>
      <timestamp>2016-10-21T14:26:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 596363030 - Original comment: **</comment>
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      <text bytes="6086" sha1="etj420pjyp6bx5x38vq074stt43ibvm" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-21 14:26:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;596363030&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=Division of the tritave (3/1) into 12 equal parts= 
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.


=A scala formatted description of the tuning= 

! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1

=Exactly analogous to meantone= 
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.



=Compositions= 
[[http://www.seraph.it/XenoTunes3.html|Instant Gamelan]] [[http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3|play]] by [[Carlo Serafini]]
[[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3|Tritave in 12]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:0:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Division of the tritave (3/1) into 12 equal parts"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:0 --&amp;gt;Division of the tritave (3/1) into 12 equal parts&amp;lt;/h1&amp;gt;
 12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for &amp;lt;a class="wiki_link" href="/Kleismic%20family#Hemikleismic"&amp;gt;hemikleismic temperament&amp;lt;/a&amp;gt;. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:2:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="A scala formatted description of the tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:2 --&amp;gt;A scala formatted description of the tuning&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
! C:\Cakewalk\scales\tritave-in-12.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
3/1 in 12&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
158.49625&amp;lt;br /&amp;gt;
316.99250&amp;lt;br /&amp;gt;
475.48875&amp;lt;br /&amp;gt;
633.98500&amp;lt;br /&amp;gt;
792.48125&amp;lt;br /&amp;gt;
950.97750&amp;lt;br /&amp;gt;
1109.47375&amp;lt;br /&amp;gt;
1267.97000&amp;lt;br /&amp;gt;
1426.46625&amp;lt;br /&amp;gt;
1584.96250&amp;lt;br /&amp;gt;
1743.45875&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:4:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Exactly analogous to meantone"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:4 --&amp;gt;Exactly analogous to meantone&amp;lt;/h1&amp;gt;
 In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Another example of a macrodiatonic scale is &amp;lt;a class="wiki_link" href="/17ed5"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:6:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:6 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3.html" rel="nofollow"&amp;gt;Instant Gamelan&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3" rel="nofollow"&amp;gt;Tritave in 12&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
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      <comment>**Imported revision 596362996 - Original comment: **</comment>
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      <text bytes="6662" sha1="akp3f0tv42fitgmohw0zor1i3oah4v2" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-21 14:26:22 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;596362996&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=Division of the tritave (3/1) into 12 equal parts= 
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.


=A scala formatted description of the tuning= 

! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1

[[media type="custom" key="10532830"]]

=Exactly analogous to meantone= 
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.



=Compositions= 
[[http://www.seraph.it/XenoTunes3.html|Instant Gamelan]] [[http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3|play]] by [[Carlo Serafini]]
[[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3|Tritave in 12]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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 12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for &amp;lt;a class="wiki_link" href="/Kleismic%20family#Hemikleismic"&amp;gt;hemikleismic temperament&amp;lt;/a&amp;gt;. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="A scala formatted description of the tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;A scala formatted description of the tuning&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
! C:\Cakewalk\scales\tritave-in-12.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
3/1 in 12&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
158.49625&amp;lt;br /&amp;gt;
316.99250&amp;lt;br /&amp;gt;
475.48875&amp;lt;br /&amp;gt;
633.98500&amp;lt;br /&amp;gt;
792.48125&amp;lt;br /&amp;gt;
950.97750&amp;lt;br /&amp;gt;
1109.47375&amp;lt;br /&amp;gt;
1267.97000&amp;lt;br /&amp;gt;
1426.46625&amp;lt;br /&amp;gt;
1584.96250&amp;lt;br /&amp;gt;
1743.45875&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Exactly analogous to meantone"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Exactly analogous to meantone&amp;lt;/h1&amp;gt;
 In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Another example of a macrodiatonic scale is &amp;lt;a class="wiki_link" href="/17ed5"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3.html" rel="nofollow"&amp;gt;Instant Gamelan&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3" rel="nofollow"&amp;gt;Tritave in 12&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>akp3f0tv42fitgmohw0zor1i3oah4v2</sha1>
    </revision>
    <revision>
      <id>4631</id>
      <timestamp>2016-10-13T23:08:30Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 595308032 - Original comment: **</comment>
      <origin>4631</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6716" sha1="0g2obnwjs6zk29nr3rx86h5e3ntrbss" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-13 23:08:30 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;595308032&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=Division of the tritave (3/1) into 12 equal parts= 
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.


=A scala formatted description of the tuning= 

! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1

[[media type="custom" key="10532830"]]

=Exactly analogous to meantone= 
In octave land, these simple temperaments, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.



=Compositions= 
[[http://www.seraph.it/XenoTunes3.html|Instant Gamelan]] [[http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3|play]] by [[Carlo Serafini]]
[[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3|Tritave in 12]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Division of the tritave (3/1) into 12 equal parts"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Division of the tritave (3/1) into 12 equal parts&amp;lt;/h1&amp;gt;
 12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for &amp;lt;a class="wiki_link" href="/Kleismic%20family#Hemikleismic"&amp;gt;hemikleismic temperament&amp;lt;/a&amp;gt;. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="A scala formatted description of the tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;A scala formatted description of the tuning&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
! C:\Cakewalk\scales\tritave-in-12.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
3/1 in 12&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
158.49625&amp;lt;br /&amp;gt;
316.99250&amp;lt;br /&amp;gt;
475.48875&amp;lt;br /&amp;gt;
633.98500&amp;lt;br /&amp;gt;
792.48125&amp;lt;br /&amp;gt;
950.97750&amp;lt;br /&amp;gt;
1109.47375&amp;lt;br /&amp;gt;
1267.97000&amp;lt;br /&amp;gt;
1426.46625&amp;lt;br /&amp;gt;
1584.96250&amp;lt;br /&amp;gt;
1743.45875&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10532830?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10532830&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Exactly analogous to meantone"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Exactly analogous to meantone&amp;lt;/h1&amp;gt;
 In octave land, these simple temperaments, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Another example of a macrodiatonic scale is &amp;lt;a class="wiki_link" href="/17ed5"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3.html" rel="nofollow"&amp;gt;Instant Gamelan&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3" rel="nofollow"&amp;gt;Tritave in 12&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>0g2obnwjs6zk29nr3rx86h5e3ntrbss</sha1>
    </revision>
    <revision>
      <id>4632</id>
      <timestamp>2016-10-03T16:45:31Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 593933388 - Original comment: **</comment>
      <origin>4632</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6760" sha1="stm5ehc9i9swi3qbn3mhfv8ow6pxxe1" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-03 16:45:31 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;593933388&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=Division of the tritave (3/1) into 12 equal parts= 
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.


=A scala formatted description of the tuning= 

! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1

[[media type="custom" key="10532830"]]

=Exactly analogous to meantone= 
In octave land, these simple temperaments, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and [[tel:130321/130000|130321/130000]] commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.



=Compositions= 
[[http://www.seraph.it/XenoTunes3.html|Instant Gamelan]] [[http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3|play]] by [[Carlo Serafini]]
[[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3|Tritave in 12]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Division of the tritave (3/1) into 12 equal parts"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Division of the tritave (3/1) into 12 equal parts&amp;lt;/h1&amp;gt;
 12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for &amp;lt;a class="wiki_link" href="/Kleismic%20family#Hemikleismic"&amp;gt;hemikleismic temperament&amp;lt;/a&amp;gt;. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="A scala formatted description of the tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;A scala formatted description of the tuning&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
! C:\Cakewalk\scales\tritave-in-12.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
3/1 in 12&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
158.49625&amp;lt;br /&amp;gt;
316.99250&amp;lt;br /&amp;gt;
475.48875&amp;lt;br /&amp;gt;
633.98500&amp;lt;br /&amp;gt;
792.48125&amp;lt;br /&amp;gt;
950.97750&amp;lt;br /&amp;gt;
1109.47375&amp;lt;br /&amp;gt;
1267.97000&amp;lt;br /&amp;gt;
1426.46625&amp;lt;br /&amp;gt;
1584.96250&amp;lt;br /&amp;gt;
1743.45875&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10532830?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10532830&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Exactly analogous to meantone"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Exactly analogous to meantone&amp;lt;/h1&amp;gt;
 In octave land, these simple temperaments, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and [[tel:130321/130000|130321/130000]] commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Another example of a macrodiatonic scale is &amp;lt;a class="wiki_link" href="/17ed5"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3.html" rel="nofollow"&amp;gt;Instant Gamelan&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3" rel="nofollow"&amp;gt;Tritave in 12&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>stm5ehc9i9swi3qbn3mhfv8ow6pxxe1</sha1>
    </revision>
    <revision>
      <id>4633</id>
      <timestamp>2016-10-03T16:40:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 593932792 - Original comment: **</comment>
      <origin>4633</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6714" sha1="t7b4njep0ujagx89ypq8tjehsewttgq" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2016-10-03 16:40:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;593932792&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=Division of the tritave (3/1) into 12 equal parts= 
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.


=A scala formatted description of the tuning= 

! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1

[[media type="custom" key="10532830"]]

=Exactly analogous to meantone= 
In octave land, these simple temperaments, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 3.13.17.19/10 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.



=Compositions= 
[[http://www.seraph.it/XenoTunes3.html|Instant Gamelan]] [[http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3|play]] by [[Carlo Serafini]]
[[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3|Tritave in 12]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Division of the tritave (3/1) into 12 equal parts"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Division of the tritave (3/1) into 12 equal parts&amp;lt;/h1&amp;gt;
 12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for &amp;lt;a class="wiki_link" href="/Kleismic%20family#Hemikleismic"&amp;gt;hemikleismic temperament&amp;lt;/a&amp;gt;. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="A scala formatted description of the tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;A scala formatted description of the tuning&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
! C:\Cakewalk\scales\tritave-in-12.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
3/1 in 12&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
158.49625&amp;lt;br /&amp;gt;
316.99250&amp;lt;br /&amp;gt;
475.48875&amp;lt;br /&amp;gt;
633.98500&amp;lt;br /&amp;gt;
792.48125&amp;lt;br /&amp;gt;
950.97750&amp;lt;br /&amp;gt;
1109.47375&amp;lt;br /&amp;gt;
1267.97000&amp;lt;br /&amp;gt;
1426.46625&amp;lt;br /&amp;gt;
1584.96250&amp;lt;br /&amp;gt;
1743.45875&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10532830?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10532830&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Exactly analogous to meantone"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Exactly analogous to meantone&amp;lt;/h1&amp;gt;
 In octave land, these simple temperaments, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 3.13.17.19/10 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Another example of a macrodiatonic scale is &amp;lt;a class="wiki_link" href="/17ed5"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3.html" rel="nofollow"&amp;gt;Instant Gamelan&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3" rel="nofollow"&amp;gt;Tritave in 12&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>t7b4njep0ujagx89ypq8tjehsewttgq</sha1>
    </revision>
    <revision>
      <id>4634</id>
      <timestamp>2011-12-30T00:47:58Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 288802419 - Original comment: **</comment>
      <origin>4634</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6590" sha1="i5hsjbrliyt98si4kenatv2kh2hwto0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2011-12-30 00:47:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;288802419&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=Division of the tritave (3/1) into 12 equal parts= 
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.


=A scala formatted description of the tuning= 

! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1

[[media type="custom" key="10532830"]]

=Exactly analogous to meantone= 
In octave land, these simple temperaments, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 3.13.17 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=13/1. Tempering out the 85293/83521 comma. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.



=Compositions= 
[[http://www.seraph.it/XenoTunes3.html|Instant Gamelan]] [[http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3|play]] by [[Carlo Serafini]]
[[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3|Tritave in 12]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Division of the tritave (3/1) into 12 equal parts"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Division of the tritave (3/1) into 12 equal parts&amp;lt;/h1&amp;gt;
 12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for &amp;lt;a class="wiki_link" href="/Kleismic%20family#Hemikleismic"&amp;gt;hemikleismic temperament&amp;lt;/a&amp;gt;. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:3:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc1"&amp;gt;&amp;lt;a name="A scala formatted description of the tuning"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:3 --&amp;gt;A scala formatted description of the tuning&amp;lt;/h1&amp;gt;
 &amp;lt;br /&amp;gt;
! C:\Cakewalk\scales\tritave-in-12.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
3/1 in 12&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
158.49625&amp;lt;br /&amp;gt;
316.99250&amp;lt;br /&amp;gt;
475.48875&amp;lt;br /&amp;gt;
633.98500&amp;lt;br /&amp;gt;
792.48125&amp;lt;br /&amp;gt;
950.97750&amp;lt;br /&amp;gt;
1109.47375&amp;lt;br /&amp;gt;
1267.97000&amp;lt;br /&amp;gt;
1426.46625&amp;lt;br /&amp;gt;
1584.96250&amp;lt;br /&amp;gt;
1743.45875&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextMediaRule:0:&amp;amp;lt;img src=&amp;amp;quot;http://www.wikispaces.com/site/embedthumbnail/custom/10532830?h=0&amp;amp;amp;w=0&amp;amp;quot; class=&amp;amp;quot;WikiMedia WikiMediaCustom&amp;amp;quot; id=&amp;amp;quot;wikitext@@media@@type=&amp;amp;amp;quot;custom&amp;amp;amp;quot; key=&amp;amp;amp;quot;10532830&amp;amp;amp;quot;&amp;amp;quot; title=&amp;amp;quot;Custom Media&amp;amp;quot;/&amp;amp;gt; --&amp;gt;&amp;lt;script type="text/javascript" src="http://mediaplayer.yahoo.com/js"&amp;gt;
&amp;lt;/script&amp;gt;&amp;lt;!-- ws:end:WikiTextMediaRule:0 --&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:5:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc2"&amp;gt;&amp;lt;a name="Exactly analogous to meantone"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:5 --&amp;gt;Exactly analogous to meantone&amp;lt;/h1&amp;gt;
 In octave land, these simple temperaments, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 3.13.17 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=13/1. Tempering out the 85293/83521 comma. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 chords.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
Another example of a macrodiatonic scale is &amp;lt;a class="wiki_link" href="/17ed5"&amp;gt;hyperpyth&amp;lt;/a&amp;gt; which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;!-- ws:start:WikiTextHeadingRule:7:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc3"&amp;gt;&amp;lt;a name="Compositions"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:7 --&amp;gt;Compositions&amp;lt;/h1&amp;gt;
 &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3.html" rel="nofollow"&amp;gt;Instant Gamelan&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3" rel="nofollow"&amp;gt;Tritave in 12&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>i5hsjbrliyt98si4kenatv2kh2hwto0</sha1>
    </revision>
    <revision>
      <id>4635</id>
      <timestamp>2011-10-07T15:40:14Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 262694422 - Original comment: **</comment>
      <origin>4635</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4865" sha1="ir0dnornmxfu55l79qznyz9gzio0rfr" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-10-07 15:40:14 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;262694422&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;=Division of the tritave (3/1) into 12 equal parts= 
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.


=A scala formatted description of the tuning=

! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1

[[media type="custom" key="10532830"]]

==Compositions== 
[[http://www.seraph.it/XenoTunes3.html|Instant Gamelan]] [[http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3|play]] by [[Carlo Serafini]]
[[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3|Tritave in 12]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12edt&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;&amp;lt;!-- ws:start:WikiTextHeadingRule:1:&amp;amp;lt;h1&amp;amp;gt; --&amp;gt;&amp;lt;h1 id="toc0"&amp;gt;&amp;lt;a name="Division of the tritave (3/1) into 12 equal parts"&amp;gt;&amp;lt;/a&amp;gt;&amp;lt;!-- ws:end:WikiTextHeadingRule:1 --&amp;gt;Division of the tritave (3/1) into 12 equal parts&amp;lt;/h1&amp;gt;
 12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for &amp;lt;a class="wiki_link" href="/Kleismic%20family#Hemikleismic"&amp;gt;hemikleismic temperament&amp;lt;/a&amp;gt;. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit.&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
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&amp;lt;br /&amp;gt;
! C:\Cakewalk\scales\tritave-in-12.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
3/1 in 12&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
158.49625&amp;lt;br /&amp;gt;
316.99250&amp;lt;br /&amp;gt;
475.48875&amp;lt;br /&amp;gt;
633.98500&amp;lt;br /&amp;gt;
792.48125&amp;lt;br /&amp;gt;
950.97750&amp;lt;br /&amp;gt;
1109.47375&amp;lt;br /&amp;gt;
1267.97000&amp;lt;br /&amp;gt;
1426.46625&amp;lt;br /&amp;gt;
1584.96250&amp;lt;br /&amp;gt;
1743.45875&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
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 &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3.html" rel="nofollow"&amp;gt;Instant Gamelan&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3" rel="nofollow"&amp;gt;Tritave in 12&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2011-09-20T11:09:37Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;vaisvil</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 256100526 - Original comment: **</comment>
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      <text bytes="3948" sha1="evmnenq2k2a1a0g1lbevehjj4xgfym0" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:vaisvil|vaisvil]] and made on &lt;tt&gt;2011-09-20 11:09:37 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;256100526&lt;/tt&gt;.&lt;br&gt;
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A scala formatted description of the tuning:

! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1

[[media type="custom" key="10532830"]]

==Compositions== 
[[http://www.seraph.it/XenoTunes3.html|Instant Gamelan]] [[http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3|play]] by [[Carlo Serafini]]
[[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3|Tritave in 12]] by [[@http://www.chrisvaisvil.com|Chris Vaisvil]]&lt;/pre&gt;&lt;/div&gt;
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 &amp;lt;br /&amp;gt;
A scala formatted description of the tuning:&amp;lt;br /&amp;gt;
&amp;lt;br /&amp;gt;
! C:\Cakewalk\scales\tritave-in-12.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
3/1 in 12&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
158.49625&amp;lt;br /&amp;gt;
316.99250&amp;lt;br /&amp;gt;
475.48875&amp;lt;br /&amp;gt;
633.98500&amp;lt;br /&amp;gt;
792.48125&amp;lt;br /&amp;gt;
950.97750&amp;lt;br /&amp;gt;
1109.47375&amp;lt;br /&amp;gt;
1267.97000&amp;lt;br /&amp;gt;
1426.46625&amp;lt;br /&amp;gt;
1584.96250&amp;lt;br /&amp;gt;
1743.45875&amp;lt;br /&amp;gt;
3/1&amp;lt;br /&amp;gt;
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 &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3.html" rel="nofollow"&amp;gt;Instant Gamelan&amp;lt;/a&amp;gt; &amp;lt;a class="wiki_link_ext" href="http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3" rel="nofollow"&amp;gt;play&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link" href="/Carlo%20Serafini"&amp;gt;Carlo Serafini&amp;lt;/a&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;a class="wiki_link_ext" href="http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3" rel="nofollow"&amp;gt;Tritave in 12&amp;lt;/a&amp;gt; by &amp;lt;a class="wiki_link_ext" href="http://www.chrisvaisvil.com" rel="nofollow" target="_blank"&amp;gt;Chris Vaisvil&amp;lt;/a&amp;gt;&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <comment>**Imported revision 239486287 - Original comment: **</comment>
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      <text bytes="2263" sha1="gvfp7y4y3rhxxigck2pnvjdw80mxix8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
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: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-30 09:07:58 UTC&lt;/tt&gt;.&lt;br&gt;
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==Compositions== 
[[http://www.seraph.it/XenoTunes3.html|Instant Gamelan]] [[http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3|play]] by [[Carlo Serafini]]&lt;/pre&gt;&lt;/div&gt;
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      <comment>**Imported revision 239486187 - Original comment: **</comment>
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      <text bytes="2066" sha1="2gfujyg6zf2mygjfz0p767h1c9tbfgd" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
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: This revision was by author [[User:xenwolf|xenwolf]] and made on &lt;tt&gt;2011-06-30 09:07:12 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;239486187&lt;/tt&gt;.&lt;br&gt;
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&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
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==Compositions== 
[[http://www.seraph.it/XenoTunes3.html|Instant Gamelan]] by [[Carlo Serafini]]&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
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      <comment>**Imported revision 16763978 - Original comment: **</comment>
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: This revision was by author [[User:hstraub|hstraub]] and made on &lt;tt&gt;2008-02-12 06:49:19 UTC&lt;/tt&gt;.&lt;br&gt;
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==Compositions== 
[[http://www.seraph.it/XenoTunes3.html|Instant Gamelan]] by Carlo Serafini &lt;/pre&gt;&lt;/div&gt;
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    <revision>
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      <timestamp>2018-09-21T07:55:36Z</timestamp>
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      <text bytes="1790" sha1="p67oky80zf2uyw38d510yvx9o3rcclt" xml:space="preserve">=Division of the tritave (3/1) into 12 equal parts=
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. [[category:macrotonal]]

=A scala formatted description of the tuning=

! C:\Cakewalk\scales\tritave-in-12.scl

!

3/1 in 12

12

!

158.49625

316.99250

475.48875

633.98500

792.48125

950.97750

1109.47375

1267.97000

1426.46625

1584.96250

1743.45875

3/1

=Exactly analogous to meantone=
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

=Compositions=
[http://www.seraph.it/XenoTunes3.html Instant Gamelan] [http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3 play] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      [[Category:edonoi]]
[[Category:edt]]
[[Category:equal]]
[[Category:listen]]</text>
      <sha1>p67oky80zf2uyw38d510yvx9o3rcclt</sha1>
    </revision>
    <revision>
      <id>39197</id>
      <parentid>33130</parentid>
      <timestamp>2018-12-29T21:34:23Z</timestamp>
      <contributor>
        <username>Spt3125</username>
        <id>16</id>
      </contributor>
      <minor/>
      <comment>&lt;pre&gt; Scala file</comment>
      <origin>39197</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1754" sha1="k9al3587l0tzqow1ymwak0eh8y4x3w6" xml:space="preserve">=Division of the tritave (3/1) into 12 equal parts=
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. [[category:macrotonal]]

=Scala file=

&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

=Exactly analogous to meantone=
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

=Compositions=
[http://www.seraph.it/XenoTunes3.html Instant Gamelan] [http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3 play] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      [[Category:edonoi]]
[[Category:edt]]
[[Category:equal]]
[[Category:listen]]</text>
      <sha1>k9al3587l0tzqow1ymwak0eh8y4x3w6</sha1>
    </revision>
    <revision>
      <id>42520</id>
      <parentid>39197</parentid>
      <timestamp>2019-11-20T23:53:56Z</timestamp>
      <contributor>
        <username>Some boi</username>
        <id>155</id>
      </contributor>
      <minor/>
      <comment>/* Compositions */  Noted a 404 error for one of the links.</comment>
      <origin>42520</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1785" sha1="h4x0dnw584xuzsmvk514ykvob83fr7j" xml:space="preserve">=Division of the tritave (3/1) into 12 equal parts=
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 
[[category:macrotonal]]

=Scala file=

&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

=Exactly analogous to meantone=
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

=Compositions=
[http://www.seraph.it/XenoTunes3.html Instant Gamelan] [http://www.seraph.it/XenoTunes3_files/instant%20gamelan.mp3 play] by [[Carlo_Serafini|Carlo Serafini]] (404 error as of 11/20/2019)

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:equal]]
[[Category:listen]]</text>
      <sha1>h4x0dnw584xuzsmvk514ykvob83fr7j</sha1>
    </revision>
    <revision>
      <id>79908</id>
      <parentid>42520</parentid>
      <timestamp>2021-10-24T14:47:02Z</timestamp>
      <contributor>
        <username>KingHyperio</username>
        <id>213</id>
      </contributor>
      <minor/>
      <origin>79908</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1695" sha1="5l11dlynetllb2w3xm3xxcr02g36pgg" xml:space="preserve">=Division of the tritave (3/1) into 12 equal parts=
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 
[[category:macrotonal]]

=Scala file=

&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

=Exactly analogous to meantone=
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

=Compositions=
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:equal]]
[[Category:listen]]</text>
      <sha1>5l11dlynetllb2w3xm3xxcr02g36pgg</sha1>
    </revision>
    <revision>
      <id>96669</id>
      <parentid>79908</parentid>
      <timestamp>2022-10-05T19:46:02Z</timestamp>
      <contributor>
        <username>Plumtree</username>
        <id>365</id>
      </contributor>
      <minor/>
      <comment>Infobox ET added; h1 headings replaced with h2</comment>
      <origin>96669</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1664" sha1="ms09jpm3lb8vfwkbuzhzbkktln7fo6a" xml:space="preserve">{{Infobox ET}}
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 
[[category:macrotonal]]

==Scala file==

&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

==Exactly analogous to meantone==
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

==Compositions==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:equal]]
[[Category:listen]]</text>
      <sha1>ms09jpm3lb8vfwkbuzhzbkktln7fo6a</sha1>
    </revision>
    <revision>
      <id>118593</id>
      <parentid>96669</parentid>
      <timestamp>2023-07-05T14:51:28Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Equal]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>118593</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1645" sha1="t6jdzsv4u340js949zteutaam46ic4w" xml:space="preserve">{{Infobox ET}}
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 
[[category:macrotonal]]

==Scala file==

&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

==Exactly analogous to meantone==
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

==Compositions==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:listen]]</text>
      <sha1>t6jdzsv4u340js949zteutaam46ic4w</sha1>
    </revision>
    <revision>
      <id>118858</id>
      <parentid>118593</parentid>
      <timestamp>2023-07-06T07:52:26Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <origin>118858</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1686" sha1="43out1liqpwnyrbddcxah784gnyonqr" xml:space="preserve">{{Infobox ET}}
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Interval table ==
{{Interval table}}

==Scala file==

&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

==Exactly analogous to meantone==
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

==Compositions==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:listen]]
[[category:macrotonal]]</text>
      <sha1>43out1liqpwnyrbddcxah784gnyonqr</sha1>
    </revision>
    <revision>
      <id>132036</id>
      <parentid>118858</parentid>
      <timestamp>2024-01-17T09:30:45Z</timestamp>
      <contributor>
        <username>Akselai</username>
        <id>437</id>
      </contributor>
      <comment>add harmonics in equal</comment>
      <origin>132036</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1739" sha1="fvgbw8iia2rx3ikhx04py1hx3mqnsgw" xml:space="preserve">{{Infobox ET}}
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Interval table ==
{{Interval table}}

== Prime harmonics ==
{{Harmonics in equal|12|3|1}}

==Scala file==

&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

==Exactly analogous to meantone==
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

==Compositions==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:listen]]
[[category:macrotonal]]</text>
      <sha1>fvgbw8iia2rx3ikhx04py1hx3mqnsgw</sha1>
    </revision>
    <revision>
      <id>132038</id>
      <parentid>132036</parentid>
      <timestamp>2024-01-17T09:39:07Z</timestamp>
      <contributor>
        <username>Akselai</username>
        <id>437</id>
      </contributor>
      <comment>/* Prime harmonics */</comment>
      <origin>132038</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1755" sha1="qkjxeb6r8h5xxxno5zdj2gkp2078vij" xml:space="preserve">{{Infobox ET}}
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Interval table ==
{{Interval table}}

== Prime harmonics ==
{{Harmonics in equal|12|3|1|intervals=prime}}

==Scala file==

&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

==Exactly analogous to meantone==
In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament. Now, exactly analogous to meantone, in which (3/2)^4=5/1, here (17/9)^4=(19/10)^4=13/1, tempering out the 171/170, 85293/83521 and 130321/130000 commas. In fact, even the MOS pattern is the same for this higher limit meantone! Relish the sweet 9:13:17 and 20:27:38 chords.

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which is found in the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

==Compositions==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:listen]]
[[category:macrotonal]]</text>
      <sha1>qkjxeb6r8h5xxxno5zdj2gkp2078vij</sha1>
    </revision>
    <revision>
      <id>133531</id>
      <parentid>132038</parentid>
      <timestamp>2024-01-30T02:58:42Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <comment>/* Macrodiatonic and macromeantone */Rework this section (I'm keeping the original information I removed at the bottom in case it's useful)</comment>
      <origin>133531</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2341" sha1="t6bmgt8w2f9u6e3vbha4n61xt07p0aa" xml:space="preserve">{{Infobox ET}}
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Interval table ==
{{Interval table}}

== Prime harmonics ==
{{Harmonics in equal|12|3|1|intervals=prime}}

==Scala file==

&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

==Macrodiatonic and macromeantone==
12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] scale ([[5L 2s (3/1-equivalent)|5L 2s&lt;3/1&gt;]]). This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable-for example, the "perfect fifth" is inflated to the size of a major seventh. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 subgroup, equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/]] octave-reduced (although this is not a completely exact stretching of mentone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

In octave land, 12edo handles the 2.3.5 subgroup and 11edo handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament.

==Compositions==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:listen]]
[[category:macrotonal]]</text>
      <sha1>t6bmgt8w2f9u6e3vbha4n61xt07p0aa</sha1>
    </revision>
    <revision>
      <id>133698</id>
      <parentid>133531</parentid>
      <timestamp>2024-01-30T08:50:35Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>/* Macrodiatonic and macromeantone */  Added link to 11edo</comment>
      <origin>133698</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2345" sha1="pd1vwg78g2wdo1jcg435o7ghs8dzt1u" xml:space="preserve">{{Infobox ET}}
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Interval table ==
{{Interval table}}

== Prime harmonics ==
{{Harmonics in equal|12|3|1|intervals=prime}}

==Scala file==

&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

==Macrodiatonic and macromeantone==
12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] scale ([[5L 2s (3/1-equivalent)|5L 2s&lt;3/1&gt;]]). This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable-for example, the "perfect fifth" is inflated to the size of a major seventh. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 subgroup, equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/]] octave-reduced (although this is not a completely exact stretching of mentone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament.

==Compositions==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:listen]]
[[category:macrotonal]]</text>
      <sha1>pd1vwg78g2wdo1jcg435o7ghs8dzt1u</sha1>
    </revision>
    <revision>
      <id>137791</id>
      <parentid>133698</parentid>
      <timestamp>2024-03-01T17:50:39Z</timestamp>
      <contributor>
        <username>CompactStar</username>
        <id>388</id>
      </contributor>
      <comment>Moved this paragraph which I think makes llittle sense anyways</comment>
      <origin>137791</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2358" sha1="0jyjbraf8q0tmi3rg6bjs80mhmxeou4" xml:space="preserve">{{Infobox ET}}
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Interval table ==
{{Interval table}}

== Prime harmonics ==
{{Harmonics in equal|12|3|1|intervals=prime}}

==Scala file==

&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

==Theory==
In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament.

===Macrodiatonic and macromeantone===
12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] scale ([[5L 2s (3/1-equivalent)|5L 2s&lt;3/1&gt;]]). This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable-for example, the "perfect fifth" is inflated to the size of a major seventh. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 subgroup, equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/]] octave-reduced (although this is not a completely exact stretching of mentone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

==Compositions==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:listen]]
[[category:macrotonal]]</text>
      <sha1>0jyjbraf8q0tmi3rg6bjs80mhmxeou4</sha1>
    </revision>
    <revision>
      <id>150701</id>
      <parentid>137791</parentid>
      <timestamp>2024-08-15T03:34:38Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>/* Theory */</comment>
      <origin>150701</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2360" sha1="h2159nw5l94pplwpe224dy3vogcjeod" xml:space="preserve">{{Infobox ET}}
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Interval table ==
{{Interval table}}

== Prime harmonics ==
{{Harmonics in equal|12|3|1|intervals=prime}}

==Scala file==

&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

==Theory==
In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup - ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen-Pierce) and 12edt handles the 2.3.5.13.17.19 -- AND! it is a multiple of 4edt which is the simplest BP equal temperament.

===Macrodiatonic and macromeantone===
12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] scale ([[5L 2s (3/1-equivalent)|5L 2s&lt;3/1&gt;]]). This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable-for example, the "perfect fifth" is inflated to the size of a major seventh. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 subgroup, equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/4]] octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

==Compositions==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:listen]]
[[category:macrotonal]]</text>
      <sha1>h2159nw5l94pplwpe224dy3vogcjeod</sha1>
    </revision>
    <revision>
      <id>183063</id>
      <parentid>150701</parentid>
      <timestamp>2025-02-25T18:26:17Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183063</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2370" sha1="3qskdj9sjlew7dp1ng5j6kgy6yvybp7" xml:space="preserve">{{Infobox ET}}
12edt divides 3, the tritave, into 12 equal parts of 158.496 cents each, corresponding to 7.571 edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Interval table ==
{{Interval table}}

== Prime harmonics ==
{{Harmonics in equal|12|3|1|intervals=prime}}

== Scala file ==
&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

== Theory ==
In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament.

=== Macrodiatonic and macromeantone ===
12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] scale ([[5L 2s (3/1-equivalent)|5L 2s&lt;3/1&gt;]]). This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable—for example, the "perfect fifth" is inflated to the size of a major seventh. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 subgroup, equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/4]] octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

== Compositions ==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:listen]]
[[category:macrotonal]]</text>
      <sha1>3qskdj9sjlew7dp1ng5j6kgy6yvybp7</sha1>
    </revision>
    <revision>
      <id>184582</id>
      <parentid>183063</parentid>
      <timestamp>2025-03-04T20:48:22Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>184582</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2320" sha1="sqcjlppclyaixahae9d4862hsb0lgzm" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

12edt corresponds to 7.571&amp;nbsp;edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Prime harmonics ==
{{Harmonics in equal|12|3|1|intervals=prime}}

== Theory ==
In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament.

=== Macrodiatonic and macromeantone ===
12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] {{sl|5L 2s}} scale. This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable, since, for example, the [[generator]] is now the size of a major seventh instead of a perfect fifth. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 [[subgroup]], equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/4]] octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

== Interval table ==
{{Interval table}}

== Scala file ==
&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

== Compositions ==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:listen]]
[[category:macrotonal]]</text>
      <sha1>sqcjlppclyaixahae9d4862hsb0lgzm</sha1>
    </revision>
    <revision>
      <id>185047</id>
      <parentid>184582</parentid>
      <timestamp>2025-03-08T00:26:02Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>Replace {{[[template:scale link|scale link]]}} with {{[[template:mos scalesig|mos scalesig]]}}</comment>
      <origin>185047</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2335" sha1="82t541fxul6iudk1ifxm4wism797c9q" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

12edt corresponds to 7.571&amp;nbsp;edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Prime harmonics ==
{{Harmonics in equal|12|3|1|intervals=prime}}

== Theory ==
In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament.

=== Macrodiatonic and macromeantone ===
12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] {{mos scalesig|5L 2s&lt;3/1&gt;}} scale. This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable, since, for example, the [[generator]] is now the size of a major seventh instead of a perfect fifth. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 [[subgroup]], equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/4]] octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

== Interval table ==
{{Interval table}}

== Scala file ==
&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

== Compositions ==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:edt]]
[[Category:listen]]
[[category:macrotonal]]</text>
      <sha1>82t541fxul6iudk1ifxm4wism797c9q</sha1>
    </revision>
    <revision>
      <id>205579</id>
      <parentid>185047</parentid>
      <timestamp>2025-07-31T15:31:26Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Edt]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>205579</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2318" sha1="45mx0dn71xnwwruw5o2qg8xep7jt7de" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

12edt corresponds to 7.571&amp;nbsp;edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Prime harmonics ==
{{Harmonics in equal|12|3|1|intervals=prime}}

== Theory ==
In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament.

=== Macrodiatonic and macromeantone ===
12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] {{mos scalesig|5L 2s&lt;3/1&gt;}} scale. This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable, since, for example, the [[generator]] is now the size of a major seventh instead of a perfect fifth. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 [[subgroup]], equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/4]] octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

== Interval table ==
{{Interval table}}

== Scala file ==
&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

== Compositions ==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:edonoi]]
[[Category:listen]]
[[category:macrotonal]]</text>
      <sha1>45mx0dn71xnwwruw5o2qg8xep7jt7de</sha1>
    </revision>
    <revision>
      <id>205940</id>
      <parentid>205579</parentid>
      <timestamp>2025-08-01T19:20:37Z</timestamp>
      <contributor>
        <username>Fredg999 category edits</username>
        <id>342</id>
      </contributor>
      <minor/>
      <comment>Removing from [[Category:Edonoi]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>205940</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2298" sha1="ognhtts4f78gv5sndxnkply8gljlfwn" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

12edt corresponds to 7.571&amp;nbsp;edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Prime harmonics ==
{{Harmonics in equal|12|3|1|intervals=prime}}

== Theory ==
In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament.

=== Macrodiatonic and macromeantone ===
12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic and microdiatonic|macrodiatonic]] {{mos scalesig|5L 2s&lt;3/1&gt;}} scale. This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable, since, for example, the [[generator]] is now the size of a major seventh instead of a perfect fifth. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 [[subgroup]], equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/4]] octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

== Interval table ==
{{Interval table}}

== Scala file ==
&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

== Compositions ==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:listen]]
[[category:macrotonal]]</text>
      <sha1>ognhtts4f78gv5sndxnkply8gljlfwn</sha1>
    </revision>
    <revision>
      <id>211508</id>
      <parentid>205940</parentid>
      <timestamp>2025-09-29T13:09:21Z</timestamp>
      <contributor>
        <username>Dummy index</username>
        <id>318</id>
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      <comment>/* Theory */ fix link</comment>
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      <format>text/x-wiki</format>
      <text bytes="2266" sha1="sd7i4ztrw76nko1j28hwv1d9rp3yl4k" xml:space="preserve">{{Infobox ET}}
{{ED intro}}

12edt corresponds to 7.571&amp;nbsp;edo, and can be used as a generator chain for [[Kleismic_family#Hemikleismic|hemikleismic temperament]]. From a no-twos point of view, it tempers out 49/45 and 27/25 in the 7-limit, and 1331/1125 and 1331/1225 in the 11-limit. 

== Prime harmonics ==
{{Harmonics in equal|12|3|1|intervals=prime}}

== Theory ==
In octave land, 12edo handles the 2.3.5 subgroup and [[11edo]] handles the 2.7.11 subgroup—ie. meantone and orgone temperaments. In tritave land however, 13edt handles the 3.5.7 territory (Bohlen–Pierce) and 12edt handles the 2.3.5.13.17.19—and, it is a multiple of 4edt which is the simplest BP equal temperament.

=== Macrodiatonic and macromeantone ===
12edt can be viewed as a version of [[12edo]] with octaves stretched out to [[3/1|tritaves]], so it contains an extremely stretched diatonic scale or [[macrodiatonic]] {{mos scalesig|5L 2s&lt;3/1&gt;}} scale. This scale has an identical structure to diatonic, but with everything stretched out to be unrecognizable, since, for example, the [[generator]] is now the size of a major seventh instead of a perfect fifth. The stretched perfect fifth can be approximated by [[17/9]] and the stretched major third by [[13/9]]. This gives rise to a "macromeantone" temperament which operates in the 3.13.17 [[subgroup]], equating 4 [[17/9]] to [[13/9]] tritave-reduced, rather than 4 [[3/2]] to [[5/4]] octave-reduced (although this is not a completely exact stretching of meantone, unlike some macromeantones like [[meansquared]] which repeats at [[4/1]]).

Another example of a macrodiatonic scale is [[17ed5|hyperpyth]] which repeats at the fifth harmonic and is based on the 5:9:13:(17):(21) chord.

== Interval table ==
{{Interval table}}

== Scala file ==
&lt;pre&gt;
! C:\Cakewalk\scales\tritave-in-12.scl
!
3/1 in 12
12
!
158.49625
316.99250
475.48875
633.98500
792.48125
950.97750
1109.47375
1267.97000
1426.46625
1584.96250
1743.45875
3/1
&lt;/pre&gt;

== Compositions ==
[https://archive.org/details/InstantGamelan Instant Gamelan] by [[Carlo_Serafini|Carlo Serafini]]

[http://micro.soonlabel.com/tritave_in_12/tritavein12_cleaned.mp3 Tritave in 12] by [http://www.chrisvaisvil.com Chris Vaisvil]      
[[Category:listen]]
[[category:macrotonal]]</text>
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  </page>
  <page>
    <title>12highschool1-rodan</title>
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    <revision>
      <id>137</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
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      <text bytes="284" sha1="h7o5y30i77yx4sfm9wvkhhwcfyeqxdy" xml:space="preserve">! 12highschool1-rodan.scl

!

12highschool1 tempered in 13-limit POTE-tuned rodan

! generators: -18, -17, -15, -14, -3, -1, 0, 3, 6, 14, 17, 20

12

!

82.76793

206.89283

317.25007

386.19634

496.55359

579.32152

703.44641

813.80366

882.74993

965.51786

1089.64275

1200.00000</text>
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    <revision>
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      <timestamp>2012-11-20T12:12:41Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 384412666 - Original comment: **</comment>
      <origin>4644</origin>
      <model>wikitext</model>
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      <text bytes="2007" sha1="q46micpy58b693wqfnktcrhzmrbqmsv" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-11-20 12:12:41 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;384412666&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12highschool1-rodan.scl
!
12highschool1 tempered in 13-limit POTE-tuned rodan
! generators: -18, -17, -15, -14, -3, -1, 0, 3, 6, 14, 17, 20
 12
!
 82.76793
 206.89283
 317.25007
 386.19634
 496.55359
 579.32152
 703.44641
 813.80366
 882.74993
 965.51786
 1089.64275
 1200.00000
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12highschool1-rodan&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12highschool1-rodan.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
12highschool1 tempered in 13-limit POTE-tuned rodan&amp;lt;br /&amp;gt;
! generators: -18, -17, -15, -14, -3, -1, 0, 3, 6, 14, 17, 20&amp;lt;br /&amp;gt;
 12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
 82.76793&amp;lt;br /&amp;gt;
 206.89283&amp;lt;br /&amp;gt;
 317.25007&amp;lt;br /&amp;gt;
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 703.44641&amp;lt;br /&amp;gt;
 813.80366&amp;lt;br /&amp;gt;
 882.74993&amp;lt;br /&amp;gt;
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 1200.00000&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <timestamp>2022-08-05T02:06:41Z</timestamp>
      <contributor>
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        <id>308</id>
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      <minor/>
      <comment>Cleanup, misc. edits, categories</comment>
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      <text bytes="453" sha1="kxj7ys2wgzrbh5myclrh8b43910debt" xml:space="preserve">&lt;pre&gt;
! 12highschool1-rodan.scl
!
12highschool1 tempered in 13-limit POTE-tuned rodan
! generators: -18, -17, -15, -14, -3, -1, 0, 3, 6, 14, 17, 20
12
!
82.76793
206.89283
317.25007
386.19634
496.55359
579.32152
703.44641
813.80366
882.74993
965.51786
1089.64275
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool1]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Highschool]]
[[Category:Rodan]]
[[Category:Pages with Scala files]]</text>
      <sha1>kxj7ys2wgzrbh5myclrh8b43910debt</sha1>
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    <revision>
      <id>186537</id>
      <parentid>93933</parentid>
      <timestamp>2025-03-17T04:02:29Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
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      <text bytes="714" sha1="lnh0a1tfszbvu4vu1jv8a2qrld29i5q" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].

This scale is tempered to [[rodan]] temperament.&lt;pre&gt;
! 12highschool1-rodan.scl
!
12highschool1 tempered in 13-limit POTE-tuned rodan
! generators: -18, -17, -15, -14, -3, -1, 0, 3, 6, 14, 17, 20
12
!
82.76793
206.89283
317.25007
386.19634
496.55359
579.32152
703.44641
813.80366
882.74993
965.51786
1089.64275
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool1]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Highschool]]
[[Category:Rodan]]
[[Category:Pages with Scala files]]</text>
      <sha1>lnh0a1tfszbvu4vu1jv8a2qrld29i5q</sha1>
    </revision>
    <revision>
      <id>200473</id>
      <parentid>186537</parentid>
      <timestamp>2025-06-23T23:32:01Z</timestamp>
      <contributor>
        <username>Hkm</username>
        <id>551</id>
      </contributor>
      <origin>200473</origin>
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      <format>text/x-wiki</format>
      <text bytes="763" sha1="3zny3fup3oglb10eq5wkyqunod8jz6l" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].

This scale is tempered to [[rodan]] temperament.&lt;pre&gt;
! 12highschool1-rodan.scl
!
12highschool1 tempered in 13-limit POTE-tuned rodan
! generators: -18, -17, -15, -14, -3, -1, 0, 3, 6, 14, 17, 20
12
!
82.76793
206.89283
317.25007
386.19634
496.55359
579.32152
703.44641
813.80366
882.74993
965.51786
1089.64275
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool1]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Highschool]]
[[Category:Rodan]]
[[Category:Pages with Scala files]]
[[Category:Pages with mostly numerical content]]</text>
      <sha1>3zny3fup3oglb10eq5wkyqunod8jz6l</sha1>
    </revision>
    <revision>
      <id>200551</id>
      <parentid>200473</parentid>
      <timestamp>2025-06-24T00:22:41Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Pages with mostly numerical content]]" to ""</comment>
      <origin>200551</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="714" sha1="lnh0a1tfszbvu4vu1jv8a2qrld29i5q" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].

This scale is tempered to [[rodan]] temperament.&lt;pre&gt;
! 12highschool1-rodan.scl
!
12highschool1 tempered in 13-limit POTE-tuned rodan
! generators: -18, -17, -15, -14, -3, -1, 0, 3, 6, 14, 17, 20
12
!
82.76793
206.89283
317.25007
386.19634
496.55359
579.32152
703.44641
813.80366
882.74993
965.51786
1089.64275
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool1]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Highschool]]
[[Category:Rodan]]
[[Category:Pages with Scala files]]</text>
      <sha1>lnh0a1tfszbvu4vu1jv8a2qrld29i5q</sha1>
    </revision>
    <revision>
      <id>200966</id>
      <parentid>200551</parentid>
      <timestamp>2025-06-24T00:33:57Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered scales]]" to "[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>200966</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="763" sha1="b01xgalgnv5vuuujs8y4t537sazegea" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].

This scale is tempered to [[rodan]] temperament.&lt;pre&gt;
! 12highschool1-rodan.scl
!
12highschool1 tempered in 13-limit POTE-tuned rodan
! generators: -18, -17, -15, -14, -3, -1, 0, 3, 6, 14, 17, 20
12
!
82.76793
206.89283
317.25007
386.19634
496.55359
579.32152
703.44641
813.80366
882.74993
965.51786
1089.64275
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool1]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Highschool]]
[[Category:Rodan]]
[[Category:Pages with Scala files]]</text>
      <sha1>b01xgalgnv5vuuujs8y4t537sazegea</sha1>
    </revision>
    <revision>
      <id>201495</id>
      <parentid>200966</parentid>
      <timestamp>2025-06-24T02:17:51Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered scales]]↵[[Category:Pages with mostly numerical content]]" to "[[Category:Tempered_scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>201495</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="763" sha1="spw9v5j98fqaj0e114870qs1u5bv6nj" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].

This scale is tempered to [[rodan]] temperament.&lt;pre&gt;
! 12highschool1-rodan.scl
!
12highschool1 tempered in 13-limit POTE-tuned rodan
! generators: -18, -17, -15, -14, -3, -1, 0, 3, 6, 14, 17, 20
12
!
82.76793
206.89283
317.25007
386.19634
496.55359
579.32152
703.44641
813.80366
882.74993
965.51786
1089.64275
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool1]]

[[Category:12-tone scales]]
[[Category:Tempered_scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Highschool]]
[[Category:Rodan]]
[[Category:Pages with Scala files]]</text>
      <sha1>spw9v5j98fqaj0e114870qs1u5bv6nj</sha1>
    </revision>
    <revision>
      <id>202821</id>
      <parentid>201495</parentid>
      <timestamp>2025-06-24T02:54:37Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered_scales]]↵[[Category:Pages with mostly numerical content]]" to "[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>202821</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="763" sha1="b01xgalgnv5vuuujs8y4t537sazegea" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].

This scale is tempered to [[rodan]] temperament.&lt;pre&gt;
! 12highschool1-rodan.scl
!
12highschool1 tempered in 13-limit POTE-tuned rodan
! generators: -18, -17, -15, -14, -3, -1, 0, 3, 6, 14, 17, 20
12
!
82.76793
206.89283
317.25007
386.19634
496.55359
579.32152
703.44641
813.80366
882.74993
965.51786
1089.64275
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool1]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Highschool]]
[[Category:Rodan]]
[[Category:Pages with Scala files]]</text>
      <sha1>b01xgalgnv5vuuujs8y4t537sazegea</sha1>
    </revision>
  </page>
  <page>
    <title>12highschool1</title>
    <ns>0</ns>
    <id>138</id>
    <revision>
      <id>138</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
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      <format>text/x-wiki</format>
      <text bytes="356" sha1="0bpsgdc8goic8ni9zgeapeqsehlqi7w" xml:space="preserve">! 12highschool1.scl

First 12-note Highschool scale

12

! dominant: sLsLssLsLsLs; august: sLLLLsLLLsLL; diminished: ssLsLssLsssL

! pajara: LLLsLLLLsLLL; meantone: sLLsLsLLssLL; augene: sLssssLsssLs

!

21/20 

9/8 

6/5 

5/4 

4/3 

7/5 

3/2 

8/5 

5/3 

7/4 

15/8 

2
[[Category:12-tone]]
[[Category:highschool]]
[[Category:just]]
[[Category:scale]]</text>
      <sha1>0bpsgdc8goic8ni9zgeapeqsehlqi7w</sha1>
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    <revision>
      <id>4640</id>
      <timestamp>2012-11-25T17:33:58Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 385833612 - Original comment: **</comment>
      <origin>4640</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1954" sha1="p3p1rhyd3398frjbwyqkvqbvsj0phfa" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-11-25 17:33:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;385833612&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12highschool1.scl
First 12-note Highschool scale
12
! dominant: sLsLssLsLsLs; august: sLLLLsLLLsLL; diminished: ssLsLssLsssL
! pajara: LLLsLLLLsLLL; meantone: sLLsLsLLssLL; augene: sLssssLsssLs
!
21/20 
9/8 
6/5 
5/4 
4/3 
7/5 
3/2 
8/5 
5/3 
7/4 
15/8 
2&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12highschool1&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12highschool1.scl&amp;lt;br /&amp;gt;
First 12-note Highschool scale&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
! dominant: sLsLssLsLsLs; august: sLLLLsLLLsLL; diminished: ssLsLssLsssL&amp;lt;br /&amp;gt;
! pajara: LLLsLLLLsLLL; meantone: sLLsLsLLssLL; augene: sLssssLsssLs&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
21/20 &amp;lt;br /&amp;gt;
9/8 &amp;lt;br /&amp;gt;
6/5 &amp;lt;br /&amp;gt;
5/4 &amp;lt;br /&amp;gt;
4/3 &amp;lt;br /&amp;gt;
7/5 &amp;lt;br /&amp;gt;
3/2 &amp;lt;br /&amp;gt;
8/5 &amp;lt;br /&amp;gt;
5/3 &amp;lt;br /&amp;gt;
7/4 &amp;lt;br /&amp;gt;
15/8 &amp;lt;br /&amp;gt;
2&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>p3p1rhyd3398frjbwyqkvqbvsj0phfa</sha1>
    </revision>
    <revision>
      <id>4641</id>
      <timestamp>2012-05-23T20:59:04Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 338962142 - Original comment: Reverted to Feb 13, 2011 5:21 pm: It's best to stick to standard Scala format**</comment>
      <origin>4641</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1723" sha1="8mhtnosnhq3mva13v6sj5fvck21j36i" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-05-23 20:59:04 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;338962142&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;Reverted to Feb 13, 2011 5:21 pm: It's best to stick to standard Scala format&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12highschool1.scl
First 12-note Highschool scale
12
!
21/20 
9/8 
6/5 
5/4 
4/3 
7/5 
3/2 
8/5 
5/3 
7/4 
15/8 
2&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12highschool1&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12highschool1.scl&amp;lt;br /&amp;gt;
First 12-note Highschool scale&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
21/20 &amp;lt;br /&amp;gt;
9/8 &amp;lt;br /&amp;gt;
6/5 &amp;lt;br /&amp;gt;
5/4 &amp;lt;br /&amp;gt;
4/3 &amp;lt;br /&amp;gt;
7/5 &amp;lt;br /&amp;gt;
3/2 &amp;lt;br /&amp;gt;
8/5 &amp;lt;br /&amp;gt;
5/3 &amp;lt;br /&amp;gt;
7/4 &amp;lt;br /&amp;gt;
15/8 &amp;lt;br /&amp;gt;
2&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>8mhtnosnhq3mva13v6sj5fvck21j36i</sha1>
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    <revision>
      <id>4642</id>
      <timestamp>2012-05-23T00:45:08Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;guest</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 338616534 - Original comment: **</comment>
      <origin>4642</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1600" sha1="adlweina4pqydwre61bist5kzkib92c" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:guest|guest]] and made on &lt;tt&gt;2012-05-23 00:45:08 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;338616534&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12highschool1.scl
First 12-note Highschool scale
12
!
1
9/8
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12highschool1&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12highschool1.scl&amp;lt;br /&amp;gt;
First 12-note Highschool scale&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
1&amp;lt;br /&amp;gt;
9/8&amp;lt;br /&amp;gt;
6/5&amp;lt;br /&amp;gt;
5/4&amp;lt;br /&amp;gt;
4/3&amp;lt;br /&amp;gt;
7/5&amp;lt;br /&amp;gt;
3/2&amp;lt;br /&amp;gt;
8/5&amp;lt;br /&amp;gt;
5/3&amp;lt;br /&amp;gt;
7/4&amp;lt;br /&amp;gt;
15/8&amp;lt;br /&amp;gt;
2&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>adlweina4pqydwre61bist5kzkib92c</sha1>
    </revision>
    <revision>
      <id>4643</id>
      <timestamp>2011-02-13T20:21:11Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 201412694 - Original comment: **</comment>
      <origin>4643</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1646" sha1="aqgtr66q6bv8hsxbo0dzvi4tx62kam3" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-02-13 20:21:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;201412694&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12highschool1.scl
First 12-note Highschool scale
12
!
21/20 
9/8 
6/5 
5/4 
4/3 
7/5 
3/2 
8/5 
5/3 
7/4 
15/8 
2&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12highschool1&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12highschool1.scl&amp;lt;br /&amp;gt;
First 12-note Highschool scale&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
21/20 &amp;lt;br /&amp;gt;
9/8 &amp;lt;br /&amp;gt;
6/5 &amp;lt;br /&amp;gt;
5/4 &amp;lt;br /&amp;gt;
4/3 &amp;lt;br /&amp;gt;
7/5 &amp;lt;br /&amp;gt;
3/2 &amp;lt;br /&amp;gt;
8/5 &amp;lt;br /&amp;gt;
5/3 &amp;lt;br /&amp;gt;
7/4 &amp;lt;br /&amp;gt;
15/8 &amp;lt;br /&amp;gt;
2&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>aqgtr66q6bv8hsxbo0dzvi4tx62kam3</sha1>
    </revision>
    <revision>
      <id>54685</id>
      <parentid>138</parentid>
      <timestamp>2020-12-04T22:25:11Z</timestamp>
      <contributor>
        <username>Keenan Pepper</username>
        <id>25</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:12-tone]] to [[Category:12-tone scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54685</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="363" sha1="90xkfevex7di7piwnvhn60hjvz253gi" xml:space="preserve">! 12highschool1.scl

First 12-note Highschool scale

12

! dominant: sLsLssLsLsLs; august: sLLLLsLLLsLL; diminished: ssLsLssLsssL

! pajara: LLLsLLLLsLLL; meantone: sLLsLsLLssLL; augene: sLssssLsssLs

!

21/20 

9/8 

6/5 

5/4 

4/3 

7/5 

3/2 

8/5 

5/3 

7/4 

15/8 

2
[[Category:12-tone scales]]
[[Category:highschool]]
[[Category:just]]
[[Category:scale]]</text>
      <sha1>90xkfevex7di7piwnvhn60hjvz253gi</sha1>
    </revision>
    <revision>
      <id>55510</id>
      <parentid>54685</parentid>
      <timestamp>2020-12-05T13:03:56Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>recat</comment>
      <origin>55510</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="363" sha1="1c5agcfbe9gut1cljd0jhtcmb1xvsxa" xml:space="preserve">! 12highschool1.scl

First 12-note Highschool scale

12

! dominant: sLsLssLsLsLs; august: sLLLLsLLLsLL; diminished: ssLsLssLsssL

! pajara: LLLsLLLLsLLL; meantone: sLLsLsLLssLL; augene: sLssssLsssLs

!

21/20 

9/8 

6/5 

5/4 

4/3 

7/5 

3/2 

8/5 

5/3 

7/4 

15/8 

2

[[Category:12-tone scales]]
[[Category:highschool]]
[[Category:Just intonation scales]]</text>
      <sha1>1c5agcfbe9gut1cljd0jhtcmb1xvsxa</sha1>
    </revision>
    <revision>
      <id>61990</id>
      <parentid>55510</parentid>
      <timestamp>2021-02-20T11:40:34Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>formatting, recat</comment>
      <origin>61990</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="395" sha1="ongf7auk001qxiy3rmczg9i87hpeo14" xml:space="preserve">&lt;pre&gt;
! 12highschool1.scl
First 12-note Highschool scale
12
! dominant: sLsLssLsLsLs; august: sLLLLsLLLsLL; diminished: ssLsLssLsssL
! pajara: LLLsLLLLsLLL; meantone: sLLsLsLLssLL; augene: sLssssLsssLs
!
21/20 
9/8 
6/5 
5/4 
4/3 
7/5 
3/2 
8/5 
5/3 
7/4 
15/8 
2
&lt;/pre&gt;

[[Category:12-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with Scala files]]</text>
      <sha1>ongf7auk001qxiy3rmczg9i87hpeo14</sha1>
    </revision>
    <revision>
      <id>186543</id>
      <parentid>61990</parentid>
      <timestamp>2025-03-17T04:04:56Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>186543</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="606" sha1="b64yaykvc4reg60cjfbbi0ux3q2gt5r" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].&lt;pre&gt;
! 12highschool1.scl
First 12-note Highschool scale
12
! dominant: sLsLssLsLsLs; august: sLLLLsLLLsLL; diminished: ssLsLssLsssL
! pajara: LLLsLLLLsLLL; meantone: sLLsLsLLssLL; augene: sLssssLsssLs
!
21/20 
9/8 
6/5 
5/4 
4/3 
7/5 
3/2 
8/5 
5/3 
7/4 
15/8 
2
&lt;/pre&gt;

[[Category:12-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with Scala files]]</text>
      <sha1>b64yaykvc4reg60cjfbbi0ux3q2gt5r</sha1>
    </revision>
    <revision>
      <id>200943</id>
      <parentid>186543</parentid>
      <timestamp>2025-06-24T00:33:07Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Just intonation scales]]" to "[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>200943</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="655" sha1="psfavp3okphn02j5w6y4cms4ibv2j1u" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].&lt;pre&gt;
! 12highschool1.scl
First 12-note Highschool scale
12
! dominant: sLsLssLsLsLs; august: sLLLLsLLLsLL; diminished: ssLsLssLsssL
! pajara: LLLsLLLLsLLL; meantone: sLLsLsLLssLL; augene: sLssssLsssLs
!
21/20 
9/8 
6/5 
5/4 
4/3 
7/5 
3/2 
8/5 
5/3 
7/4 
15/8 
2
&lt;/pre&gt;

[[Category:12-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Pages with Scala files]]</text>
      <sha1>psfavp3okphn02j5w6y4cms4ibv2j1u</sha1>
    </revision>
  </page>
  <page>
    <title>12highschool2-miracle</title>
    <ns>0</ns>
    <id>139</id>
    <revision>
      <id>139</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>139</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="285" sha1="qsrj7bboj03zmlfdzbitec4msugfyus" xml:space="preserve">! 12highschool2-miracle.scl

!

12highschool2 tempered in 11-limit POTE-tuned miracle

! generators: -13, -7, -6, -5, -2, -1, 0, 1, 6, 7, 12, 13

12

!

116.63274

199.59291

316.22566

383.57080

500.20354

616.83629

699.79646

816.42920

883.77434

966.73451

1083.36726

1200.00000</text>
      <sha1>qsrj7bboj03zmlfdzbitec4msugfyus</sha1>
    </revision>
    <revision>
      <id>4646</id>
      <timestamp>2012-11-20T12:13:28Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 384413020 - Original comment: **</comment>
      <origin>4646</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2011" sha1="cirgcu9ueuoj95n4cixg9gywr5q1sfw" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2012-11-20 12:13:28 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;384413020&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12highschool2-miracle.scl
!
12highschool2 tempered in 11-limit POTE-tuned miracle
! generators: -13, -7, -6, -5, -2, -1, 0, 1, 6, 7, 12, 13
 12
!
 116.63274
 199.59291
 316.22566
 383.57080
 500.20354
 616.83629
 699.79646
 816.42920
 883.77434
 966.73451
 1083.36726
 1200.00000
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12highschool2-miracle&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12highschool2-miracle.scl&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
12highschool2 tempered in 11-limit POTE-tuned miracle&amp;lt;br /&amp;gt;
! generators: -13, -7, -6, -5, -2, -1, 0, 1, 6, 7, 12, 13&amp;lt;br /&amp;gt;
 12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
 116.63274&amp;lt;br /&amp;gt;
 199.59291&amp;lt;br /&amp;gt;
 316.22566&amp;lt;br /&amp;gt;
 383.57080&amp;lt;br /&amp;gt;
 500.20354&amp;lt;br /&amp;gt;
 616.83629&amp;lt;br /&amp;gt;
 699.79646&amp;lt;br /&amp;gt;
 816.42920&amp;lt;br /&amp;gt;
 883.77434&amp;lt;br /&amp;gt;
 966.73451&amp;lt;br /&amp;gt;
 1083.36726&amp;lt;br /&amp;gt;
 1200.00000&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>cirgcu9ueuoj95n4cixg9gywr5q1sfw</sha1>
    </revision>
    <revision>
      <id>93934</id>
      <parentid>139</parentid>
      <timestamp>2022-08-05T02:07:21Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <minor/>
      <comment>Cleanup, misc. edits, categories</comment>
      <origin>93934</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="456" sha1="kggnllhla295bl5ur152mkvi56guukx" xml:space="preserve">&lt;pre&gt;
! 12highschool2-miracle.scl
!
12highschool2 tempered in 11-limit POTE-tuned miracle
! generators: -13, -7, -6, -5, -2, -1, 0, 1, 6, 7, 12, 13
12
!
116.63274
199.59291
316.22566
383.57080
500.20354
616.83629
699.79646
816.42920
883.77434
966.73451
1083.36726
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool2]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Highschool]]
[[Category:Miracle]]
[[Category:Pages with Scala files]]</text>
      <sha1>kggnllhla295bl5ur152mkvi56guukx</sha1>
    </revision>
    <revision>
      <id>186539</id>
      <parentid>93934</parentid>
      <timestamp>2025-03-17T04:02:55Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>186539</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="719" sha1="lis8706zt8mqe82hw9l6zbd531y2ybf" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].

This scale is tempered to [[miracle]] temperament.&lt;pre&gt;
! 12highschool2-miracle.scl
!
12highschool2 tempered in 11-limit POTE-tuned miracle
! generators: -13, -7, -6, -5, -2, -1, 0, 1, 6, 7, 12, 13
12
!
116.63274
199.59291
316.22566
383.57080
500.20354
616.83629
699.79646
816.42920
883.77434
966.73451
1083.36726
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool2]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Highschool]]
[[Category:Miracle]]
[[Category:Pages with Scala files]]</text>
      <sha1>lis8706zt8mqe82hw9l6zbd531y2ybf</sha1>
    </revision>
    <revision>
      <id>200474</id>
      <parentid>186539</parentid>
      <timestamp>2025-06-23T23:32:12Z</timestamp>
      <contributor>
        <username>Hkm</username>
        <id>551</id>
      </contributor>
      <origin>200474</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="768" sha1="h1odr5tyrrqk4jwidqyaq8w25dzfx68" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].

This scale is tempered to [[miracle]] temperament.&lt;pre&gt;
! 12highschool2-miracle.scl
!
12highschool2 tempered in 11-limit POTE-tuned miracle
! generators: -13, -7, -6, -5, -2, -1, 0, 1, 6, 7, 12, 13
12
!
116.63274
199.59291
316.22566
383.57080
500.20354
616.83629
699.79646
816.42920
883.77434
966.73451
1083.36726
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool2]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Highschool]]
[[Category:Miracle]]
[[Category:Pages with Scala files]]
[[Category:Pages with mostly numerical content]]</text>
      <sha1>h1odr5tyrrqk4jwidqyaq8w25dzfx68</sha1>
    </revision>
    <revision>
      <id>200563</id>
      <parentid>200474</parentid>
      <timestamp>2025-06-24T00:22:48Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Pages with mostly numerical content]]" to ""</comment>
      <origin>200563</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="719" sha1="lis8706zt8mqe82hw9l6zbd531y2ybf" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].

This scale is tempered to [[miracle]] temperament.&lt;pre&gt;
! 12highschool2-miracle.scl
!
12highschool2 tempered in 11-limit POTE-tuned miracle
! generators: -13, -7, -6, -5, -2, -1, 0, 1, 6, 7, 12, 13
12
!
116.63274
199.59291
316.22566
383.57080
500.20354
616.83629
699.79646
816.42920
883.77434
966.73451
1083.36726
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool2]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Highschool]]
[[Category:Miracle]]
[[Category:Pages with Scala files]]</text>
      <sha1>lis8706zt8mqe82hw9l6zbd531y2ybf</sha1>
    </revision>
    <revision>
      <id>200640</id>
      <parentid>200563</parentid>
      <timestamp>2025-06-24T00:24:32Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered scales]]" to "[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>200640</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="768" sha1="lf7qfwi3s7u09hfymso8hr377jv58ln" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].

This scale is tempered to [[miracle]] temperament.&lt;pre&gt;
! 12highschool2-miracle.scl
!
12highschool2 tempered in 11-limit POTE-tuned miracle
! generators: -13, -7, -6, -5, -2, -1, 0, 1, 6, 7, 12, 13
12
!
116.63274
199.59291
316.22566
383.57080
500.20354
616.83629
699.79646
816.42920
883.77434
966.73451
1083.36726
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool2]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Highschool]]
[[Category:Miracle]]
[[Category:Pages with Scala files]]</text>
      <sha1>lf7qfwi3s7u09hfymso8hr377jv58ln</sha1>
    </revision>
    <revision>
      <id>201422</id>
      <parentid>200640</parentid>
      <timestamp>2025-06-24T02:16:26Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered scales]]↵[[Category:Pages with mostly numerical content]]" to "[[Category:Tempered_scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>201422</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="768" sha1="aqgvkfhdwc3771xhdnorzijrgs41xi2" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].

This scale is tempered to [[miracle]] temperament.&lt;pre&gt;
! 12highschool2-miracle.scl
!
12highschool2 tempered in 11-limit POTE-tuned miracle
! generators: -13, -7, -6, -5, -2, -1, 0, 1, 6, 7, 12, 13
12
!
116.63274
199.59291
316.22566
383.57080
500.20354
616.83629
699.79646
816.42920
883.77434
966.73451
1083.36726
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool2]]

[[Category:12-tone scales]]
[[Category:Tempered_scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Highschool]]
[[Category:Miracle]]
[[Category:Pages with Scala files]]</text>
      <sha1>aqgvkfhdwc3771xhdnorzijrgs41xi2</sha1>
    </revision>
    <revision>
      <id>202812</id>
      <parentid>201422</parentid>
      <timestamp>2025-06-24T02:54:26Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Tempered_scales]]↵[[Category:Pages with mostly numerical content]]" to "[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>202812</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="768" sha1="lf7qfwi3s7u09hfymso8hr377jv58ln" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].

This scale is tempered to [[miracle]] temperament.&lt;pre&gt;
! 12highschool2-miracle.scl
!
12highschool2 tempered in 11-limit POTE-tuned miracle
! generators: -13, -7, -6, -5, -2, -1, 0, 1, 6, 7, 12, 13
12
!
116.63274
199.59291
316.22566
383.57080
500.20354
616.83629
699.79646
816.42920
883.77434
966.73451
1083.36726
1200.00000
&lt;/pre&gt;

== See also ==
* [[12highschool2]]

[[Category:12-tone scales]]
[[Category:Tempered scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Highschool]]
[[Category:Miracle]]
[[Category:Pages with Scala files]]</text>
      <sha1>lf7qfwi3s7u09hfymso8hr377jv58ln</sha1>
    </revision>
  </page>
  <page>
    <title>12highschool2</title>
    <ns>0</ns>
    <id>140</id>
    <revision>
      <id>140</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>140</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="214" sha1="dpvme355t3i1ul1bkzlph39jw2ja7ix" xml:space="preserve">! 12highschool2.scl

Second 12-note Highschool scale

12

!

15/14 

9/8 

6/5 

5/4 

4/3 

10/7 

3/2 

8/5 

5/3 

7/4 

15/8 

2
[[Category:12-tone]]
[[Category:highschool]]
[[Category:just]]
[[Category:scale]]</text>
      <sha1>dpvme355t3i1ul1bkzlph39jw2ja7ix</sha1>
    </revision>
    <revision>
      <id>4645</id>
      <timestamp>2011-02-13T20:22:09Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 201412986 - Original comment: **</comment>
      <origin>4645</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1650" sha1="g7ze3abzybcmp2zq0kqvhe3cy6y6r9y" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-02-13 20:22:09 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;201412986&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12highschool2.scl
Second 12-note Highschool scale
12
!
15/14 
9/8 
6/5 
5/4 
4/3 
10/7 
3/2 
8/5 
5/3 
7/4 
15/8 
2&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12highschool2&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12highschool2.scl&amp;lt;br /&amp;gt;
Second 12-note Highschool scale&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
15/14 &amp;lt;br /&amp;gt;
9/8 &amp;lt;br /&amp;gt;
6/5 &amp;lt;br /&amp;gt;
5/4 &amp;lt;br /&amp;gt;
4/3 &amp;lt;br /&amp;gt;
10/7 &amp;lt;br /&amp;gt;
3/2 &amp;lt;br /&amp;gt;
8/5 &amp;lt;br /&amp;gt;
5/3 &amp;lt;br /&amp;gt;
7/4 &amp;lt;br /&amp;gt;
15/8 &amp;lt;br /&amp;gt;
2&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>g7ze3abzybcmp2zq0kqvhe3cy6y6r9y</sha1>
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    <revision>
      <id>54688</id>
      <parentid>140</parentid>
      <timestamp>2020-12-04T22:25:11Z</timestamp>
      <contributor>
        <username>Keenan Pepper</username>
        <id>25</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:12-tone]] to [[Category:12-tone scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
      <origin>54688</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="221" sha1="t3f49p11xrzziowthmvu1beplq6pq08" xml:space="preserve">! 12highschool2.scl

Second 12-note Highschool scale

12

!

15/14 

9/8 

6/5 

5/4 

4/3 

10/7 

3/2 

8/5 

5/3 

7/4 

15/8 

2
[[Category:12-tone scales]]
[[Category:highschool]]
[[Category:just]]
[[Category:scale]]</text>
      <sha1>t3f49p11xrzziowthmvu1beplq6pq08</sha1>
    </revision>
    <revision>
      <id>55511</id>
      <parentid>54688</parentid>
      <timestamp>2020-12-05T13:04:27Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>recat</comment>
      <origin>55511</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="221" sha1="280kh0l4empis8d90clw2l0905o23fe" xml:space="preserve">! 12highschool2.scl

Second 12-note Highschool scale

12

!

15/14 

9/8 

6/5 

5/4 

4/3 

10/7 

3/2 

8/5 

5/3 

7/4 

15/8 

2

[[Category:12-tone scales]]
[[Category:highschool]]
[[Category:Just intonation scales]]</text>
      <sha1>280kh0l4empis8d90clw2l0905o23fe</sha1>
    </revision>
    <revision>
      <id>61991</id>
      <parentid>55511</parentid>
      <timestamp>2021-02-20T11:41:08Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
      </contributor>
      <minor/>
      <comment>formatting, recat</comment>
      <origin>61991</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="255" sha1="5bq8io0nrokuxtlu3ysyara3kcsidjf" xml:space="preserve">&lt;pre&gt;
! 12highschool2.scl
Second 12-note Highschool scale
12
!
15/14 
9/8 
6/5 
5/4 
4/3 
10/7 
3/2 
8/5 
5/3 
7/4 
15/8 
2
&lt;/pre&gt;

[[Category:12-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with Scala files]]</text>
      <sha1>5bq8io0nrokuxtlu3ysyara3kcsidjf</sha1>
    </revision>
    <revision>
      <id>186538</id>
      <parentid>61991</parentid>
      <timestamp>2025-03-17T04:02:38Z</timestamp>
      <contributor>
        <username>VectorGraphics</username>
        <id>430</id>
      </contributor>
      <origin>186538</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="466" sha1="rtjyddumdrrl8epiagkvzsgo4xzfr55" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].&lt;pre&gt;
! 12highschool2.scl
Second 12-note Highschool scale
12
!
15/14 
9/8 
6/5 
5/4 
4/3 
10/7 
3/2 
8/5 
5/3 
7/4 
15/8 
2
&lt;/pre&gt;

[[Category:12-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with Scala files]]</text>
      <sha1>rtjyddumdrrl8epiagkvzsgo4xzfr55</sha1>
    </revision>
    <revision>
      <id>200764</id>
      <parentid>186538</parentid>
      <timestamp>2025-06-24T00:27:30Z</timestamp>
      <contributor>
        <username>Lériendil</username>
        <id>480</id>
      </contributor>
      <minor/>
      <comment>Text replacement - "[[Category:Just intonation scales]]" to "[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]"</comment>
      <origin>200764</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="515" sha1="dhv2pwlgcgw0gko0k8ukhue23hbvyqg" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].&lt;pre&gt;
! 12highschool2.scl
Second 12-note Highschool scale
12
!
15/14 
9/8 
6/5 
5/4 
4/3 
10/7 
3/2 
8/5 
5/3 
7/4 
15/8 
2
&lt;/pre&gt;

[[Category:12-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Pages with Scala files]]</text>
      <sha1>dhv2pwlgcgw0gko0k8ukhue23hbvyqg</sha1>
    </revision>
  </page>
  <page>
    <title>12highschool3</title>
    <ns>0</ns>
    <id>141</id>
    <revision>
      <id>141</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>141</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="203" sha1="1vunjvz6cu5uxhsnmhphl9vwsxp9yer" xml:space="preserve">! 12highschool3.scl

Third 12-note Highschool scale

12

!

16/15

8/7

6/5

5/4

4/3

7/5

3/2

8/5

5/3

7/4

15/8

2/1
[[Category:12-tone]]
[[Category:highschool]]
[[Category:just]]
[[Category:scale]]</text>
      <sha1>1vunjvz6cu5uxhsnmhphl9vwsxp9yer</sha1>
    </revision>
    <revision>
      <id>4647</id>
      <timestamp>2011-07-07T13:42:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;genewardsmith</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 240390149 - Original comment: **</comment>
      <origin>4647</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1629" sha1="k8kop147wnmesgzvzdlc38pg25n6dw8" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-07-07 13:42:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;240390149&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12highschool3.scl
Third 12-note Highschool scale
12
!
16/15
8/7
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2/1
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12highschool3&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12highschool3.scl&amp;lt;br /&amp;gt;
Third 12-note Highschool scale&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
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4/3&amp;lt;br /&amp;gt;
7/5&amp;lt;br /&amp;gt;
3/2&amp;lt;br /&amp;gt;
8/5&amp;lt;br /&amp;gt;
5/3&amp;lt;br /&amp;gt;
7/4&amp;lt;br /&amp;gt;
15/8&amp;lt;br /&amp;gt;
2/1&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>k8kop147wnmesgzvzdlc38pg25n6dw8</sha1>
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    <revision>
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      <timestamp>2020-12-04T22:25:11Z</timestamp>
      <contributor>
        <username>Keenan Pepper</username>
        <id>25</id>
      </contributor>
      <minor/>
      <comment>Moving from [[Category:12-tone]] to [[Category:12-tone scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
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      <text bytes="210" sha1="g79bq5ke4n7kgc2b5tk6acbq959kl7g" xml:space="preserve">! 12highschool3.scl

Third 12-note Highschool scale

12

!

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[[Category:12-tone scales]]
[[Category:highschool]]
[[Category:just]]
[[Category:scale]]</text>
      <sha1>g79bq5ke4n7kgc2b5tk6acbq959kl7g</sha1>
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    <revision>
      <id>55512</id>
      <parentid>54686</parentid>
      <timestamp>2020-12-05T13:04:42Z</timestamp>
      <contributor>
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        <id>6</id>
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      <minor/>
      <comment>recat</comment>
      <origin>55512</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="210" sha1="4o0lstgop9lus98emzi6zhr8gelw87l" xml:space="preserve">! 12highschool3.scl

Third 12-note Highschool scale

12

!

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[[Category:12-tone scales]]
[[Category:highschool]]
[[Category:Just intonation scales]]</text>
      <sha1>4o0lstgop9lus98emzi6zhr8gelw87l</sha1>
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    <revision>
      <id>61992</id>
      <parentid>55512</parentid>
      <timestamp>2021-02-20T11:41:44Z</timestamp>
      <contributor>
        <username>Xenwolf</username>
        <id>6</id>
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      <minor/>
      <comment>formatting, recat</comment>
      <origin>61992</origin>
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      <text bytes="244" sha1="9m1532lrdf8c04qw8gv8oh7d3n3hq1h" xml:space="preserve">&lt;pre&gt;
! 12highschool3.scl
Third 12-note Highschool scale
12
!
16/15
8/7
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2/1
&lt;/pre&gt;

[[Category:12-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with Scala files]]</text>
      <sha1>9m1532lrdf8c04qw8gv8oh7d3n3hq1h</sha1>
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    <revision>
      <id>186540</id>
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      <text bytes="455" sha1="poksm3yjv202imzvhjluv9aw9pybm8a" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].&lt;pre&gt;
! 12highschool3.scl
Third 12-note Highschool scale
12
!
16/15
8/7
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2/1
&lt;/pre&gt;

[[Category:12-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with Scala files]]</text>
      <sha1>poksm3yjv202imzvhjluv9aw9pybm8a</sha1>
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    <revision>
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      <parentid>186540</parentid>
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        <id>480</id>
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      <comment>Text replacement - "[[Category:Just intonation scales]]" to "[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]"</comment>
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      <text bytes="504" sha1="mwp0xiejex0g55iyrrxqbs6c7yzxr07" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].&lt;pre&gt;
! 12highschool3.scl
Third 12-note Highschool scale
12
!
16/15
8/7
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2/1
&lt;/pre&gt;

[[Category:12-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Pages with Scala files]]</text>
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  <page>
    <title>12highschool4</title>
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    <revision>
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      <timestamp>2018-07-17T00:00:00Z</timestamp>
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      <text bytes="205" sha1="9wwrhqs4q52z9lszmhf91hoovwws0fp" xml:space="preserve">! 12highschool4.scl

Fourth 12-note Highschool scale

12

!

16/15

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[[Category:12-tone]]
[[Category:highschool]]
[[Category:just]]
[[Category:scale]]</text>
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    <revision>
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      <comment>**Imported revision 240390245 - Original comment: **</comment>
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      <text bytes="1633" sha1="k6q6u2emnxbm1cbe29tro8t17b7oegs" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on &lt;tt&gt;2011-07-07 13:42:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;240390245&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;! 12highschool4.scl
Fourth 12-note Highschool scale
12
!
16/15
8/7
6/5
5/4
4/3
10/7
3/2
8/5
5/3
7/4
15/8
2/1
&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12highschool4&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;! 12highschool4.scl&amp;lt;br /&amp;gt;
Fourth 12-note Highschool scale&amp;lt;br /&amp;gt;
12&amp;lt;br /&amp;gt;
!&amp;lt;br /&amp;gt;
16/15&amp;lt;br /&amp;gt;
8/7&amp;lt;br /&amp;gt;
6/5&amp;lt;br /&amp;gt;
5/4&amp;lt;br /&amp;gt;
4/3&amp;lt;br /&amp;gt;
10/7&amp;lt;br /&amp;gt;
3/2&amp;lt;br /&amp;gt;
8/5&amp;lt;br /&amp;gt;
5/3&amp;lt;br /&amp;gt;
7/4&amp;lt;br /&amp;gt;
15/8&amp;lt;br /&amp;gt;
2/1&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
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      <contributor>
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        <id>25</id>
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      <minor/>
      <comment>Moving from [[Category:12-tone]] to [[Category:12-tone scales]] using [[c:Help:Cat-a-lot|Cat-a-lot]]</comment>
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      <text bytes="212" sha1="13zbb0kor326ywn9p9irvgj0ggslnwv" xml:space="preserve">! 12highschool4.scl

Fourth 12-note Highschool scale

12

!

16/15

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3/2

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15/8

2/1
[[Category:12-tone scales]]
[[Category:highschool]]
[[Category:just]]
[[Category:scale]]</text>
      <sha1>13zbb0kor326ywn9p9irvgj0ggslnwv</sha1>
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    <revision>
      <id>55513</id>
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      <timestamp>2020-12-05T13:05:01Z</timestamp>
      <contributor>
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      <minor/>
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      <text bytes="212" sha1="8npna5ytm1zj8lv9w2miy0c072jk5pr" xml:space="preserve">! 12highschool4.scl

Fourth 12-note Highschool scale

12

!

16/15

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2/1

[[Category:12-tone scales]]
[[Category:highschool]]
[[Category:Just intonation scales]]</text>
      <sha1>8npna5ytm1zj8lv9w2miy0c072jk5pr</sha1>
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      <text bytes="246" sha1="ftm3hwtlo2bp1woe3ku83kqf10yuluz" xml:space="preserve">&lt;pre&gt;
! 12highschool4.scl
Fourth 12-note Highschool scale
12
!
16/15
8/7
6/5
5/4
4/3
10/7
3/2
8/5
5/3
7/4
15/8
2/1
&lt;/pre&gt;

[[Category:12-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with Scala files]]</text>
      <sha1>ftm3hwtlo2bp1woe3ku83kqf10yuluz</sha1>
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    <revision>
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      <text bytes="457" sha1="q7tf2e4blpz6f5whg6stnhl8bomhavo" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].&lt;pre&gt;
! 12highschool4.scl
Fourth 12-note Highschool scale
12
!
16/15
8/7
6/5
5/4
4/3
10/7
3/2
8/5
5/3
7/4
15/8
2/1
&lt;/pre&gt;

[[Category:12-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with Scala files]]</text>
      <sha1>q7tf2e4blpz6f5whg6stnhl8bomhavo</sha1>
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    <revision>
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[[Category:Pages with mostly numerical content]]"</comment>
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      <text bytes="506" sha1="lrqoyb1hxpw8hpfyt2c2rh36ryf2zfs" xml:space="preserve">A 12-note just intonation scale made by Gene Ward Smith when he was in high school, based on the properties of superparticular intervals with square and triangular numerators. Read more at [[Highschool scales]].&lt;pre&gt;
! 12highschool4.scl
Fourth 12-note Highschool scale
12
!
16/15
8/7
6/5
5/4
4/3
10/7
3/2
8/5
5/3
7/4
15/8
2/1
&lt;/pre&gt;

[[Category:12-tone scales]]
[[Category:Highschool]]
[[Category:Just intonation scales]]
[[Category:Pages with mostly numerical content]]
[[Category:Pages with Scala files]]</text>
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  </page>
  <page>
    <title>12L 10s</title>
    <ns>0</ns>
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    <revision>
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      <timestamp>2018-07-17T00:00:00Z</timestamp>
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      <text bytes="1269" sha1="kjlvzw941w2m97p0230omyuta8ofcm3" xml:space="preserve">This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109.091 cents), making 5/4 one period minus two generators,

{| class="wikitable"
|-
| | 2/22
| | 
| | 
| | 
| | 
| | 109.091
|-
| | 
| | 
| | 
| | 
| | 9/100
| | 108
|-
| | 
| | 
| | 
| | 7/78
| | 
| | 107.692
|-
| | 
| | 
| | 
| | 
| | 12/134
| | 107.463
|-
| | 
| | 
| | 5/56
| | 
| | 
| | 107.143
|-
| | 
| | 
| | 
| | 
| | 
| | 106.9325
|-
| | 
| | 
| | 
| | 
| | 13/146
| | 106.849
|-
| | 
| | 
| | 
| | 
| | 
| | 106.799
|-
| | 
| | 
| | 
| | 8/90
| | 
| | 106.667
|-
| | 
| | 
| | 
| | 
| | 
| | 106.497
|-
| | 
| | 
| | 
| | 
| | 11/124
| | 106.452
|-
| | 
| | 3/34
| | 
| | 
| | 
| | 105.882
|-
| | 
| | 
| | 
| | 
| | 10/114
| | 105.263
|-
| | 
| | 
| | 
| | 7/80
| | 
| | 105
|-
| | 
| | 
| | 
| | 
| | 
| | 104.829
|-
| | 
| | 
| | 
| | 
| | 11/126
| | 104.762
|-
| | 
| | 
| | 
| | 
| | 
| | 104.693
|-
| | 
| | 
| | 4/46
| | 
| | 
| | 104.348
|-
| | 
| | 
| | 
| | 
| | 
| | 104.193
|-
| | 
| | 
| | 
| | 
| | 9/104
| | 103.846
|-
| | 
| | 
| | 
| | 5/58
| | 
| | 103.448
|-
| | 
| | 
| | 
| | 
| | 6/70
| | 102.857
|-
| | 1/12
| | 
| | 
| | 
| | 
| | 100
|}</text>
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      <comment>**Imported revision 566526957 - Original comment: **</comment>
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      <text bytes="9624" sha1="cdil2gkzq8gum9c2aqvysarapzsk59h" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-15 15:31:58 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566526957&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109.091 cents), making 5/4 one period minus two generators,
|| 2/22 ||   ||   ||   ||   || 109.091 ||
||   ||   ||   ||   || 9/100 || 108 ||
||   ||   ||   || 7/78 ||   || 107.692 ||
||   ||   ||   ||   || 12/134 || 107.463 ||
||   ||   || 5/56 ||   ||   || 107.143 ||
||   ||   ||   ||   ||   || 106.9325 ||
||   ||   ||   ||   || 13/146 || 106.849 ||
||   ||   ||   ||   ||   || 106.799 ||
||   ||   ||   || 8/90 ||   || 106.667 ||
||   ||   ||   ||   ||   || 106.497 ||
||   ||   ||   ||   || 11/124 || 106.452 ||
||   || 3/34 ||   ||   ||   || 105.882 ||
||   ||   ||   ||   || 10/114 || 105.263 ||
||   ||   ||   || 7/80 ||   || 105 ||
||   ||   ||   ||   ||   || 104.829 ||
||   ||   ||   ||   || 11/126 || 104.762 ||
||   ||   ||   ||   ||   || 104.693 ||
||   ||   || 4/46 ||   ||   || 104.348 ||
||   ||   ||   ||   ||   || 104.193 ||
||   ||   ||   ||   || 9/104 || 103.846 ||
||   ||   ||   || 5/58 ||   || 103.448 ||
||   ||   ||   ||   || 6/70 || 102.857 ||
|| 1/12 ||   ||   ||   ||   || 100 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 10s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109.091 cents), making 5/4 one period minus two generators,&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;2/22&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;1/12&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>cdil2gkzq8gum9c2aqvysarapzsk59h</sha1>
    </revision>
    <revision>
      <id>4650</id>
      <timestamp>2015-10-28T17:54:20Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 564262057 - Original comment: **</comment>
      <origin>4650</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9296" sha1="f9j2sclnqe8d5i2o488xzvmt9g8n7um" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-10-28 17:54:20 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;564262057&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109.091 cents), making 5/4 one period minus two generators,
|| 2/22 ||   ||   ||   ||   || 109.091 ||
||   ||   ||   ||   || 9/100 || 108 ||
||   ||   ||   || 7/78 ||   || 107.692 ||
||   ||   ||   ||   || 12/134 || 107.463 ||
||   ||   || 5/56 ||   ||   || 107.143 ||
||   ||   ||   ||   ||   || 106.9325 ||
||   ||   ||   ||   || 13/146 || 106.849 ||
||   ||   ||   ||   ||   || 106.799 ||
||   ||   ||   || 8/90 ||   || 106.667 ||
||   ||   ||   ||   || 11/124 || 106.452 ||
||   || 3/34 ||   ||   ||   || 105.882 ||
||   ||   ||   ||   || 10/114 || 105.263 ||
||   ||   ||   || 7/80 ||   || 105 ||
||   ||   ||   ||   ||   || 104.829 ||
||   ||   ||   ||   || 11/126 || 104.762 ||
||   ||   ||   ||   ||   || 104.693 ||
||   ||   || 4/46 ||   ||   || 104.348 ||
||   ||   ||   ||   ||   || 104.193 ||
||   ||   ||   ||   || 9/104 || 103.846 ||
||   ||   ||   || 5/58 ||   || 103.448 ||
||   ||   ||   ||   || 6/70 || 102.857 ||
|| 1/12 ||   ||   ||   ||   || 100 ||
849&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 10s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109.091 cents), making 5/4 one period minus two generators,&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;2/22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;9/104&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.846&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;102.857&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;1/12&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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849&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>f9j2sclnqe8d5i2o488xzvmt9g8n7um</sha1>
    </revision>
    <revision>
      <id>4651</id>
      <timestamp>2015-10-28T17:54:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 564262007 - Original comment: **</comment>
      <origin>4651</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9296" sha1="6w99i5rkrrhnmyf9bqbixf7yujoz6sy" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-10-28 17:54:00 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;564262007&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109.091 cents), making 5/4 one period minus two generators,
|| 2/22 ||   ||   ||   ||   || 109.091 ||
||   ||   ||   ||   || 9/100 || 108 ||
||   ||   ||   || 7/78 ||   || 107.692 ||
||   ||   ||   ||   || 12/134 || 107.463 ||
||   ||   || 5/56 ||   ||   || 107.143 ||
||   ||   ||   ||   ||   || 106.9325 ||
||   ||   ||   ||   || 13/146 || 106.849 ||
||   ||   ||   ||   ||   || 196.799 ||
||   ||   ||   || 8/90 ||   || 106.667 ||
||   ||   ||   ||   || 11/124 || 106.452 ||
||   || 3/34 ||   ||   ||   || 105.882 ||
||   ||   ||   ||   || 10/114 || 105.263 ||
||   ||   ||   || 7/80 ||   || 105 ||
||   ||   ||   ||   ||   || 104.829 ||
||   ||   ||   ||   || 11/126 || 104.762 ||
||   ||   ||   ||   ||   || 104.693 ||
||   ||   || 4/46 ||   ||   || 104.348 ||
||   ||   ||   ||   ||   || 104.193 ||
||   ||   ||   ||   || 9/104 || 103.846 ||
||   ||   ||   || 5/58 ||   || 103.448 ||
||   ||   ||   ||   || 6/70 || 102.857 ||
|| 1/12 ||   ||   ||   ||   || 100 ||
849&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 10s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109.091 cents), making 5/4 one period minus two generators,&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;2/22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107.463&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107.143&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106.849&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;104.762&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;4/46&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/104&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.846&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103.448&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102.857&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

849&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>6w99i5rkrrhnmyf9bqbixf7yujoz6sy</sha1>
    </revision>
    <revision>
      <id>4652</id>
      <timestamp>2015-05-24T14:17:12Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 552016326 - Original comment: **</comment>
      <origin>4652</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8330" sha1="mp3ufdqhhnoer4y6o0445lq4sibeqad" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-24 14:17:12 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;552016326&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109 1/11 cents), making 5/4 one period minus two generators,
|| 2/22 ||   ||   ||   ||   || 109 1/11 ||
||   ||   ||   ||   || 9/100 || 108 ||
||   ||   ||   || 7/78 ||   || 107 9/13 ||
||   ||   ||   ||   || 12/134 || 107 31/67 ||
||   ||   || 5/56 ||   ||   || 107 1/7 ||
||   ||   ||   ||   || 13/146 || 106 62/73 ||
||   ||   ||   || 8/90 ||   || 106 2/3 ||
||   ||   ||   ||   || 11/124 || 106 14/31 ||
||   || 3/34 ||   ||   ||   || 105 15/17 ||
||   ||   ||   ||   || 10/114 || 105 5/19 ||
||   ||   ||   || 7/80 ||   || 105 ||
||   ||   ||   ||   || 11/126 || 104 16/21 ||
||   ||   ||   ||   ||   || 104.692701 ||
||   ||   || 4/46 ||   ||   || 104 8/23 ||
||   ||   ||   ||   ||   || 104.19295 ||
||   ||   ||   ||   || 9/104 || 103 11/13 ||
||   ||   ||   || 5/58 ||   || 103 13/29 ||
||   ||   ||   ||   || 6/70 || 102 6/7 ||
|| 1/12 ||   ||   ||   ||   || 100 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 10s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109 1/11 cents), making 5/4 one period minus two generators,&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109 1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;108&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/78&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107 9/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;12/134&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107 31/67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107 1/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/146&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 62/73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/124&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 14/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105 15/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/114&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105 5/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/126&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104 16/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.692701&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104 8/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.19295&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/104&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 11/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 13/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;6/70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 6/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;1/12&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>mp3ufdqhhnoer4y6o0445lq4sibeqad</sha1>
    </revision>
    <revision>
      <id>4653</id>
      <timestamp>2015-02-13T15:31:39Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540927400 - Original comment: **</comment>
      <origin>4653</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8332" sha1="9xcbxbe29xnwc6lk70ifv3y44jmtov5" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-13 15:31:39 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540927400&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109 1/11 cents), making 5/4 one period minus two generators,
|| 2/22 ||   ||   ||   ||   || 109 1/11 ||
||   ||   ||   ||   || 9/100 || 108 ||
||   ||   ||   || 7/78 ||   || 107 9/13 ||
||   ||   ||   ||   || 12/134 || 107 31/67 ||
||   ||   || 5/56 ||   ||   || 107 1/7 ||
||   ||   ||   ||   || 13/146 || 106 62/73 ||
||   ||   ||   || 8/90 ||   || 106 2/3 ||
||   ||   ||   ||   || 11/124 || 106 14/31 ||
||   || 3/34 ||   ||   ||   || 105 15/17 ||
||   ||   ||   ||   || 10/114 || 105 5/19 ||
||   ||   ||   || 7/80 ||   || 105 ||
||   ||   ||   ||   || 11/126 || 104 16/21 ||
||   ||   ||   ||   ||   || 104.379355 ||
||   ||   || 4/46 ||   ||   || 104 8/23 ||
||   ||   ||   ||   ||   || 104.316461 ||
||   ||   ||   ||   || 9/104 || 103 11/13 ||
||   ||   ||   || 5/58 ||   || 103 13/29 ||
||   ||   ||   ||   || 6/70 || 102 6/7 ||
|| 1/12 ||   ||   ||   ||   || 100 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 10s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109 1/11 cents), making 5/4 one period minus two generators,&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109 1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;108&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/78&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107 9/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;12/134&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107 31/67&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/56&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107 1/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;13/146&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;106 62/73&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;8/90&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/124&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 14/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;105 15/17&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/114&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;105 5/19&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;104 16/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.379355&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104 8/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104.316461&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;9/104&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 11/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 13/29&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 6/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9xcbxbe29xnwc6lk70ifv3y44jmtov5</sha1>
    </revision>
    <revision>
      <id>4654</id>
      <timestamp>2012-11-14T23:33:26Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 382647624 - Original comment: **</comment>
      <origin>4654</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7650" sha1="giwk7uqkbf3xtxj9iuvafxxbvkvj27w" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-11-14 23:33:26 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;382647624&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109 1/11 cents), making 5/4 one period minus two generators,
|| 2/22 ||   ||   ||   ||   || 109 1/11 ||
||   ||   ||   ||   || 9/100 || 108 ||
||   ||   ||   || 7/78 ||   || 107 9/13 ||
||   ||   ||   ||   || 12/134 || 107 31/67 ||
||   ||   || 5/56 ||   ||   || 107 1/7 ||
||   ||   ||   ||   || 13/146 || 106 62/73 ||
||   ||   ||   || 8/90 ||   || 106 2/3 ||
||   ||   ||   ||   || 11/124 || 106 14/31 ||
||   || 3/34 ||   ||   ||   || 105 15/17 ||
||   ||   ||   ||   || 10/114 || 105 5/19 ||
||   ||   ||   || 7/80 ||   || 105 ||
||   ||   ||   ||   || 11/126 || 104 16/21 ||
||   ||   || 4/46 ||   ||   || 104 8/23 ||
||   ||   ||   ||   || 9/104 || 103 11/13 ||
||   ||   ||   || 5/58 ||   || 103 13/29 ||
||   ||   ||   ||   || 6/70 || 102 6/7 ||
|| 1/12 ||   ||   ||   ||   || 100 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 10s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS where the pattern 6L 5s repeats at the half octave. It is the quartertone scale of Diaschismic temperament and its generator is a diatonic semitone no larger than 2/22edo~1/11edo (109 1/11 cents), making 5/4 one period minus two generators,&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;2/22&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;109 1/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;108&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/78&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107 9/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;12/134&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107 31/67&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/56&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;107 1/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/146&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 62/73&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/90&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 2/3&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/124&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;106 14/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;3/34&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105 15/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/114&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105 5/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/80&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;105&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/126&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104 16/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;4/46&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;104 8/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/104&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 11/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/58&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;103 13/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;6/70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;102 6/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>giwk7uqkbf3xtxj9iuvafxxbvkvj27w</sha1>
    </revision>
    <revision>
      <id>94399</id>
      <parentid>143</parentid>
      <timestamp>2022-08-16T04:10:45Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox, name still missing, misc. edits</comment>
      <origin>94399</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1190" sha1="tt25xdj62wc5zctxz1751uo2btjsngi" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 2
| nLargeSteps = 12
| nSmallSteps = 10
| Equalized = 2
| Paucitonic = 1
| Pattern = LLsLsLsLsLsLLsLsLsLsLs
}}
The '''12L 10s''' (or '''6L 5s&lt;1\2&gt;''') [[MOS scale]] contains two [[6L 5s]] patterns per octave.

It is the quartertone scale of [[diaschismic]] temperament and its generator is a diatonic semitone no larger than 1\11 (109.091{{cent}}), making 5/4 one period minus two generators,

== Scale tree ==
{| class="wikitable"
|-
| 2/22
| 
| 
| 
| 
| 109.091
|-
| 
| 
| 
| 
| 9/100
| 108
|-
| 
| 
| 
| 7/78
| 
| 107.692
|-
| 
| 
| 
| 
| 12/134
| 107.463
|-
| 
| 
| 5/56
| 
| 
| 107.143
|-
| 
| 
| 
| 
| 
| 106.9325
|-
| 
| 
| 
| 
| 13/146
| 106.849
|-
| 
| 
| 
| 
| 
| 106.799
|-
| 
| 
| 
| 8/90
| 
| 106.667
|-
| 
| 
| 
| 
| 
| 106.497
|-
| 
| 
| 
| 
| 11/124
| 106.452
|-
| 
| 3/34
| 
| 
| 
| 105.882
|-
| 
| 
| 
| 
| 10/114
| 105.263
|-
| 
| 
| 
| 7/80
| 
| 105
|-
| 
| 
| 
| 
| 
| 104.829
|-
| 
| 
| 
| 
| 11/126
| 104.762
|-
| 
| 
| 
| 
| 
| 104.693
|-
| 
| 
| 4/46
| 
| 
| 104.348
|-
| 
| 
| 
| 
| 
| 104.193
|-
| 
| 
| 
| 
| 9/104
| 103.846
|-
| 
| 
| 
| 5/58
| 
| 103.448
|-
| 
| 
| 
| 
| 6/70
| 102.857
|-
| 1/12
| 
| 
| 
| 
| 100
|}</text>
      <sha1>tt25xdj62wc5zctxz1751uo2btjsngi</sha1>
    </revision>
    <revision>
      <id>94511</id>
      <parentid>94399</parentid>
      <timestamp>2022-08-17T09:21:26Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94511</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1189" sha1="tsr9fbgjc08u3w3zv04nj3zn1hhbr4k" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 2
| nLargeSteps = 12
| nSmallSteps = 10
| Equalized = 2
| Collapsed = 1
| Pattern = LLsLsLsLsLsLLsLsLsLsLs
}}
The '''12L 10s''' (or '''6L 5s&lt;1\2&gt;''') [[MOS scale]] contains two [[6L 5s]] patterns per octave.

It is the quartertone scale of [[diaschismic]] temperament and its generator is a diatonic semitone no larger than 1\11 (109.091{{cent}}), making 5/4 one period minus two generators,

== Scale tree ==
{| class="wikitable"
|-
| 2/22
| 
| 
| 
| 
| 109.091
|-
| 
| 
| 
| 
| 9/100
| 108
|-
| 
| 
| 
| 7/78
| 
| 107.692
|-
| 
| 
| 
| 
| 12/134
| 107.463
|-
| 
| 
| 5/56
| 
| 
| 107.143
|-
| 
| 
| 
| 
| 
| 106.9325
|-
| 
| 
| 
| 
| 13/146
| 106.849
|-
| 
| 
| 
| 
| 
| 106.799
|-
| 
| 
| 
| 8/90
| 
| 106.667
|-
| 
| 
| 
| 
| 
| 106.497
|-
| 
| 
| 
| 
| 11/124
| 106.452
|-
| 
| 3/34
| 
| 
| 
| 105.882
|-
| 
| 
| 
| 
| 10/114
| 105.263
|-
| 
| 
| 
| 7/80
| 
| 105
|-
| 
| 
| 
| 
| 
| 104.829
|-
| 
| 
| 
| 
| 11/126
| 104.762
|-
| 
| 
| 
| 
| 
| 104.693
|-
| 
| 
| 4/46
| 
| 
| 104.348
|-
| 
| 
| 
| 
| 
| 104.193
|-
| 
| 
| 
| 
| 9/104
| 103.846
|-
| 
| 
| 
| 5/58
| 
| 103.448
|-
| 
| 
| 
| 
| 6/70
| 102.857
|-
| 1/12
| 
| 
| 
| 
| 100
|}</text>
      <sha1>tsr9fbgjc08u3w3zv04nj3zn1hhbr4k</sha1>
    </revision>
    <revision>
      <id>133822</id>
      <parentid>94511</parentid>
      <timestamp>2024-01-31T04:34:50Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>+modes</comment>
      <origin>133822</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1216" sha1="05fzemkuybwj1b97lsfp2t55ue22a2y" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 2
| nLargeSteps = 12
| nSmallSteps = 10
| Equalized = 2
| Collapsed = 1
| Pattern = LLsLsLsLsLsLLsLsLsLsLs
}}
The '''12L 10s''' (or '''6L 5s&lt;1\2&gt;''') [[MOS scale]] contains two [[6L 5s]] patterns per octave.

It is the quartertone scale of [[diaschismic]] temperament and its generator is a diatonic semitone no larger than 1\11 (109.091{{cent}}), making 5/4 one period minus two generators,

== Modes ==
{{MOS modes}}

== Scale tree ==
{| class="wikitable"
|-
| 2/22
| 
| 
| 
| 
| 109.091
|-
| 
| 
| 
| 
| 9/100
| 108
|-
| 
| 
| 
| 7/78
| 
| 107.692
|-
| 
| 
| 
| 
| 12/134
| 107.463
|-
| 
| 
| 5/56
| 
| 
| 107.143
|-
| 
| 
| 
| 
| 
| 106.9325
|-
| 
| 
| 
| 
| 13/146
| 106.849
|-
| 
| 
| 
| 
| 
| 106.799
|-
| 
| 
| 
| 8/90
| 
| 106.667
|-
| 
| 
| 
| 
| 
| 106.497
|-
| 
| 
| 
| 
| 11/124
| 106.452
|-
| 
| 3/34
| 
| 
| 
| 105.882
|-
| 
| 
| 
| 
| 10/114
| 105.263
|-
| 
| 
| 
| 7/80
| 
| 105
|-
| 
| 
| 
| 
| 
| 104.829
|-
| 
| 
| 
| 
| 11/126
| 104.762
|-
| 
| 
| 
| 
| 
| 104.693
|-
| 
| 
| 4/46
| 
| 
| 104.348
|-
| 
| 
| 
| 
| 
| 104.193
|-
| 
| 
| 
| 
| 9/104
| 103.846
|-
| 
| 
| 
| 5/58
| 
| 103.448
|-
| 
| 
| 
| 
| 6/70
| 102.857
|-
| 1/12
| 
| 
| 
| 
| 100
|}</text>
      <sha1>05fzemkuybwj1b97lsfp2t55ue22a2y</sha1>
    </revision>
    <revision>
      <id>136583</id>
      <parentid>133822</parentid>
      <timestamp>2024-02-22T04:55:11Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Mos intro</comment>
      <origin>136583</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1132" sha1="frakl6we6stb3lcuelu5xi4d2t532it" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 2
| nLargeSteps = 12
| nSmallSteps = 10
| Equalized = 2
| Collapsed = 1
| Pattern = LLsLsLsLsLsLLsLsLsLsLs
}}
{{MOS intro}}

It is the quartertone scale of [[diaschismic]] temperament and its generator is a diatonic semitone no larger than 1\11 (109.091{{cent}}), making 5/4 one period minus two generators,

== Modes ==
{{MOS modes}}

== Scale tree ==
{| class="wikitable"
|-
| 2/22
| 
| 
| 
| 
| 109.091
|-
| 
| 
| 
| 
| 9/100
| 108
|-
| 
| 
| 
| 7/78
| 
| 107.692
|-
| 
| 
| 
| 
| 12/134
| 107.463
|-
| 
| 
| 5/56
| 
| 
| 107.143
|-
| 
| 
| 
| 
| 
| 106.9325
|-
| 
| 
| 
| 
| 13/146
| 106.849
|-
| 
| 
| 
| 
| 
| 106.799
|-
| 
| 
| 
| 8/90
| 
| 106.667
|-
| 
| 
| 
| 
| 
| 106.497
|-
| 
| 
| 
| 
| 11/124
| 106.452
|-
| 
| 3/34
| 
| 
| 
| 105.882
|-
| 
| 
| 
| 
| 10/114
| 105.263
|-
| 
| 
| 
| 7/80
| 
| 105
|-
| 
| 
| 
| 
| 
| 104.829
|-
| 
| 
| 
| 
| 11/126
| 104.762
|-
| 
| 
| 
| 
| 
| 104.693
|-
| 
| 
| 4/46
| 
| 
| 104.348
|-
| 
| 
| 
| 
| 
| 104.193
|-
| 
| 
| 
| 
| 9/104
| 103.846
|-
| 
| 
| 
| 5/58
| 
| 103.448
|-
| 
| 
| 
| 
| 6/70
| 102.857
|-
| 1/12
| 
| 
| 
| 
| 100
|}</text>
      <sha1>frakl6we6stb3lcuelu5xi4d2t532it</sha1>
    </revision>
    <revision>
      <id>136584</id>
      <parentid>136583</parentid>
      <timestamp>2024-02-22T04:55:21Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <minor/>
      <comment>a space</comment>
      <origin>136584</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1131" sha1="7v6mdplitvzflx1px3pgf1gr0v2hrma" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 2
| nLargeSteps = 12
| nSmallSteps = 10
| Equalized = 2
| Collapsed = 1
| Pattern = LLsLsLsLsLsLLsLsLsLsLs
}}
{{MOS intro}}
It is the quartertone scale of [[diaschismic]] temperament and its generator is a diatonic semitone no larger than 1\11 (109.091{{cent}}), making 5/4 one period minus two generators,

== Modes ==
{{MOS modes}}

== Scale tree ==
{| class="wikitable"
|-
| 2/22
| 
| 
| 
| 
| 109.091
|-
| 
| 
| 
| 
| 9/100
| 108
|-
| 
| 
| 
| 7/78
| 
| 107.692
|-
| 
| 
| 
| 
| 12/134
| 107.463
|-
| 
| 
| 5/56
| 
| 
| 107.143
|-
| 
| 
| 
| 
| 
| 106.9325
|-
| 
| 
| 
| 
| 13/146
| 106.849
|-
| 
| 
| 
| 
| 
| 106.799
|-
| 
| 
| 
| 8/90
| 
| 106.667
|-
| 
| 
| 
| 
| 
| 106.497
|-
| 
| 
| 
| 
| 11/124
| 106.452
|-
| 
| 3/34
| 
| 
| 
| 105.882
|-
| 
| 
| 
| 
| 10/114
| 105.263
|-
| 
| 
| 
| 7/80
| 
| 105
|-
| 
| 
| 
| 
| 
| 104.829
|-
| 
| 
| 
| 
| 11/126
| 104.762
|-
| 
| 
| 
| 
| 
| 104.693
|-
| 
| 
| 4/46
| 
| 
| 104.348
|-
| 
| 
| 
| 
| 
| 104.193
|-
| 
| 
| 
| 
| 9/104
| 103.846
|-
| 
| 
| 
| 5/58
| 
| 103.448
|-
| 
| 
| 
| 
| 6/70
| 102.857
|-
| 1/12
| 
| 
| 
| 
| 100
|}</text>
      <sha1>7v6mdplitvzflx1px3pgf1gr0v2hrma</sha1>
    </revision>
    <revision>
      <id>137565</id>
      <parentid>136584</parentid>
      <timestamp>2024-02-29T19:54:43Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Replaced scale tree, rewrote rest of intro</comment>
      <origin>137565</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="454" sha1="t0sk4my4h7nxtegp562g0yhuzdxy77a" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 2
| nLargeSteps = 12
| nSmallSteps = 10
| Equalized = 2
| Collapsed = 1
| Pattern = LLsLsLsLsLsLLsLsLsLsLs
}}
{{MOS intro}}
This scale is the quartertone scale of [[diaschismic]] temperament with dark generators above 100¢ (around the upper range of a [[diatonic semitone]]). This maps [[5/4]] (386¢) to one period (600¢) minus two dark generators.

== Modes ==
{{MOS modes}}

== Scale tree ==
{{Scale tree}}</text>
      <sha1>t0sk4my4h7nxtegp562g0yhuzdxy77a</sha1>
    </revision>
    <revision>
      <id>137566</id>
      <parentid>137565</parentid>
      <timestamp>2024-02-29T19:56:00Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>/* Scale tree */ Added entries to scale tree</comment>
      <origin>137566</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="484" sha1="nputfjh1wchbd3qp16sbhqllffnh8v2" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 2
| nLargeSteps = 12
| nSmallSteps = 10
| Equalized = 2
| Collapsed = 1
| Pattern = LLsLsLsLsLsLLsLsLsLsLs
}}
{{MOS intro}}
This scale is the quartertone scale of [[diaschismic]] temperament with dark generators above 100¢ (around the upper range of a [[diatonic semitone]]). This maps [[5/4]] (386¢) to one period (600¢) minus two dark generators.

== Modes ==
{{MOS modes}}

== Scale tree ==
{{Scale tree|Comments=7/2: [[Diaschismic]]}}</text>
      <sha1>nputfjh1wchbd3qp16sbhqllffnh8v2</sha1>
    </revision>
    <revision>
      <id>170579</id>
      <parentid>137566</parentid>
      <timestamp>2024-12-15T07:06:24Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>== Intervals == {{MOS intervals}}</comment>
      <origin>170579</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="537" sha1="4n9dpwe9ab4o28f7irwcvfyniexsbyx" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 2
| nLargeSteps = 12
| nSmallSteps = 10
| Equalized = 2
| Collapsed = 1
| Pattern = LLsLsLsLsLsLLsLsLsLsLs
}}
{{MOS intro}}

This scale is the quartertone scale of [[diaschismic]] temperament with dark generators above 100¢ (around the upper range of a [[diatonic semitone]]). This maps [[5/4]] (386¢) to one period (600¢) minus two dark generators.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree|Comments=7/2: [[Diaschismic]]}}

{{todo|expand}}</text>
      <sha1>4n9dpwe9ab4o28f7irwcvfyniexsbyx</sha1>
    </revision>
    <revision>
      <id>172807</id>
      <parentid>170579</parentid>
      <timestamp>2024-12-24T05:32:32Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>Links</comment>
      <origin>172807</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="545" sha1="teyntvlbnhyjlzo05sx2v09f9lt4irm" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 2
| nLargeSteps = 12
| nSmallSteps = 10
| Equalized = 2
| Collapsed = 1
| Pattern = LLsLsLsLsLsLLsLsLsLsLs
}}
{{MOS intro}}

This scale is the [[quartertone]] scale of [[diaschismic]] temperament with [[dark]] generators above 100¢ (around the upper range of a [[diatonic semitone]]). This maps [[5/4]] (386¢) to one period (600¢) minus two dark generators.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree|Comments=7/2: [[Diaschismic]]}}

{{todo|expand}}</text>
      <sha1>teyntvlbnhyjlzo05sx2v09f9lt4irm</sha1>
    </revision>
    <revision>
      <id>183733</id>
      <parentid>172807</parentid>
      <timestamp>2025-02-28T16:42:17Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183733</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="555" sha1="suqy132zh2w4j3comnlx23pc2padt9z" xml:space="preserve">{{Infobox MOS
| Other names = 
| Periods = 2
| nLargeSteps = 12
| nSmallSteps = 10
| Equalized = 2
| Collapsed = 1
| Pattern = LLsLsLsLsLsLLsLsLsLsLs
}}
{{MOS intro}}

This scale is the [[quartertone]] scale of [[diaschismic]] temperament with [[dark]] generators above 100¢ (around the upper range of a [[diatonic semitone]]). This maps [[5/4]] (386{{c}}) to one period (600{{c}}) minus two dark generators.

== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 7/2 = [[Diaschismic]]
}}

{{todo|expand}}</text>
      <sha1>suqy132zh2w4j3comnlx23pc2padt9z</sha1>
    </revision>
  </page>
  <page>
    <title>12L 17s</title>
    <ns>0</ns>
    <id>144</id>
    <revision>
      <id>144</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>144</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1320" sha1="70gkblxjgu78k3hegej9rgxt8glh6im" xml:space="preserve">This is the MOS which splits the small steps 1-2-1-2-1-2-1-1-2-1-1-2 between the large steps. Its generator is a near-perfect fourth of no less than 5/12edo (500 cents), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.

{| class="wikitable"
|-
| | 12/29
| | 
| | 
| | 
| | 
| | 496.552
|-
| | 
| | 
| | 
| | 
| | 53/128
| | 496.875
|-
| | 
| | 
| | 
| | 41/99
| | 
| | 496.97
|-
| | 
| | 
| | 
| | 
| | 70/169
| | 497.041
|-
| | 
| | 
| | 29/70
| | 
| | 
| | 497.143
|-
| | 
| | 
| | 
| | 
| | 
| | 497.211
|-
| | 
| | 
| | 
| | 
| | 75/181
| | 497.238
|-
| | 
| | 
| | 
| | 
| | 
| | 497.254
|-
| | 
| | 
| | 
| | 46/111
| | 
| | 497.297
|-
| | 
| | 
| | 
| | 
| | 
| | 497.353
|-
| | 
| | 
| | 
| | 
| | 63/152
| | 497.368
|-
| | 
| | 17/41
| | 
| | 
| | 
| | 497.561
|-
| | 
| | 
| | 
| | 
| | 56/135
| | 497.778
|-
| | 
| | 
| | 
| | 39/94
| | 
| | 497.842
|-
| | 
| | 
| | 
| | 
| | 
| | 497.935
|-
| | 
| | 
| | 
| | 
| | 61/147
| | 497.959
|-
| | 
| | 
| | 
| | 
| | 
| | 497.985
|-
| | 
| | 
| | 22/53
| | 
| | 
| | 498.113
|-
| | 
| | 
| | 
| | 
| | 
| | 498.178
|-
| | 
| | 
| | 
| | 
| | 49/118
| | 498.305
|-
| | 
| | 
| | 
| | 27/65
| | 
| | 498.4615
|-
| | 
| | 
| | 
| | 
| | 32/77
| | 498.701
|-
| | 5/12
| | 
| | 
| | 
| | 
| | 500
|}</text>
      <sha1>70gkblxjgu78k3hegej9rgxt8glh6im</sha1>
    </revision>
    <revision>
      <id>4655</id>
      <timestamp>2015-11-15T17:06:32Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 566535471 - Original comment: **</comment>
      <origin>4655</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9744" sha1="a590i6lcxqr05djfi1a0v1er3z1a2bb" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-15 17:06:32 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566535471&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits the small steps 1-2-1-2-1-2-1-1-2-1-1-2 between the large steps. Its generator is a near-perfect fourth of no less than 5/12edo (500 cents), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.
|| 12/29 ||   ||   ||   ||   || 496.552 ||
||   ||   ||   ||   || 53/128 || 496.875 ||
||   ||   ||   || 41/99 ||   || 496.97 ||
||   ||   ||   ||   || 70/169 || 497.041 ||
||   ||   || 29/70 ||   ||   || 497.143 ||
||   ||   ||   ||   ||   || 497.211 ||
||   ||   ||   ||   || 75/181 || 497.238 ||
||   ||   ||   ||   ||   || 497.254 ||
||   ||   ||   || 46/111 ||   || 497.297 ||
||   ||   ||   ||   ||   || 497.353 ||
||   ||   ||   ||   || 63/152 || 497.368 ||
||   || 17/41 ||   ||   ||   || 497.561 ||
||   ||   ||   ||   || 56/135 || 497.778 ||
||   ||   ||   || 39/94 ||   || 497.842 ||
||   ||   ||   ||   ||   || 497.935 ||
||   ||   ||   ||   || 61/147 || 497.959 ||
||   ||   ||   ||   ||   || 497.985 ||
||   ||   || 22/53 ||   ||   || 498.113 ||
||   ||   ||   ||   ||   || 498.178 ||
||   ||   ||   ||   || 49/118 || 498.305 ||
||   ||   ||   || 27/65 ||   || 498.4615 ||
||   ||   ||   ||   || 32/77 || 498.701 ||
|| 5/12 ||   ||   ||   ||   || 500 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 17s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits the small steps 1-2-1-2-1-2-1-1-2-1-1-2 between the large steps. Its generator is a near-perfect fourth of no less than 5/12edo (500 cents), and 53edo falls near the beginning of its boundary of &amp;amp;quot;practicality&amp;amp;quot; and its harmonic entropy minimum of exactly 4/3.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;496.552&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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      <sha1>a590i6lcxqr05djfi1a0v1er3z1a2bb</sha1>
    </revision>
    <revision>
      <id>4656</id>
      <timestamp>2015-10-31T12:30:51Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 564565013 - Original comment: **</comment>
      <origin>4656</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9074" sha1="5kq4wzpprxon4bizp5654yo0w2tud2i" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-10-31 12:30:51 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;564565013&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits the small steps 1-2-1-2-1-2-1-1-2-1-1-2 between the large steps. Its generator is a near-perfect fourth of no less than 5/12edo (500 cents), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.
|| 12/29 ||   ||   ||   ||   || 496.552 ||
||   ||   ||   ||   || 53/128 || 496.875 ||
||   ||   ||   || 41/99 ||   || 496.97 ||
||   ||   ||   ||   || 70/169 || 497.041 ||
||   ||   || 29/70 ||   ||   || 497.143 ||
||   ||   ||   ||   ||   || 497.211 ||
||   ||   ||   ||   || 75/181 || 497.238 ||
||   ||   ||   ||   ||   || 497.254 ||
||   ||   ||   || 46/111 ||   || 497.297 ||
||   ||   ||   ||   || 63/152 || 497.368 ||
||   || 17/41 ||   ||   ||   || 497.561 ||
||   ||   ||   ||   || 56/135 || 497.778 ||
||   ||   ||   || 39/94 ||   || 497.842 ||
||   ||   ||   ||   || 61/147 || 497.959 ||
||   ||   ||   ||   ||   || 497.985 ||
||   ||   || 22/53 ||   ||   || 498.113 ||
||   ||   ||   ||   ||   || 498.178 ||
||   ||   ||   ||   || 49/118 || 498.305 ||
||   ||   ||   || 27/65 ||   || 498.4615 ||
||   ||   ||   ||   || 32/77 || 498.701 ||
|| 5/12 ||   ||   ||   ||   || 500 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 17s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits the small steps 1-2-1-2-1-2-1-1-2-1-1-2 between the large steps. Its generator is a near-perfect fourth of no less than 5/12edo (500 cents), and 53edo falls near the beginning of its boundary of &amp;amp;quot;practicality&amp;amp;quot; and its harmonic entropy minimum of exactly 4/3.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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      <sha1>5kq4wzpprxon4bizp5654yo0w2tud2i</sha1>
    </revision>
    <revision>
      <id>4657</id>
      <timestamp>2015-05-24T14:55:50Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 552018370 - Original comment: **</comment>
      <origin>4657</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8466" sha1="e76twwstf9wsdfnl3x5641ebfr1by1r" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-05-24 14:55:50 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;552018370&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits the small steps 1-2-1-2-1-2-1-1-2-1-1-2 between the large steps. Its generator is a near-perfect fourth of no less than 5/12edo (500 cents), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.
|| 12/29 ||   ||   ||   ||   || 496 16/29 ||
||   ||   ||   ||   || 53/128 || 496.875 ||
||   ||   ||   || 41/99 ||   || 496 32/99 ||
||   ||   ||   ||   || 70/169 || 497 7/169 ||
||   ||   || 29/70 ||   ||   || 497 1/7 ||
||   ||   ||   ||   || 75/181 || 497 43/181 ||
||   ||   ||   || 46/111 ||   || 497 11/37 ||
||   ||   ||   ||   || 63/152 || 497 7/19 ||
||   || 17/41 ||   ||   ||   || 497 23/41 ||
||   ||   ||   ||   || 56/135 || 497 7/9 ||
||   ||   ||   || 39/94 ||   || 497 41/47 ||
||   ||   ||   ||   || 61/147 || 497 47/49 ||
||   ||   ||   ||   ||   || 497.9846545 ||
||   ||   || 22/53 ||   ||   || 498 6/53 ||
||   ||   ||   ||   ||   || 498.171817 ||
||   ||   ||   ||   || 49/118 || 498 18/59 ||
||   ||   ||   || 27/65 ||   || 498 6/13 ||
||   ||   ||   ||   || 32/77 || 498 54/77 ||
|| 5/12 ||   ||   ||   ||   || 500 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 17s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits the small steps 1-2-1-2-1-2-1-1-2-1-1-2 between the large steps. Its generator is a near-perfect fourth of no less than 5/12edo (500 cents), and 53edo falls near the beginning of its boundary of &amp;amp;quot;practicality&amp;amp;quot; and its harmonic entropy minimum of exactly 4/3.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;498 18/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>e76twwstf9wsdfnl3x5641ebfr1by1r</sha1>
    </revision>
    <revision>
      <id>4658</id>
      <timestamp>2015-02-14T17:18:52Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 540983892 - Original comment: **</comment>
      <origin>4658</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8466" sha1="9yte688ux9tt0pgnulpeavnz95r7eyx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-02-14 17:18:52 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;540983892&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits the small steps 1-2-1-2-1-2-1-1-2-1-1-2 between the large steps. Its generator is a near-perfect fourth of no less than 5/12edo (500 cents), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.
|| 12/29 ||   ||   ||   ||   || 496 16/29 ||
||   ||   ||   ||   || 53/128 || 496.875 ||
||   ||   ||   || 41/99 ||   || 496 32/99 ||
||   ||   ||   ||   || 70/169 || 497 7/169 ||
||   ||   || 29/70 ||   ||   || 497 1/7 ||
||   ||   ||   ||   || 75/181 || 497 43/181 ||
||   ||   ||   || 46/111 ||   || 497 11/37 ||
||   ||   ||   ||   || 63/152 || 497 7/19 ||
||   || 17/41 ||   ||   ||   || 497 23/41 ||
||   ||   ||   ||   || 56/135 || 497 7/9 ||
||   ||   ||   || 39/94 ||   || 497 41/47 ||
||   ||   ||   ||   || 61/147 || 497 47/49 ||
||   ||   ||   ||   ||   || 498.1013435 ||
||   ||   || 22/53 ||   ||   || 498 6/53 ||
||   ||   ||   ||   ||   || 498.125032 ||
||   ||   ||   ||   || 49/118 || 498 18/59 ||
||   ||   ||   || 27/65 ||   || 498 6/13 ||
||   ||   ||   ||   || 32/77 || 498 54/77 ||
|| 5/12 ||   ||   ||   ||   || 500 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 17s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits the small steps 1-2-1-2-1-2-1-1-2-1-1-2 between the large steps. Its generator is a near-perfect fourth of no less than 5/12edo (500 cents), and 53edo falls near the beginning of its boundary of &amp;amp;quot;practicality&amp;amp;quot; and its harmonic entropy minimum of exactly 4/3.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
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        &amp;lt;td&amp;gt;12/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;496 16/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;496.875&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;497 11/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;63/152&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;497 7/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;497 47/49&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;498.1013435&amp;lt;br /&amp;gt;
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    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22/53&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;498 6/53&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/118&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;498 18/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;27/65&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;498 6/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;498 54/77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;5/12&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;500&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9yte688ux9tt0pgnulpeavnz95r7eyx</sha1>
    </revision>
    <revision>
      <id>4659</id>
      <timestamp>2012-11-19T00:14:19Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 383880040 - Original comment: **</comment>
      <origin>4659</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="7782" sha1="dbjeyoob4akvwb8pdv1i57ypimls17g" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2012-11-19 00:14:19 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;383880040&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This is the MOS which splits the small steps 1-2-1-2-1-2-1-1-2-1-1-2 between the large steps. Its generator is a near-perfect fourth of no less than 5/12edo (500 cents), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.
|| 12/29 ||   ||   ||   ||   || 496 16/29 ||
||   ||   ||   ||   || 53/128 || 496.875 ||
||   ||   ||   || 41/99 ||   || 496 32/99 ||
||   ||   ||   ||   || 70/169 || 497 7/169 ||
||   ||   || 29/70 ||   ||   || 497 1/7 ||
||   ||   ||   ||   || 75/181 || 497 43/181 ||
||   ||   ||   || 46/111 ||   || 497 11/37 ||
||   ||   ||   ||   || 63/152 || 497 7/19 ||
||   || 17/41 ||   ||   ||   || 497 23/41 ||
||   ||   ||   ||   || 56/135 || 497 7/9 ||
||   ||   ||   || 39/94 ||   || 497 41/47 ||
||   ||   ||   ||   || 61/147 || 497 47/49 ||
||   ||   || 22/53 ||   ||   || 498 6/53 ||
||   ||   ||   ||   || 49/118 || 498 18/59 ||
||   ||   ||   || 27/65 ||   || 498 6/13 ||
||   ||   ||   ||   || 32/77 || 498 54/77 ||
|| 5/12 ||   ||   ||   ||   || 500 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 17s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This is the MOS which splits the small steps 1-2-1-2-1-2-1-1-2-1-1-2 between the large steps. Its generator is a near-perfect fourth of no less than 5/12edo (500 cents), and 53edo falls near the beginning of its boundary of &amp;amp;quot;practicality&amp;amp;quot; and its harmonic entropy minimum of exactly 4/3.&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;12/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;29/70&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;497 1/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;46/111&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;497 11/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;63/152&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;497 7/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;497 23/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;497 7/9&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;39/94&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;497 41/47&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;61/147&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;497 47/49&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;49/118&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;498 18/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;498 54/77&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/12&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>dbjeyoob4akvwb8pdv1i57ypimls17g</sha1>
    </revision>
    <revision>
      <id>94451</id>
      <parentid>144</parentid>
      <timestamp>2022-08-16T20:35:32Z</timestamp>
      <contributor>
        <username>Fredg999</username>
        <id>308</id>
      </contributor>
      <comment>+Infobox, name still missing, misc. edits</comment>
      <origin>94451</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1153" sha1="c551p05v2d494w621ku47mg5e9usbct" xml:space="preserve">{{Infobox MOS
| Other names =
| Periods = 1
| nLargeSteps = 12
| nSmallSteps = 17
| Equalized = 12
| Paucitonic = 5
| Pattern = LsLsLssLsLssLsLsLssLsLssLsLss
}}
The '''12L 17s''' [[MOS scale]] splits the small steps 1-1-2-1-2-1-1-2-1-2-1-2 between the large steps. Its [[chroma-positive]] generator is a near-perfect fourth of no more than 5\12 (500{{cent}}), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.

== Scale tree ==
{| class="wikitable"
|-
| 12/29
|
|
|
|
| 496.552
|-
|
|
|
|
| 53/128
| 496.875
|-
|
|
|
| 41/99
|
| 496.97
|-
|
|
|
|
| 70/169
| 497.041
|-
|
|
| 29/70
|
|
| 497.143
|-
|
|
|
|
|
| 497.211
|-
|
|
|
|
| 75/181
| 497.238
|-
|
|
|
|
|
| 497.254
|-
|
|
|
| 46/111
|
| 497.297
|-
|
|
|
|
|
| 497.353
|-
|
|
|
|
| 63/152
| 497.368
|-
|
| 17/41
|
|
|
| 497.561
|-
|
|
|
|
| 56/135
| 497.778
|-
|
|
|
| 39/94
|
| 497.842
|-
|
|
|
|
|
| 497.935
|-
|
|
|
|
| 61/147
| 497.959
|-
|
|
|
|
|
| 497.985
|-
|
|
| 22/53
|
|
| 498.113
|-
|
|
|
|
|
| 498.178
|-
|
|
|
|
| 49/118
| 498.305
|-
|
|
|
| 27/65
|
| 498.4615
|-
|
|
|
|
| 32/77
| 498.701
|-
| 5/12
|
|
|
|
| 500
|}</text>
      <sha1>c551p05v2d494w621ku47mg5e9usbct</sha1>
    </revision>
    <revision>
      <id>94512</id>
      <parentid>94451</parentid>
      <timestamp>2022-08-17T09:21:46Z</timestamp>
      <contributor>
        <username>Inthar</username>
        <id>76</id>
      </contributor>
      <origin>94512</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1152" sha1="3n0o2gs0g39vpvjt7boncut0zqwi9jz" xml:space="preserve">{{Infobox MOS
| Other names =
| Periods = 1
| nLargeSteps = 12
| nSmallSteps = 17
| Equalized = 12
| Collapsed = 5
| Pattern = LsLsLssLsLssLsLsLssLsLssLsLss
}}
The '''12L 17s''' [[MOS scale]] splits the small steps 1-1-2-1-2-1-1-2-1-2-1-2 between the large steps. Its [[chroma-positive]] generator is a near-perfect fourth of no more than 5\12 (500{{cent}}), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.

== Scale tree ==
{| class="wikitable"
|-
| 12/29
|
|
|
|
| 496.552
|-
|
|
|
|
| 53/128
| 496.875
|-
|
|
|
| 41/99
|
| 496.97
|-
|
|
|
|
| 70/169
| 497.041
|-
|
|
| 29/70
|
|
| 497.143
|-
|
|
|
|
|
| 497.211
|-
|
|
|
|
| 75/181
| 497.238
|-
|
|
|
|
|
| 497.254
|-
|
|
|
| 46/111
|
| 497.297
|-
|
|
|
|
|
| 497.353
|-
|
|
|
|
| 63/152
| 497.368
|-
|
| 17/41
|
|
|
| 497.561
|-
|
|
|
|
| 56/135
| 497.778
|-
|
|
|
| 39/94
|
| 497.842
|-
|
|
|
|
|
| 497.935
|-
|
|
|
|
| 61/147
| 497.959
|-
|
|
|
|
|
| 497.985
|-
|
|
| 22/53
|
|
| 498.113
|-
|
|
|
|
|
| 498.178
|-
|
|
|
|
| 49/118
| 498.305
|-
|
|
|
| 27/65
|
| 498.4615
|-
|
|
|
|
| 32/77
| 498.701
|-
| 5/12
|
|
|
|
| 500
|}</text>
      <sha1>3n0o2gs0g39vpvjt7boncut0zqwi9jz</sha1>
    </revision>
    <revision>
      <id>134225</id>
      <parentid>94512</parentid>
      <timestamp>2024-02-03T21:06:22Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>Mos intro</comment>
      <origin>134225</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="918" sha1="r3wpp5vmhqaldqikoqherw1dzi18t8b" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
Its [[chroma-positive]] generator is a near-perfect fourth of no more than 5\12 (500{{cent}}), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.

== Scale tree ==
{| class="wikitable"
|-
| 12/29
|
|
|
|
| 496.552
|-
|
|
|
|
| 53/128
| 496.875
|-
|
|
|
| 41/99
|
| 496.97
|-
|
|
|
|
| 70/169
| 497.041
|-
|
|
| 29/70
|
|
| 497.143
|-
|
|
|
|
|
| 497.211
|-
|
|
|
|
| 75/181
| 497.238
|-
|
|
|
|
|
| 497.254
|-
|
|
|
| 46/111
|
| 497.297
|-
|
|
|
|
|
| 497.353
|-
|
|
|
|
| 63/152
| 497.368
|-
|
| 17/41
|
|
|
| 497.561
|-
|
|
|
|
| 56/135
| 497.778
|-
|
|
|
| 39/94
|
| 497.842
|-
|
|
|
|
|
| 497.935
|-
|
|
|
|
| 61/147
| 497.959
|-
|
|
|
|
|
| 497.985
|-
|
|
| 22/53
|
|
| 498.113
|-
|
|
|
|
|
| 498.178
|-
|
|
|
|
| 49/118
| 498.305
|-
|
|
|
| 27/65
|
| 498.4615
|-
|
|
|
|
| 32/77
| 498.701
|-
| 5/12
|
|
|
|
| 500
|}</text>
      <sha1>r3wpp5vmhqaldqikoqherw1dzi18t8b</sha1>
    </revision>
    <revision>
      <id>137552</id>
      <parentid>134225</parentid>
      <timestamp>2024-02-29T19:21:23Z</timestamp>
      <contributor>
        <username>Ganaram inukshuk</username>
        <id>324</id>
      </contributor>
      <comment>/* Scale tree */ Replaced scale tree</comment>
      <origin>137552</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="275" sha1="4vxo1m4d93s94jsc4mn4k02h57b1qgk" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
Its [[chroma-positive]] generator is a near-perfect fourth of no more than 5\12 (500{{cent}}), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.

== Scale tree ==
{{Scale tree}}</text>
      <sha1>4vxo1m4d93s94jsc4mn4k02h57b1qgk</sha1>
    </revision>
    <revision>
      <id>168278</id>
      <parentid>137552</parentid>
      <timestamp>2024-12-02T14:16:45Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>/* Scale tree */ +++</comment>
      <origin>168278</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="592" sha1="7667n5a24goksiqoecdlnd7s17d3i2h" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
Its [[chroma-positive]] generator is a near-perfect fourth of no more than 5\12 (500{{cent}}), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.

== Scale tree ==
{{Scale tree|Comments=
4/3: [[Undecental]]; 
10/7: Argent tuning (497.056¢); 
13/8: Unnamed golden tuning (497.254¢); 
7/4: [[Kwai]]; 
9/4: [[Cotoneum]]; 
5/2: [[Garibaldi]] / [[cassandra]]; 
11/4: [[Pythagorean tuning]] (498.045¢); 
3/1: Garibaldi / [[helenus]]; 
10/3: [[Pontiac]]; 
6/1: [[Grackle]], ↓ [[gracecordial]]; 
}}</text>
      <sha1>7667n5a24goksiqoecdlnd7s17d3i2h</sha1>
    </revision>
    <revision>
      <id>170580</id>
      <parentid>168278</parentid>
      <timestamp>2024-12-15T07:06:53Z</timestamp>
      <contributor>
        <username>BudjarnLambeth</username>
        <id>359</id>
      </contributor>
      <minor/>
      <comment>== Intervals == {{MOS intervals}}</comment>
      <origin>170580</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="644" sha1="4nm3wxa5pkj0i5x761le7xb47bq7qqw" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
Its [[chroma-positive]] generator is a near-perfect fourth of no more than 5\12 (500{{cent}}), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{Scale tree|Comments=
4/3: [[Undecental]]; 
10/7: Argent tuning (497.056¢); 
13/8: Unnamed golden tuning (497.254¢); 
7/4: [[Kwai]]; 
9/4: [[Cotoneum]]; 
5/2: [[Garibaldi]] / [[cassandra]]; 
11/4: [[Pythagorean tuning]] (498.045¢); 
3/1: Garibaldi / [[helenus]]; 
10/3: [[Pontiac]]; 
6/1: [[Grackle]], ↓ [[gracecordial]]; 
}}

{{todo|expand}}</text>
      <sha1>4nm3wxa5pkj0i5x761le7xb47bq7qqw</sha1>
    </revision>
    <revision>
      <id>183734</id>
      <parentid>170580</parentid>
      <timestamp>2025-02-28T16:42:55Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <origin>183734</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="663" sha1="839brsgljc3otv0pl0060uf8g9gx17d" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
Its [[chroma-positive]] generator is a near-perfect fourth of no more than 5\12 (500{{cent}}), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3.

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 4/3 = [[Undecental]] 
| 10/7 = Argent tuning (497.056¢) 
| 13/8 = Unnamed golden tuning (497.254¢) 
| 7/4 = [[Kwai]] 
| 9/4 = [[Cotoneum]] 
| 5/2 = [[Garibaldi]] / [[cassandra]] 
| 11/4 = [[Pythagorean tuning]] (498.045¢) 
| 3/1 = Garibaldi / [[helenus]] 
| 10/3 = [[Pontiac]] 
| 6/1 = [[Grackle]], ↓ [[gracecordial]] 
}}

{{todo|expand}}</text>
      <sha1>839brsgljc3otv0pl0060uf8g9gx17d</sha1>
    </revision>
    <revision>
      <id>218769</id>
      <parentid>183734</parentid>
      <timestamp>2025-12-08T22:49:52Z</timestamp>
      <contributor>
        <username>Eufalesio</username>
        <id>581</id>
      </contributor>
      <comment>Proposed adding the hardness of Pythagorean tuning with this MOS and subsequent ones</comment>
      <origin>218769</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="735" sha1="2mf775iivvqrajo6i240hq58rnx830p" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
Its [[chroma-positive]] generator is a near-perfect fourth of no more than 5\12 (500{{cent}}), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3. [[Pythagorean tuning]] generates this scale, with a hardness of 2.8459.

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 4/3 = [[Undecental]] 
| 10/7 = Argent tuning (497.056¢) 
| 13/8 = Unnamed golden tuning (497.254¢) 
| 7/4 = [[Kwai]] 
| 9/4 = [[Cotoneum]] 
| 5/2 = [[Garibaldi]] / [[cassandra]] 
| 11/4 = [[Pythagorean tuning]] (498.045¢) 
| 3/1 = Garibaldi / [[helenus]] 
| 10/3 = [[Pontiac]] 
| 6/1 = [[Grackle]], ↓ [[gracecordial]] 
}}

{{todo|expand}}</text>
      <sha1>2mf775iivvqrajo6i240hq58rnx830p</sha1>
    </revision>
    <revision>
      <id>218772</id>
      <parentid>218769</parentid>
      <timestamp>2025-12-08T23:42:04Z</timestamp>
      <contributor>
        <username>ArrowHead294</username>
        <id>448</id>
      </contributor>
      <minor/>
      <comment>/* Scale tree */</comment>
      <origin>218772</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="749" sha1="7cok6twcyiyh9fzk28t6z2z8irpsfv7" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
Its [[chroma-positive]] generator is a near-perfect fourth of no more than 5\12 (500{{cent}}), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3. [[Pythagorean tuning]] generates this scale, with a hardness of 2.8459.

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 4/3 = [[Undecental]] 
| 10/7 = Argent tuning (497.056{{c}}) 
| 13/8 = Unnamed golden tuning (497.254{{c}}) 
| 7/4 = [[Kwai]] 
| 9/4 = [[Cotoneum]] 
| 5/2 = [[Garibaldi]] / [[cassandra]] 
| 11/4 = [[Pythagorean tuning]] (498.045{{c}}) 
| 3/1 = Garibaldi / [[helenus]] 
| 10/3 = [[Pontiac]] 
| 6/1 = [[Grackle]], ↓&amp;nbsp;[[gracecordial]] 
}}

{{todo|expand}}</text>
      <sha1>7cok6twcyiyh9fzk28t6z2z8irpsfv7</sha1>
    </revision>
    <revision>
      <id>224884</id>
      <parentid>218772</parentid>
      <timestamp>2026-02-27T17:15:59Z</timestamp>
      <contributor>
        <username>Eufalesio</username>
        <id>581</id>
      </contributor>
      <comment>More temps in scale tree</comment>
      <origin>224884</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="849" sha1="dr4rsj23tcufhjt1ls1a03nqlagwfix" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
Its [[chroma-positive]] generator is a near-perfect fourth of no more than 5\12 (500{{cent}}), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3. [[Pythagorean tuning]] generates this scale, with a hardness of 2.8459. It is known as '''pythagotonic'''.

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 4/3 = [[Undecental]] 
| 10/7 = Argent tuning (497.056{{c}}) 
| 13/8 = Unnamed golden tuning (497.254{{c}}) 
| 7/4 = [[Kwai]] 
| 9/4 = [[Cotoneum]] 
| 5/2 = [[Garibaldi]] / [[cassandra]] 
| 11/4 = [[Pythagorean tuning]] (498.045{{c}}) 
| 3/1 = Garibaldi / [[helenus]] 
| 10/3 = [[Pontiac]] 
| 6/1 = [[Grackle]], ↓&amp;nbsp;[[gracecordial]] 
|1/1=[[Mystery]]|1/0=[[Compton]] / [[Catler]]|2/1=[[Countercomp]]}}

{{todo|expand}}</text>
      <sha1>dr4rsj23tcufhjt1ls1a03nqlagwfix</sha1>
    </revision>
    <revision>
      <id>224887</id>
      <parentid>224884</parentid>
      <timestamp>2026-02-27T17:32:09Z</timestamp>
      <contributor>
        <username>FloraC</username>
        <id>186</id>
      </contributor>
      <comment>No fractional-octave temps here</comment>
      <origin>224887</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="791" sha1="3spjupk87hr2nsqzezr5kjt044vacfv" xml:space="preserve">{{Infobox MOS}}
{{MOS intro}}
Its [[chroma-positive]] generator is a near-perfect fourth of no more than 5\12 (500{{cent}}), and 53edo falls near the beginning of its boundary of "practicality" and its harmonic entropy minimum of exactly 4/3. [[Pythagorean tuning]] generates this scale, with a hardness of 2.8459. It is known as '''pythagotonic''' in [[TAMNAMS Extension]].

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 4/3 = [[Undecental]] 
| 10/7 = Argent tuning (497.056{{c}}) 
| 13/8 = Unnamed golden tuning (497.254{{c}}) 
| 7/4 = [[Kwai]] 
| 9/4 = [[Cotoneum]] 
| 5/2 = [[Garibaldi]] / [[cassandra]] 
| 11/4 = [[Pythagorean tuning]] (498.045{{c}}) 
| 3/1 = Garibaldi / [[helenus]] 
| 10/3 = [[Pontiac]] 
| 6/1 = [[Grackle]], ↓&amp;nbsp;[[gracecordial]]
}}</text>
      <sha1>3spjupk87hr2nsqzezr5kjt044vacfv</sha1>
    </revision>
  </page>
  <page>
    <title>12L 1s</title>
    <ns>0</ns>
    <id>145</id>
    <revision>
      <id>145</id>
      <timestamp>2018-07-17T00:00:00Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;FREEZE</username>
        <id>0</id>
      </contributor>
      <origin>145</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10091" sha1="2bd8ctbgl5qr0ndpc537q03gnsnnsjw" xml:space="preserve">This MOS, the Grumpy tridecatonic, apparently belongs to no particularly important temperament. However, it becomes a compressed 12ed scale when you ignore the octave (this obviously does not work when the generator is very near 12edo &lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&gt;(within -7/24¢ of it)&lt;/span&gt;&lt;span style="font-size: 13px; line-height: 1.5;"&gt;, for the 13th degree of the scale registers as identical to the octave for human listeners, and it becomes indistinct from 13edo or the Happy dodecatonic ([[1L_11s|1L 11s]]) in the 1.75¢ above 1/13edo because the large and small steps register as identical to one another for human listeners).&lt;/span&gt;

{| class="wikitable"
|-
! colspan="3" | Generator
! | &lt;span style="font-size: 12.8000001907349px;"&gt;Cents&lt;/span&gt;
! | &lt;span style="font-size: 12.8000001907349px;"&gt;12g&lt;/span&gt;
|-
| | 1/13
| | 
| | 
| | 92.308
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1107.692&lt;/span&gt;
|-
| | 5/64
| | 
| | 
| | 93.75
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1125&lt;/span&gt;
|-
| | 
| | 9/115
| | 
| | 93.913
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1126.9565&lt;/span&gt;
|-
| | 
| | 13/166
| | 
| | 93.976
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1127.711&lt;/span&gt;
|-
| | 
| | 17/217
| | 
| | 94.009
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1128.111&lt;/span&gt;
|-
| | 4/51
| | 
| | 
| | 94.118
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1129.412&lt;/span&gt;
|-
| | 
| | 15/191
| | 
| | 94.241
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1130.89&lt;/span&gt;
|-
| | 
| | 11/140
| | 
| | 94.296
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1131 3/7&lt;/span&gt;
|-
| | 
| | 7/89
| | 
| | 94.382
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1132.584&lt;/span&gt;
|-
| | 
| | 10/127
| | 
| | 94.448
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1133.858&lt;/span&gt;
|-
| | 
| | 13/165
| | 
| | 94.5455
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1134.5455&lt;/span&gt;
|-
| | 
| | 16/203
| | 
| | 94.581
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1134.975&lt;/span&gt;
|-
| | 
| | 19/241
| | 
| | 94.606
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1135.27&lt;/span&gt;
|-
| | 3/38
| | 
| | 
| | 94.737
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1136.842&lt;/span&gt;
|-
| | 
| | 26/329
| | 
| | 94.8875
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1137.6505&lt;/span&gt;
|-
| | 
| | 23/291
| | 
| | 94.845
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1138.1443&lt;/span&gt;
|-
| | 
| | 20/253
| | 
| | 94.862
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1138.34&lt;/span&gt;
|-
| | 
| | 17/215
| | 
| | 94.884
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1138.605&lt;/span&gt;
|-
| | 
| | 14/177
| | 
| | 94.915
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1138.983&lt;/span&gt;
|-
| | 
| | 
| | 
| | 94.962
| | 1139.545
|-
| | 
| | 11/139
| | 
| | 94.964
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1139.568&lt;/span&gt;
|-
| | 
| | 8/101
| | 
| | 95.0495
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1140.594&lt;/span&gt;
|-
| | 
| | 
| | 
| | 95.102
| | 1141.224
|-
| | 
| | 
| | 13/164
| | 95.122
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1141.463&lt;/span&gt;
|-
| | 
| | 5/63
| | 
| | 95.238
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1142.714&lt;/span&gt;
|-
| | 
| | 
| | 17/214
| | 95.374
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1143.486&lt;/span&gt;
|-
| | 
| | 
| | 12/151
| | 95.362
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1144.371&lt;/span&gt;
|-
| | 
| | 
| | 
| | 95.41
| | 1144.915
|-
| | 
| | 7/88
| | 
| | 95,4545
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1145.4545&lt;/span&gt;
|-
| | 
| | 9/113
| | 
| | 95.575
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1146.903&lt;/span&gt;
|-
| | 
| | 11/138
| | 
| | 95.652
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1147.826&lt;/span&gt;
|-
| | 
| | 13/163
| | 
| | 95.7055
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1148.466&lt;/span&gt;
|-
| | 
| | 15/188
| | 
| | 95.745
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1148.936&lt;/span&gt;
|-
| | 
| | 17/213
| | 
| | 95.775
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1149.296&lt;/span&gt;
|-
| | 
| | 19/238
| | 
| | 95.798
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1149.58&lt;/span&gt;
|-
| | 
| | 21/263
| | 
| | 95.8175
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1149.81&lt;/span&gt;
|-
| | 
| | 23/288
| | 
| | 95.833
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1150&lt;/span&gt;
|-
| | 
| | 25/313
| | 
| | 95.847
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1150.16&lt;/span&gt;
|-
| | 
| | 27/338
| | 
| | 95.858
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1150.296&lt;/span&gt;
|-
| | 
| | 29/363
| | 
| | 95.868
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1150.467&lt;/span&gt;
|-
| | 
| | 31/388
| | 
| | 95.876
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1150.5155&lt;/span&gt;
|-
| | 
| | 33/413
| | 
| | 95.884
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1150.605&lt;/span&gt;
|-
| | 
| | 35/438
| | 
| | 95.89
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1150.685&lt;/span&gt;
|-
| | 
| | 37/463
| | 
| | 95.896
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1150.75&lt;/span&gt;
|-
| | 
| | 39/488
| | 
| | 95.902
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1150.82&lt;/span&gt;
|-
| | 
| | 41/513
| | 
| | 95.906
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1150.877&lt;/span&gt;
|-
| | 
| | 43/538
| | 
| | 95.911
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1150.929&lt;/span&gt;
|-
| | 
| | 45/563
| | 
| | 95.915
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1150.977&lt;/span&gt;
|-
| | 
| | 47/588
| | 
| | 95.918
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1151.02&lt;/span&gt;
|-
| | 2/25
| | 
| | 
| | 96
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1152&lt;/span&gt;
|-
| | 
| | 25/312
| | 
| | 96.154
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1153.846&lt;/span&gt;
|-
| | 
| | 23/287
| | 
| | 96.167
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1154.007&lt;/span&gt;
|-
| | 
| | 21/262
| | 
| | 96.183
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1154.1985&lt;/span&gt;
|-
| | 
| | 19/237
| | 
| | 96.2025
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1154,43&lt;/span&gt;
|-
| | 
| | 17/212
| | 
| | 96.226
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1154.717&lt;/span&gt;
|-
| | 
| | 15/187
| | 
| | 96.257
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1155.08&lt;/span&gt;
|-
| | 
| | 13/162
| | 
| | 96.296
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1155.556&lt;/span&gt;
|-
| | 
| | 11/137
| | 
| | 96.35
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1156.204&lt;/span&gt;
|-
| | 
| | 9/112
| | 
| | 96.429
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1157.143&lt;/span&gt;
|-
| | 
| | 7/87
| | 
| | 96.552
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1158.621&lt;/span&gt;
|-
| | 
| | 
| | 12/149
| | 96.644
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1159.7315&lt;/span&gt;
|-
| | 
| | 
| | 17/211
| | 96.6825
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1160.278&lt;/span&gt;
|-
| | 
| | 5/62
| | 
| | 96.774
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1161.29&lt;/span&gt;
|-
| | 
| | 
| | 13/161
| | 96.894
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1162.733&lt;/span&gt;
|-
| | 
| | 
| | 
| | 96.915
| | 1162.982
|-
| | 
| | 8/99
| | 
| | 96.97
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1163.636&lt;/span&gt;
|-
| | 
| | 11/136
| | 
| | 97.059
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1164.706&lt;/span&gt;
|-
| | 
| | 
| | 
| | 97.0255
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1164.306&lt;/span&gt;
|-
| | 
| | 14/173
| | 
| | 97.11
| | &lt;span style="font-size: 12.8000001907349px;"&gt;116&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&gt;5.318&lt;/span&gt;
|-
| | 
| | 17/210
| | 
| | 97.143
| | &lt;span style="font-size: 12.8000001907349px;"&gt;116&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&gt;5.714&lt;/span&gt;
|-
| | 
| | 20/247
| | 
| | 97.166
| | &lt;span style="font-size: 12.8000001907349px;"&gt;116&lt;/span&gt;&lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&gt;5.992&lt;/span&gt;
|-
| | 
| | 23/284
| | 
| | 97.&lt;span style="line-height: 1.5;"&gt;183&lt;/span&gt;
| | 116&lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&gt;6.197&lt;/span&gt;
|-
| | 3/37
| | 
| | 
| | 97.297
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1167.568&lt;/span&gt;
|-
| | 
| | 25/308
| | 
| | 97.403
| | 1168&lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&gt;.831&lt;/span&gt;
|-
| | 
| | 
| | 
| | 97.416
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1168.9915&lt;/span&gt;
|-
| | 
| | 22/271
| | 
| | 97.417
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1169.004&lt;/span&gt;
|-
| | 
| | 19/234
| | 
| | 97.436
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1169.231&lt;/span&gt;
|-
| | 
| | 16/197
| | 
| | 97.462
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1169.543&lt;/span&gt;
|-
| | 
| | 13/160
| | 
| | 97.5
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1170&lt;/span&gt;
|-
| | 
| | 10/123
| | 
| | 97.561
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1170.731&lt;/span&gt;
|-
| | 
| | 
| | 17/209
| | 97.608
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1171.292&lt;/span&gt;
|-
| | 
| | 7/86
| | 
| | 97.674
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1172.093&lt;/span&gt;
|-
| | 
| | 11/135
| | 
| | 97,778
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1173.333&lt;/span&gt;
|-
| | 
| | 15/184
| | 
| | 97.826
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1173.913&lt;/span&gt;
|-
| | 
| | 19/233
| | 
| | 97.854
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1174.249&lt;/span&gt;
|-
| | 4/49
| | 
| | 
| | 97.959
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1175.51&lt;/span&gt;
|-
| | 
| | 17/208
| | 
| | 98.077
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1176.923&lt;/span&gt;
|-
| | 
| | 13/159
| | 
| | 98.113
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1177.3585&lt;/span&gt;
|-
| | 
| | 9/110
| | 
| | 98.182
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1178.182&lt;/span&gt;
|-
| | 
| | 14/171
| | 
| | 98.246
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1178.947&lt;/span&gt;
|-
| | 
| | 19/232
| | 
| | 98.276
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1179.31&lt;/span&gt;
|-
| | 5/61
| | 
| | 
| | 98.361
| | &lt;span style="font-size: 12.8000001907349px;"&gt;1180.323&lt;/span&gt;
|-
| | 1/12
| | 
| | 
| | 100
| | 1200
|}</text>
      <sha1>2bd8ctbgl5qr0ndpc537q03gnsnnsjw</sha1>
    </revision>
    <revision>
      <id>4660</id>
      <timestamp>2015-11-13T16:10:11Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 566390699 - Original comment: **</comment>
      <origin>4660</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="43894" sha1="d4zxsmh7fy6pkxklrou6d4jrvlzf5yx" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-11-13 16:10:11 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;566390699&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, the Grumpy tridecatonic, apparently belongs to no particularly important temperament. However, it becomes a compressed 12ed scale when you ignore the octave (this obviously does not work when the generator is very near 12edo &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(within -7/24¢ of it)&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 13px; line-height: 1.5;"&amp;gt;, for the 13th degree of the scale registers as identical to the octave for human listeners, and it becomes indistinct from 13edo or the Happy dodecatonic ([[1L 11s]]) in the 1.75¢ above 1/13edo because the large and small steps register as identical to one another for human listeners).&amp;lt;/span&amp;gt;
||||||~ Generator ||~ &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;Cents&amp;lt;/span&amp;gt; ||~ &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;12g&amp;lt;/span&amp;gt; ||
|| 1/13 ||   ||   || 92.308 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1107.692&amp;lt;/span&amp;gt; ||
|| 5/64 ||   ||   || 93.75 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1125&amp;lt;/span&amp;gt; ||
||   || 9/115 ||   || 93.913 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1126.9565&amp;lt;/span&amp;gt; ||
||   || 13/166 ||   || 93.976 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1127.711&amp;lt;/span&amp;gt; ||
||   || 17/217 ||   || 94.009 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1128.111&amp;lt;/span&amp;gt; ||
|| 4/51 ||   ||   || 94.118 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1129.412&amp;lt;/span&amp;gt; ||
||   || 15/191 ||   || 94.241 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1130.89&amp;lt;/span&amp;gt; ||
||   || 11/140 ||   || 94.296 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1131 3/7&amp;lt;/span&amp;gt; ||
||   || 7/89 ||   || 94.382 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1132.584&amp;lt;/span&amp;gt; ||
||   || 10/127 ||   || 94.448 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1133.858&amp;lt;/span&amp;gt; ||
||   || 13/165 ||   || 94.5455 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1134.5455&amp;lt;/span&amp;gt; ||
||   || 16/203 ||   || 94.581 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1134.975&amp;lt;/span&amp;gt; ||
||   || 19/241 ||   || 94.606 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1135.27&amp;lt;/span&amp;gt; ||
|| 3/38 ||   ||   || 94.737 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1136.842&amp;lt;/span&amp;gt; ||
||   || 26/329 ||   || 94.8875 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1137.6505&amp;lt;/span&amp;gt; ||
||   || 23/291 ||   || 94.845 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138.1443&amp;lt;/span&amp;gt; ||
||   || 20/253 ||   || 94.862 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138.34&amp;lt;/span&amp;gt; ||
||   || 17/215 ||   || 94.884 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138.605&amp;lt;/span&amp;gt; ||
||   || 14/177 ||   || 94.915 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138.983&amp;lt;/span&amp;gt; ||
||   ||   ||   || 94.962 || 1139.545 ||
||   || 11/139 ||   || 94.964 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1139.568&amp;lt;/span&amp;gt; ||
||   || 8/101 ||   || 95.0495 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1140.594&amp;lt;/span&amp;gt; ||
||   ||   ||   || 95.102 || 1141.224 ||
||   ||   || 13/164 || 95.122 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1141.463&amp;lt;/span&amp;gt; ||
||   || 5/63 ||   || 95.238 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1142.714&amp;lt;/span&amp;gt; ||
||   ||   || 17/214 || 95.374 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1143.486&amp;lt;/span&amp;gt; ||
||   ||   || 12/151 || 95.362 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1144.371&amp;lt;/span&amp;gt; ||
||   ||   ||   || 95.41 || 1144.915 ||
||   || 7/88 ||   || 95,4545 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1145.4545&amp;lt;/span&amp;gt; ||
||   || 9/113 ||   || 95.575 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1146.903&amp;lt;/span&amp;gt; ||
||   || 11/138 ||   || 95.652 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1147.826&amp;lt;/span&amp;gt; ||
||   || 13/163 ||   || 95.7055 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1148.466&amp;lt;/span&amp;gt; ||
||   || 15/188 ||   || 95.745 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1148.936&amp;lt;/span&amp;gt; ||
||   || 17/213 ||   || 95.775 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1149.296&amp;lt;/span&amp;gt; ||
||   || 19/238 ||   || 95.798 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1149.58&amp;lt;/span&amp;gt; ||
||   || 21/263 ||   || 95.8175 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1149.81&amp;lt;/span&amp;gt; ||
||   || 23/288 ||   || 95.833 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150&amp;lt;/span&amp;gt; ||
||   || 25/313 ||   || 95.847 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.16&amp;lt;/span&amp;gt; ||
||   || 27/338 ||   || 95.858 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.296&amp;lt;/span&amp;gt; ||
||   || 29/363 ||   || 95.868 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.467&amp;lt;/span&amp;gt; ||
||   || 31/388 ||   || 95.876 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.5155&amp;lt;/span&amp;gt; ||
||   || 33/413 ||   || 95.884 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.605&amp;lt;/span&amp;gt; ||
||   || 35/438 ||   || 95.89 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.685&amp;lt;/span&amp;gt; ||
||   || 37/463 ||   || 95.896 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.75&amp;lt;/span&amp;gt; ||
||   || 39/488 ||   || 95.902 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.82&amp;lt;/span&amp;gt; ||
||   || 41/513 ||   || 95.906 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.877&amp;lt;/span&amp;gt; ||
||   || 43/538 ||   || 95.911 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.929&amp;lt;/span&amp;gt; ||
||   || 45/563 ||   || 95.915 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.977&amp;lt;/span&amp;gt; ||
||   || 47/588 ||   || 95.918 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1151.02&amp;lt;/span&amp;gt; ||
|| 2/25 ||   ||   || 96 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1152&amp;lt;/span&amp;gt; ||
||   || 25/312 ||   || 96.154 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1153.846&amp;lt;/span&amp;gt; ||
||   || 23/287 ||   || 96.167 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1154.007&amp;lt;/span&amp;gt; ||
||   || 21/262 ||   || 96.183 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1154.1985&amp;lt;/span&amp;gt; ||
||   || 19/237 ||   || 96.2025 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1154,43&amp;lt;/span&amp;gt; ||
||   || 17/212 ||   || 96.226 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1154.717&amp;lt;/span&amp;gt; ||
||   || 15/187 ||   || 96.257 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1155.08&amp;lt;/span&amp;gt; ||
||   || 13/162 ||   || 96.296 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1155.556&amp;lt;/span&amp;gt; ||
||   || 11/137 ||   || 96.35 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1156.204&amp;lt;/span&amp;gt; ||
||   || 9/112 ||   || 96.429 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1157.143&amp;lt;/span&amp;gt; ||
||   || 7/87 ||   || 96.552 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1158.621&amp;lt;/span&amp;gt; ||
||   ||   || 12/149 || 96.644 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1159.7315&amp;lt;/span&amp;gt; ||
||   ||   || 17/211 || 96.6825 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1160.278&amp;lt;/span&amp;gt; ||
||   || 5/62 ||   || 96.774 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1161.29&amp;lt;/span&amp;gt; ||
||   ||   || 13/161 || 96.894 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1162.733&amp;lt;/span&amp;gt; ||
||   ||   ||   || 96.915 || 1162.982 ||
||   || 8/99 ||   || 96.97 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1163.636&amp;lt;/span&amp;gt; ||
||   || 11/136 ||   || 97.059 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1164.706&amp;lt;/span&amp;gt; ||
||   ||   ||   || 97.0255 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1164.306&amp;lt;/span&amp;gt; ||
||   || 14/173 ||   || 97.11 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;116&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;5.318&amp;lt;/span&amp;gt; ||
||   || 17/210 ||   || 97.143 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;116&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;5.714&amp;lt;/span&amp;gt; ||
||   || 20/247 ||   || 97.166 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;116&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;5.992&amp;lt;/span&amp;gt; ||
||   || 23/284 ||   || 97.&amp;lt;span style="line-height: 1.5;"&amp;gt;183&amp;lt;/span&amp;gt; || 116&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;6.197&amp;lt;/span&amp;gt; ||
|| 3/37 ||   ||   || 97.297 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1167.568&amp;lt;/span&amp;gt; ||
||   || 25/308 ||   || 97.403 || 1168&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;.831&amp;lt;/span&amp;gt; ||
||   ||   ||   || 97.416 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1168.9915&amp;lt;/span&amp;gt; ||
||   || 22/271 ||   || 97.417 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1169.004&amp;lt;/span&amp;gt; ||
||   || 19/234 ||   || 97.436 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1169.231&amp;lt;/span&amp;gt; ||
||   || 16/197 ||   || 97.462 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1169.543&amp;lt;/span&amp;gt; ||
||   || 13/160 ||   || 97.5 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1170&amp;lt;/span&amp;gt; ||
||   || 10/123 ||   || 97.561 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1170.731&amp;lt;/span&amp;gt; ||
||   ||   || 17/209 || 97.608 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1171.292&amp;lt;/span&amp;gt; ||
||   || 7/86 ||   || 97.674 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1172.093&amp;lt;/span&amp;gt; ||
||   || 11/135 ||   || 97,778 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1173.333&amp;lt;/span&amp;gt; ||
||   || 15/184 ||   || 97.826 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1173.913&amp;lt;/span&amp;gt; ||
||   || 19/233 ||   || 97.854 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1174.249&amp;lt;/span&amp;gt; ||
|| 4/49 ||   ||   || 97.959 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1175.51&amp;lt;/span&amp;gt; ||
||   || 17/208 ||   || 98.077 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1176.923&amp;lt;/span&amp;gt; ||
||   || 13/159 ||   || 98.113 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1177.3585&amp;lt;/span&amp;gt; ||
||   || 9/110 ||   || 98.182 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1178.182&amp;lt;/span&amp;gt; ||
||   || 14/171 ||   || 98.246 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1178.947&amp;lt;/span&amp;gt; ||
||   || 19/232 ||   || 98.276 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1179.31&amp;lt;/span&amp;gt; ||
|| 5/61 ||   ||   || 98.361 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1180.323&amp;lt;/span&amp;gt; ||
|| 1/12 ||   ||   || 100 || 1200 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 1s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, the Grumpy tridecatonic, apparently belongs to no particularly important temperament. However, it becomes a compressed 12ed scale when you ignore the octave (this obviously does not work when the generator is very near 12edo &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(within -7/24¢ of it)&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 13px; line-height: 1.5;"&amp;gt;, for the 13th degree of the scale registers as identical to the octave for human listeners, and it becomes indistinct from 13edo or the Happy dodecatonic (&amp;lt;a class="wiki_link" href="/1L%2011s"&amp;gt;1L 11s&amp;lt;/a&amp;gt;) in the 1.75¢ above 1/13edo because the large and small steps register as identical to one another for human listeners).&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th colspan="3"&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;Cents&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;12g&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92.308&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1107.692&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1125&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/115&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.913&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1126.9565&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/166&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.976&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1127.711&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/217&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.009&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1128.111&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.118&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1129.412&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/191&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.241&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1130.89&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/140&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.296&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1131 3/7&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/89&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.382&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1132.584&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.448&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1133.858&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/165&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.5455&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1134.5455&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/203&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.581&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1134.975&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/241&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.606&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1135.27&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/38&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.737&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1136.842&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;26/329&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.8875&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1137.6505&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23/291&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.845&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138.1443&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20/253&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.862&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138.34&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/215&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.884&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138.605&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/177&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.915&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138.983&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.962&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1139.545&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/139&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.964&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1139.568&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;95.0495&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1140.594&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;98.246&amp;lt;br /&amp;gt;
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&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>d4zxsmh7fy6pkxklrou6d4jrvlzf5yx</sha1>
    </revision>
    <revision>
      <id>4661</id>
      <timestamp>2015-06-11T22:21:19Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 553733888 - Original comment: **</comment>
      <origin>4661</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="43402" sha1="9hctqfhs7i2ym6vvi3da7wl9sb7e4ry" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-06-11 22:21:19 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;553733888&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, the Grumpy tridecatonic, apparently belongs to no particularly important temperament. However, it becomes a compressed 12ed scale when you ignore the octave (this obviously does not work when the generator is very near 12edo &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(within -7/24¢ of it)&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 13px; line-height: 1.5;"&amp;gt;, for the 13th degree of the scale registers as identical to the octave for human listeners, and it becomes indistinct from 13edo or the Happy dodecatonic ([[1L 11s]]) in the 1.75¢ above 1/13edo because the large and small steps register as identical to one another for human listeners).&amp;lt;/span&amp;gt;
||||||~ Generator ||~ &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;Cents&amp;lt;/span&amp;gt; ||~ &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;12g&amp;lt;/span&amp;gt; ||
|| 1/13 ||   ||   || 92 4/13 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1107 9/13&amp;lt;/span&amp;gt; ||
|| 5/64 ||   ||   || 93.75 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1125&amp;lt;/span&amp;gt; ||
||   || 9/115 ||   || 93 21/23 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1126 22/23&amp;lt;/span&amp;gt; ||
||   || 13/166 ||   || 93 81/83 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1127 59/83&amp;lt;/span&amp;gt; ||
||   || 17/217 ||   || 94.009 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1128.111&amp;lt;/span&amp;gt; ||
|| 4/51 ||   ||   || 94 2/17 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1129 7/17&amp;lt;/span&amp;gt; ||
||   || 15/191 ||   || 94.241 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1130.89&amp;lt;/span&amp;gt; ||
||   || 11/140 ||   || 94 2/7 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1131 3/7&amp;lt;/span&amp;gt; ||
||   || 7/89 ||   || 94.382 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1132.584&amp;lt;/span&amp;gt; ||
||   || 10/127 ||   || 94.448 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1133.858&amp;lt;/span&amp;gt; ||
||   || 13/165 ||   || 94 6/11 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1134 6/11&amp;lt;/span&amp;gt; ||
||   || 16/203 ||   || 94.581 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1134.975&amp;lt;/span&amp;gt; ||
||   || 19/241 ||   || 94.606 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1135.27&amp;lt;/span&amp;gt; ||
|| 3/38 ||   ||   || 94 14/19 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1136 16/19&amp;lt;/span&amp;gt; ||
||   || 26/329 ||   || 94.8875 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1137.6505&amp;lt;/span&amp;gt; ||
||   || 23/291 ||   || 94 82/97 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138 14/97&amp;lt;/span&amp;gt; ||
||   || 20/253 ||   || 94.862 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138.34&amp;lt;/span&amp;gt; ||
||   || 17/215 ||   || 94 38/43 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138 26/43&amp;lt;/span&amp;gt; ||
||   || 14/177 ||   || 94 54/59 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138 58/59&amp;lt;/span&amp;gt; ||
||   || 11/139 ||   || 94.964 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1139.568&amp;lt;/span&amp;gt; ||
||   || 8/101 ||   || 95.0495 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1140.594&amp;lt;/span&amp;gt; ||
||   ||   || 13/164 || 95 5/41 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1141 19/41&amp;lt;/span&amp;gt; ||
||   || 5/63 ||   || 95 5/21 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1142 6/7&amp;lt;/span&amp;gt; ||
||   ||   || 17/214 || 95.374 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1143.486&amp;lt;/span&amp;gt; ||
||   ||   || 12/151 || 95.4305 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1144.385&amp;lt;/span&amp;gt; ||
||   || 7/88 ||   || 95 5/11 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1145 5/11&amp;lt;/span&amp;gt; ||
||   || 9/113 ||   || 95.575 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1146.903&amp;lt;/span&amp;gt; ||
||   || 11/138 ||   || 95 15/23 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1147 19/23&amp;lt;/span&amp;gt; ||
||   || 13/163 ||   || 95.7055 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1148.466&amp;lt;/span&amp;gt; ||
||   || 15/188 ||   || 95 35/47 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1148 44/47&amp;lt;/span&amp;gt; ||
||   || 17/213 ||   || 95 55/71 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1149 21/71&amp;lt;/span&amp;gt; ||
||   || 19/238 ||   || 95.798 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1149.58&amp;lt;/span&amp;gt; ||
||   || 21/263 ||   || 95.8175 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1149.81&amp;lt;/span&amp;gt; ||
||   || 23/288 ||   || 95 5/6 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150&amp;lt;/span&amp;gt; ||
||   || 25/313 ||   || 95.847 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.16&amp;lt;/span&amp;gt; ||
||   || 27/338 ||   || 95.858 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.296&amp;lt;/span&amp;gt; ||
||   || 29/363 ||   || 95.868 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.467&amp;lt;/span&amp;gt; ||
||   || 31/388 ||   || 95 85/97 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150 50/97&amp;lt;/span&amp;gt; ||
||   || 33/413 ||   || 95.884 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.605&amp;lt;/span&amp;gt; ||
||   || 35/438 ||   || 95 65/73 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150 50/73&amp;lt;/span&amp;gt; ||
||   || 37/463 ||   || 95.896 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.75&amp;lt;/span&amp;gt; ||
||   || 39/488 ||   || 95 55/61 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150 50/61&amp;lt;/span&amp;gt; ||
||   || 41/513 ||   || 95.906 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.877&amp;lt;/span&amp;gt; ||
||   || 43/538 ||   || 95.911 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.929&amp;lt;/span&amp;gt; ||
||   || 45/563 ||   || 95.915 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.977&amp;lt;/span&amp;gt; ||
||   || 47/588 ||   || 95 45/49 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1151 1/49&amp;lt;/span&amp;gt; ||
|| 2/25 ||   ||   || 96 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1152&amp;lt;/span&amp;gt; ||
||   || 25/312 ||   || 96 2/13 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1153 11/13&amp;lt;/span&amp;gt; ||
||   || 23/287 ||   || 96.167 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1154.007&amp;lt;/span&amp;gt; ||
||   || 21/262 ||   || 96.183 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1154.1985&amp;lt;/span&amp;gt; ||
||   || 19/237 ||   || 96 16/79 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1154 34/79&amp;lt;/span&amp;gt; ||
||   || 17/212 ||   || 96 12/53 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1154 38/53&amp;lt;/span&amp;gt; ||
||   || 15/187 ||   || 96.257 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1155.08&amp;lt;/span&amp;gt; ||
||   || 13/162 ||   || 96 8/27 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1155 5/9&amp;lt;/span&amp;gt; ||
||   || 11/137 ||   || 96.35 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1156.204&amp;lt;/span&amp;gt; ||
||   || 9/112 ||   || 96 3/7 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1157 1/7&amp;lt;/span&amp;gt; ||
||   || 7/87 ||   || 96 16/29 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1158 18/29&amp;lt;/span&amp;gt; ||
||   ||   || 12/149 || 96.644 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1159.7315&amp;lt;/span&amp;gt; ||
||   ||   || 17/211 || 96.6825 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1160.278&amp;lt;/span&amp;gt; ||
||   || 5/62 ||   || 96 24/31 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1161 9/31&amp;lt;/span&amp;gt; ||
||   ||   || 13/161 || 96.708 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1162.819&amp;lt;/span&amp;gt; ||
||   || 8/99 ||   || 96 32/33 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1163 7/11&amp;lt;/span&amp;gt; ||
||   || 11/136 ||   || 97 1/17 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1164 12/17&amp;lt;/span&amp;gt; ||
||   ||   ||   || 97.0255 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1164.306&amp;lt;/span&amp;gt; ||
||   || 14/173 ||   || 97 &amp;lt;span style="line-height: 1.5;"&amp;gt;19/173&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;116&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;5 55/173&amp;lt;/span&amp;gt; ||
||   || 17/210 ||   || 97 &amp;lt;span style="line-height: 1.5;"&amp;gt;1/7&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;116&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;5 5/7&amp;lt;/span&amp;gt; ||
||   || 20/247 ||   || 97 &amp;lt;span style="line-height: 1.5;"&amp;gt;41/247&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;116&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;5 245/247&amp;lt;/span&amp;gt; ||
||   || 23/284 ||   || 97 &amp;lt;span style="line-height: 1.5;"&amp;gt;13/71&amp;lt;/span&amp;gt; || 116&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;6 14/71&amp;lt;/span&amp;gt; ||
|| 3/37 ||   ||   || 97 11/37 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1167 21/37&amp;lt;/span&amp;gt; ||
||   || 25/308 ||   || 97 &amp;lt;span style="line-height: 1.5;"&amp;gt;31/76&amp;lt;/span&amp;gt; || 1168&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt; 8/19&amp;lt;/span&amp;gt; ||
||   ||   ||   || 97.416 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1168.9915&amp;lt;/span&amp;gt; ||
||   || 22/271 ||   || 97.417 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1169.004&amp;lt;/span&amp;gt; ||
||   || 19/234 ||   || 97 17/39 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1169 3/13&amp;lt;/span&amp;gt; ||
||   || 16/197 ||   || 97.462 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1169.543&amp;lt;/span&amp;gt; ||
||   || 13/160 ||   || 97.5 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1170&amp;lt;/span&amp;gt; ||
||   || 10/123 ||   || 97 23/41 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1170 30/41&amp;lt;/span&amp;gt; ||
||   ||   || 17/209 || 97.608 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1171.292&amp;lt;/span&amp;gt; ||
||   || 7/86 ||   || 97 29/43 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1172 4/43&amp;lt;/span&amp;gt; ||
||   || 11/135 ||   || 97 7/9 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1173 1/3&amp;lt;/span&amp;gt; ||
||   || 15/184 ||   || 97 19/23 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1173 21/23&amp;lt;/span&amp;gt; ||
||   || 19/233 ||   || 97.854 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1174.249&amp;lt;/span&amp;gt; ||
|| 4/49 ||   ||   || 97 47/49 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1175 25/49&amp;lt;/span&amp;gt; ||
||   || 17/208 ||   || 98 1/13 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1176 12/13&amp;lt;/span&amp;gt; ||
||   || 13/159 ||   || 98 6/53 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1177 19/53&amp;lt;/span&amp;gt; ||
||   || 9/110 ||   || 98 2/11 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1178 2/11&amp;lt;/span&amp;gt; ||
||   || 14/171 ||   || 98 14/57 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1178 18/19&amp;lt;/span&amp;gt; ||
||   || 19/232 ||   || 98 4/29 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1179 9/29&amp;lt;/span&amp;gt; ||
|| 5/61 ||   ||   || 98 22/61 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1180 20/61&amp;lt;/span&amp;gt; ||
|| 1/12 ||   ||   || 100 || 1200 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 1s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, the Grumpy tridecatonic, apparently belongs to no particularly important temperament. However, it becomes a compressed 12ed scale when you ignore the octave (this obviously does not work when the generator is very near 12edo &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(within -7/24¢ of it)&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 13px; line-height: 1.5;"&amp;gt;, for the 13th degree of the scale registers as identical to the octave for human listeners, and it becomes indistinct from 13edo or the Happy dodecatonic (&amp;lt;a class="wiki_link" href="/1L%2011s"&amp;gt;1L 11s&amp;lt;/a&amp;gt;) in the 1.75¢ above 1/13edo because the large and small steps register as identical to one another for human listeners).&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;


&amp;lt;table class="wiki_table"&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;th colspan="3"&amp;gt;Generator&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;Cents&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
        &amp;lt;th&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;12g&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/th&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;1/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;92 4/13&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1107 9/13&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;5/64&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93.75&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1125&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;9/115&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93 21/23&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1126 22/23&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/166&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;93 81/83&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1127 59/83&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/217&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.009&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1128.111&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;4/51&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94 2/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1129 7/17&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;15/191&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.241&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1130.89&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/140&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94 2/7&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1131 3/7&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;7/89&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.382&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1132.584&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;10/127&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.448&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1133.858&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/165&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94 6/11&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1134 6/11&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/203&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.581&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1134.975&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/241&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.606&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1135.27&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/38&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94 14/19&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1136 16/19&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;26/329&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.8875&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1137.6505&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23/291&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94 82/97&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138 14/97&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20/253&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.862&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138.34&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/215&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94 38/43&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138 26/43&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/177&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94 54/59&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138 58/59&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/139&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;94.964&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1139.568&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/101&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;95.0495&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1140.594&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;13/164&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;95 5/41&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1141 19/41&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;5/63&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;95 5/21&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1142 6/7&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/214&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;95.374&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1143.486&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;12/151&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;95.4305&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1144.385&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96 16/29&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1158 18/29&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;12/149&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;17/211&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96 24/31&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1161 9/31&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
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        &amp;lt;td&amp;gt;13/161&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1162.819&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;8/99&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;96 32/33&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1163 7/11&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;11/136&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97 1/17&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1164 12/17&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1164.306&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;14/173&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97 &amp;lt;span style="line-height: 1.5;"&amp;gt;19/173&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;116&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;5 55/173&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;17/210&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97 &amp;lt;span style="line-height: 1.5;"&amp;gt;1/7&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;116&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;5 5/7&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;20/247&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97 &amp;lt;span style="line-height: 1.5;"&amp;gt;41/247&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;116&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;5 245/247&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;23/284&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97 &amp;lt;span style="line-height: 1.5;"&amp;gt;13/71&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;116&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;6 14/71&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;3/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97 11/37&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1167 21/37&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;25/308&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97 &amp;lt;span style="line-height: 1.5;"&amp;gt;31/76&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1168&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt; 8/19&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97.416&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1168.9915&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;22/271&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1169.004&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;19/234&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;97 17/39&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1169 3/13&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;16/197&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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        &amp;lt;td&amp;gt;13/160&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
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&amp;lt;/td&amp;gt;
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    &amp;lt;tr&amp;gt;
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        &amp;lt;td&amp;gt;10/123&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1170 30/41&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
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    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;17/209&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1171.292&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;7/86&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;97 29/43&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1172 4/43&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
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    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
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        &amp;lt;td&amp;gt;11/135&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1173 1/3&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
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    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;15/184&amp;lt;br /&amp;gt;
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    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;19/233&amp;lt;br /&amp;gt;
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    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1175 25/49&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
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    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;17/208&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;98 1/13&amp;lt;br /&amp;gt;
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    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;13/159&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;98 6/53&amp;lt;br /&amp;gt;
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    &amp;lt;/tr&amp;gt;
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        &amp;lt;td&amp;gt;9/110&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1178 2/11&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
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    &amp;lt;/tr&amp;gt;
    &amp;lt;tr&amp;gt;
        &amp;lt;td&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;14/171&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;98 14/57&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1178 18/19&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;98 4/29&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1179 9/29&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1180 20/61&amp;lt;/span&amp;gt;&amp;lt;br /&amp;gt;
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        &amp;lt;td&amp;gt;100&amp;lt;br /&amp;gt;
&amp;lt;/td&amp;gt;
        &amp;lt;td&amp;gt;1200&amp;lt;br /&amp;gt;
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&amp;lt;/table&amp;gt;

&amp;lt;/body&amp;gt;&amp;lt;/html&amp;gt;&lt;/pre&gt;&lt;/div&gt;</text>
      <sha1>9hctqfhs7i2ym6vvi3da7wl9sb7e4ry</sha1>
    </revision>
    <revision>
      <id>4662</id>
      <timestamp>2015-06-11T22:19:49Z</timestamp>
      <contributor>
        <username>Wikispaces&gt;JosephRuhf</username>
        <id>0</id>
      </contributor>
      <comment>**Imported revision 553733812 - Original comment: **</comment>
      <origin>4662</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="43450" sha1="ot8i9oies7ksuumykqznyguk31r7igb" xml:space="preserve">&lt;h2&gt;IMPORTED REVISION FROM WIKISPACES&lt;/h2&gt;
This is an imported revision from Wikispaces. The revision metadata is included below for reference:&lt;br&gt;
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on &lt;tt&gt;2015-06-11 22:19:49 UTC&lt;/tt&gt;.&lt;br&gt;
: The original revision id was &lt;tt&gt;553733812&lt;/tt&gt;.&lt;br&gt;
: The revision comment was: &lt;tt&gt;&lt;/tt&gt;&lt;br&gt;
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.&lt;br&gt;
&lt;h4&gt;Original Wikitext content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"&gt;This MOS, the Grumpy tridecatonic, apparently belongs to no particularly important temperament. However, it becomes a compressed 12ed scale when you ignore the octave (this obviously does not work when the generator is very near 12edo &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(within -7/24¢ of it)&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 13px; line-height: 1.5;"&amp;gt;, for the 13th degree of the scale registers as identical to the octave for human listeners, and it becomes indistinct from 13edo or the Happy dodecatonic ([[1L 11s]]) in the 1.75¢ above 1/13edo because the large and small steps register as identical to one another for human listeners).&amp;lt;/span&amp;gt;
||||||~ Generator ||~ &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;Cents&amp;lt;/span&amp;gt; ||~ &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;12g&amp;lt;/span&amp;gt; ||
|| 1/13 ||   ||   || 92 4/13 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1107 9/13&amp;lt;/span&amp;gt; ||
|| 5/64 ||   ||   || 93.75 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1125&amp;lt;/span&amp;gt; ||
||   || 9/115 ||   || 93 21/23 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1126 22/23&amp;lt;/span&amp;gt; ||
||   || 13/166 ||   || 93 81/83 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1127 59/83&amp;lt;/span&amp;gt; ||
||   || 17/217 ||   || 94.009 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1128.111&amp;lt;/span&amp;gt; ||
|| 4/51 ||   ||   || 94 2/17 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1129 7/17&amp;lt;/span&amp;gt; ||
||   || 15/191 ||   || 94.241 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1130.89&amp;lt;/span&amp;gt; ||
||   || 11/140 ||   || 94 2/7 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1131 3/7&amp;lt;/span&amp;gt; ||
||   || 7/89 ||   || 94.382 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1132.584&amp;lt;/span&amp;gt; ||
||   || 10/127 ||   || 94.448 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1133.858&amp;lt;/span&amp;gt; ||
||   || 13/165 ||   || 94 6/11 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1134 6/11&amp;lt;/span&amp;gt; ||
||   || 16/203 ||   || 94.581 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1134.975&amp;lt;/span&amp;gt; ||
||   || 19/241 ||   || 94.606 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1135.27&amp;lt;/span&amp;gt; ||
|| 3/38 ||   ||   || 94 14/19 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1136 16/19&amp;lt;/span&amp;gt; ||
||   || 26/329 ||   || 94.8875 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1137.6505&amp;lt;/span&amp;gt; ||
||   || 23/291 ||   || 94 82/97 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138 14/97&amp;lt;/span&amp;gt; ||
||   || 20/253 ||   || 94.862 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138.34&amp;lt;/span&amp;gt; ||
||   || 17/215 ||   || 94 38/43 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138 26/43&amp;lt;/span&amp;gt; ||
||   || 14/177 ||   || 94 54/59 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1138 58/59&amp;lt;/span&amp;gt; ||
||   || 11/139 ||   || 94.964 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1139.568&amp;lt;/span&amp;gt; ||
||   || 8/101 ||   || 95 5/101 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;[[tel:1140 60/101|1140 60/101]]&amp;lt;/span&amp;gt; ||
||   ||   || 13/164 || 95 5/41 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1141 19/41&amp;lt;/span&amp;gt; ||
||   || 5/63 ||   || 95 5/21 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1142 6/7&amp;lt;/span&amp;gt; ||
||   ||   || 17/214 || 95.374 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1143.486&amp;lt;/span&amp;gt; ||
||   ||   || 12/151 || 95.4305 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1144.385&amp;lt;/span&amp;gt; ||
||   || 7/88 ||   || 95 5/11 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1145 5/11&amp;lt;/span&amp;gt; ||
||   || 9/113 ||   || 95.575 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1146.903&amp;lt;/span&amp;gt; ||
||   || 11/138 ||   || 95 15/23 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1147 19/23&amp;lt;/span&amp;gt; ||
||   || 13/163 ||   || 95.7055 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1148.466&amp;lt;/span&amp;gt; ||
||   || 15/188 ||   || 95 35/47 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1148 44/47&amp;lt;/span&amp;gt; ||
||   || 17/213 ||   || 95 55/71 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1149 21/71&amp;lt;/span&amp;gt; ||
||   || 19/238 ||   || 95.798 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1149.58&amp;lt;/span&amp;gt; ||
||   || 21/263 ||   || 95.8175 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1149.81&amp;lt;/span&amp;gt; ||
||   || 23/288 ||   || 95 5/6 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150&amp;lt;/span&amp;gt; ||
||   || 25/313 ||   || 95.847 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.16&amp;lt;/span&amp;gt; ||
||   || 27/338 ||   || 95.858 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.296&amp;lt;/span&amp;gt; ||
||   || 29/363 ||   || 95.868 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.467&amp;lt;/span&amp;gt; ||
||   || 31/388 ||   || 95 85/97 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150 50/97&amp;lt;/span&amp;gt; ||
||   || 33/413 ||   || 95.884 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.605&amp;lt;/span&amp;gt; ||
||   || 35/438 ||   || 95 65/73 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150 50/73&amp;lt;/span&amp;gt; ||
||   || 37/463 ||   || 95.896 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.75&amp;lt;/span&amp;gt; ||
||   || 39/488 ||   || 95 55/61 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150 50/61&amp;lt;/span&amp;gt; ||
||   || 41/513 ||   || 95.906 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.877&amp;lt;/span&amp;gt; ||
||   || 43/538 ||   || 95.911 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.929&amp;lt;/span&amp;gt; ||
||   || 45/563 ||   || 95.915 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1150.977&amp;lt;/span&amp;gt; ||
||   || 47/588 ||   || 95 45/49 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1151 1/49&amp;lt;/span&amp;gt; ||
|| 2/25 ||   ||   || 96 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1152&amp;lt;/span&amp;gt; ||
||   || 25/312 ||   || 96 2/13 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1153 11/13&amp;lt;/span&amp;gt; ||
||   || 23/287 ||   || 96.167 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1154.007&amp;lt;/span&amp;gt; ||
||   || 21/262 ||   || 96.183 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1154.1985&amp;lt;/span&amp;gt; ||
||   || 19/237 ||   || 96 16/79 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1154 34/79&amp;lt;/span&amp;gt; ||
||   || 17/212 ||   || 96 12/53 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1154 38/53&amp;lt;/span&amp;gt; ||
||   || 15/187 ||   || 96.257 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1155.08&amp;lt;/span&amp;gt; ||
||   || 13/162 ||   || 96 8/27 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1155 5/9&amp;lt;/span&amp;gt; ||
||   || 11/137 ||   || 96.35 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1156.204&amp;lt;/span&amp;gt; ||
||   || 9/112 ||   || 96 3/7 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1157 1/7&amp;lt;/span&amp;gt; ||
||   || 7/87 ||   || 96 16/29 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1158 18/29&amp;lt;/span&amp;gt; ||
||   ||   || 12/149 || 96.644 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1159.7315&amp;lt;/span&amp;gt; ||
||   ||   || 17/211 || 96.6825 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1160.278&amp;lt;/span&amp;gt; ||
||   || 5/62 ||   || 96 24/31 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1161 9/31&amp;lt;/span&amp;gt; ||
||   ||   || 13/161 || 96.708 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1162.819&amp;lt;/span&amp;gt; ||
||   || 8/99 ||   || 96 32/33 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1163 7/11&amp;lt;/span&amp;gt; ||
||   || 11/136 ||   || 97 1/17 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1164 12/17&amp;lt;/span&amp;gt; ||
||   ||   ||   || 97.0255 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1164.306&amp;lt;/span&amp;gt; ||
||   || 14/173 ||   || 97 &amp;lt;span style="line-height: 1.5;"&amp;gt;19/173&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;116&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;5 55/173&amp;lt;/span&amp;gt; ||
||   || 17/210 ||   || 97 &amp;lt;span style="line-height: 1.5;"&amp;gt;1/7&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;116&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;5 5/7&amp;lt;/span&amp;gt; ||
||   || 20/247 ||   || 97 &amp;lt;span style="line-height: 1.5;"&amp;gt;41/247&amp;lt;/span&amp;gt; || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;116&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;5 245/247&amp;lt;/span&amp;gt; ||
||   || 23/284 ||   || 97 &amp;lt;span style="line-height: 1.5;"&amp;gt;13/71&amp;lt;/span&amp;gt; || 116&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt;6 14/71&amp;lt;/span&amp;gt; ||
|| 3/37 ||   ||   || 97 11/37 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1167 21/37&amp;lt;/span&amp;gt; ||
||   || 25/308 ||   || 97 &amp;lt;span style="line-height: 1.5;"&amp;gt;31/76&amp;lt;/span&amp;gt; || 1168&amp;lt;span style="font-size: 12.8000001907349px; line-height: 1.5;"&amp;gt; 8/19&amp;lt;/span&amp;gt; ||
||   ||   ||   || 97.416 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1168.9915&amp;lt;/span&amp;gt; ||
||   || 22/271 ||   || 97.417 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1169.004&amp;lt;/span&amp;gt; ||
||   || 19/234 ||   || 97 17/39 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1169 3/13&amp;lt;/span&amp;gt; ||
||   || 16/197 ||   || 97.462 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1169.543&amp;lt;/span&amp;gt; ||
||   || 13/160 ||   || 97.5 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1170&amp;lt;/span&amp;gt; ||
||   || 10/123 ||   || 97 23/41 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1170 30/41&amp;lt;/span&amp;gt; ||
||   ||   || 17/209 || 97.608 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1171.292&amp;lt;/span&amp;gt; ||
||   || 7/86 ||   || 97 29/43 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1172 4/43&amp;lt;/span&amp;gt; ||
||   || 11/135 ||   || 97 7/9 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1173 1/3&amp;lt;/span&amp;gt; ||
||   || 15/184 ||   || 97 19/23 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1173 21/23&amp;lt;/span&amp;gt; ||
||   || 19/233 ||   || 97.854 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1174.249&amp;lt;/span&amp;gt; ||
|| 4/49 ||   ||   || 97 47/49 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1175 25/49&amp;lt;/span&amp;gt; ||
||   || 17/208 ||   || 98 1/13 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1176 12/13&amp;lt;/span&amp;gt; ||
||   || 13/159 ||   || 98 6/53 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1177 19/53&amp;lt;/span&amp;gt; ||
||   || 9/110 ||   || 98 2/11 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1178 2/11&amp;lt;/span&amp;gt; ||
||   || 14/171 ||   || 98 14/57 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1178 18/19&amp;lt;/span&amp;gt; ||
||   || 19/232 ||   || 98 4/29 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1179 9/29&amp;lt;/span&amp;gt; ||
|| 5/61 ||   ||   || 98 22/61 || &amp;lt;span style="font-size: 12.8000001907349px;"&amp;gt;1180 20/61&amp;lt;/span&amp;gt; ||
|| 1/12 ||   ||   || 100 || 1200 ||&lt;/pre&gt;&lt;/div&gt;
&lt;h4&gt;Original HTML content:&lt;/h4&gt;
&lt;div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"&gt;&lt;pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"&gt;&amp;lt;html&amp;gt;&amp;lt;head&amp;gt;&amp;lt;title&amp;gt;12L 1s&amp;lt;/title&amp;gt;&amp;lt;/head&amp;gt;&amp;lt;body&amp;gt;This MOS, the Grumpy tridecatonic, apparently belongs to no particularly important temperament. However, it becomes a compressed 12ed scale when you ignore the octave (this obviously does not work when the generator is very near 12edo &amp;lt;span style="font-size: 12.8000001907349px; line-height: 15.6000003814697px;"&amp;gt;(within -7/24¢ of it)&amp;lt;/span&amp;gt;&amp;lt;span style="font-size: 13px; line-height: 1.5;"&amp;gt;, for the 13th degree of the scale registers as identical to the octave for human listeners, and it becomes indistinct from 13ed